From 50ec5ab6558cc831d49c67e8b2553322287717d6 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 10:08:04 +0200
Subject: [PATCH 01/79] examples: adjoint boundary sensitivities; SIMSOPT
 analytic gradient

Add the implicit-function adjoint that turns VMEC++ into a
gradient-providing equilibrium component for SIMSOPT, the original goal.

vmecpp_adjoint.py: for a converged fixed-boundary equilibrium F_I(x)=0,
the boundary sensitivity of a scalar objective J follows from
H_II lambda = dJ/dx_I, dJ/dx_B = dJ/dx_B - (dF_I/dx_B)^T lambda, with H
the symmetric Hessian of the augmented functional. It is matrix-free via
hessian_vector_product and apply_preconditioner (the SPD interior system
is solved with preconditioned CG). One Hessian solve gives the whole
boundary gradient, versus one equilibrium re-solve per boundary DOF for
finite differences.

simsopt_vmec_gradient.py: VmecEnergy wraps this as a SIMSOPT Optimizable
whose dJ is the adjoint gradient, plus a gradient-cost benchmark.

Verified: the adjoint gradient matches brute-force re-solve finite
differences (rel 2.4e-4) and the SIMSOPT Optimizable's dJ matches finite
differences of J (rel ~1e-6). On solovev (ns=11, 18 boundary DOFs) the
adjoint boundary gradient costs 762 force evaluations versus 9112 for
finite differences (12x), and the gap grows with the boundary DOF count.
---
 examples/simsopt_vmec_gradient.py | 154 +++++++++++++++++++++++++++
 examples/vmecpp_adjoint.py        | 169 ++++++++++++++++++++++++++++++
 tests/test_adjoint.py             |  51 +++++++++
 tests/test_simsopt_gradient.py    |  57 ++++++++++
 4 files changed, 431 insertions(+)
 create mode 100644 examples/simsopt_vmec_gradient.py
 create mode 100644 examples/vmecpp_adjoint.py
 create mode 100644 tests/test_adjoint.py
 create mode 100644 tests/test_simsopt_gradient.py

diff --git a/examples/simsopt_vmec_gradient.py b/examples/simsopt_vmec_gradient.py
new file mode 100644
index 000000000..2240d2a66
--- /dev/null
+++ b/examples/simsopt_vmec_gradient.py
@@ -0,0 +1,154 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Use VMEC++ as a gradient-providing equilibrium component for SIMSOPT.
+
+A SIMSOPT optimization over a plasma boundary normally differentiates VMEC by
+finite differences: one equilibrium re-solve per boundary degree of freedom and
+per outer iteration. VMEC++ instead provides the boundary gradient analytically
+through the implicit-function adjoint (``vmecpp_adjoint.boundary_gradient``): one
+extra Hessian solve, independent of the number of boundary DOFs.
+
+``VmecEnergy`` wraps that as a SIMSOPT ``Optimizable`` whose objective is the MHD
+energy of the converged equilibrium and whose ``dJ`` is the adjoint gradient.
+``optimize_to_target`` runs a gradient-based optimization of the boundary toward
+a target energy, with the analytic gradient or with finite differences, and
+reports the cost (forward-model evaluations counted inside VMEC++, outer
+iterations, wall time) so the two can be compared.
+"""
+
+from __future__ import annotations
+
+import time
+from dataclasses import dataclass
+from pathlib import Path
+
+import numpy as np
+from vmecpp_adjoint import (
+    DEFAULT_INPUT,
+    boundary_gradient,
+    finite_difference_boundary_gradient,
+    make_model,
+    mhd_energy,
+    partition,
+    solve_interior,
+)
+
+
+class VmecBoundaryProblem:
+    """Equilibrium energy and its boundary gradient, cached per boundary state."""
+
+    def __init__(self, input_path: Path = DEFAULT_INPUT, ns: int = 11):
+        self.model = make_model(input_path, ns)
+        self.model.solve()
+        self.ns = ns
+        self.interior, self.boundary = partition(self.model, ns)
+        self._x_full = np.asarray(self.model.get_state(), float).copy()
+        self._cached_p = self._x_full[self.boundary].copy()
+
+    @property
+    def x0(self):
+        return self._x_full[self.boundary].copy()
+
+    def _resolve(self, p):
+        p = np.asarray(p, float)
+        if not np.array_equal(p, self._cached_p):
+            self._x_full = solve_interior(
+                self.model, self._x_full, self.interior, self.boundary, p
+            )
+            self._cached_p = p.copy()
+
+    def value(self, p):
+        self._resolve(p)
+        self.model.set_state(np.ascontiguousarray(self._x_full))
+        self.model.evaluate(2, 2, False)
+        return self.model.mhd_energy
+
+    def gradient(self, p):
+        self._resolve(p)
+        return boundary_gradient(
+            self.model, self._x_full, self.interior, self.boundary, mhd_energy
+        )
+
+
+def make_simsopt_optimizable(problem: VmecBoundaryProblem):
+    """Wrap the problem as a SIMSOPT Optimizable exposing an analytic gradient."""
+    # Imported lazily so the rest of the module (and the gradient benchmark) work
+    # without SIMSOPT installed.
+    from simsopt._core import Optimizable  # noqa: PLC0415
+    from simsopt._core.derivative import Derivative, derivative_dec  # noqa: PLC0415
+
+    class VmecEnergy(Optimizable):
+        def __init__(self):
+            x0 = problem.x0
+            super().__init__(x0=x0, names=[f"boundary{i}" for i in range(x0.size)])
+
+        def J(self):
+            return problem.value(self.local_full_x)
+
+        @derivative_dec
+        def dJ(self):
+            return Derivative({self: problem.gradient(self.local_full_x)})
+
+    return VmecEnergy()
+
+
+@dataclass
+class GradResult:
+    method: str
+    force_evals: int
+    seconds: float
+    gradient: np.ndarray
+
+
+def gradient_cost(input_path: Path = DEFAULT_INPUT, ns: int = 11, analytic=True):
+    """Cost of one full boundary gradient at the converged equilibrium.
+
+    This is what an external optimizer pays per iteration. The analytic adjoint
+    needs one Hessian solve regardless of the number of boundary DOFs; finite
+    differences re-converge the equilibrium twice per boundary DOF.
+    """
+    problem = VmecBoundaryProblem(input_path, ns)
+    x_star = problem._x_full.copy()
+    interior, boundary = problem.interior, problem.boundary
+    problem.model.reset_force_eval_count()
+    t0 = time.perf_counter()
+    if analytic:
+        g_dict = None
+        g = boundary_gradient(problem.model, x_star, interior, boundary, mhd_energy)
+    else:
+        g_dict = finite_difference_boundary_gradient(
+            problem.model,
+            x_star,
+            interior,
+            boundary,
+            mhd_energy,
+            range(boundary.size),
+        )
+        g = np.array([g_dict[j] for j in range(boundary.size)])
+    return GradResult(
+        "analytic adjoint" if analytic else "finite differences",
+        problem.model.force_eval_count,
+        time.perf_counter() - t0,
+        g,
+    )
+
+
+def main():
+    analytic = gradient_cost(analytic=True)
+    fd = gradient_cost(analytic=False)
+    rel = np.linalg.norm(analytic.gradient - fd.gradient) / np.linalg.norm(fd.gradient)
+    n_boundary = analytic.gradient.size
+    print(f"boundary gradient cost ({n_boundary} boundary DOFs, solovev ns=11)\n")
+    print(f"{'method':20s} {'F-evals':>9s} {'time[s]':>8s}")
+    for r in (fd, analytic):
+        print(f"{r.method:20s} {r.force_evals:9d} {r.seconds:8.2f}")
+    print(
+        f"\nspeedup (force evals): {fd.force_evals / max(analytic.force_evals, 1):.1f}x"
+        f"   gradient agreement: {rel:.1e}"
+    )
+
+
+if __name__ == "__main__":
+    main()
diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
new file mode 100644
index 000000000..e4b27f3bf
--- /dev/null
+++ b/examples/vmecpp_adjoint.py
@@ -0,0 +1,169 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Adjoint sensitivity of a converged VMEC++ equilibrium to its boundary.
+
+A fixed-boundary equilibrium satisfies the interior force balance F_I(x) = 0,
+where x is the decomposed internal-basis state and F is the gradient of VMEC's
+augmented functional. The outermost flux surface (the boundary) is the last
+radial block of the state and is held fixed during the solve. For a scalar
+objective J(x), the sensitivity to the boundary degrees of freedom follows from
+the implicit function theorem:
+
+    dJ/dx_B = dJ/dx_B|_x - (dF_I/dx_B)^T lambda,   H_II lambda = dJ/dx_I,
+
+with H = dF/dx the (symmetric) Hessian of the augmented functional. Every
+operator is matrix-free and already exposed by VmecModel: the Hessian-vector
+product (``hessian_vector_product``) and the preconditioner
+(``apply_preconditioner``), used to solve the adjoint system. Only one Hessian
+solve is needed for the full boundary gradient, versus one equilibrium re-solve
+per boundary degree of freedom for finite differences.
+"""
+
+from __future__ import annotations
+
+from pathlib import Path
+
+import numpy as np
+from scipy.sparse.linalg import LinearOperator, cg
+
+try:
+    from vmecpp.cpp import _vmecpp
+except ImportError:
+    import _vmecpp
+
+DEFAULT_INPUT = (
+    Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+)
+
+
+def make_model(input_path: Path = DEFAULT_INPUT, ns: int = 11):
+    return _vmecpp.VmecModel.create(_vmecpp.VmecINDATA.from_file(str(input_path)), ns)
+
+
+def partition(model, ns: int):
+    """Indices of the interior (free) and boundary (LCFS) state components."""
+    k = model.mpol * (model.ntor + 1)
+    n = np.asarray(model.get_state()).size
+    per_span = ns * k
+    n_span = n // per_span
+    boundary = []
+    for s in range(n_span):
+        boundary.extend(range(s * per_span + (ns - 1) * k, s * per_span + ns * k))
+    boundary = np.array(sorted(boundary))
+    interior = np.setdiff1d(np.arange(n), boundary)
+    return interior, boundary
+
+
+def _raw_force(model, x):
+    model.set_state(np.ascontiguousarray(x))
+    model.evaluate(2, 2, False)
+    return np.asarray(model.get_forces(), float)
+
+
+def _interior_operators(model, x, interior):
+    # The caller must set the base state to x and assemble the preconditioner
+    # (evaluate(2, 2, True)) before using these. hessian_vector_product uses the
+    # current state as its base point and restores it, so no per-matvec state
+    # update is needed: that keeps each Hessian matvec at two force evaluations.
+    n = x.size
+    ni = interior.size
+
+    def hii(vi):
+        v = np.zeros(n)
+        v[interior] = vi
+        return np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)[
+            interior
+        ]
+
+    def mii(bi):
+        v = np.zeros(n)
+        v[interior] = bi
+        return np.asarray(model.apply_preconditioner(np.ascontiguousarray(v)), float)[
+            interior
+        ]
+
+    return (
+        LinearOperator((ni, ni), matvec=hii),
+        LinearOperator((ni, ni), matvec=mii),
+    )
+
+
+def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=40):
+    """Converge the interior to force balance with the boundary held fixed."""
+    x = np.asarray(x0, float).copy()
+    x[boundary] = x_boundary
+    for _ in range(max_newton):
+        f = _raw_force(model, x)
+        if np.linalg.norm(f[interior]) < tol:
+            break
+        model.set_state(np.ascontiguousarray(x))
+        model.evaluate(2, 2, True)  # assemble preconditioner + set base state
+        h_op, m_op = _interior_operators(model, x, interior)
+        dxi, _ = cg(h_op, -f[interior], M=m_op, rtol=1e-4, maxiter=200)
+        x[interior] += dxi
+    return x
+
+
+def objective_state_gradient(model, x, objective, h=1e-6):
+    """Partial derivative dJ/dx at fixed state, by central finite differences."""
+    n = x.size
+    g = np.zeros(n)
+    for i in range(n):
+        xp = x.copy()
+        xp[i] += h
+        model.set_state(np.ascontiguousarray(xp))
+        model.evaluate(2, 2, False)
+        jp = objective(model)
+        xm = x.copy()
+        xm[i] -= h
+        model.set_state(np.ascontiguousarray(xm))
+        model.evaluate(2, 2, False)
+        jm = objective(model)
+        g[i] = (jp - jm) / (2 * h)
+    return g
+
+
+def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6):
+    """Adjoint gradient dJ/dx_B at the converged equilibrium x_star."""
+    n = x_star.size
+    dj = objective_state_gradient(model, x_star, objective, h)
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, True)  # assemble preconditioner + set base state to x_star
+    h_op, m_op = _interior_operators(model, x_star, interior)
+    lam, _ = cg(h_op, dj[interior], M=m_op, rtol=1e-7, maxiter=400)
+    embedded = np.zeros(n)
+    embedded[interior] = lam
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, False)
+    coupling = np.asarray(
+        model.hessian_vector_product(np.ascontiguousarray(embedded)), float
+    )[boundary]
+    return dj[boundary] - coupling
+
+
+def finite_difference_boundary_gradient(
+    model, x_star, interior, boundary, objective, dofs, h=1e-5
+):
+    """Reference gradient: re-solve the interior for each perturbed boundary DOF."""
+    g = {}
+    for j in dofs:
+        xbp = x_star[boundary].copy()
+        xbp[j] += h
+        xp = solve_interior(model, x_star, interior, boundary, xbp)
+        model.set_state(np.ascontiguousarray(xp))
+        model.evaluate(2, 2, False)
+        jp = objective(model)
+        xbm = x_star[boundary].copy()
+        xbm[j] -= h
+        xm = solve_interior(model, x_star, interior, boundary, xbm)
+        model.set_state(np.ascontiguousarray(xm))
+        model.evaluate(2, 2, False)
+        jm = objective(model)
+        g[j] = (jp - jm) / (2 * h)
+    return g
+
+
+def mhd_energy(model):
+    return model.mhd_energy
diff --git a/tests/test_adjoint.py b/tests/test_adjoint.py
new file mode 100644
index 000000000..0d740d6be
--- /dev/null
+++ b/tests/test_adjoint.py
@@ -0,0 +1,51 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""The adjoint boundary gradient matches brute-force finite differences.
+
+dJ/d(boundary) from one Hessian solve (implicit-function adjoint) agrees with the
+reference gradient obtained by re-converging the interior equilibrium for each
+perturbed boundary degree of freedom. J here is the MHD energy of the converged
+equilibrium.
+"""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+
+sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
+from vmecpp_adjoint import (
+    boundary_gradient,
+    finite_difference_boundary_gradient,
+    make_model,
+    mhd_energy,
+    partition,
+)
+
+
+def test_adjoint_matches_finite_difference():
+    ns = 11
+    model = make_model(ns=ns)
+    model.solve()
+    x_star = np.asarray(model.get_state(), float).copy()
+    interior, boundary = partition(model, ns)
+
+    g_adjoint = boundary_gradient(model, x_star, interior, boundary, mhd_energy)
+
+    # Reference on a representative subset (each requires interior re-solves).
+    dofs = [0, 2, 9]
+    g_fd = finite_difference_boundary_gradient(
+        model, x_star, interior, boundary, mhd_energy, dofs
+    )
+
+    scale = max(np.linalg.norm(g_adjoint), 1e-30)
+    for j in dofs:
+        assert abs(g_adjoint[j] - g_fd[j]) < 1e-3 * scale
+
+
+if __name__ == "__main__":
+    import pytest
+
+    raise SystemExit(pytest.main([__file__, "-v"]))
diff --git a/tests/test_simsopt_gradient.py b/tests/test_simsopt_gradient.py
new file mode 100644
index 000000000..4080079fe
--- /dev/null
+++ b/tests/test_simsopt_gradient.py
@@ -0,0 +1,57 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""VMEC++ exposes an analytic boundary gradient to SIMSOPT.
+
+The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches
+finite differences of its objective, and computing it is much cheaper than the
+conventional finite-difference boundary gradient (which re-solves the
+equilibrium per boundary degree of freedom).
+"""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+import pytest
+
+sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
+
+pytest.importorskip("simsopt")
+
+from simsopt_vmec_gradient import (
+    VmecBoundaryProblem,
+    gradient_cost,
+    make_simsopt_optimizable,
+)
+
+
+def test_simsopt_optimizable_gradient_matches_fd():
+    problem = VmecBoundaryProblem(ns=11)
+    opt = make_simsopt_optimizable(problem)
+    g = np.asarray(opt.dJ(), float)
+
+    p0 = np.asarray(opt.local_full_x, float)
+    h = 1e-5
+    scale = max(np.linalg.norm(g), 1e-30)
+    for j in (0, 2, 9):
+        pp = p0.copy()
+        pp[j] += h
+        opt.local_full_x = pp
+        jp = opt.J()
+        pm = p0.copy()
+        pm[j] -= h
+        opt.local_full_x = pm
+        jm = opt.J()
+        opt.local_full_x = p0
+        assert abs(g[j] - (jp - jm) / (2 * h)) < 1e-3 * scale
+
+
+def test_adjoint_gradient_cheaper_than_finite_difference():
+    analytic = gradient_cost(analytic=True)
+    fd = gradient_cost(analytic=False)
+    # Same gradient, far fewer force evaluations (advantage grows with #DOFs).
+    rel = np.linalg.norm(analytic.gradient - fd.gradient) / np.linalg.norm(fd.gradient)
+    assert rel < 1e-2
+    assert analytic.force_evals < fd.force_evals

From 8a319b663928a83b7a55d349fbe12fdc5c93e0d0 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 10:52:21 +0200
Subject: [PATCH 02/79] examples: fix adjoint on 3D (GMRES for indefinite
 Hessian) + globalize

Two correctness fixes for stiff 3D equilibria (cth_like):

- VMEC's augmented-Lagrangian Hessian is symmetric *indefinite* (the lambda
  constraint makes it a saddle, not a minimum), so CG silently gives the
  wrong adjoint there. Use GMRES, which handles indefinite systems, for the
  H_II solve and the interior Newton solve. With a loose, restarted tolerance
  the adjoint solve stays cheap.
- Add a backtracking line search to solve_interior so the interior re-solve
  (used by the SIMSOPT wrapper and the finite-difference reference) converges
  on 3D instead of overshooting.

Verified with a directional-derivative check against a re-converged
finite-difference reference: solovev 1.5e-4, cth_like 2.2e-2 relative; both
previously agreed only in 2D. Boundary-gradient cost on solovev: 626 force
evaluations (analytic adjoint) versus 10460 (finite differences).
---
 examples/vmecpp_adjoint.py | 29 ++++++++++++++++++++++-------
 1 file changed, 22 insertions(+), 7 deletions(-)

diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index e4b27f3bf..a85ea2006 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -26,7 +26,7 @@
 from pathlib import Path
 
 import numpy as np
-from scipy.sparse.linalg import LinearOperator, cg
+from scipy.sparse.linalg import LinearOperator, gmres
 
 try:
     from vmecpp.cpp import _vmecpp
@@ -90,19 +90,34 @@ def mii(bi):
     )
 
 
-def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=40):
-    """Converge the interior to force balance with the boundary held fixed."""
+def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=80):
+    """Converge the interior to force balance with the boundary held fixed.
+
+    Preconditioned Newton-Krylov on the interior residual with a backtracking
+    line search; the line search is required for stiff 3D equilibria, where the
+    full Newton step overshoots.
+    """
     x = np.asarray(x0, float).copy()
     x[boundary] = x_boundary
     for _ in range(max_newton):
         f = _raw_force(model, x)
-        if np.linalg.norm(f[interior]) < tol:
+        norm0 = np.linalg.norm(f[interior])
+        if norm0 < tol:
             break
         model.set_state(np.ascontiguousarray(x))
         model.evaluate(2, 2, True)  # assemble preconditioner + set base state
         h_op, m_op = _interior_operators(model, x, interior)
-        dxi, _ = cg(h_op, -f[interior], M=m_op, rtol=1e-4, maxiter=200)
-        x[interior] += dxi
+        dxi, _ = gmres(h_op, -f[interior], M=m_op, rtol=1e-4, maxiter=300)
+        alpha = 1.0
+        for _ in range(30):
+            xt = x.copy()
+            xt[interior] += alpha * dxi
+            if np.linalg.norm(_raw_force(model, xt)[interior]) < norm0:
+                break
+            alpha *= 0.5
+        else:
+            break  # no decrease found; stop
+        x[interior] += alpha * dxi
     return x
 
 
@@ -132,7 +147,7 @@ def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6):
     model.set_state(np.ascontiguousarray(x_star))
     model.evaluate(2, 2, True)  # assemble preconditioner + set base state to x_star
     h_op, m_op = _interior_operators(model, x_star, interior)
-    lam, _ = cg(h_op, dj[interior], M=m_op, rtol=1e-7, maxiter=400)
+    lam, _ = gmres(h_op, dj[interior], M=m_op, rtol=1e-6, restart=100, maxiter=30)
     embedded = np.zeros(n)
     embedded[interior] = lam
     model.set_state(np.ascontiguousarray(x_star))

From 11cfee695206c3aedccf5aa5ded86809be73e76d Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 13:05:20 +0200
Subject: [PATCH 03/79] ideal_mhd_model: share the local force-density
 composition

Move the composed local force map g (MHD force chain + hybrid lambda force)
into local_force_composition.h (ComputeLocalForceDensity), parameterized by
the radial partition offsets so the same composition serves the Enzyme
autodiff test and the exact Hessian-vector product over the live model state.

The autodiff test now calls the shared composition; forward/reverse vs
finite-diff and forward/reverse agreement are unchanged (2.55e-8, 4e-15).
---
 .../common/enzyme/local_force_hessian_test.cc | 135 +++-----------
 .../ideal_mhd_model/local_force_composition.h | 164 ++++++++++++++++++
 2 files changed, 185 insertions(+), 114 deletions(-)
 create mode 100644 src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h

diff --git a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
index ad41be232..27613b032 100644
--- a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
+++ b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
@@ -31,13 +31,7 @@
 #include <vector>
 
 #include "vmecpp/common/enzyme/enzyme.h"
-#include "vmecpp/vmec/ideal_mhd_model/bco_kernel.h"
-#include "vmecpp/vmec/ideal_mhd_model/bcontra_kernel.h"
-#include "vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h"
-#include "vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h"
-#include "vmecpp/vmec/ideal_mhd_model/metric_kernel.h"
-#include "vmecpp/vmec/ideal_mhd_model/mhdforce_kernel.h"
-#include "vmecpp/vmec/ideal_mhd_model/pressure_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
 
 // Problem dimensions and the constant (non-differentiated) context.
 struct Ctx {
@@ -61,113 +55,26 @@ enum { kGeomBlocks = 16, kForceBlocks = 16 };
 // g: geometry -> force density, composing the MHD and lambda-force kernels.
 __attribute__((noinline)) void LocalForce(const double* geom, double* work,
                                            double* force, const Ctx* c) {
-  const int nF = c->nFull;
-  const int nH = c->nHalf;
-  const int nZnT = c->nZnT, nsH = c->nsH;
-  // geometry blocks
-  const double* r1e = geom + 0 * nF;
-  const double* r1o = geom + 1 * nF;
-  const double* z1e = geom + 2 * nF;
-  const double* z1o = geom + 3 * nF;
-  const double* rue = geom + 4 * nF;
-  const double* ruo = geom + 5 * nF;
-  const double* zue = geom + 6 * nF;
-  const double* zuo = geom + 7 * nF;
-  const double* rve = geom + 8 * nF;
-  const double* rvo = geom + 9 * nF;
-  const double* zve = geom + 10 * nF;
-  const double* zvo = geom + 11 * nF;
-  const double* lue = geom + 12 * nF;
-  const double* luo = geom + 13 * nF;
-  const double* lve = geom + 14 * nF;
-  const double* lvo = geom + 15 * nF;
-  // half-grid intermediates from work
-  double* p = work;
-  double* r12 = p; p += nH;
-  double* ru12 = p; p += nH;
-  double* zu12 = p; p += nH;
-  double* rs = p; p += nH;
-  double* zs = p; p += nH;
-  double* tau = p; p += nH;
-  double* gsqrt = p; p += nH;
-  double* guu = p; p += nH;
-  double* guv = p; p += nH;
-  double* gvv = p; p += nH;
-  double* bsupu = p; p += nH;
-  double* bsupv = p; p += nH;
-  double* bsubu = p; p += nH;
-  double* bsubv = p; p += nH;
-  double* tp = p; p += nH;
-  // per-nZnT scratch for the force kernel (26 blocks)
-  double* sc = p;  // 26 * nZnT
-
-  vmecpp::ComputeHalfGridJacobian(r1e, r1o, z1e, z1o, rue, ruo, zue, zuo, c->sqrtSH,
-                          c->deltaS, /*dSHalfDsInterp=*/0.25, nZnT, 0, 0, nsH,
-                          r12, ru12, zu12, rs, zs, tau);
-  vmecpp::ComputeMetricElements(r1e, r1o, rue, ruo, zue, zuo, rve, rvo, zve, zvo,
-                                tau, r12, c->sqrtSF, c->sqrtSH, c->lthreed, nZnT,
-                                0, 0, nsH, gsqrt, guu, guv, gvv);
-  vmecpp::ComputeBsupContra(lue, luo, lve, lvo, gsqrt, c->sqrtSH, c->lthreed,
-                            nZnT, 0, 0, nsH, bsupu, bsupv);
-  for (int jH = 0; jH < nsH; ++jH) {
-    for (int kl = 0; kl < nZnT; ++kl) {
-      const int ih = jH * nZnT + kl;
-      bsupu[ih] += c->chipH[jH] / gsqrt[ih];
-    }
-  }
-  vmecpp::ComputeBCo(guu, guv, gvv, bsupu, bsupv, c->lthreed, nH, bsubu, bsubv);
-  vmecpp::ComputeMagneticPressure(bsupu, bsubu, bsupv, bsubv, nH, tp);
-  for (int jH = 0; jH < nsH; ++jH) {
-    for (int kl = 0; kl < nZnT; ++kl) tp[jH * nZnT + kl] += c->presH[jH];
-  }
-  double* s = sc;
-  double* P_i = s; s += nZnT; double* rup_i = s; s += nZnT;
-  double* zup_i = s; s += nZnT; double* rsp_i = s; s += nZnT;
-  double* zsp_i = s; s += nZnT; double* taup_i = s; s += nZnT;
-  double* gbubu_i = s; s += nZnT; double* gbubv_i = s; s += nZnT;
-  double* gbvbv_i = s; s += nZnT; double* P_o = s; s += nZnT;
-  double* rup_o = s; s += nZnT; double* zup_o = s; s += nZnT;
-  double* rsp_o = s; s += nZnT; double* zsp_o = s; s += nZnT;
-  double* taup_o = s; s += nZnT; double* gbubu_o = s; s += nZnT;
-  double* gbubv_o = s; s += nZnT; double* gbvbv_o = s; s += nZnT;
-  double* P_avg = s; s += nZnT; double* P_wavg = s; s += nZnT;
-  double* gbubu_avg = s; s += nZnT; double* gbubu_wavg = s; s += nZnT;
-  double* gbvbv_avg = s; s += nZnT; double* gbvbv_wavg = s; s += nZnT;
-  double* gbubv_avg = s; s += nZnT; double* gbubv_wavg = s; s += nZnT;
-  // lambda-force radial-sweep scratch (carried inside half-grid point)
-  double* bsubu_i = s; s += nZnT; double* bsubv_i = s; s += nZnT;
-  double* gvv_gsqrt_i = s; s += nZnT; double* guv_bsupu_i = s; s += nZnT;
-  double* armn_e = force + 0 * nH;
-  double* armn_o = force + 1 * nH;
-  double* azmn_e = force + 2 * nH;
-  double* azmn_o = force + 3 * nH;
-  double* brmn_e = force + 4 * nH;
-  double* brmn_o = force + 5 * nH;
-  double* bzmn_e = force + 6 * nH;
-  double* bzmn_o = force + 7 * nH;
-  double* crmn_e = force + 8 * nH;
-  double* crmn_o = force + 9 * nH;
-  double* czmn_e = force + 10 * nH;
-  double* czmn_o = force + 11 * nH;
-  vmecpp::ComputeMHDForceDensity(
-      r1e, r1o, rue, ruo, zue, zuo, z1o, rve, rvo, zve, zvo, r12, ru12, zu12, rs,
-      zs, tau, tp, gsqrt, bsupu, bsupv, c->sqrtSF, c->sqrtSH, P_i, rup_i, zup_i,
-      rsp_i, zsp_i, taup_i, gbubu_i, gbubv_i, gbvbv_i, P_o, rup_o, zup_o, rsp_o,
-      zsp_o, taup_o, gbubu_o, gbubv_o, gbvbv_o, P_avg, P_wavg, gbubu_avg,
-      gbubu_wavg, gbvbv_avg, gbvbv_wavg, gbubv_avg, gbubv_wavg, c->deltaS, nZnT,
-      /*nsMinF=*/0, 0, 0, nsH, /*jMaxRZ=*/nsH, c->lthreed, armn_e, armn_o,
-      azmn_e, azmn_o, brmn_e, brmn_o, bzmn_e, bzmn_o, crmn_e, crmn_o, czmn_e,
-      czmn_o);
-  // lambda force (blmn_e/o, clmn_e/o) from the shared kernel
-  double* blmn_e = force + 12 * nH;
-  double* blmn_o = force + 13 * nH;
-  double* clmn_e = force + 14 * nH;
-  double* clmn_o = force + 15 * nH;
-  vmecpp::ComputeHybridLambdaForce(
-      bsubu, bsubv, gvv, gsqrt, guv, bsupu, lue, luo, c->sqrtSH, c->sqrtSF,
-      c->radialBlending, c->lamscale, c->lthreed, nZnT, /*nsMinF=*/0,
-      /*nsMinF1=*/0, /*nsMinH=*/0, nsH, /*nsMaxFIncludingLcfs=*/nsH, bsubu_i,
-      bsubv_i, gvv_gsqrt_i, guv_bsupu_i, blmn_e, blmn_o, clmn_e, clmn_o);
+  vmecpp::LocalForceComposition lc;
+  lc.nZnT = c->nZnT;
+  lc.geom_stride = c->nFull;
+  lc.force_stride = c->nHalf;
+  lc.nsMinF = 0;
+  lc.nsMinF1 = 0;
+  lc.nsMinH = 0;
+  lc.nsMaxH = c->nsH;
+  lc.jMaxRZ = c->nsH;
+  lc.nsMaxFIncludingLcfs = c->nsH;
+  lc.sqrtSF = c->sqrtSF;
+  lc.sqrtSH = c->sqrtSH;
+  lc.chipH = c->chipH;
+  lc.presH = c->presH;
+  lc.radialBlending = c->radialBlending;
+  lc.deltaS = c->deltaS;
+  lc.dSHalfDsInterp = 0.25;
+  lc.lamscale = c->lamscale;
+  lc.lthreed = c->lthreed;
+  vmecpp::ComputeLocalForceDensity(geom, work, force, lc);
 }
 
 // Scalar objective L = 0.5 ||force||^2; work and force are caller-owned scratch.
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
new file mode 100644
index 000000000..cb670d455
--- /dev/null
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -0,0 +1,164 @@
+// SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+// <info@proximafusion.com>
+//
+// SPDX-License-Identifier: MIT
+#ifndef VMECPP_VMEC_IDEAL_MHD_MODEL_LOCAL_FORCE_COMPOSITION_H_
+#define VMECPP_VMEC_IDEAL_MHD_MODEL_LOCAL_FORCE_COMPOSITION_H_
+
+#include "vmecpp/vmec/ideal_mhd_model/bco_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/bcontra_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/metric_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/mhdforce_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/pressure_kernel.h"
+
+namespace vmecpp {
+
+// Composition of the local force-density chain (MHD force + hybrid lambda force)
+// as a single allocation-free map g: real-space geometry -> real-space force
+// density. This is the nonlinear core of VMEC's force; the spectral transforms
+// around it are linear and applied separately. Shared between the Enzyme
+// autodiff validation and the exact Hessian-vector product. The spectral-
+// condensation constraint force is handled by the caller (it carries a linear
+// Fourier bandpass), so g here is the MHD-plus-lambda part of the force.
+//
+// Geometry layout (each block GeomStride doubles, index (jF-nsMinF1)*nZnT):
+//   r1_e r1_o z1_e z1_o ru_e ru_o zu_e zu_o rv_e rv_o zv_e zv_o lu_e lu_o lv_e lv_o
+// Force layout (each block ForceStride doubles): the 12 MHD densities then
+//   blmn_e blmn_o clmn_e clmn_o.
+struct LocalForceComposition {
+  int nZnT;
+  int geom_stride;   // doubles per geometry block (>= (nsMaxF1-nsMinF1)*nZnT)
+  int force_stride;  // doubles per force block (>= (nsMaxFIncludingLcfs-nsMinF)*nZnT)
+  int nsMinF, nsMinF1, nsMinH, nsMaxH;
+  int jMaxRZ;                 // MHD force surfaces: [nsMinF, jMaxRZ)
+  int nsMaxFIncludingLcfs;    // lambda force surfaces: [nsMinF, nsMaxFIncludingLcfs)
+  const double* sqrtSF;       // index jF-nsMinF1
+  const double* sqrtSH;       // index jH-nsMinH
+  const double* chipH;        // index jH-nsMinH (frozen for ncurr==0)
+  const double* presH;        // index jH-nsMinH
+  const double* radialBlending;  // index jF-nsMinF1
+  double deltaS;
+  double dSHalfDsInterp;
+  double lamscale;
+  bool lthreed;
+};
+
+// work must hold 15*nHalf + 30*nZnT doubles, where nHalf=(nsMaxH-nsMinH)*nZnT.
+inline void ComputeLocalForceDensity(const double* geom, double* work,
+                                     double* force,
+                                     const LocalForceComposition& c) {
+  const int nZnT = c.nZnT;
+  const int gS = c.geom_stride;
+  const int fS = c.force_stride;
+  const int nH = (c.nsMaxH - c.nsMinH) * nZnT;
+  const double* r1e = geom + 0 * gS;
+  const double* r1o = geom + 1 * gS;
+  const double* z1e = geom + 2 * gS;
+  const double* z1o = geom + 3 * gS;
+  const double* rue = geom + 4 * gS;
+  const double* ruo = geom + 5 * gS;
+  const double* zue = geom + 6 * gS;
+  const double* zuo = geom + 7 * gS;
+  const double* rve = geom + 8 * gS;
+  const double* rvo = geom + 9 * gS;
+  const double* zve = geom + 10 * gS;
+  const double* zvo = geom + 11 * gS;
+  const double* lue = geom + 12 * gS;
+  const double* luo = geom + 13 * gS;
+  const double* lve = geom + 14 * gS;
+  const double* lvo = geom + 15 * gS;
+
+  double* p = work;
+  double* r12 = p; p += nH;
+  double* ru12 = p; p += nH;
+  double* zu12 = p; p += nH;
+  double* rs = p; p += nH;
+  double* zs = p; p += nH;
+  double* tau = p; p += nH;
+  double* gsqrt = p; p += nH;
+  double* guu = p; p += nH;
+  double* guv = p; p += nH;
+  double* gvv = p; p += nH;
+  double* bsupu = p; p += nH;
+  double* bsupv = p; p += nH;
+  double* bsubu = p; p += nH;
+  double* bsubv = p; p += nH;
+  double* tp = p; p += nH;
+  double* s = p;  // 30 * nZnT
+
+  ComputeHalfGridJacobian(r1e, r1o, z1e, z1o, rue, ruo, zue, zuo, c.sqrtSH,
+                          c.deltaS, c.dSHalfDsInterp, nZnT, c.nsMinF1, c.nsMinH,
+                          c.nsMaxH, r12, ru12, zu12, rs, zs, tau);
+  ComputeMetricElements(r1e, r1o, rue, ruo, zue, zuo, rve, rvo, zve, zvo, tau,
+                        r12, c.sqrtSF, c.sqrtSH, c.lthreed, nZnT, c.nsMinF1,
+                        c.nsMinH, c.nsMaxH, gsqrt, guu, guv, gvv);
+  ComputeBsupContra(lue, luo, lve, lvo, gsqrt, c.sqrtSH, c.lthreed, nZnT,
+                    c.nsMinF1, c.nsMinH, c.nsMaxH, bsupu, bsupv);
+  for (int jH = c.nsMinH; jH < c.nsMaxH; ++jH) {
+    for (int kl = 0; kl < nZnT; ++kl) {
+      const int ih = (jH - c.nsMinH) * nZnT + kl;
+      bsupu[ih] += c.chipH[jH - c.nsMinH] / gsqrt[ih];
+    }
+  }
+  ComputeBCo(guu, guv, gvv, bsupu, bsupv, c.lthreed, nH, bsubu, bsubv);
+  ComputeMagneticPressure(bsupu, bsubu, bsupv, bsubv, nH, tp);
+  for (int jH = c.nsMinH; jH < c.nsMaxH; ++jH) {
+    for (int kl = 0; kl < nZnT; ++kl)
+      tp[(jH - c.nsMinH) * nZnT + kl] += c.presH[jH - c.nsMinH];
+  }
+
+  double* P_i = s; s += nZnT; double* rup_i = s; s += nZnT;
+  double* zup_i = s; s += nZnT; double* rsp_i = s; s += nZnT;
+  double* zsp_i = s; s += nZnT; double* taup_i = s; s += nZnT;
+  double* gbubu_i = s; s += nZnT; double* gbubv_i = s; s += nZnT;
+  double* gbvbv_i = s; s += nZnT; double* P_o = s; s += nZnT;
+  double* rup_o = s; s += nZnT; double* zup_o = s; s += nZnT;
+  double* rsp_o = s; s += nZnT; double* zsp_o = s; s += nZnT;
+  double* taup_o = s; s += nZnT; double* gbubu_o = s; s += nZnT;
+  double* gbubv_o = s; s += nZnT; double* gbvbv_o = s; s += nZnT;
+  double* P_avg = s; s += nZnT; double* P_wavg = s; s += nZnT;
+  double* gbubu_avg = s; s += nZnT; double* gbubu_wavg = s; s += nZnT;
+  double* gbvbv_avg = s; s += nZnT; double* gbvbv_wavg = s; s += nZnT;
+  double* gbubv_avg = s; s += nZnT; double* gbubv_wavg = s; s += nZnT;
+  double* bsubu_i = s; s += nZnT; double* bsubv_i = s; s += nZnT;
+  double* gvv_gsqrt_i = s; s += nZnT; double* guv_bsupu_i = s; s += nZnT;
+
+  double* armn_e = force + 0 * fS;
+  double* armn_o = force + 1 * fS;
+  double* azmn_e = force + 2 * fS;
+  double* azmn_o = force + 3 * fS;
+  double* brmn_e = force + 4 * fS;
+  double* brmn_o = force + 5 * fS;
+  double* bzmn_e = force + 6 * fS;
+  double* bzmn_o = force + 7 * fS;
+  double* crmn_e = force + 8 * fS;
+  double* crmn_o = force + 9 * fS;
+  double* czmn_e = force + 10 * fS;
+  double* czmn_o = force + 11 * fS;
+  ComputeMHDForceDensity(
+      r1e, r1o, rue, ruo, zue, zuo, z1o, rve, rvo, zve, zvo, r12, ru12, zu12, rs,
+      zs, tau, tp, gsqrt, bsupu, bsupv, c.sqrtSF, c.sqrtSH, P_i, rup_i, zup_i,
+      rsp_i, zsp_i, taup_i, gbubu_i, gbubv_i, gbvbv_i, P_o, rup_o, zup_o, rsp_o,
+      zsp_o, taup_o, gbubu_o, gbubv_o, gbvbv_o, P_avg, P_wavg, gbubu_avg,
+      gbubu_wavg, gbvbv_avg, gbvbv_wavg, gbubv_avg, gbubv_wavg, c.deltaS, nZnT,
+      c.nsMinF, c.nsMinF1, c.nsMinH, c.nsMaxH, c.jMaxRZ, c.lthreed, armn_e,
+      armn_o, azmn_e, azmn_o, brmn_e, brmn_o, bzmn_e, bzmn_o, crmn_e, crmn_o,
+      czmn_e, czmn_o);
+
+  double* blmn_e = force + 12 * fS;
+  double* blmn_o = force + 13 * fS;
+  double* clmn_e = force + 14 * fS;
+  double* clmn_o = force + 15 * fS;
+  ComputeHybridLambdaForce(bsubu, bsubv, gvv, gsqrt, guv, bsupu, lue, luo,
+                           c.sqrtSH, c.sqrtSF, c.radialBlending, c.lamscale,
+                           c.lthreed, nZnT, c.nsMinF, c.nsMinF1, c.nsMinH,
+                           c.nsMaxH, c.nsMaxFIncludingLcfs, bsubu_i, bsubv_i,
+                           gvv_gsqrt_i, guv_bsupu_i, blmn_e, blmn_o, clmn_e,
+                           clmn_o);
+}
+
+}  // namespace vmecpp
+
+#endif  // VMECPP_VMEC_IDEAL_MHD_MODEL_LOCAL_FORCE_COMPOSITION_H_

From f72c2812c0c1ef4a23e668da32f3b85ca51f1e80 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 13:13:16 +0200
Subject: [PATCH 04/79] enzyme: plugin JVP entry point for the exact force
 Hessian

Add exact_force_jvp.cc/.h: ExactForceDensityJvp wraps one Enzyme forward
pass over ComputeLocalForceDensity, returning the force-density tangent for
a geometry tangent. This translation unit is compiled with the Clang/Enzyme
plugin and is the single nonlinear pass the exact Hessian-vector product
calls; the rest of VMEC++ stays normally compiled and wraps it with the
linear spectral transforms.

The autodiff test now also validates this standalone entry point against a
finite difference of the composition's force-density output.
---
 CMakeLists.txt                                |  3 +-
 .../common/enzyme/local_force_hessian_test.cc | 42 ++++++++++++++++++-
 .../vmec/ideal_mhd_model/exact_force_jvp.cc   | 27 ++++++++++++
 .../vmec/ideal_mhd_model/exact_force_jvp.h    | 23 ++++++++++
 4 files changed, 92 insertions(+), 3 deletions(-)
 create mode 100644 src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
 create mode 100644 src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h

diff --git a/CMakeLists.txt b/CMakeLists.txt
index 813c0fe9c..b4fb0ac43 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -217,7 +217,8 @@ if(VMECPP_ENABLE_ENZYME)
   # Exact Hessian of the composed local force map (all six force-chain kernels):
   # forward/reverse Jacobian vs finite differences, and JVP cost vs FD-HVP.
   add_executable(local_force_hessian_test
-    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc)
+    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
+    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc)
   target_compile_options(local_force_hessian_test PRIVATE
     -O2 -fplugin=${VMECPP_ENZYME_PLUGIN})
   add_test(NAME local_force_hessian COMMAND local_force_hessian_test)
diff --git a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
index 27613b032..a052b9f4b 100644
--- a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
+++ b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
@@ -31,6 +31,7 @@
 #include <vector>
 
 #include "vmecpp/common/enzyme/enzyme.h"
+#include "vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h"
 #include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
 
 // Problem dimensions and the constant (non-differentiated) context.
@@ -74,7 +75,7 @@ __attribute__((noinline)) void LocalForce(const double* geom, double* work,
   lc.dSHalfDsInterp = 0.25;
   lc.lamscale = c->lamscale;
   lc.lthreed = c->lthreed;
-  vmecpp::ComputeLocalForceDensity(geom, work, force, lc);
+  vmecpp::ComputeLocalForceDensity(geom, work, force, &lc);
 }
 
 // Scalar objective L = 0.5 ||force||^2; work and force are caller-owned scratch.
@@ -176,9 +177,46 @@ int main() {
   printf("  cost: exact forward JVP %.1f us/pass vs FD-HVP %.1f us (2 evals)\n",
          us_ad, us_fd);
 
+  // Validate the standalone plugin JVP wrapper (exact_force_jvp.cc, the same
+  // entry point the exact Hessian-vector product calls) against a finite
+  // difference of the composition's force-density output.
+  vmecpp::LocalForceComposition lc;
+  lc.nZnT = c.nZnT;
+  lc.geom_stride = c.nFull;
+  lc.force_stride = c.nHalf;
+  lc.nsMinF = 0;
+  lc.nsMinF1 = 0;
+  lc.nsMinH = 0;
+  lc.nsMaxH = c.nsH;
+  lc.jMaxRZ = c.nsH;
+  lc.nsMaxFIncludingLcfs = c.nsH;
+  lc.sqrtSF = c.sqrtSF;
+  lc.sqrtSH = c.sqrtSH;
+  lc.chipH = c.chipH;
+  lc.presH = c.presH;
+  lc.radialBlending = c.radialBlending;
+  lc.deltaS = c.deltaS;
+  lc.dSHalfDsInterp = 0.25;
+  lc.lamscale = c.lamscale;
+  lc.lthreed = c.lthreed;
+  std::vector<double> jf(nFc, 0.0), jdf(nFc, 0.0), jw(nW, 0.0), jdw(nW, 0.0);
+  vmecpp::ExactForceDensityJvp(geom.data(), v.data(), jw.data(), jdw.data(),
+                               jf.data(), jdf.data(), &lc);
+  std::vector<double> fpf(nFc, 0.0), fmf(nFc, 0.0), fw(nW, 0.0);
+  vmecpp::ComputeLocalForceDensity(gp.data(), fw.data(), fpf.data(), &lc);
+  vmecpp::ComputeLocalForceDensity(gm.data(), fw.data(), fmf.data(), &lc);
+  double jvp_err = 0.0, jvp_scale = 1e-300;
+  for (int i = 0; i < nFc; ++i) {
+    const double fd = (fpf[i] - fmf[i]) / (2 * h);
+    jvp_err = std::max(jvp_err, std::fabs(jdf[i] - fd));
+    jvp_scale = std::max(jvp_scale, std::fabs(fd));
+  }
+  printf("  plugin JVP wrapper vs finite-diff : %.2e\n", jvp_err / jvp_scale);
+
   const bool ok = std::fabs(drev - dfd) < 1e-5 * scale &&
                   std::fabs(dfwd - dfd) < 1e-5 * scale &&
-                  std::fabs(dfwd - drev) < 1e-9 * (std::fabs(drev) + 1e-300);
+                  std::fabs(dfwd - drev) < 1e-9 * (std::fabs(drev) + 1e-300) &&
+                  jvp_err < 1e-5 * jvp_scale;
   printf("%s\n", ok ? "PASS" : "FAIL");
   return ok ? 0 : 1;
 }
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
new file mode 100644
index 000000000..e4b40a7d0
--- /dev/null
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
@@ -0,0 +1,27 @@
+// SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+// <info@proximafusion.com>
+//
+// SPDX-License-Identifier: MIT
+
+// Forward-mode Jacobian-vector product of the local force-density composition.
+// This translation unit is compiled with the Clang/Enzyme plugin so that the
+// __enzyme_fwddiff call differentiates ComputeLocalForceDensity. The rest of
+// VMEC++ is compiled normally; the exact Hessian-vector product calls into here
+// for the single nonlinear pass and wraps it with the linear spectral
+// transforms.
+
+#include "vmecpp/common/enzyme/enzyme.h"
+#include "vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h"
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
+
+namespace vmecpp {
+
+void ExactForceDensityJvp(const double* geom, const double* dgeom, double* work,
+                          double* dwork, double* force, double* dforce,
+                          const LocalForceComposition* c) {
+  __enzyme_fwddiff<void>((void*)ComputeLocalForceDensity, enzyme_dup, geom,
+                         dgeom, enzyme_dup, work, dwork, enzyme_dup, force,
+                         dforce, enzyme_const, c);
+}
+
+}  // namespace vmecpp
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h
new file mode 100644
index 000000000..66e3c46be
--- /dev/null
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h
@@ -0,0 +1,23 @@
+// SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+// <info@proximafusion.com>
+//
+// SPDX-License-Identifier: MIT
+#ifndef VMECPP_VMEC_IDEAL_MHD_MODEL_EXACT_FORCE_JVP_H_
+#define VMECPP_VMEC_IDEAL_MHD_MODEL_EXACT_FORCE_JVP_H_
+
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
+
+namespace vmecpp {
+
+// Forward-mode Jacobian-vector product of ComputeLocalForceDensity: given the
+// geometry primal and tangent, returns the force-density tangent in dforce in
+// one Enzyme forward pass. work/dwork and force/dforce are caller-owned scratch
+// sized as ComputeLocalForceDensity requires. Defined in exact_force_jvp.cc,
+// which is compiled with the Clang/Enzyme plugin.
+void ExactForceDensityJvp(const double* geom, const double* dgeom, double* work,
+                          double* dwork, double* force, double* dforce,
+                          const LocalForceComposition* c);
+
+}  // namespace vmecpp
+
+#endif  // VMECPP_VMEC_IDEAL_MHD_MODEL_EXACT_FORCE_JVP_H_

From 58bc8d0de917c38fc197f193d849027c88a9fc71 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 13:17:59 +0200
Subject: [PATCH 05/79] ideal_mhd_model: exact force Hessian-vector product
 wrapper

Add IdealMhdModel::applyExactForceJacobian: given the packed real-space
geometry primal and a geometry tangent, differentiate the MHD-plus-lambda
force density by one Enzyme forward pass (ExactForceDensityJvp), scatter the
tangent into the real-space force members, then apply the linear forward
transform and preconditioner decomposition (forcesToFourier, decomposeInto,
m1Constraint, zeroZForceForM1) exactly as the tail of update() does.

The geometry tangent is supplied by the caller via the linearity of
geometryFromFourier (geom(x+v) - geom(x)), so the only nonlinear step is the
single Enzyme pass. The constraint force (a linear Fourier bandpass over a
nonlinear product) is omitted here and added separately.

Compiles under clang-21; end-to-end exact-vs-finite-difference validation of
the assembled Hessian-vector product against VmecModel runs once the internal
solver stack (VmecModel HVP + preconditioner) is merged with this kernel
stack.
---
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   | 76 +++++++++++++++++++
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    | 12 +++
 2 files changed, 88 insertions(+)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 396b46680..6455cf25d 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -23,7 +23,9 @@
 #include "vmecpp/vmec/ideal_mhd_model/bco_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/bcontra_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h"
 #include "vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
 #include "vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/metric_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/mhdforce_kernel.h"
@@ -2091,6 +2093,80 @@ void IdealMhdModel::assembleTotalForces() {
                       fzcon_e.data(), fzcon_o.data());
 }
 
+void IdealMhdModel::applyExactForceJacobian(const double* geomP,
+                                            const double* dgeom, int geom_stride,
+                                            FourierForces& m_physical_f,
+                                            FourierForces& m_decomposed_hv) {
+  LocalForceComposition comp;
+  comp.nZnT = s_.nZnT;
+  comp.geom_stride = geom_stride;
+  const int nForce = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
+  comp.force_stride = nForce;
+  comp.nsMinF = r_.nsMinF;
+  comp.nsMinF1 = r_.nsMinF1;
+  comp.nsMinH = r_.nsMinH;
+  comp.nsMaxH = r_.nsMaxH;
+  comp.jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns - 1);
+  comp.nsMaxFIncludingLcfs = r_.nsMaxFIncludingLcfs;
+  comp.sqrtSF = m_p_.sqrtSF.data();
+  comp.sqrtSH = m_p_.sqrtSH.data();
+  comp.chipH = m_p_.chipH.data();
+  comp.presH = m_p_.presH.data();
+  comp.radialBlending = m_p_.radialBlending.data();
+  comp.deltaS = m_fc_.deltaS;
+  comp.dSHalfDsInterp = dSHalfDsInterp;
+  comp.lamscale = constants_.lamscale;
+  comp.lthreed = s_.lthreed;
+
+  const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
+  std::vector<double> work(15 * nH + 30 * s_.nZnT, 0.0);
+  std::vector<double> dwork(work.size(), 0.0);
+  std::vector<double> force(16 * nForce, 0.0);
+  std::vector<double> dforce(16 * nForce, 0.0);
+
+  // single nonlinear forward pass: J_g . (T v)
+  ExactForceDensityJvp(geomP, dgeom, work.data(), dwork.data(), force.data(),
+                       dforce.data(), &comp);
+
+  // scatter the force-density tangent into the real-space force members
+  auto put = [&](int block, Eigen::VectorXd& dst) {
+    const int n = static_cast<int>(dst.size());
+    for (int i = 0; i < n; ++i) {
+      dst[i] = dforce[block * nForce + i];
+    }
+  };
+  put(0, armn_e);
+  put(1, armn_o);
+  put(2, azmn_e);
+  put(3, azmn_o);
+  put(4, brmn_e);
+  put(5, brmn_o);
+  put(6, bzmn_e);
+  put(7, bzmn_o);
+  put(12, blmn_e);
+  put(13, blmn_o);
+  if (s_.lthreed) {
+    put(8, crmn_e);
+    put(9, crmn_o);
+    put(10, czmn_e);
+    put(11, czmn_o);
+    put(14, clmn_e);
+    put(15, clmn_o);
+  }
+  // constraint and free-boundary force tangents are not part of this pass
+  frcon_e.setZero();
+  frcon_o.setZero();
+  fzcon_e.setZero();
+  fzcon_o.setZero();
+
+  // linear forward transform and preconditioner decomposition, mirroring the
+  // tail of update()
+  forcesToFourier(m_physical_f);
+  m_physical_f.decomposeInto(m_decomposed_hv, m_p_.scalxc);
+  m_decomposed_hv.m1Constraint(1.0 / std::numbers::sqrt2);
+  m_decomposed_hv.zeroZForceForM1();
+}
+
 void IdealMhdModel::forcesToFourier(FourierForces& m_physical_f) {
   // symmetric contribution is always needed
   if (s_.lthreed) {
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index 71bc5b1eb..14c69fa35 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -155,6 +155,18 @@ class IdealMhdModel {
   // Fourier space.
   void forcesToFourier(FourierForces& m_physical_f);
 
+  // Exact Hessian-vector product of the local force chain. Given the packed
+  // real-space geometry primal geomP and a geometry tangent dgeom (each
+  // geom_stride doubles per block, see local_force_composition.h), differentiate
+  // the MHD-plus-lambda force density by one Enzyme forward pass, then apply the
+  // linear forward transform and preconditioner decomposition to obtain the
+  // decomposed force tangent in m_decomposed_hv. The constraint force is omitted
+  // here (it carries a linear bandpass handled separately). Used by the exact
+  // internal Newton-Krylov Hessian-vector product.
+  void applyExactForceJacobian(const double* geomP, const double* dgeom,
+                               int geom_stride, FourierForces& m_physical_f,
+                               FourierForces& m_decomposed_hv);
+
   // Computes the forward-DFT of forces for the 3D (Stellarator) case.
   // Dispatching dft_ForcesToFourier_3d_symm
   void dft_ForcesToFourier_3d_symm(FourierForces& m_physical_f);

From 63fbf6a3e8a75ddd9c6d7f5f21347adecf3a157a Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 13:23:34 +0200
Subject: [PATCH 06/79] pybind: exact Hessian-vector product end-to-end (MHD +
 lambda)

Wire the exact force Hessian-vector product through to Python:
VmecModel.exact_hessian_vector_product(v) computes H v = T^T J_g T v with one
Enzyme forward pass over the local force-density composition, using the
linearity of geometryFromFourier for the exact geometry tangent
(geom(x+v) - geom(x)) and the existing forward transform + decomposition for
the output. exact_force_jvp.cc is compiled into the core library with the
Enzyme plugin; VMECPP_ENABLE_ENZYME guards the callers so the default
plugin-free build is unchanged.

Built into the extension with clang-21 + Enzyme. The result is 96% cosine
aligned with the finite-difference HVP on solovev; the remaining difference is
the spectral-condensation constraint force, which is omitted from this pass and
added next (it carries a linear Fourier bandpass over a nonlinear product).
---
 CMakeLists.txt                                | 12 +++++
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   |  2 +
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 46 +++++++++++++++++++
 3 files changed, 60 insertions(+)

diff --git a/CMakeLists.txt b/CMakeLists.txt
index b4fb0ac43..9bfeb25df 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -222,4 +222,16 @@ if(VMECPP_ENABLE_ENZYME)
   target_compile_options(local_force_hessian_test PRIVATE
     -O2 -fplugin=${VMECPP_ENZYME_PLUGIN})
   add_test(NAME local_force_hessian COMMAND local_force_hessian_test)
+
+  # Wire the exact force Hessian-vector product into the core library and the
+  # Python extension: the single Enzyme forward pass lives in exact_force_jvp.cc
+  # (compiled with the plugin), the rest of the core stays normally compiled.
+  # VMECPP_ENABLE_ENZYME guards the callers so the default (plugin-free) build is
+  # unaffected.
+  target_sources(vmecpp_core PRIVATE
+    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc)
+  set_source_files_properties(
+    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
+    PROPERTIES COMPILE_OPTIONS "-O2;-fplugin=${VMECPP_ENZYME_PLUGIN}")
+  target_compile_definitions(vmecpp_core PUBLIC VMECPP_ENABLE_ENZYME)
 endif()
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 6455cf25d..e59a980a1 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -2093,6 +2093,7 @@ void IdealMhdModel::assembleTotalForces() {
                       fzcon_e.data(), fzcon_o.data());
 }
 
+#ifdef VMECPP_ENABLE_ENZYME
 void IdealMhdModel::applyExactForceJacobian(const double* geomP,
                                             const double* dgeom, int geom_stride,
                                             FourierForces& m_physical_f,
@@ -2166,6 +2167,7 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   m_decomposed_hv.m1Constraint(1.0 / std::numbers::sqrt2);
   m_decomposed_hv.zeroZForceForM1();
 }
+#endif  // VMECPP_ENABLE_ENZYME
 
 void IdealMhdModel::forcesToFourier(FourierForces& m_physical_f) {
   // symmetric contribution is always needed
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index 766b9d8de..e348ff755 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -416,6 +416,48 @@ class VmecModel {
     return (fp - fm) / (2.0 * eps);
   }
 
+#ifdef VMECPP_ENABLE_ENZYME
+  // Exact Hessian-vector product H v = T^T J_g T v of the augmented functional's
+  // MHD-plus-lambda part, computed with one Enzyme forward pass through the local
+  // force-density composition (J_g) wrapped by the linear spectral transforms.
+  // The geometry tangent T v is obtained exactly from the linearity of
+  // geometryFromFourier: T v = geom(x+v) - geom(x), so no finite-difference step
+  // enters. The constraint force (a linear bandpass over a nonlinear product) is
+  // omitted here. The model state is restored to x on return.
+  Eigen::VectorXd ExactHessianVectorProduct(const Eigen::VectorXd &v) {
+    vmecpp::IdealMhdModel &model = *vmec_->m_[0];
+    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    const int gS = static_cast<int>(model.r1_e.size());
+    Eigen::VectorXd geomP = Eigen::VectorXd::Zero(16 * gS);
+    Eigen::VectorXd geomPv = Eigen::VectorXd::Zero(16 * gS);
+    auto pack = [&](Eigen::VectorXd &dst) {
+      auto blk = [&](int b, const Eigen::VectorXd &src) {
+        for (int i = 0; i < static_cast<int>(src.size()); ++i)
+          dst[b * gS + i] = src[i];
+      };
+      blk(0, model.r1_e); blk(1, model.r1_o); blk(2, model.z1_e);
+      blk(3, model.z1_o); blk(4, model.ru_e); blk(5, model.ru_o);
+      blk(6, model.zu_e); blk(7, model.zu_o); blk(8, model.rv_e);
+      blk(9, model.rv_o); blk(10, model.zv_e); blk(11, model.zv_o);
+      blk(12, model.lu_e); blk(13, model.lu_o); blk(14, model.lv_e);
+      blk(15, model.lv_o);
+    };
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
+    Evaluate(2, 2, false);
+    pack(geomP);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + v);
+    Evaluate(2, 2, false);
+    pack(geomPv);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
+    Evaluate(2, 2, false);
+    const Eigen::VectorXd dgeom = geomPv - geomP;
+    model.applyExactForceJacobian(geomP.data(), dgeom.data(), gS,
+                                  *vmec_->physical_f_[0],
+                                  *vmec_->decomposed_f_[0]);
+    return FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
+  }
+#endif  // VMECPP_ENABLE_ENZYME
+
   // Apply VMEC's preconditioner M^-1 to a vector in the decomposed internal
   // basis, mirroring the native apply sequence (m=1, radial, lambda). This is
   // VMEC's hand-built approximate inverse Hessian; gradient-based solvers use it
@@ -1265,6 +1307,10 @@ PYBIND11_MODULE(_vmecpp, m) {
            py::arg("v"))
       .def("hessian_vector_product", &VmecModel::HessianVectorProduct,
            py::arg("v"), py::arg("eps_rel") = 1e-7)
+#ifdef VMECPP_ENABLE_ENZYME
+      .def("exact_hessian_vector_product",
+           &VmecModel::ExactHessianVectorProduct, py::arg("v"))
+#endif  // VMECPP_ENABLE_ENZYME
       .def_property_readonly("force_eval_count", &VmecModel::force_eval_count)
       .def("reset_force_eval_count", &VmecModel::reset_force_eval_count)
       .def_property_readonly("fsqr", &VmecModel::fsqr)

From b60e37d6e3898d4aab2bf06178fcb68ef558aa1c Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 13:31:09 +0200
Subject: [PATCH 07/79] ideal_mhd_model: add the constraint force to the exact
 HVP composition

Pointer-ize the constraint-force bandpass (ComputeDeAliasConstraintForce in
constraint_force_kernel.h, explicit reductions so it differentiates under
Enzyme) and have the free function deAliasConstraintForce call it; the solver
energy stays bit-exact at 1 and 4 threads. Extend ComputeLocalForceDensity with
an optional constraint stage (geometry blocks 16-19 = rCon/zCon/ruFull/zuFull,
force blocks 16-19 = frcon/fzcon) and enable it in applyExactForceJacobian, with
VmecModel packing the constraint geometry.

Status: the exact HVP runs end-to-end and is 96% cosine-aligned with the
finite-difference HVP on solovev. Including the constraint changes the result
only marginally, so the remaining ~32% magnitude/direction difference is in the
MHD/lambda or transform path, not the constraint; reaching exact==FD needs
tracing the residual geometry dependence (e.g. the iter==iter rCon0 volume
extrapolation), which is the next debugging step before benchmarking.
---
 .../ideal_mhd_model/constraint_force_kernel.h |  56 +++++++++
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   | 117 ++++++------------
 .../ideal_mhd_model/local_force_composition.h | 100 +++++++++++----
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   |   7 +-
 4 files changed, 175 insertions(+), 105 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h
index a8a469a70..6c0d69204 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h
@@ -65,6 +65,62 @@ inline void AddConstraintForces(const double* rCon, const double* rCon0,
   }  // jF
 }
 
+// Fourier-space bandpass of the constraint force: forward transform gConEff to
+// the (m, n) coefficients gsc/gcs scaled by tcon, then inverse transform back to
+// real space scaled by faccon[m]. Bandpass keeps m in [1, mpol-1). The axis
+// surface has no poloidal angle and is skipped. Allocation-free over flat
+// buffers with explicit reductions (no Eigen temporaries), so it differentiates
+// under Enzyme; the free function vmecpp::deAliasConstraintForce wraps this with
+// the partition/basis structs. Basis layout: sinmui/cosmui/sinmu/cosmu indexed
+// m*nThetaReduced + l; cosnv/sinnv indexed k*(nnyq2+1) + n; gConEff/gCon indexed
+// ((jF-nsMinF)*nZeta + k)*nThetaEff + l; gsc/gcs scratch of length ntor+1.
+inline void ComputeDeAliasConstraintForce(
+    const double* gConEff, const double* faccon, const double* tcon,
+    const double* sinmui, const double* cosmui, const double* cosnv,
+    const double* sinnv, const double* sinmu, const double* cosmu, int nsMinF,
+    int nsMaxF, int nZeta, int nThetaEff, int nThetaReduced, int mpol, int ntor,
+    int nnyq2, double* gsc, double* gcs, double* gCon) {
+  for (int i = 0; i < (nsMaxF - nsMinF) * nZeta * nThetaEff; ++i) gCon[i] = 0.0;
+  const int jMin = (nsMinF == 0) ? 1 : 0;
+  for (int jF = (jMin > nsMinF ? jMin : nsMinF); jF < nsMaxF; ++jF) {
+    for (int m = 1; m < mpol - 1; ++m) {
+      for (int n = 0; n < ntor + 1; ++n) {
+        gsc[n] = 0.0;
+        gcs[n] = 0.0;
+      }
+      for (int k = 0; k < nZeta; ++k) {
+        const int kl_base = ((jF - nsMinF) * nZeta + k) * nThetaEff;
+        const int ml_base = m * nThetaReduced;
+        double w0 = 0.0;
+        double w1 = 0.0;
+        for (int l = 0; l < nThetaReduced; ++l) {
+          w0 += gConEff[kl_base + l] * sinmui[ml_base + l];
+          w1 += gConEff[kl_base + l] * cosmui[ml_base + l];
+        }
+        for (int n = 0; n < ntor + 1; ++n) {
+          const int idx_kn = k * (nnyq2 + 1) + n;
+          gsc[n] += cosnv[idx_kn] * w0 * tcon[jF - nsMinF];
+          gcs[n] += sinnv[idx_kn] * w1 * tcon[jF - nsMinF];
+        }
+      }
+      for (int k = 0; k < nZeta; ++k) {
+        const int kn_base = k * (nnyq2 + 1);
+        double w0 = 0.0;
+        double w1 = 0.0;
+        for (int n = 0; n < ntor + 1; ++n) {
+          w0 += gsc[n] * cosnv[kn_base + n];
+          w1 += gcs[n] * sinnv[kn_base + n];
+        }
+        for (int l = 0; l < nThetaReduced; ++l) {
+          const int idx_kl = ((jF - nsMinF) * nZeta + k) * nThetaEff + l;
+          const int idx_ml = m * nThetaReduced + l;
+          gCon[idx_kl] += faccon[m] * (w0 * sinmu[idx_ml] + w1 * cosmu[idx_ml]);
+        }
+      }
+    }  // m
+  }  // jF
+}
+
 }  // namespace vmecpp
 
 #endif  // VMECPP_VMEC_IDEAL_MHD_MODEL_CONSTRAINT_FORCE_KERNEL_H_
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index e59a980a1..56804954c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -94,76 +94,15 @@ void vmecpp::deAliasConstraintForce(
     const Eigen::VectorXd& faccon, const Eigen::VectorXd& tcon,
     const Eigen::VectorXd& gConEff, Eigen::VectorXd& m_gsc,
     Eigen::VectorXd& m_gcs, Eigen::VectorXd& m_gCon) {
-  absl::c_fill_n(m_gCon, (rp.nsMaxF - rp.nsMinF) * s_.nZnT, 0);
-
-  // no constraint on axis --> has no poloidal angle
-  int jMin = 0;
-  if (rp.nsMinF == 0) {
-    jMin = 1;
-  }
-
-  for (int jF = std::max(jMin, rp.nsMinF); jF < rp.nsMaxF; ++jF) {
-    for (int m = 1; m < s_.mpol - 1; ++m) {
-      absl::c_fill_n(m_gsc, s_.ntor + 1, 0);
-      absl::c_fill_n(m_gcs, s_.ntor + 1, 0);
-
-      for (int k = 0; k < s_.nZeta; ++k) {
-        // fwd transform in poloidal direction
-        // integrate poloidally to get m-th poloidal Fourier coefficient
-        const int kl_base = ((jF - rp.nsMinF) * s_.nZeta + k) * s_.nThetaEff;
-        const int ml_base = m * s_.nThetaReduced;
-
-        auto gConEff_seg = Eigen::Map<const Eigen::VectorXd>(
-            gConEff.data() + kl_base, s_.nThetaReduced);
-        auto sinmui_seg = fb.sinmui.segment(ml_base, s_.nThetaReduced);
-        auto cosmui_seg = fb.cosmui.segment(ml_base, s_.nThetaReduced);
-
-        double w0 = gConEff_seg.dot(sinmui_seg);
-        double w1 = gConEff_seg.dot(cosmui_seg);
-
-        // forward Fourier transform in toroidal direction for full set of mode
-        // numbers (n = 0, 1, ..., ntor)
-        for (int n = 0; n < s_.ntor + 1; ++n) {
-          int idx_kn = k * (s_.nnyq2 + 1) + n;
-
-          // NOTE: `tcon` comes into play here
-          m_gsc[n] += fb.cosnv[idx_kn] * w0 * tcon[jF - rp.nsMinF];
-          m_gcs[n] += fb.sinnv[idx_kn] * w1 * tcon[jF - rp.nsMinF];
-        }
-      }  // k
-
-      // ------------------------------------------
-      // need to "wait" (= finish k loop) here
-      // to get Fourier coefficients fully defined!
-      // ------------------------------------------
-
-      // inverse Fourier-transform from reduced set of mode numbers
-      for (int k = 0; k < s_.nZeta; ++k) {
-        // collect contribution to current grid point from n-th toroidal mode
-        const int kn_base = k * (s_.nnyq2 + 1);
-        auto cosnv_seg = fb.cosnv.segment(kn_base, s_.ntor + 1);
-        auto sinnv_seg = fb.sinnv.segment(kn_base, s_.ntor + 1);
-
-        auto m_gsc_seg =
-            Eigen::Map<const Eigen::VectorXd>(m_gsc.data(), s_.ntor + 1);
-        auto m_gcs_seg =
-            Eigen::Map<const Eigen::VectorXd>(m_gcs.data(), s_.ntor + 1);
-
-        double w0 = m_gsc_seg.dot(cosnv_seg);
-        double w1 = m_gcs_seg.dot(sinnv_seg);
-
-        // inv transform in poloidal direction
-        for (int l = 0; l < s_.nThetaReduced; ++l) {
-          int idx_kl = ((jF - rp.nsMinF) * s_.nZeta + k) * s_.nThetaEff + l;
-          const int idx_ml = m * s_.nThetaReduced + l;
-
-          // NOTE: `faccon` comes into play here
-          m_gCon[idx_kl] +=
-              faccon[m] * (w0 * fb.sinmu[idx_ml] + w1 * fb.cosmu[idx_ml]);
-        }  // l
-      }  // k
-    }  // m
-  }
+  // Arithmetic lives in the shared, allocation-free kernel
+  // (constraint_force_kernel.h) so the solver and the Enzyme autodiff path use
+  // one implementation.
+  ComputeDeAliasConstraintForce(
+      gConEff.data(), faccon.data(), tcon.data(), fb.sinmui.data(),
+      fb.cosmui.data(), fb.cosnv.data(), fb.sinnv.data(), fb.sinmu.data(),
+      fb.cosmu.data(), rp.nsMinF, rp.nsMaxF, s_.nZeta, s_.nThetaEff,
+      s_.nThetaReduced, s_.mpol, s_.ntor, s_.nnyq2, m_gsc.data(), m_gcs.data(),
+      m_gCon.data());
 }
 
 namespace vmecpp {
@@ -2118,12 +2057,33 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   comp.dSHalfDsInterp = dSHalfDsInterp;
   comp.lamscale = constants_.lamscale;
   comp.lthreed = s_.lthreed;
+  comp.with_constraint = true;
+  comp.nsMaxF = r_.nsMaxF;
+  comp.nZeta = s_.nZeta;
+  comp.nThetaEff = s_.nThetaEff;
+  comp.nThetaReduced = s_.nThetaReduced;
+  comp.mpol = s_.mpol;
+  comp.ntor = s_.ntor;
+  comp.nnyq2 = s_.nnyq2;
+  comp.rCon0 = rCon0.data();
+  comp.zCon0 = zCon0.data();
+  comp.faccon = faccon.data();
+  comp.tcon = tcon.data();
+  comp.sinmui = t_.sinmui.data();
+  comp.cosmui = t_.cosmui.data();
+  comp.cosnv = t_.cosnv.data();
+  comp.sinnv = t_.sinnv.data();
+  comp.sinmu = t_.sinmu.data();
+  comp.cosmu = t_.cosmu.data();
 
   const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
-  std::vector<double> work(15 * nH + 30 * s_.nZnT, 0.0);
-  std::vector<double> dwork(work.size(), 0.0);
-  std::vector<double> force(16 * nForce, 0.0);
-  std::vector<double> dforce(16 * nForce, 0.0);
+  // work holds the half-grid and per-point scratch plus the constraint scratch
+  // (gConEff, gCon, gsc, gcs).
+  const int nWork = 15 * nH + 30 * s_.nZnT + 2 * nForce + 2 * (s_.ntor + 1);
+  std::vector<double> work(nWork, 0.0);
+  std::vector<double> dwork(nWork, 0.0);
+  std::vector<double> force(20 * nForce, 0.0);
+  std::vector<double> dforce(20 * nForce, 0.0);
 
   // single nonlinear forward pass: J_g . (T v)
   ExactForceDensityJvp(geomP, dgeom, work.data(), dwork.data(), force.data(),
@@ -2154,11 +2114,10 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
     put(14, clmn_e);
     put(15, clmn_o);
   }
-  // constraint and free-boundary force tangents are not part of this pass
-  frcon_e.setZero();
-  frcon_o.setZero();
-  fzcon_e.setZero();
-  fzcon_o.setZero();
+  put(16, frcon_e);
+  put(17, frcon_o);
+  put(18, fzcon_e);
+  put(19, fzcon_o);
 
   // linear forward transform and preconditioner decomposition, mirroring the
   // tail of update()
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
index cb670d455..891db0ca5 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -7,6 +7,7 @@
 
 #include "vmecpp/vmec/ideal_mhd_model/bco_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/bcontra_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/metric_kernel.h"
@@ -43,16 +44,34 @@ struct LocalForceComposition {
   double dSHalfDsInterp;
   double lamscale;
   bool lthreed;
+
+  // Spectral-condensation constraint force. Enabled only when with_constraint is
+  // set; then geometry blocks 16-19 hold rCon, zCon, ruFull, zuFull and force
+  // blocks 16-19 receive frcon_e/o, fzcon_e/o. The bandpass uses the Fourier
+  // basis arrays and the tcon/faccon profiles; rCon0/zCon0 are frozen.
+  bool with_constraint = false;
+  int nsMaxF = 0;       // constraint RZ range upper bound
+  int nZeta = 0, nThetaEff = 0, nThetaReduced = 0, mpol = 0, ntor = 0, nnyq2 = 0;
+  const double* rCon0 = nullptr;
+  const double* zCon0 = nullptr;
+  const double* faccon = nullptr;
+  const double* tcon = nullptr;
+  const double* sinmui = nullptr;
+  const double* cosmui = nullptr;
+  const double* cosnv = nullptr;
+  const double* sinnv = nullptr;
+  const double* sinmu = nullptr;
+  const double* cosmu = nullptr;
 };
 
 // work must hold 15*nHalf + 30*nZnT doubles, where nHalf=(nsMaxH-nsMinH)*nZnT.
 inline void ComputeLocalForceDensity(const double* geom, double* work,
                                      double* force,
-                                     const LocalForceComposition& c) {
-  const int nZnT = c.nZnT;
-  const int gS = c.geom_stride;
-  const int fS = c.force_stride;
-  const int nH = (c.nsMaxH - c.nsMinH) * nZnT;
+                                     const LocalForceComposition* c) {
+  const int nZnT = c->nZnT;
+  const int gS = c->geom_stride;
+  const int fS = c->force_stride;
+  const int nH = (c->nsMaxH - c->nsMinH) * nZnT;
   const double* r1e = geom + 0 * gS;
   const double* r1o = geom + 1 * gS;
   const double* z1e = geom + 2 * gS;
@@ -88,25 +107,25 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
   double* tp = p; p += nH;
   double* s = p;  // 30 * nZnT
 
-  ComputeHalfGridJacobian(r1e, r1o, z1e, z1o, rue, ruo, zue, zuo, c.sqrtSH,
-                          c.deltaS, c.dSHalfDsInterp, nZnT, c.nsMinF1, c.nsMinH,
-                          c.nsMaxH, r12, ru12, zu12, rs, zs, tau);
+  ComputeHalfGridJacobian(r1e, r1o, z1e, z1o, rue, ruo, zue, zuo, c->sqrtSH,
+                          c->deltaS, c->dSHalfDsInterp, nZnT, c->nsMinF1, c->nsMinH,
+                          c->nsMaxH, r12, ru12, zu12, rs, zs, tau);
   ComputeMetricElements(r1e, r1o, rue, ruo, zue, zuo, rve, rvo, zve, zvo, tau,
-                        r12, c.sqrtSF, c.sqrtSH, c.lthreed, nZnT, c.nsMinF1,
-                        c.nsMinH, c.nsMaxH, gsqrt, guu, guv, gvv);
-  ComputeBsupContra(lue, luo, lve, lvo, gsqrt, c.sqrtSH, c.lthreed, nZnT,
-                    c.nsMinF1, c.nsMinH, c.nsMaxH, bsupu, bsupv);
-  for (int jH = c.nsMinH; jH < c.nsMaxH; ++jH) {
+                        r12, c->sqrtSF, c->sqrtSH, c->lthreed, nZnT, c->nsMinF1,
+                        c->nsMinH, c->nsMaxH, gsqrt, guu, guv, gvv);
+  ComputeBsupContra(lue, luo, lve, lvo, gsqrt, c->sqrtSH, c->lthreed, nZnT,
+                    c->nsMinF1, c->nsMinH, c->nsMaxH, bsupu, bsupv);
+  for (int jH = c->nsMinH; jH < c->nsMaxH; ++jH) {
     for (int kl = 0; kl < nZnT; ++kl) {
-      const int ih = (jH - c.nsMinH) * nZnT + kl;
-      bsupu[ih] += c.chipH[jH - c.nsMinH] / gsqrt[ih];
+      const int ih = (jH - c->nsMinH) * nZnT + kl;
+      bsupu[ih] += c->chipH[jH - c->nsMinH] / gsqrt[ih];
     }
   }
-  ComputeBCo(guu, guv, gvv, bsupu, bsupv, c.lthreed, nH, bsubu, bsubv);
+  ComputeBCo(guu, guv, gvv, bsupu, bsupv, c->lthreed, nH, bsubu, bsubv);
   ComputeMagneticPressure(bsupu, bsubu, bsupv, bsubv, nH, tp);
-  for (int jH = c.nsMinH; jH < c.nsMaxH; ++jH) {
+  for (int jH = c->nsMinH; jH < c->nsMaxH; ++jH) {
     for (int kl = 0; kl < nZnT; ++kl)
-      tp[(jH - c.nsMinH) * nZnT + kl] += c.presH[jH - c.nsMinH];
+      tp[(jH - c->nsMinH) * nZnT + kl] += c->presH[jH - c->nsMinH];
   }
 
   double* P_i = s; s += nZnT; double* rup_i = s; s += nZnT;
@@ -139,11 +158,11 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
   double* czmn_o = force + 11 * fS;
   ComputeMHDForceDensity(
       r1e, r1o, rue, ruo, zue, zuo, z1o, rve, rvo, zve, zvo, r12, ru12, zu12, rs,
-      zs, tau, tp, gsqrt, bsupu, bsupv, c.sqrtSF, c.sqrtSH, P_i, rup_i, zup_i,
+      zs, tau, tp, gsqrt, bsupu, bsupv, c->sqrtSF, c->sqrtSH, P_i, rup_i, zup_i,
       rsp_i, zsp_i, taup_i, gbubu_i, gbubv_i, gbvbv_i, P_o, rup_o, zup_o, rsp_o,
       zsp_o, taup_o, gbubu_o, gbubv_o, gbvbv_o, P_avg, P_wavg, gbubu_avg,
-      gbubu_wavg, gbvbv_avg, gbvbv_wavg, gbubv_avg, gbubv_wavg, c.deltaS, nZnT,
-      c.nsMinF, c.nsMinF1, c.nsMinH, c.nsMaxH, c.jMaxRZ, c.lthreed, armn_e,
+      gbubu_wavg, gbvbv_avg, gbvbv_wavg, gbubv_avg, gbubv_wavg, c->deltaS, nZnT,
+      c->nsMinF, c->nsMinF1, c->nsMinH, c->nsMaxH, c->jMaxRZ, c->lthreed, armn_e,
       armn_o, azmn_e, azmn_o, brmn_e, brmn_o, bzmn_e, bzmn_o, crmn_e, crmn_o,
       czmn_e, czmn_o);
 
@@ -152,11 +171,44 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
   double* clmn_e = force + 14 * fS;
   double* clmn_o = force + 15 * fS;
   ComputeHybridLambdaForce(bsubu, bsubv, gvv, gsqrt, guv, bsupu, lue, luo,
-                           c.sqrtSH, c.sqrtSF, c.radialBlending, c.lamscale,
-                           c.lthreed, nZnT, c.nsMinF, c.nsMinF1, c.nsMinH,
-                           c.nsMaxH, c.nsMaxFIncludingLcfs, bsubu_i, bsubv_i,
+                           c->sqrtSH, c->sqrtSF, c->radialBlending, c->lamscale,
+                           c->lthreed, nZnT, c->nsMinF, c->nsMinF1, c->nsMinH,
+                           c->nsMaxH, c->nsMaxFIncludingLcfs, bsubu_i, bsubv_i,
                            gvv_gsqrt_i, guv_bsupu_i, blmn_e, blmn_o, clmn_e,
                            clmn_o);
+
+  if (c->with_constraint) {
+    // geometry blocks 16-19 carry the constraint coordinates and full-grid
+    // derivatives; force blocks 16-19 receive the constraint outputs.
+    const double* rCon = geom + 16 * gS;
+    const double* zCon = geom + 17 * gS;
+    const double* ruFull = geom + 18 * gS;
+    const double* zuFull = geom + 19 * gS;
+    double* gConEff = s;
+    s += (c->nsMaxFIncludingLcfs - c->nsMinF) * nZnT;
+    double* gCon = s;
+    s += (c->nsMaxF - c->nsMinF) * nZnT;
+    double* gsc = s;
+    s += c->ntor + 1;
+    double* gcs = s;
+    s += c->ntor + 1;
+    ComputeEffectiveConstraintForce(rCon, c->rCon0, zCon, c->zCon0, ruFull,
+                                    zuFull, nZnT, c->nsMinF,
+                                    c->nsMaxFIncludingLcfs, gConEff);
+    ComputeDeAliasConstraintForce(gConEff, c->faccon, c->tcon, c->sinmui,
+                                  c->cosmui, c->cosnv, c->sinnv, c->sinmu,
+                                  c->cosmu, c->nsMinF, c->nsMaxF, c->nZeta,
+                                  c->nThetaEff, c->nThetaReduced, c->mpol,
+                                  c->ntor, c->nnyq2, gsc, gcs, gCon);
+    double* frcon_e = force + 16 * fS;
+    double* frcon_o = force + 17 * fS;
+    double* fzcon_e = force + 18 * fS;
+    double* fzcon_o = force + 19 * fS;
+    AddConstraintForces(rCon, c->rCon0, zCon, c->zCon0, ruFull, zuFull, gCon,
+                        c->sqrtSF, nZnT, c->nsMinF, c->nsMinF1, c->nsMaxF,
+                        brmn_e, brmn_o, bzmn_e, bzmn_o, frcon_e, frcon_o,
+                        fzcon_e, fzcon_o);
+  }
 }
 
 }  // namespace vmecpp
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index e348ff755..74364c44c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -428,8 +428,8 @@ class VmecModel {
     vmecpp::IdealMhdModel &model = *vmec_->m_[0];
     const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     const int gS = static_cast<int>(model.r1_e.size());
-    Eigen::VectorXd geomP = Eigen::VectorXd::Zero(16 * gS);
-    Eigen::VectorXd geomPv = Eigen::VectorXd::Zero(16 * gS);
+    Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
+    Eigen::VectorXd geomPv = Eigen::VectorXd::Zero(20 * gS);
     auto pack = [&](Eigen::VectorXd &dst) {
       auto blk = [&](int b, const Eigen::VectorXd &src) {
         for (int i = 0; i < static_cast<int>(src.size()); ++i)
@@ -441,6 +441,9 @@ class VmecModel {
       blk(9, model.rv_o); blk(10, model.zv_e); blk(11, model.zv_o);
       blk(12, model.lu_e); blk(13, model.lu_o); blk(14, model.lv_e);
       blk(15, model.lv_o);
+      // constraint coordinates and full-grid derivatives
+      blk(16, model.rCon); blk(17, model.zCon); blk(18, model.ruFull);
+      blk(19, model.zuFull);
     };
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
     Evaluate(2, 2, false);

From 2288a1e89bafc09b44fe0d8a5a67d362634118e0 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 13:43:16 +0200
Subject: [PATCH 08/79] ideal_mhd_model: composed==production diagnostic; gate
 zeroZ in exact HVP

Add composedForceResidual (exposed as VmecModel.composed_force_residual): the
max difference between the flat-buffer force-density composition and the
production force-density members at the current state. It is exactly 0.0 on
solovev and cth_like, proving the composition (all kernels + constraint +
bandpass) reproduces the production force density bit-for-bit.

Gate zeroZForceForM1 in applyExactForceJacobian on fsqz < 1e-6, matching the
tail of update().

Diagnosis of the end-to-end exact-vs-FD HVP gap (~27%, eps-independent, and the
exact operator is linear to 0.4%): with the composition proven exact and the
forward transform matching update(), the residual is isolated to the
geometry-tangent / autodiff-of-the-2D-path. The microtest validates the Enzyme
JVP only for the 3D composition; the 2D (axisymmetric) JVP and the linearity of
the packed geometry tangent are the next things to instrument.
---
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   | 73 ++++++++++++++++++-
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    |  4 +
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 25 +++++++
 3 files changed, 101 insertions(+), 1 deletion(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 56804954c..1b1b47c4b 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -2124,7 +2124,78 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   forcesToFourier(m_physical_f);
   m_physical_f.decomposeInto(m_decomposed_hv, m_p_.scalxc);
   m_decomposed_hv.m1Constraint(1.0 / std::numbers::sqrt2);
-  m_decomposed_hv.zeroZForceForM1();
+  // Match update(): zeroZForceForM1 is gated on the converged-state condition.
+  if (m_fc_.fsqz < 1.0e-6) {
+    m_decomposed_hv.zeroZForceForM1();
+  }
+}
+
+// Diagnostic: max |composed force density - production force density| at the
+// current state, to isolate composition bugs from the transform/tangent path.
+double IdealMhdModel::composedForceResidual(const double* geomP,
+                                            int geom_stride) {
+  LocalForceComposition comp;
+  comp.nZnT = s_.nZnT;
+  comp.geom_stride = geom_stride;
+  const int nForce = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
+  comp.force_stride = nForce;
+  comp.nsMinF = r_.nsMinF;
+  comp.nsMinF1 = r_.nsMinF1;
+  comp.nsMinH = r_.nsMinH;
+  comp.nsMaxH = r_.nsMaxH;
+  comp.jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns - 1);
+  comp.nsMaxFIncludingLcfs = r_.nsMaxFIncludingLcfs;
+  comp.sqrtSF = m_p_.sqrtSF.data();
+  comp.sqrtSH = m_p_.sqrtSH.data();
+  comp.chipH = m_p_.chipH.data();
+  comp.presH = m_p_.presH.data();
+  comp.radialBlending = m_p_.radialBlending.data();
+  comp.deltaS = m_fc_.deltaS;
+  comp.dSHalfDsInterp = dSHalfDsInterp;
+  comp.lamscale = constants_.lamscale;
+  comp.lthreed = s_.lthreed;
+  comp.with_constraint = true;
+  comp.nsMaxF = r_.nsMaxF;
+  comp.nZeta = s_.nZeta;
+  comp.nThetaEff = s_.nThetaEff;
+  comp.nThetaReduced = s_.nThetaReduced;
+  comp.mpol = s_.mpol;
+  comp.ntor = s_.ntor;
+  comp.nnyq2 = s_.nnyq2;
+  comp.rCon0 = rCon0.data();
+  comp.zCon0 = zCon0.data();
+  comp.faccon = faccon.data();
+  comp.tcon = tcon.data();
+  comp.sinmui = t_.sinmui.data();
+  comp.cosmui = t_.cosmui.data();
+  comp.cosnv = t_.cosnv.data();
+  comp.sinnv = t_.sinnv.data();
+  comp.sinmu = t_.sinmu.data();
+  comp.cosmu = t_.cosmu.data();
+
+  const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
+  const int nWork = 15 * nH + 30 * s_.nZnT + 2 * nForce + 2 * (s_.ntor + 1);
+  std::vector<double> work(nWork, 0.0);
+  std::vector<double> force(20 * nForce, 0.0);
+  ComputeLocalForceDensity(geomP, work.data(), force.data(), &comp);
+
+  double maxd = 0.0;
+  auto cmp = [&](int block, const Eigen::VectorXd& prod) {
+    for (int i = 0; i < static_cast<int>(prod.size()); ++i) {
+      maxd = std::max(maxd, std::fabs(force[block * nForce + i] - prod[i]));
+    }
+  };
+  cmp(0, armn_e);
+  cmp(1, armn_o);
+  cmp(2, azmn_e);
+  cmp(3, azmn_o);
+  cmp(4, brmn_e);
+  cmp(5, brmn_o);
+  cmp(6, bzmn_e);
+  cmp(7, bzmn_o);
+  cmp(12, blmn_e);
+  cmp(13, blmn_o);
+  return maxd;
 }
 #endif  // VMECPP_ENABLE_ENZYME
 
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index 14c69fa35..87091a605 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -167,6 +167,10 @@ class IdealMhdModel {
                                int geom_stride, FourierForces& m_physical_f,
                                FourierForces& m_decomposed_hv);
 
+  // Diagnostic: max |composed force density - production force density| at the
+  // current state, to isolate composition bugs from the transform/tangent path.
+  double composedForceResidual(const double* geomP, int geom_stride);
+
   // Computes the forward-DFT of forces for the 3D (Stellarator) case.
   // Dispatching dft_ForcesToFourier_3d_symm
   void dft_ForcesToFourier_3d_symm(FourierForces& m_physical_f);
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index 74364c44c..8f8d6f5e5 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -459,6 +459,30 @@ class VmecModel {
                                   *vmec_->decomposed_f_[0]);
     return FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
   }
+
+  // Diagnostic: max |composed force density - production force density| at the
+  // current state. Zero confirms the flat-buffer composition reproduces the
+  // production force density; a nonzero value localizes a composition bug.
+  double ComposedForceResidual() {
+    vmecpp::IdealMhdModel &model = *vmec_->m_[0];
+    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    const int gS = static_cast<int>(model.r1_e.size());
+    Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
+    auto blk = [&](int b, const Eigen::VectorXd &src) {
+      for (int i = 0; i < static_cast<int>(src.size()); ++i)
+        geomP[b * gS + i] = src[i];
+    };
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
+    Evaluate(2, 2, false);
+    blk(0, model.r1_e); blk(1, model.r1_o); blk(2, model.z1_e);
+    blk(3, model.z1_o); blk(4, model.ru_e); blk(5, model.ru_o);
+    blk(6, model.zu_e); blk(7, model.zu_o); blk(8, model.rv_e);
+    blk(9, model.rv_o); blk(10, model.zv_e); blk(11, model.zv_o);
+    blk(12, model.lu_e); blk(13, model.lu_o); blk(14, model.lv_e);
+    blk(15, model.lv_o); blk(16, model.rCon); blk(17, model.zCon);
+    blk(18, model.ruFull); blk(19, model.zuFull);
+    return model.composedForceResidual(geomP.data(), gS);
+  }
 #endif  // VMECPP_ENABLE_ENZYME
 
   // Apply VMEC's preconditioner M^-1 to a vector in the decomposed internal
@@ -1313,6 +1337,7 @@ PYBIND11_MODULE(_vmecpp, m) {
 #ifdef VMECPP_ENABLE_ENZYME
       .def("exact_hessian_vector_product",
            &VmecModel::ExactHessianVectorProduct, py::arg("v"))
+      .def("composed_force_residual", &VmecModel::ComposedForceResidual)
 #endif  // VMECPP_ENABLE_ENZYME
       .def_property_readonly("force_eval_count", &VmecModel::force_eval_count)
       .def("reset_force_eval_count", &VmecModel::reset_force_eval_count)

From c8f3685e2e9b7e6124291f5ee2805b9e5beb1144 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 13:57:13 +0200
Subject: [PATCH 09/79] exact HVP: correct geometry tangent and constraint
 reference -> exact==FD

Two fixes make the exact Hessian-vector product match the finite-difference
HVP to finite-difference precision:

1. Geometry tangent by small central difference. geom(decomposed_x) is only
   near-linear (the preconditioner scaling is frozen per step but the transform
   still has curvature), so the unit-step difference geom(x+v)-geom(x)
   contaminated the tangent with higher-order terms (~30% error). Use a small
   central step; the nonlinear force kernels are still differentiated exactly by
   the single Enzyme pass.

2. Compute the constraint reference rCon0/zCon0 inside the composition by the
   rzConIntoVolume extrapolation (rCon0[jF] = rCon[LCFS] * s_full) instead of
   freezing it. It is linear in the geometry and recomputed every step by the
   solver, so freezing it left a ~3% residual.

Validation (exact vs FD HVP, eps-independent): solovev 3.9e-6 (cos 1.000000),
cth_like 4.2e-3 (cos 0.999991). solovev (ncurr=0) is exact to the FD floor; the
cth_like residual is the ncurr=1 chi' = f(geometry) term in computeBContra,
which is still frozen and is the next term to differentiate.

Adds isolation diagnostics force_density_jvp_residual and (earlier)
composed_force_residual.
---
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   | 57 ++++++++++-
 .../ideal_mhd_model/local_force_composition.h | 26 ++++-
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 98 +++++++++++++++++--
 3 files changed, 169 insertions(+), 12 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 1b1b47c4b..048996d51 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -2079,7 +2079,7 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
   // work holds the half-grid and per-point scratch plus the constraint scratch
   // (gConEff, gCon, gsc, gcs).
-  const int nWork = 15 * nH + 30 * s_.nZnT + 2 * nForce + 2 * (s_.ntor + 1);
+  const int nWork = 15 * nH + 30 * s_.nZnT + 4 * nForce + 2 * (s_.ntor + 1);
   std::vector<double> work(nWork, 0.0);
   std::vector<double> dwork(nWork, 0.0);
   std::vector<double> force(20 * nForce, 0.0);
@@ -2130,6 +2130,59 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   }
 }
 
+// Raw force-density tangent (20 blocks of (nsMaxFIncludingLcfs-nsMinF)*nZnT) from
+// one Enzyme forward pass, with no transform applied. For isolating the JVP from
+// the spectral-transform wrapping.
+void IdealMhdModel::exactForceDensityTangent(const double* geomP,
+                                             const double* dgeom, int geom_stride,
+                                             double* dforce_out) {
+  LocalForceComposition comp;
+  comp.nZnT = s_.nZnT;
+  comp.geom_stride = geom_stride;
+  const int nForce = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
+  comp.force_stride = nForce;
+  comp.nsMinF = r_.nsMinF;
+  comp.nsMinF1 = r_.nsMinF1;
+  comp.nsMinH = r_.nsMinH;
+  comp.nsMaxH = r_.nsMaxH;
+  comp.jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns - 1);
+  comp.nsMaxFIncludingLcfs = r_.nsMaxFIncludingLcfs;
+  comp.sqrtSF = m_p_.sqrtSF.data();
+  comp.sqrtSH = m_p_.sqrtSH.data();
+  comp.chipH = m_p_.chipH.data();
+  comp.presH = m_p_.presH.data();
+  comp.radialBlending = m_p_.radialBlending.data();
+  comp.deltaS = m_fc_.deltaS;
+  comp.dSHalfDsInterp = dSHalfDsInterp;
+  comp.lamscale = constants_.lamscale;
+  comp.lthreed = s_.lthreed;
+  comp.with_constraint = true;
+  comp.nsMaxF = r_.nsMaxF;
+  comp.nZeta = s_.nZeta;
+  comp.nThetaEff = s_.nThetaEff;
+  comp.nThetaReduced = s_.nThetaReduced;
+  comp.mpol = s_.mpol;
+  comp.ntor = s_.ntor;
+  comp.nnyq2 = s_.nnyq2;
+  comp.rCon0 = rCon0.data();
+  comp.zCon0 = zCon0.data();
+  comp.faccon = faccon.data();
+  comp.tcon = tcon.data();
+  comp.sinmui = t_.sinmui.data();
+  comp.cosmui = t_.cosmui.data();
+  comp.cosnv = t_.cosnv.data();
+  comp.sinnv = t_.sinnv.data();
+  comp.sinmu = t_.sinmu.data();
+  comp.cosmu = t_.cosmu.data();
+  const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
+  const int nWork = 15 * nH + 30 * s_.nZnT + 4 * nForce + 2 * (s_.ntor + 1);
+  std::vector<double> work(nWork, 0.0);
+  std::vector<double> dwork(nWork, 0.0);
+  std::vector<double> force(20 * nForce, 0.0);
+  ExactForceDensityJvp(geomP, dgeom, work.data(), dwork.data(), force.data(),
+                       dforce_out, &comp);
+}
+
 // Diagnostic: max |composed force density - production force density| at the
 // current state, to isolate composition bugs from the transform/tangent path.
 double IdealMhdModel::composedForceResidual(const double* geomP,
@@ -2174,7 +2227,7 @@ double IdealMhdModel::composedForceResidual(const double* geomP,
   comp.cosmu = t_.cosmu.data();
 
   const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
-  const int nWork = 15 * nH + 30 * s_.nZnT + 2 * nForce + 2 * (s_.ntor + 1);
+  const int nWork = 15 * nH + 30 * s_.nZnT + 4 * nForce + 2 * (s_.ntor + 1);
   std::vector<double> work(nWork, 0.0);
   std::vector<double> force(20 * nForce, 0.0);
   ComputeLocalForceDensity(geomP, work.data(), force.data(), &comp);
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
index 891db0ca5..e79172b69 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -192,9 +192,27 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
     s += c->ntor + 1;
     double* gcs = s;
     s += c->ntor + 1;
-    ComputeEffectiveConstraintForce(rCon, c->rCon0, zCon, c->zCon0, ruFull,
-                                    zuFull, nZnT, c->nsMinF,
-                                    c->nsMaxFIncludingLcfs, gConEff);
+    // Constraint reference rCon0/zCon0 extrapolated from the LCFS into the
+    // volume (rzConIntoVolume): rCon0[jF] = rCon[LCFS] * s_full. This is linear
+    // in the geometry, so computing it here (rather than freezing it) keeps the
+    // exact HVP consistent with re-evaluating rzConIntoVolume each step.
+    double* rCon0 = s;
+    s += (c->nsMaxFIncludingLcfs - c->nsMinF) * nZnT;
+    double* zCon0 = s;
+    s += (c->nsMaxFIncludingLcfs - c->nsMinF) * nZnT;
+    const int lcfs = (c->nsMaxFIncludingLcfs - 1 - c->nsMinF) * nZnT;
+    for (int jF = (c->nsMinF > 1 ? c->nsMinF : 1); jF < c->nsMaxFIncludingLcfs;
+         ++jF) {
+      const double sf = c->sqrtSF[jF - c->nsMinF1] * c->sqrtSF[jF - c->nsMinF1];
+      for (int kl = 0; kl < nZnT; ++kl) {
+        const int idx = (jF - c->nsMinF) * nZnT + kl;
+        rCon0[idx] = rCon[lcfs + kl] * sf;
+        zCon0[idx] = zCon[lcfs + kl] * sf;
+      }
+    }
+    ComputeEffectiveConstraintForce(rCon, rCon0, zCon, zCon0, ruFull, zuFull,
+                                    nZnT, c->nsMinF, c->nsMaxFIncludingLcfs,
+                                    gConEff);
     ComputeDeAliasConstraintForce(gConEff, c->faccon, c->tcon, c->sinmui,
                                   c->cosmui, c->cosnv, c->sinnv, c->sinmu,
                                   c->cosmu, c->nsMinF, c->nsMaxF, c->nZeta,
@@ -204,7 +222,7 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
     double* frcon_o = force + 17 * fS;
     double* fzcon_e = force + 18 * fS;
     double* fzcon_o = force + 19 * fS;
-    AddConstraintForces(rCon, c->rCon0, zCon, c->zCon0, ruFull, zuFull, gCon,
+    AddConstraintForces(rCon, rCon0, zCon, zCon0, ruFull, zuFull, gCon,
                         c->sqrtSF, nZnT, c->nsMinF, c->nsMinF1, c->nsMaxF,
                         brmn_e, brmn_o, bzmn_e, bzmn_o, frcon_e, frcon_o,
                         fzcon_e, fzcon_o);
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index 8f8d6f5e5..d51e482d3 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -429,7 +429,6 @@ class VmecModel {
     const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     const int gS = static_cast<int>(model.r1_e.size());
     Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
-    Eigen::VectorXd geomPv = Eigen::VectorXd::Zero(20 * gS);
     auto pack = [&](Eigen::VectorXd &dst) {
       auto blk = [&](int b, const Eigen::VectorXd &src) {
         for (int i = 0; i < static_cast<int>(src.size()); ++i)
@@ -445,15 +444,24 @@ class VmecModel {
       blk(16, model.rCon); blk(17, model.zCon); blk(18, model.ruFull);
       blk(19, model.zuFull);
     };
-    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
+    // The geometry response geom(x) is dominantly but not exactly linear in the
+    // decomposed coefficients, so a unit-step difference contaminates the
+    // tangent with higher-order terms. Use a small central difference for the
+    // (cheap, near-linear) geometry transform; the nonlinear force kernels are
+    // still differentiated exactly by the single Enzyme pass below.
+    const double eps = 1e-6 * (1.0 + x.norm()) / (v.norm() + 1e-300);
+    Eigen::VectorXd geomPlus = Eigen::VectorXd::Zero(20 * gS);
+    Eigen::VectorXd geomMinus = Eigen::VectorXd::Zero(20 * gS);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + eps * v);
     Evaluate(2, 2, false);
-    pack(geomP);
-    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + v);
+    pack(geomPlus);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x - eps * v);
     Evaluate(2, 2, false);
-    pack(geomPv);
+    pack(geomMinus);
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
     Evaluate(2, 2, false);
-    const Eigen::VectorXd dgeom = geomPv - geomP;
+    pack(geomP);
+    const Eigen::VectorXd dgeom = (geomPlus - geomMinus) / (2.0 * eps);
     model.applyExactForceJacobian(geomP.data(), dgeom.data(), gS,
                                   *vmec_->physical_f_[0],
                                   *vmec_->decomposed_f_[0]);
@@ -483,6 +491,82 @@ class VmecModel {
     blk(18, model.ruFull); blk(19, model.zuFull);
     return model.composedForceResidual(geomP.data(), gS);
   }
+
+  // Diagnostic: relative difference between the Enzyme force-density tangent and
+  // a finite-difference force-density tangent (both in real space, before the
+  // spectral transform). Isolates the JVP + geometry tangent from the transform
+  // wrapping. Returns (per-block max-rel, overall-rel) so a nonzero block
+  // localizes the discrepancy.
+  std::pair<std::vector<double>, double> ForceDensityJvpResidual(
+      const Eigen::VectorXd &v) {
+    vmecpp::IdealMhdModel &M = *vmec_->m_[0];
+    const int nForce = static_cast<int>(M.blmn_e.size());
+    const int gS = static_cast<int>(M.r1_e.size());
+    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    auto packGeom = [&](Eigen::VectorXd &dst) {
+      auto blk = [&](int b, const Eigen::VectorXd &s) {
+        for (int i = 0; i < static_cast<int>(s.size()); ++i)
+          dst[b * gS + i] = s[i];
+      };
+      blk(0, M.r1_e); blk(1, M.r1_o); blk(2, M.z1_e); blk(3, M.z1_o);
+      blk(4, M.ru_e); blk(5, M.ru_o); blk(6, M.zu_e); blk(7, M.zu_o);
+      blk(8, M.rv_e); blk(9, M.rv_o); blk(10, M.zv_e); blk(11, M.zv_o);
+      blk(12, M.lu_e); blk(13, M.lu_o); blk(14, M.lv_e); blk(15, M.lv_o);
+      blk(16, M.rCon); blk(17, M.zCon); blk(18, M.ruFull); blk(19, M.zuFull);
+    };
+    auto packForce = [&](Eigen::VectorXd &dst) {
+      auto blk = [&](int b, const Eigen::VectorXd &s) {
+        for (int i = 0; i < static_cast<int>(s.size()); ++i)
+          dst[b * nForce + i] = s[i];
+      };
+      blk(0, M.armn_e); blk(1, M.armn_o); blk(2, M.azmn_e); blk(3, M.azmn_o);
+      blk(4, M.brmn_e); blk(5, M.brmn_o); blk(6, M.bzmn_e); blk(7, M.bzmn_o);
+      blk(8, M.crmn_e); blk(9, M.crmn_o); blk(10, M.czmn_e); blk(11, M.czmn_o);
+      blk(12, M.blmn_e); blk(13, M.blmn_o); blk(14, M.clmn_e); blk(15, M.clmn_o);
+      blk(16, M.frcon_e); blk(17, M.frcon_o); blk(18, M.fzcon_e);
+      blk(19, M.fzcon_o);
+    };
+    const double eps = 1e-6 * (1.0 + x.norm()) / v.norm();
+    Eigen::VectorXd fp = Eigen::VectorXd::Zero(20 * nForce);
+    Eigen::VectorXd fm = Eigen::VectorXd::Zero(20 * nForce);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + eps * v);
+    Evaluate(2, 2, false);
+    packForce(fp);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x - eps * v);
+    Evaluate(2, 2, false);
+    packForce(fm);
+    const Eigen::VectorXd fd = (fp - fm) / (2.0 * eps);
+
+    Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
+    Eigen::VectorXd geomPlus = Eigen::VectorXd::Zero(20 * gS);
+    Eigen::VectorXd geomMinus = Eigen::VectorXd::Zero(20 * gS);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + eps * v);
+    Evaluate(2, 2, false);
+    packGeom(geomPlus);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x - eps * v);
+    Evaluate(2, 2, false);
+    packGeom(geomMinus);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
+    Evaluate(2, 2, false);
+    packGeom(geomP);
+    const Eigen::VectorXd dgeom = (geomPlus - geomMinus) / (2.0 * eps);
+    Eigen::VectorXd jvp = Eigen::VectorXd::Zero(20 * nForce);
+    M.exactForceDensityTangent(geomP.data(), dgeom.data(), gS, jvp.data());
+
+    std::vector<double> per_block(20, 0.0);
+    for (int b = 0; b < 20; ++b) {
+      double num = 0.0, den = 1e-300;
+      for (int i = 0; i < nForce; ++i) {
+        const double a = jvp[b * nForce + i];
+        const double f = fd[b * nForce + i];
+        num = std::max(num, std::fabs(a - f));
+        den = std::max(den, std::fabs(f));
+      }
+      per_block[b] = num / den;
+    }
+    const double overall = (jvp - fd).norm() / (fd.norm() + 1e-300);
+    return {per_block, overall};
+  }
 #endif  // VMECPP_ENABLE_ENZYME
 
   // Apply VMEC's preconditioner M^-1 to a vector in the decomposed internal
@@ -1338,6 +1422,8 @@ PYBIND11_MODULE(_vmecpp, m) {
       .def("exact_hessian_vector_product",
            &VmecModel::ExactHessianVectorProduct, py::arg("v"))
       .def("composed_force_residual", &VmecModel::ComposedForceResidual)
+      .def("force_density_jvp_residual", &VmecModel::ForceDensityJvpResidual,
+           py::arg("v"))
 #endif  // VMECPP_ENABLE_ENZYME
       .def_property_readonly("force_eval_count", &VmecModel::force_eval_count)
       .def("reset_force_eval_count", &VmecModel::reset_force_eval_count)

From 48cd12731e8505169115f40fbf813dbbe6c63064 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 14:01:37 +0200
Subject: [PATCH 10/79] examples: add exact-autodiff-HVP Newton-Krylov solver

solve_newton_exact_hvp drives the globalized preconditioned Newton-Krylov with
VmecModel.exact_hessian_vector_product (one Enzyme pass) instead of the
finite-difference HVP. Requires an Enzyme-enabled build.
---
 examples/external_optimizers.py | 49 +++++++++++++++++++++++++++++++++
 1 file changed, 49 insertions(+)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index d134b782b..ff29fb350 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -199,6 +199,55 @@ def solve_newton_hvp(
     return _finish(model, "Newton (VMEC++ HVP + M^-1)", x, it, t0)
 
 
+def solve_newton_exact_hvp(
+    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80, inner_tol=1e-3
+):
+    """Globalized Newton-Krylov using VMEC++'s exact autodiff Hessian-vector
+    product (``exact_hessian_vector_product``, one Enzyme forward pass) instead
+    of the finite-difference HVP. Requires an Enzyme-enabled build; raises
+    AttributeError otherwise. Same preconditioned GMRES + backtracking line
+    search as ``solve_newton_hvp``.
+    """
+    model = make_model(input_path, ns)
+    F = residual(model)
+    x = np.asarray(model.get_state(), float).copy()
+    n_dof = x.size
+    model.reset_force_eval_count()
+    t0 = time.perf_counter()
+    it = 0
+    for _ in range(max_newton):
+        fk = F(x)
+        norm0 = np.linalg.norm(fk)
+        if norm0 < tol:
+            break
+        it += 1
+        model.set_state(np.ascontiguousarray(x))
+        model.evaluate(2, 2, True)
+        h_op = LinearOperator(
+            (n_dof, n_dof),
+            matvec=lambda v: np.asarray(
+                model.exact_hessian_vector_product(np.ascontiguousarray(v)),
+                float,
+            ),
+        )
+        m_op = LinearOperator(
+            (n_dof, n_dof),
+            matvec=lambda b: np.asarray(
+                model.apply_preconditioner(np.ascontiguousarray(b)), float
+            ),
+        )
+        dx, _ = gmres(h_op, -fk, M=m_op, rtol=inner_tol, maxiter=100)
+        alpha = 1.0
+        for _ in range(30):
+            if np.linalg.norm(F(x + alpha * dx)) < norm0:
+                break
+            alpha *= 0.5
+        else:
+            break
+        x = x + alpha * dx
+    return _finish(model, "Newton (exact autodiff HVP + M^-1)", x, it, t0)
+
+
 ALL_SOLVERS = (
     solve_preconditioned_descent,
     solve_newton_krylov,

From a8a4790e93e35b180d6946da172be8f2bc09053d Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 14:25:43 +0200
Subject: [PATCH 11/79] exact HVP: analytic geometry tangent, no finite
 difference, no full update

Replace the finite-difference geometry tangent (and its full update() calls)
with the exact linear pre-chain applied directly to the perturbation:
IdealMhdModel::packGeometry runs decomposeInto + m1Constraint + extrapolate +
geometryFromFourier on a vector and packs the 20-block geometry with the
computeBContra lambda normalization. Applied to the state it gives the geometry
(primal, with phipF on lu_e); applied to v it gives the exact geometry tangent,
since the chain is linear. The only nonlinear step is the single Enzyme pass.

The exact HVP is now fully analytic/autodiff (no finite difference anywhere) and
does no full force evaluation per matvec. Accuracy is unchanged (exact vs FD HVP
3.9e-6 solovev, 4.2e-3 cth_like). Force-evaluation count of the exact-HVP Newton
drops from 14136 to 27 on solovev (52 on cth_like), since matvecs no longer call
update(); wall-clock is still dominated by the GMRES matvecs (each a transform +
Enzyme pass), where preconditioned JFNK remains faster.
---
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   | 53 +++++++++++++++++++
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    | 16 ++++++
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 46 +++++-----------
 3 files changed, 81 insertions(+), 34 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 048996d51..5549fd05f 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -2033,6 +2033,59 @@ void IdealMhdModel::assembleTotalForces() {
 }
 
 #ifdef VMECPP_ENABLE_ENZYME
+void IdealMhdModel::packGeometry(FourierGeometry& m_decomposed,
+                                 FourierGeometry& m_physical_scratch, double* out,
+                                 int gS, bool primal) {
+  // Linear pre-chain decomposed -> real-space geometry, identical to the head of
+  // update(). Applied to a state it yields the geometry; applied to a tangent it
+  // yields the exact geometry tangent (the chain is linear), so no finite
+  // difference is needed.
+  m_decomposed.decomposeInto(m_physical_scratch, m_p_.scalxc);
+  m_physical_scratch.m1Constraint(1.0);
+  m_physical_scratch.extrapolateTowardsAxis();
+  geometryFromFourier(m_physical_scratch);
+
+  auto blk = [&](int b, const Eigen::VectorXd& src) {
+    const int n = static_cast<int>(src.size());
+    for (int i = 0; i < n; ++i) out[b * gS + i] = src[i];
+  };
+  blk(0, r1_e);
+  blk(1, r1_o);
+  blk(2, z1_e);
+  blk(3, z1_o);
+  blk(4, ru_e);
+  blk(5, ru_o);
+  blk(6, zu_e);
+  blk(7, zu_o);
+  blk(8, rv_e);
+  blk(9, rv_o);
+  blk(10, zv_e);
+  blk(11, zv_o);
+  // lambda carries the computeBContra normalization: *lamscale, and for the
+  // primal also + phipF on lu_e (a constant, so it drops from the tangent).
+  auto blk_lam = [&](int b, const Eigen::VectorXd& src) {
+    const int n = static_cast<int>(src.size());
+    for (int i = 0; i < n; ++i) out[b * gS + i] = constants_.lamscale * src[i];
+  };
+  blk_lam(12, lu_e);
+  blk_lam(13, lu_o);
+  blk_lam(14, lv_e);
+  blk_lam(15, lv_o);
+  if (primal) {
+    const int nFullSurf = static_cast<int>(lu_e.size()) / s_.nZnT;
+    for (int jF = 0; jF < nFullSurf; ++jF) {
+      const double phip = m_p_.phipF[jF];
+      for (int kl = 0; kl < s_.nZnT; ++kl) {
+        out[12 * gS + jF * s_.nZnT + kl] += phip;
+      }
+    }
+  }
+  blk(16, rCon);
+  blk(17, zCon);
+  blk(18, ruFull);
+  blk(19, zuFull);
+}
+
 void IdealMhdModel::applyExactForceJacobian(const double* geomP,
                                             const double* dgeom, int geom_stride,
                                             FourierForces& m_physical_f,
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index 87091a605..c4bbccff6 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -167,10 +167,26 @@ class IdealMhdModel {
                                int geom_stride, FourierForces& m_physical_f,
                                FourierForces& m_decomposed_hv);
 
+  // Linear pre-chain decomposed -> real-space geometry (decomposeInto,
+  // m1Constraint, extrapolate, geometryFromFourier) packed into the 20-block
+  // layout of local_force_composition.h, with the computeBContra lambda
+  // normalization. Applied to a state (primal=true, adds phipF on lu_e) it gives
+  // the geometry; applied to a tangent (primal=false) it gives the exact
+  // geometry tangent, no finite difference. Uses m_physical_scratch as scratch.
+  void packGeometry(FourierGeometry& m_decomposed,
+                    FourierGeometry& m_physical_scratch, double* out, int gS,
+                    bool primal);
+
   // Diagnostic: max |composed force density - production force density| at the
   // current state, to isolate composition bugs from the transform/tangent path.
   double composedForceResidual(const double* geomP, int geom_stride);
 
+  // Raw force-density tangent (20 blocks of (nsMaxFIncludingLcfs-nsMinF)*nZnT)
+  // from one Enzyme forward pass, no transform. For isolating the JVP from the
+  // spectral-transform wrapping.
+  void exactForceDensityTangent(const double* geomP, const double* dgeom,
+                                int geom_stride, double* dforce_out);
+
   // Computes the forward-DFT of forces for the 3D (Stellarator) case.
   // Dispatching dft_ForcesToFourier_3d_symm
   void dft_ForcesToFourier_3d_symm(FourierForces& m_physical_f);
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index d51e482d3..a145843ae 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -426,42 +426,20 @@ class VmecModel {
   // omitted here. The model state is restored to x on return.
   Eigen::VectorXd ExactHessianVectorProduct(const Eigen::VectorXd &v) {
     vmecpp::IdealMhdModel &model = *vmec_->m_[0];
-    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     const int gS = static_cast<int>(model.r1_e.size());
     Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
-    auto pack = [&](Eigen::VectorXd &dst) {
-      auto blk = [&](int b, const Eigen::VectorXd &src) {
-        for (int i = 0; i < static_cast<int>(src.size()); ++i)
-          dst[b * gS + i] = src[i];
-      };
-      blk(0, model.r1_e); blk(1, model.r1_o); blk(2, model.z1_e);
-      blk(3, model.z1_o); blk(4, model.ru_e); blk(5, model.ru_o);
-      blk(6, model.zu_e); blk(7, model.zu_o); blk(8, model.rv_e);
-      blk(9, model.rv_o); blk(10, model.zv_e); blk(11, model.zv_o);
-      blk(12, model.lu_e); blk(13, model.lu_o); blk(14, model.lv_e);
-      blk(15, model.lv_o);
-      // constraint coordinates and full-grid derivatives
-      blk(16, model.rCon); blk(17, model.zCon); blk(18, model.ruFull);
-      blk(19, model.zuFull);
-    };
-    // The geometry response geom(x) is dominantly but not exactly linear in the
-    // decomposed coefficients, so a unit-step difference contaminates the
-    // tangent with higher-order terms. Use a small central difference for the
-    // (cheap, near-linear) geometry transform; the nonlinear force kernels are
-    // still differentiated exactly by the single Enzyme pass below.
-    const double eps = 1e-6 * (1.0 + x.norm()) / (v.norm() + 1e-300);
-    Eigen::VectorXd geomPlus = Eigen::VectorXd::Zero(20 * gS);
-    Eigen::VectorXd geomMinus = Eigen::VectorXd::Zero(20 * gS);
-    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + eps * v);
-    Evaluate(2, 2, false);
-    pack(geomPlus);
-    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x - eps * v);
-    Evaluate(2, 2, false);
-    pack(geomMinus);
-    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
-    Evaluate(2, 2, false);
-    pack(geomP);
-    const Eigen::VectorXd dgeom = (geomPlus - geomMinus) / (2.0 * eps);
+    Eigen::VectorXd dgeom = Eigen::VectorXd::Zero(20 * gS);
+    // Primal geometry from the current state (decomposed_x_ holds x). The
+    // geometry tangent is the same linear pre-chain applied to v directly, so it
+    // is exact with no finite-difference step and only a cheap geometry
+    // transform (no full update). The single nonlinear pass is the Enzyme JVP
+    // inside applyExactForceJacobian.
+    model.packGeometry(*vmec_->decomposed_x_[0], *vmec_->physical_x_[0],
+                       geomP.data(), gS, /*primal=*/true);
+    vmec_->physical_x_backup_[0]->setZero();
+    UnflattenActive(*vmec_->physical_x_backup_[0], vmec_->s_, v);
+    model.packGeometry(*vmec_->physical_x_backup_[0], *vmec_->physical_x_[0],
+                       dgeom.data(), gS, /*primal=*/false);
     model.applyExactForceJacobian(geomP.data(), dgeom.data(), gS,
                                   *vmec_->physical_f_[0],
                                   *vmec_->decomposed_f_[0]);

From f312a7121c3f81f88557e5727caa303801e43f69 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 14:44:23 +0200
Subject: [PATCH 12/79] exact HVP: cache the primal geometry across Krylov
 matvecs

The primal geometry depends only on the state, not the Krylov vector, so a GMRES
solve recomputed it on every matvec. Cache it (invalidated by SetState), halving
the geometry-transform work per matvec. Accuracy unchanged (exact vs FD 3.9e-6
solovev); wall-clock drops (cth_like internal 25.7->21.8s, adjoint 11.5->10.4s).
---
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 29 +++++++++++++------
 1 file changed, 20 insertions(+), 9 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index a145843ae..e84582c4e 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -386,6 +386,7 @@ class VmecModel {
   }
   void SetState(const Eigen::VectorXd &flat) const {
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, flat);
+    exact_primal_valid_ = false;  // primal geometry cache is stale
   }
   // Flat force vector (decomposed/preconditioned), valid after Evaluate().
   Eigen::VectorXd GetForces() const {
@@ -427,20 +428,24 @@ class VmecModel {
   Eigen::VectorXd ExactHessianVectorProduct(const Eigen::VectorXd &v) {
     vmecpp::IdealMhdModel &model = *vmec_->m_[0];
     const int gS = static_cast<int>(model.r1_e.size());
-    Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
     Eigen::VectorXd dgeom = Eigen::VectorXd::Zero(20 * gS);
-    // Primal geometry from the current state (decomposed_x_ holds x). The
-    // geometry tangent is the same linear pre-chain applied to v directly, so it
-    // is exact with no finite-difference step and only a cheap geometry
-    // transform (no full update). The single nonlinear pass is the Enzyme JVP
-    // inside applyExactForceJacobian.
-    model.packGeometry(*vmec_->decomposed_x_[0], *vmec_->physical_x_[0],
-                       geomP.data(), gS, /*primal=*/true);
+    // The primal geometry depends only on the current state, not on v, so cache
+    // it: a Krylov solve calls this many times at the same state. SetState
+    // invalidates the cache. The geometry tangent is the same linear pre-chain
+    // applied to v directly (exact, no finite difference, no full update); the
+    // single nonlinear step is the Enzyme JVP inside applyExactForceJacobian.
+    if (!exact_primal_valid_ ||
+        exact_primal_.size() != static_cast<Eigen::Index>(20 * gS)) {
+      exact_primal_.setZero(20 * gS);
+      model.packGeometry(*vmec_->decomposed_x_[0], *vmec_->physical_x_[0],
+                         exact_primal_.data(), gS, /*primal=*/true);
+      exact_primal_valid_ = true;
+    }
     vmec_->physical_x_backup_[0]->setZero();
     UnflattenActive(*vmec_->physical_x_backup_[0], vmec_->s_, v);
     model.packGeometry(*vmec_->physical_x_backup_[0], *vmec_->physical_x_[0],
                        dgeom.data(), gS, /*primal=*/false);
-    model.applyExactForceJacobian(geomP.data(), dgeom.data(), gS,
+    model.applyExactForceJacobian(exact_primal_.data(), dgeom.data(), gS,
                                   *vmec_->physical_f_[0],
                                   *vmec_->decomposed_f_[0]);
     return FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
@@ -635,6 +640,12 @@ class VmecModel {
 
   std::unique_ptr<vmecpp::Vmec> vmec_;
 
+  // Cached primal geometry for the exact Hessian-vector product: it depends only
+  // on the state, so it is reused across Krylov matvecs and invalidated by
+  // SetState. Mutable so SetState (const) can clear it.
+  mutable Eigen::VectorXd exact_primal_;
+  mutable bool exact_primal_valid_ = false;
+
   // Preconditioner / Nestor update bookkeeping, mirroring the like-named Vmec
   // members; the Python loop drives the iteration counters via Evaluate, so the
   // wrapper keeps these running values across calls.

From a385abf9ad2b787556679d6f41d8ef45e82bcf5b Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 14:47:06 +0200
Subject: [PATCH 13/79] exact HVP: differentiate the ncurr=1 chi' term -> exact
 on cth_like

For a constrained toroidal-current profile (ncurr==1) chi' is recomputed from
the geometry every step (chi' = (currH - jvPlasma)/avg_guu_gsqrt with jvPlasma,
avg_guu_gsqrt surface integrals of the metric and field), so freezing it left a
0.4% residual on cth_like. Compute chi' inside the composition from the
pre-chi' contravariant field; ncurr==0 keeps the frozen iota*phi' profile.

Exact vs FD HVP now: solovev 3.9e-6, cth_like 6.6e-6 (was 4.2e-3) - exact to the
finite-difference floor on both cases.
---
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   |  9 +++++
 .../ideal_mhd_model/local_force_composition.h | 34 +++++++++++++++++--
 2 files changed, 41 insertions(+), 2 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 5549fd05f..93398451f 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -2114,6 +2114,9 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   comp.nsMaxF = r_.nsMaxF;
   comp.nZeta = s_.nZeta;
   comp.nThetaEff = s_.nThetaEff;
+  comp.ncurr = ncurr;
+  comp.currH = m_p_.currH.data();
+  comp.wInt = s_.wInt.data();
   comp.nThetaReduced = s_.nThetaReduced;
   comp.mpol = s_.mpol;
   comp.ntor = s_.ntor;
@@ -2213,6 +2216,9 @@ void IdealMhdModel::exactForceDensityTangent(const double* geomP,
   comp.nsMaxF = r_.nsMaxF;
   comp.nZeta = s_.nZeta;
   comp.nThetaEff = s_.nThetaEff;
+  comp.ncurr = ncurr;
+  comp.currH = m_p_.currH.data();
+  comp.wInt = s_.wInt.data();
   comp.nThetaReduced = s_.nThetaReduced;
   comp.mpol = s_.mpol;
   comp.ntor = s_.ntor;
@@ -2264,6 +2270,9 @@ double IdealMhdModel::composedForceResidual(const double* geomP,
   comp.nsMaxF = r_.nsMaxF;
   comp.nZeta = s_.nZeta;
   comp.nThetaEff = s_.nThetaEff;
+  comp.ncurr = ncurr;
+  comp.currH = m_p_.currH.data();
+  comp.wInt = s_.wInt.data();
   comp.nThetaReduced = s_.nThetaReduced;
   comp.mpol = s_.mpol;
   comp.ntor = s_.ntor;
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
index e79172b69..37d6d557b 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -45,13 +45,21 @@ struct LocalForceComposition {
   double lamscale;
   bool lthreed;
 
+  // Constrained-current profile (ncurr==1): chi' is a function of geometry,
+  // recomputed each step, so it is differentiated in place. ncurr==0 uses the
+  // frozen chipH above.
+  int ncurr = 0;
+  int nThetaEff = 0;
+  const double* currH = nullptr;  // index jH-nsMinH
+  const double* wInt = nullptr;   // index kl % nThetaEff
+
   // Spectral-condensation constraint force. Enabled only when with_constraint is
   // set; then geometry blocks 16-19 hold rCon, zCon, ruFull, zuFull and force
   // blocks 16-19 receive frcon_e/o, fzcon_e/o. The bandpass uses the Fourier
   // basis arrays and the tcon/faccon profiles; rCon0/zCon0 are frozen.
   bool with_constraint = false;
   int nsMaxF = 0;       // constraint RZ range upper bound
-  int nZeta = 0, nThetaEff = 0, nThetaReduced = 0, mpol = 0, ntor = 0, nnyq2 = 0;
+  int nZeta = 0, nThetaReduced = 0, mpol = 0, ntor = 0, nnyq2 = 0;
   const double* rCon0 = nullptr;
   const double* zCon0 = nullptr;
   const double* faccon = nullptr;
@@ -116,9 +124,31 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
   ComputeBsupContra(lue, luo, lve, lvo, gsqrt, c->sqrtSH, c->lthreed, nZnT,
                     c->nsMinF1, c->nsMinH, c->nsMaxH, bsupu, bsupv);
   for (int jH = c->nsMinH; jH < c->nsMaxH; ++jH) {
+    // For a prescribed-current profile (ncurr==1), chi' is recomputed from the
+    // geometry each step (constrained toroidal current), so differentiate it
+    // here rather than freezing it. For ncurr==0 chi' = iota*phi' is a fixed
+    // profile, so use the frozen c->chipH.
+    double chip = c->chipH[jH - c->nsMinH];
+    if (c->ncurr == 1) {
+      double jvPlasma = 0.0;
+      double avg_guu_gsqrt = 0.0;
+      for (int kl = 0; kl < nZnT; ++kl) {
+        const int ih = (jH - c->nsMinH) * nZnT + kl;
+        const int l = kl % c->nThetaEff;
+        if (c->lthreed) {
+          jvPlasma += (guu[ih] * bsupu[ih] + guv[ih] * bsupv[ih]) * c->wInt[l];
+        } else {
+          jvPlasma += guu[ih] * bsupu[ih] * c->wInt[l];
+        }
+        avg_guu_gsqrt += guu[ih] / gsqrt[ih] * c->wInt[l];
+      }
+      if (avg_guu_gsqrt != 0.0) {
+        chip = (c->currH[jH - c->nsMinH] - jvPlasma) / avg_guu_gsqrt;
+      }
+    }
     for (int kl = 0; kl < nZnT; ++kl) {
       const int ih = (jH - c->nsMinH) * nZnT + kl;
-      bsupu[ih] += c->chipH[jH - c->nsMinH] / gsqrt[ih];
+      bsupu[ih] += chip / gsqrt[ih];
     }
   }
   ComputeBCo(guu, guv, gvv, bsupu, bsupv, c->lthreed, nH, bsubu, bsubv);

From 3ab1fc68edd0b596e28d853be3498d6d8b8b588b Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 15:02:35 +0200
Subject: [PATCH 14/79] examples: reproducible benchmarks for the exact HVP and
 adjoint gradient

benchmark_exact_hvp.py: preconditioned JFNK vs FD-HVP vs exact-HVP Newton-Krylov
(internal solver). benchmark_adjoint_gradient.py: FD-over-boundary vs FD-HVP
adjoint vs exact-HVP adjoint (external/SIMSOPT boundary gradient). These produce
the performance tables in the PR descriptions; the exact-HVP rows need an
Enzyme-enabled build.
---
 examples/benchmark_adjoint_gradient.py | 91 ++++++++++++++++++++++++++
 examples/benchmark_exact_hvp.py        | 42 ++++++++++++
 2 files changed, 133 insertions(+)
 create mode 100644 examples/benchmark_adjoint_gradient.py
 create mode 100644 examples/benchmark_exact_hvp.py

diff --git a/examples/benchmark_adjoint_gradient.py b/examples/benchmark_adjoint_gradient.py
new file mode 100644
index 000000000..6734dc0a1
--- /dev/null
+++ b/examples/benchmark_adjoint_gradient.py
@@ -0,0 +1,91 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Benchmark the boundary-shape gradient dJ/dx_B used by external optimizers.
+
+Three ways to get the same gradient: finite difference over boundary DOFs (the
+stock approach, re-solving VMEC per DOF), the implicit-function adjoint with the
+finite-difference HVP, and the adjoint with the exact autodiff HVP. Reports force
+evaluations, wall time, and agreement vs the FD reference. The exact-HVP path
+needs an Enzyme-enabled build.
+
+Run: python examples/benchmark_adjoint_gradient.py
+"""
+
+import time
+from pathlib import Path
+
+import numpy as np
+from scipy.sparse.linalg import LinearOperator, gmres
+
+import vmecpp_adjoint as va
+
+DATA = Path(__file__).resolve().parent / "data"
+CASES = [("solovev", DATA / "solovev.json", 11),
+         ("cth_like", DATA / "cth_like_fixed_bdy.json", 11)]
+OBJ = va.mhd_energy
+
+
+def adjoint_grad(model, x_star, interior, boundary, exact):
+    """va.boundary_gradient with a selectable HVP backend."""
+    n = x_star.size
+    dj = va.objective_state_gradient(model, x_star, OBJ)
+    hvp = (model.exact_hessian_vector_product if exact
+           else model.hessian_vector_product)
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, True)
+    ni = interior.size
+
+    def hii(vi):
+        v = np.zeros(n)
+        v[interior] = vi
+        return np.asarray(hvp(np.ascontiguousarray(v)), float)[interior]
+
+    def mii(bi):
+        v = np.zeros(n)
+        v[interior] = bi
+        return np.asarray(model.apply_preconditioner(np.ascontiguousarray(v)),
+                          float)[interior]
+
+    lam, _ = gmres(LinearOperator((ni, ni), matvec=hii), dj[interior],
+                   M=LinearOperator((ni, ni), matvec=mii), rtol=1e-6,
+                   restart=100, maxiter=30)
+    emb = np.zeros(n)
+    emb[interior] = lam
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, False)
+    return dj[boundary] - np.asarray(hvp(np.ascontiguousarray(emb)), float)[boundary]
+
+
+for case, path, ns in CASES:
+    print(f"\n=== {case} (ns={ns}) ===")
+    model = va.make_model(path, ns)
+    interior, boundary = va.partition(model, ns)
+    x0 = np.asarray(model.get_state(), float)
+    x_star = va.solve_interior(model, x0, interior, boundary, x0[boundary])
+    ndof = boundary.size
+    dofs = list(range(min(ndof, 12)))
+
+    model.reset_force_eval_count()
+    t0 = time.perf_counter()
+    g_fd = va.finite_difference_boundary_gradient(
+        model, x_star, interior, boundary, OBJ, dofs)
+    fd_evals, fd_t = model.force_eval_count, time.perf_counter() - t0
+    gfd = np.array([g_fd[j] for j in dofs])
+    full = fd_evals * ndof / len(dofs)
+
+    print(f"  boundary DOFs = {ndof} (FD reference on {len(dofs)})")
+    print(f"  {'method':28s} {'F-evals':>10s} {'time[s]':>8s} {'rel vs FD':>10s}")
+    print(f"  {'FD over boundary (all, est)':28s} {int(full):10d} "
+          f"{fd_t * ndof / len(dofs):8.2f} {'(ref)':>10s}")
+    for label, exact in [("adjoint, FD HVP", False), ("adjoint, exact HVP", True)]:
+        try:
+            model.reset_force_eval_count()
+            t0 = time.perf_counter()
+            g = adjoint_grad(model, x_star, interior, boundary, exact)
+            ev, t = model.force_eval_count, time.perf_counter() - t0
+            rel = np.linalg.norm(g[dofs] - gfd) / (np.linalg.norm(gfd) + 1e-300)
+            print(f"  {label:28s} {ev:10d} {t:8.2f} {rel:10.1e}")
+        except AttributeError:
+            print(f"  {label:28s} (needs an Enzyme-enabled build)")
diff --git a/examples/benchmark_exact_hvp.py b/examples/benchmark_exact_hvp.py
new file mode 100644
index 000000000..e7a2040f1
--- /dev/null
+++ b/examples/benchmark_exact_hvp.py
@@ -0,0 +1,42 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Benchmark the exact autodiff Hessian-vector product as an internal solver.
+
+Compares preconditioned JFNK, the finite-difference Newton-HVP, and the exact
+autodiff Newton-HVP on the same VMEC++ residual, counting force evaluations
+inside VMEC++. Requires an Enzyme-enabled build (VMECPP_ENABLE_ENZYME) for the
+exact-HVP solver; the others run on any build.
+
+Run: python -m examples.benchmark_exact_hvp  (or python examples/benchmark_exact_hvp.py)
+"""
+
+from pathlib import Path
+
+import external_optimizers as eo
+
+DATA = Path(__file__).resolve().parent / "data"
+CASES = [("solovev", DATA / "solovev.json", 11),
+         ("cth_like", DATA / "cth_like_fixed_bdy.json", 11)]
+
+SOLVERS = [
+    ("precond JFNK", eo.solve_newton_krylov_preconditioned),
+    ("Newton FD-HVP + M^-1", eo.solve_newton_hvp),
+    ("Newton exact-HVP + M^-1", eo.solve_newton_exact_hvp),
+]
+
+for case, path, ns in CASES:
+    print(f"\n=== {case} (ns={ns}) ===")
+    _, wref = eo.reference_equilibrium(path, ns)
+    print(f"native solve: W = {wref:.8e}")
+    print(f"{'optimizer':28s} {'F-evals':>8s} {'iters':>6s} {'time[s]':>8s} "
+          f"{'||F||':>10s} {'dW':>10s}")
+    for label, solver in SOLVERS:
+        try:
+            r = solver(path, ns)[1]
+            print(f"{label:28s} {r.force_evals:8d} {r.outer_iters:6d} "
+                  f"{r.seconds:8.2f} {r.residual_norm:10.1e} "
+                  f"{abs(r.energy - wref):10.1e}")
+        except AttributeError:
+            print(f"{label:28s} (needs an Enzyme-enabled build)")

From e43935be70527a170581293dba5b4abc86e86de4 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 15:18:51 +0200
Subject: [PATCH 15/79] adjoint: use the exact autodiff HVP in the boundary
 gradient (SIMSOPT)

boundary_gradient and _interior_operators take exact=True to use
exact_hessian_vector_product (no force evaluation per matvec); the SIMSOPT
VmecEnergy.gradient wrapper uses it by default. A real SIMSOPT
least_squares_serial_solve loop (minimize (energy-target)^2 over the boundary)
converges with this analytic gradient to |J-target|/target = 7e-7 in 10
iterations.
---
 examples/simsopt_vmec_gradient.py |  5 +++--
 examples/vmecpp_adjoint.py        | 31 ++++++++++++++++++-------------
 2 files changed, 21 insertions(+), 15 deletions(-)

diff --git a/examples/simsopt_vmec_gradient.py b/examples/simsopt_vmec_gradient.py
index 2240d2a66..3374ec059 100644
--- a/examples/simsopt_vmec_gradient.py
+++ b/examples/simsopt_vmec_gradient.py
@@ -65,10 +65,11 @@ def value(self, p):
         self.model.evaluate(2, 2, False)
         return self.model.mhd_energy
 
-    def gradient(self, p):
+    def gradient(self, p, exact=True):
         self._resolve(p)
         return boundary_gradient(
-            self.model, self._x_full, self.interior, self.boundary, mhd_energy
+            self.model, self._x_full, self.interior, self.boundary, mhd_energy,
+            exact=exact,
         )
 
 
diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index a85ea2006..a5cb77448 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -62,20 +62,20 @@ def _raw_force(model, x):
     return np.asarray(model.get_forces(), float)
 
 
-def _interior_operators(model, x, interior):
+def _interior_operators(model, x, interior, exact=False):
     # The caller must set the base state to x and assemble the preconditioner
-    # (evaluate(2, 2, True)) before using these. hessian_vector_product uses the
-    # current state as its base point and restores it, so no per-matvec state
-    # update is needed: that keeps each Hessian matvec at two force evaluations.
+    # (evaluate(2, 2, True)) before using these. With exact=True the analytic
+    # autodiff HVP (no force evaluation per matvec) is used; otherwise the
+    # finite-difference HVP (two force evaluations per matvec).
     n = x.size
     ni = interior.size
+    hvp = (model.exact_hessian_vector_product if exact
+           else model.hessian_vector_product)
 
     def hii(vi):
         v = np.zeros(n)
         v[interior] = vi
-        return np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)[
-            interior
-        ]
+        return np.asarray(hvp(np.ascontiguousarray(v)), float)[interior]
 
     def mii(bi):
         v = np.zeros(n)
@@ -140,21 +140,26 @@ def objective_state_gradient(model, x, objective, h=1e-6):
     return g
 
 
-def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6):
-    """Adjoint gradient dJ/dx_B at the converged equilibrium x_star."""
+def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6,
+                      exact=False):
+    """Adjoint gradient dJ/dx_B at the converged equilibrium x_star.
+
+    With exact=True the analytic autodiff Hessian-vector product is used (no
+    force evaluation per matvec); otherwise the finite-difference HVP.
+    """
     n = x_star.size
     dj = objective_state_gradient(model, x_star, objective, h)
     model.set_state(np.ascontiguousarray(x_star))
     model.evaluate(2, 2, True)  # assemble preconditioner + set base state to x_star
-    h_op, m_op = _interior_operators(model, x_star, interior)
+    h_op, m_op = _interior_operators(model, x_star, interior, exact)
     lam, _ = gmres(h_op, dj[interior], M=m_op, rtol=1e-6, restart=100, maxiter=30)
     embedded = np.zeros(n)
     embedded[interior] = lam
+    hvp = (model.exact_hessian_vector_product if exact
+           else model.hessian_vector_product)
     model.set_state(np.ascontiguousarray(x_star))
     model.evaluate(2, 2, False)
-    coupling = np.asarray(
-        model.hessian_vector_product(np.ascontiguousarray(embedded)), float
-    )[boundary]
+    coupling = np.asarray(hvp(np.ascontiguousarray(embedded)), float)[boundary]
     return dj[boundary] - coupling
 
 

From b4669a0cff510ff04b4c110bff700f81f2369331 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 15:25:12 +0200
Subject: [PATCH 16/79] optimizers: Eisenstat-Walker forcing + lgmres ->
 exact-HVP Newton beats JFNK

The inner Krylov solve dominated wall-clock: the augmented Hessian is indefinite,
so a fixed tight inner tolerance made GMRES take ~580 matvecs per Newton step
even though only ~5-10 outer steps are needed. Profiling: 8 Newton iters but 4655
matvecs, one exact HVP matvec is 0.04 ms, and only 21% of the time was in the HVP.

Fix: Eisenstat-Walker adaptive inner forcing (loose-early/tight-late) and lgmres
(recycles Krylov vectors), applied to both Newton-HVP solvers (both already
preconditioned by VMEC's M^-1). Final, fair comparison (all preconditioned,
ns=11):

  solovev:  precond JFNK 507 ev / 0.08 s ;  exact-HVP Newton  17 ev / 0.03 s
  cth_like: precond JFNK 1633 ev / 2.00 s;  exact-HVP Newton  26 ev / 1.32 s

The exact-autodiff-HVP Newton-Krylov now beats preconditioned JFNK on both cases
in both force evaluations (30x / 63x fewer) and wall-clock (2.7x / 1.5x faster).

Also add examples/simsopt_optimization_loop.py: a real SIMSOPT
least_squares_serial_solve loop driving the boundary to a target energy via the
analytic adjoint gradient (converges to |J-target|/target = 7e-7).
---
 examples/external_optimizers.py       | 46 ++++++++++++++++++---------
 examples/simsopt_optimization_loop.py | 42 ++++++++++++++++++++++++
 2 files changed, 73 insertions(+), 15 deletions(-)
 create mode 100644 examples/simsopt_optimization_loop.py

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index ff29fb350..1fd7d4860 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -34,7 +34,7 @@
 
 import numpy as np
 from scipy.optimize import newton_krylov
-from scipy.sparse.linalg import LinearOperator, gmres
+from scipy.sparse.linalg import LinearOperator, gmres, lgmres
 
 try:
     from vmecpp.cpp import _vmecpp
@@ -148,16 +148,16 @@ def solve_newton_krylov_preconditioned(input_path=DEFAULT_INPUT, ns=11, tol=1e-9
     return solve_newton_krylov(input_path, ns, tol, preconditioned=True)
 
 
-def solve_newton_hvp(
-    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80, inner_tol=1e-3
-):
-    """Globalized Newton-Krylov using VMEC++'s own Hessian-vector product.
+def solve_newton_hvp(input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80):
+    """Globalized Newton-Krylov using VMEC++'s finite-difference Hessian-vector
+    product.
 
-    Each Newton step solves H dx = -F with GMRES, where H v is
-    hessian_vector_product (the analytic force's directional derivative computed
-    inside VMEC++) and the inner solve is preconditioned by M^-1. A backtracking
-    line search on ||F|| globalizes the step, which is required on stiff 3D cases
-    where the full Newton step overshoots.
+    Each Newton step solves H dx = -F with GMRES preconditioned by M^-1 (VMEC's
+    approximate inverse Hessian), with Eisenstat-Walker adaptive inner forcing
+    and a backtracking line search. H v is hessian_vector_product (a central
+    difference of the analytic force; two force evaluations per matvec). Same
+    solver as solve_newton_exact_hvp but with the FD HVP, for a like-for-like
+    comparison of the HVP backends.
     """
     model = make_model(input_path, ns)
     F = residual(model)
@@ -166,12 +166,17 @@ def solve_newton_hvp(
     model.reset_force_eval_count()
     t0 = time.perf_counter()
     it = 0
+    prev_norm = None
+    eta = 0.5
     for _ in range(max_newton):
         fk = F(x)
         norm0 = np.linalg.norm(fk)
         if norm0 < tol:
             break
         it += 1
+        if prev_norm is not None:
+            eta = min(0.5, max(1e-4, 0.9 * (norm0 / prev_norm) ** 2))
+        prev_norm = norm0
         model.set_state(np.ascontiguousarray(x))
         model.evaluate(2, 2, True)  # assemble M at the current iterate
         h_op = LinearOperator(
@@ -186,7 +191,7 @@ def solve_newton_hvp(
                 model.apply_preconditioner(np.ascontiguousarray(b)), float
             ),
         )
-        dx, _ = gmres(h_op, -fk, M=m_op, rtol=inner_tol, maxiter=100)
+        dx, _ = lgmres(h_op, -fk, M=m_op, rtol=eta, maxiter=200)
         # Backtracking line search: accept the largest step that reduces ||F||.
         alpha = 1.0
         for _ in range(30):
@@ -200,13 +205,18 @@ def solve_newton_hvp(
 
 
 def solve_newton_exact_hvp(
-    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80, inner_tol=1e-3
+    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80
 ):
     """Globalized Newton-Krylov using VMEC++'s exact autodiff Hessian-vector
     product (``exact_hessian_vector_product``, one Enzyme forward pass) instead
     of the finite-difference HVP. Requires an Enzyme-enabled build; raises
-    AttributeError otherwise. Same preconditioned GMRES + backtracking line
-    search as ``solve_newton_hvp``.
+    AttributeError otherwise.
+
+    The inner GMRES tolerance is set adaptively (Eisenstat-Walker forcing): the
+    augmented-Lagrangian Hessian is indefinite, so solving it tightly every step
+    wastes hundreds of matvecs early on. Loose-early/tight-late forcing cuts the
+    total matvec count several-fold while preserving the asymptotic convergence,
+    which is what dominates wall-clock (each matvec is cheap but there are many).
     """
     model = make_model(input_path, ns)
     F = residual(model)
@@ -215,12 +225,18 @@ def solve_newton_exact_hvp(
     model.reset_force_eval_count()
     t0 = time.perf_counter()
     it = 0
+    prev_norm = None
+    eta = 0.5
     for _ in range(max_newton):
         fk = F(x)
         norm0 = np.linalg.norm(fk)
         if norm0 < tol:
             break
         it += 1
+        # Eisenstat-Walker choice 2 for the inner forcing term.
+        if prev_norm is not None:
+            eta = min(0.5, max(1e-4, 0.9 * (norm0 / prev_norm) ** 2))
+        prev_norm = norm0
         model.set_state(np.ascontiguousarray(x))
         model.evaluate(2, 2, True)
         h_op = LinearOperator(
@@ -236,7 +252,7 @@ def solve_newton_exact_hvp(
                 model.apply_preconditioner(np.ascontiguousarray(b)), float
             ),
         )
-        dx, _ = gmres(h_op, -fk, M=m_op, rtol=inner_tol, maxiter=100)
+        dx, _ = lgmres(h_op, -fk, M=m_op, rtol=eta, maxiter=200)
         alpha = 1.0
         for _ in range(30):
             if np.linalg.norm(F(x + alpha * dx)) < norm0:
diff --git a/examples/simsopt_optimization_loop.py b/examples/simsopt_optimization_loop.py
new file mode 100644
index 000000000..4c3f4a050
--- /dev/null
+++ b/examples/simsopt_optimization_loop.py
@@ -0,0 +1,42 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""A real SIMSOPT gradient-based optimization loop driven by VMEC++.
+
+Builds the VmecEnergy Optimizable (MHD energy of the converged equilibrium, with
+the analytic adjoint boundary gradient from the exact autodiff Hessian-vector
+product), wraps it in a SIMSOPT least-squares problem, and runs
+least_squares_serial_solve to drive the plasma boundary toward a target energy.
+Demonstrates that VMEC++ is usable as a gradient-providing component in a real
+SIMSOPT optimization loop. Requires SIMSOPT and an Enzyme-enabled VMEC++ build.
+
+Run: python examples/simsopt_optimization_loop.py
+"""
+
+import time
+
+import numpy as np
+from simsopt.objectives import LeastSquaresProblem
+from simsopt.solve import least_squares_serial_solve
+
+from simsopt_vmec_gradient import VmecBoundaryProblem, make_simsopt_optimizable
+
+problem = VmecBoundaryProblem(ns=11)
+energy = make_simsopt_optimizable(problem)
+
+j0 = energy.J()
+target = 1.03 * j0
+print(f"initial MHD energy = {j0:.6e}, target = {target:.6e}")
+
+least_squares = LeastSquaresProblem.from_tuples([(energy.J, target, 1.0)])
+
+problem.model.reset_force_eval_count()
+t0 = time.perf_counter()
+least_squares_serial_solve(least_squares, grad=True)  # analytic adjoint gradient
+seconds = time.perf_counter() - t0
+
+j_final = energy.J()
+print(f"final MHD energy   = {j_final:.6e}")
+print(f"|J - target| / target = {abs(j_final - target) / target:.2e}")
+print(f"VMEC++ force evals = {problem.model.force_eval_count}, time = {seconds:.2f}s")

From 267fdd0177429bcb24833030c56cfc9c24ea178f Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 15:46:43 +0200
Subject: [PATCH 17/79] fix CI: formatting (ruff/docformatter/clang-format) +
 SIMSOPT gradient fallback

Apply pre-commit formatting across the touched files. Make the SIMSOPT
VmecEnergy.gradient auto-detect the exact HVP: use exact_hessian_vector_product
when the extension was built with Enzyme, otherwise fall back to the
finite-difference HVP, so the default (plugin-free) build -- and
test_simsopt_gradient -- pass without an Enzyme build.
---
 examples/benchmark_adjoint_gradient.py        |  37 ++--
 examples/benchmark_exact_hvp.py               |  20 +-
 examples/data/cth_like_fixed_bdy.out.h5       | Bin 0 -> 3031398 bytes
 examples/data/solovev.out.h5                  | Bin 0 -> 571757 bytes
 examples/external_optimizers.py               |  34 ++-
 examples/simsopt_optimization_loop.py         |   2 -
 examples/simsopt_vmec_gradient.py             |  19 +-
 examples/vmecpp_adjoint.py                    |  21 +-
 .../common/enzyme/local_force_hessian_test.cc |  36 ++--
 .../ideal_mhd_model/constraint_force_kernel.h |  25 ++-
 .../vmec/ideal_mhd_model/dft_toroidal.cc      |   7 +-
 .../vmec/ideal_mhd_model/exact_force_jvp.cc   |   3 +-
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   |  30 +--
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    |  16 +-
 .../ideal_mhd_model/lambda_force_kernel.h     |  14 +-
 .../ideal_mhd_model/local_force_composition.h | 202 ++++++++++++------
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 124 +++++++----
 tests/test_adjoint.py                         |   5 +-
 tests/test_external_optimizers.py             |   8 +-
 tests/test_internal_gradient.py               |  17 +-
 tests/test_preconditioner.py                  |  10 +-
 tests/test_simsopt_gradient.py                |   8 +-
 22 files changed, 385 insertions(+), 253 deletions(-)
 create mode 100644 examples/data/cth_like_fixed_bdy.out.h5
 create mode 100644 examples/data/solovev.out.h5

diff --git a/examples/benchmark_adjoint_gradient.py b/examples/benchmark_adjoint_gradient.py
index 6734dc0a1..081533f42 100644
--- a/examples/benchmark_adjoint_gradient.py
+++ b/examples/benchmark_adjoint_gradient.py
@@ -17,13 +17,14 @@
 from pathlib import Path
 
 import numpy as np
-from scipy.sparse.linalg import LinearOperator, gmres
-
 import vmecpp_adjoint as va
+from scipy.sparse.linalg import LinearOperator, gmres
 
 DATA = Path(__file__).resolve().parent / "data"
-CASES = [("solovev", DATA / "solovev.json", 11),
-         ("cth_like", DATA / "cth_like_fixed_bdy.json", 11)]
+CASES = [
+    ("solovev", DATA / "solovev.json", 11),
+    ("cth_like", DATA / "cth_like_fixed_bdy.json", 11),
+]
 OBJ = va.mhd_energy
 
 
@@ -31,8 +32,7 @@ def adjoint_grad(model, x_star, interior, boundary, exact):
     """va.boundary_gradient with a selectable HVP backend."""
     n = x_star.size
     dj = va.objective_state_gradient(model, x_star, OBJ)
-    hvp = (model.exact_hessian_vector_product if exact
-           else model.hessian_vector_product)
+    hvp = model.exact_hessian_vector_product if exact else model.hessian_vector_product
     model.set_state(np.ascontiguousarray(x_star))
     model.evaluate(2, 2, True)
     ni = interior.size
@@ -45,12 +45,18 @@ def hii(vi):
     def mii(bi):
         v = np.zeros(n)
         v[interior] = bi
-        return np.asarray(model.apply_preconditioner(np.ascontiguousarray(v)),
-                          float)[interior]
+        return np.asarray(model.apply_preconditioner(np.ascontiguousarray(v)), float)[
+            interior
+        ]
 
-    lam, _ = gmres(LinearOperator((ni, ni), matvec=hii), dj[interior],
-                   M=LinearOperator((ni, ni), matvec=mii), rtol=1e-6,
-                   restart=100, maxiter=30)
+    lam, _ = gmres(
+        LinearOperator((ni, ni), matvec=hii),
+        dj[interior],
+        M=LinearOperator((ni, ni), matvec=mii),
+        rtol=1e-6,
+        restart=100,
+        maxiter=30,
+    )
     emb = np.zeros(n)
     emb[interior] = lam
     model.set_state(np.ascontiguousarray(x_star))
@@ -70,15 +76,18 @@ def mii(bi):
     model.reset_force_eval_count()
     t0 = time.perf_counter()
     g_fd = va.finite_difference_boundary_gradient(
-        model, x_star, interior, boundary, OBJ, dofs)
+        model, x_star, interior, boundary, OBJ, dofs
+    )
     fd_evals, fd_t = model.force_eval_count, time.perf_counter() - t0
     gfd = np.array([g_fd[j] for j in dofs])
     full = fd_evals * ndof / len(dofs)
 
     print(f"  boundary DOFs = {ndof} (FD reference on {len(dofs)})")
     print(f"  {'method':28s} {'F-evals':>10s} {'time[s]':>8s} {'rel vs FD':>10s}")
-    print(f"  {'FD over boundary (all, est)':28s} {int(full):10d} "
-          f"{fd_t * ndof / len(dofs):8.2f} {'(ref)':>10s}")
+    print(
+        f"  {'FD over boundary (all, est)':28s} {int(full):10d} "
+        f"{fd_t * ndof / len(dofs):8.2f} {'(ref)':>10s}"
+    )
     for label, exact in [("adjoint, FD HVP", False), ("adjoint, exact HVP", True)]:
         try:
             model.reset_force_eval_count()
diff --git a/examples/benchmark_exact_hvp.py b/examples/benchmark_exact_hvp.py
index e7a2040f1..24d804fe7 100644
--- a/examples/benchmark_exact_hvp.py
+++ b/examples/benchmark_exact_hvp.py
@@ -17,8 +17,10 @@
 import external_optimizers as eo
 
 DATA = Path(__file__).resolve().parent / "data"
-CASES = [("solovev", DATA / "solovev.json", 11),
-         ("cth_like", DATA / "cth_like_fixed_bdy.json", 11)]
+CASES = [
+    ("solovev", DATA / "solovev.json", 11),
+    ("cth_like", DATA / "cth_like_fixed_bdy.json", 11),
+]
 
 SOLVERS = [
     ("precond JFNK", eo.solve_newton_krylov_preconditioned),
@@ -30,13 +32,17 @@
     print(f"\n=== {case} (ns={ns}) ===")
     _, wref = eo.reference_equilibrium(path, ns)
     print(f"native solve: W = {wref:.8e}")
-    print(f"{'optimizer':28s} {'F-evals':>8s} {'iters':>6s} {'time[s]':>8s} "
-          f"{'||F||':>10s} {'dW':>10s}")
+    print(
+        f"{'optimizer':28s} {'F-evals':>8s} {'iters':>6s} {'time[s]':>8s} "
+        f"{'||F||':>10s} {'dW':>10s}"
+    )
     for label, solver in SOLVERS:
         try:
             r = solver(path, ns)[1]
-            print(f"{label:28s} {r.force_evals:8d} {r.outer_iters:6d} "
-                  f"{r.seconds:8.2f} {r.residual_norm:10.1e} "
-                  f"{abs(r.energy - wref):10.1e}")
+            print(
+                f"{label:28s} {r.force_evals:8d} {r.outer_iters:6d} "
+                f"{r.seconds:8.2f} {r.residual_norm:10.1e} "
+                f"{abs(r.energy - wref):10.1e}"
+            )
         except AttributeError:
             print(f"{label:28s} (needs an Enzyme-enabled build)")
diff --git a/examples/data/cth_like_fixed_bdy.out.h5 b/examples/data/cth_like_fixed_bdy.out.h5
new file mode 100644
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diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index 1fd7d4860..9801ef3a1 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -34,7 +34,7 @@
 
 import numpy as np
 from scipy.optimize import newton_krylov
-from scipy.sparse.linalg import LinearOperator, gmres, lgmres
+from scipy.sparse.linalg import LinearOperator, lgmres
 
 try:
     from vmecpp.cpp import _vmecpp
@@ -149,15 +149,14 @@ def solve_newton_krylov_preconditioned(input_path=DEFAULT_INPUT, ns=11, tol=1e-9
 
 
 def solve_newton_hvp(input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80):
-    """Globalized Newton-Krylov using VMEC++'s finite-difference Hessian-vector
-    product.
+    """Globalized Newton-Krylov using VMEC++'s finite-difference Hessian-vector product.
 
     Each Newton step solves H dx = -F with GMRES preconditioned by M^-1 (VMEC's
-    approximate inverse Hessian), with Eisenstat-Walker adaptive inner forcing
-    and a backtracking line search. H v is hessian_vector_product (a central
-    difference of the analytic force; two force evaluations per matvec). Same
-    solver as solve_newton_exact_hvp but with the FD HVP, for a like-for-like
-    comparison of the HVP backends.
+    approximate inverse Hessian), with Eisenstat-Walker adaptive inner forcing and a
+    backtracking line search. H v is hessian_vector_product (a central difference of the
+    analytic force; two force evaluations per matvec). Same solver as
+    solve_newton_exact_hvp but with the FD HVP, for a like-for-like comparison of the
+    HVP backends.
     """
     model = make_model(input_path, ns)
     F = residual(model)
@@ -204,19 +203,16 @@ def solve_newton_hvp(input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80):
     return _finish(model, "Newton (VMEC++ HVP + M^-1)", x, it, t0)
 
 
-def solve_newton_exact_hvp(
-    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80
-):
-    """Globalized Newton-Krylov using VMEC++'s exact autodiff Hessian-vector
-    product (``exact_hessian_vector_product``, one Enzyme forward pass) instead
-    of the finite-difference HVP. Requires an Enzyme-enabled build; raises
-    AttributeError otherwise.
+def solve_newton_exact_hvp(input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80):
+    """Globalized Newton-Krylov using VMEC++'s exact autodiff Hessian-vector product
+    (``exact_hessian_vector_product``, one Enzyme forward pass) instead of the finite-
+    difference HVP. Requires an Enzyme-enabled build; raises AttributeError otherwise.
 
     The inner GMRES tolerance is set adaptively (Eisenstat-Walker forcing): the
-    augmented-Lagrangian Hessian is indefinite, so solving it tightly every step
-    wastes hundreds of matvecs early on. Loose-early/tight-late forcing cuts the
-    total matvec count several-fold while preserving the asymptotic convergence,
-    which is what dominates wall-clock (each matvec is cheap but there are many).
+    augmented-Lagrangian Hessian is indefinite, so solving it tightly every step wastes
+    hundreds of matvecs early on. Loose-early/tight-late forcing cuts the total matvec
+    count several-fold while preserving the asymptotic convergence, which is what
+    dominates wall-clock (each matvec is cheap but there are many).
     """
     model = make_model(input_path, ns)
     F = residual(model)
diff --git a/examples/simsopt_optimization_loop.py b/examples/simsopt_optimization_loop.py
index 4c3f4a050..c42876d18 100644
--- a/examples/simsopt_optimization_loop.py
+++ b/examples/simsopt_optimization_loop.py
@@ -16,10 +16,8 @@
 
 import time
 
-import numpy as np
 from simsopt.objectives import LeastSquaresProblem
 from simsopt.solve import least_squares_serial_solve
-
 from simsopt_vmec_gradient import VmecBoundaryProblem, make_simsopt_optimizable
 
 problem = VmecBoundaryProblem(ns=11)
diff --git a/examples/simsopt_vmec_gradient.py b/examples/simsopt_vmec_gradient.py
index 3374ec059..0a1502f03 100644
--- a/examples/simsopt_vmec_gradient.py
+++ b/examples/simsopt_vmec_gradient.py
@@ -65,10 +65,19 @@ def value(self, p):
         self.model.evaluate(2, 2, False)
         return self.model.mhd_energy
 
-    def gradient(self, p, exact=True):
+    def gradient(self, p, exact=None):
+        # Use the exact autodiff HVP when the extension was built with Enzyme,
+        # otherwise fall back to the finite-difference HVP so the default build
+        # still works.
+        if exact is None:
+            exact = hasattr(self.model, "exact_hessian_vector_product")
         self._resolve(p)
         return boundary_gradient(
-            self.model, self._x_full, self.interior, self.boundary, mhd_energy,
+            self.model,
+            self._x_full,
+            self.interior,
+            self.boundary,
+            mhd_energy,
             exact=exact,
         )
 
@@ -106,9 +115,9 @@ class GradResult:
 def gradient_cost(input_path: Path = DEFAULT_INPUT, ns: int = 11, analytic=True):
     """Cost of one full boundary gradient at the converged equilibrium.
 
-    This is what an external optimizer pays per iteration. The analytic adjoint
-    needs one Hessian solve regardless of the number of boundary DOFs; finite
-    differences re-converge the equilibrium twice per boundary DOF.
+    This is what an external optimizer pays per iteration. The analytic adjoint needs
+    one Hessian solve regardless of the number of boundary DOFs; finite differences re-
+    converge the equilibrium twice per boundary DOF.
     """
     problem = VmecBoundaryProblem(input_path, ns)
     x_star = problem._x_full.copy()
diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index a5cb77448..bd34ec056 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -69,8 +69,7 @@ def _interior_operators(model, x, interior, exact=False):
     # finite-difference HVP (two force evaluations per matvec).
     n = x.size
     ni = interior.size
-    hvp = (model.exact_hessian_vector_product if exact
-           else model.hessian_vector_product)
+    hvp = model.exact_hessian_vector_product if exact else model.hessian_vector_product
 
     def hii(vi):
         v = np.zeros(n)
@@ -93,9 +92,9 @@ def mii(bi):
 def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=80):
     """Converge the interior to force balance with the boundary held fixed.
 
-    Preconditioned Newton-Krylov on the interior residual with a backtracking
-    line search; the line search is required for stiff 3D equilibria, where the
-    full Newton step overshoots.
+    Preconditioned Newton-Krylov on the interior residual with a backtracking line
+    search; the line search is required for stiff 3D equilibria, where the full Newton
+    step overshoots.
     """
     x = np.asarray(x0, float).copy()
     x[boundary] = x_boundary
@@ -140,12 +139,13 @@ def objective_state_gradient(model, x, objective, h=1e-6):
     return g
 
 
-def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6,
-                      exact=False):
+def boundary_gradient(
+    model, x_star, interior, boundary, objective, h=1e-6, exact=False
+):
     """Adjoint gradient dJ/dx_B at the converged equilibrium x_star.
 
-    With exact=True the analytic autodiff Hessian-vector product is used (no
-    force evaluation per matvec); otherwise the finite-difference HVP.
+    With exact=True the analytic autodiff Hessian-vector product is used (no force
+    evaluation per matvec); otherwise the finite-difference HVP.
     """
     n = x_star.size
     dj = objective_state_gradient(model, x_star, objective, h)
@@ -155,8 +155,7 @@ def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6,
     lam, _ = gmres(h_op, dj[interior], M=m_op, rtol=1e-6, restart=100, maxiter=30)
     embedded = np.zeros(n)
     embedded[interior] = lam
-    hvp = (model.exact_hessian_vector_product if exact
-           else model.hessian_vector_product)
+    hvp = model.exact_hessian_vector_product if exact else model.hessian_vector_product
     model.set_state(np.ascontiguousarray(x_star))
     model.evaluate(2, 2, False)
     coupling = np.asarray(hvp(np.ascontiguousarray(embedded)), float)[boundary]
diff --git a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
index a052b9f4b..450818a83 100644
--- a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
+++ b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
@@ -36,12 +36,12 @@
 
 // Problem dimensions and the constant (non-differentiated) context.
 struct Ctx {
-  int nZnT, nsH;       // half-grid surfaces; full-grid has nsH + 1
-  int nFull, nHalf;    // (nsH+1)*nZnT, nsH*nZnT
-  const double* sqrtSF;  // [nsH+1]
-  const double* sqrtSH;  // [nsH]
-  const double* chipH;   // [nsH]
-  const double* presH;   // [nsH]
+  int nZnT, nsH;                 // half-grid surfaces; full-grid has nsH + 1
+  int nFull, nHalf;              // (nsH+1)*nZnT, nsH*nZnT
+  const double* sqrtSF;          // [nsH+1]
+  const double* sqrtSH;          // [nsH]
+  const double* chipH;           // [nsH]
+  const double* presH;           // [nsH]
   const double* radialBlending;  // [nsH+1]
   double deltaS;
   double lamscale;
@@ -55,7 +55,7 @@ enum { kGeomBlocks = 16, kForceBlocks = 16 };
 
 // g: geometry -> force density, composing the MHD and lambda-force kernels.
 __attribute__((noinline)) void LocalForce(const double* geom, double* work,
-                                           double* force, const Ctx* c) {
+                                          double* force, const Ctx* c) {
   vmecpp::LocalForceComposition lc;
   lc.nZnT = c->nZnT;
   lc.geom_stride = c->nFull;
@@ -78,7 +78,8 @@ __attribute__((noinline)) void LocalForce(const double* geom, double* work,
   vmecpp::ComputeLocalForceDensity(geom, work, force, &lc);
 }
 
-// Scalar objective L = 0.5 ||force||^2; work and force are caller-owned scratch.
+// Scalar objective L = 0.5 ||force||^2; work and force are caller-owned
+// scratch.
 __attribute__((noinline)) double Loss(const double* geom, double* work,
                                       double* force, const Ctx* c) {
   LocalForce(geom, work, force, c);
@@ -141,17 +142,19 @@ int main() {
     gp[i] = geom[i] + h * v[i];
     gm[i] = geom[i] - h * v[i];
   }
-  const double dfd =
-      (Loss(gp.data(), w2.data(), f2.data(), &c) -
-       Loss(gm.data(), w2.data(), f2.data(), &c)) /
-      (2 * h);
-  const double drev = std::inner_product(grev.begin(), grev.end(), v.begin(), 0.0);
+  const double dfd = (Loss(gp.data(), w2.data(), f2.data(), &c) -
+                      Loss(gm.data(), w2.data(), f2.data(), &c)) /
+                     (2 * h);
+  const double drev =
+      std::inner_product(grev.begin(), grev.end(), v.begin(), 0.0);
   const double scale = std::fabs(dfd) + 1e-300;
 
   printf("exact Hessian of VMEC local force map (MHD + lambda kernels)\n");
   printf("  geom dofs=%d  force outputs=%d\n", nG, nFc);
-  printf("  reverse dL.v vs finite-diff : %.2e\n", std::fabs(drev - dfd) / scale);
-  printf("  forward dL.v vs finite-diff : %.2e\n", std::fabs(dfwd - dfd) / scale);
+  printf("  reverse dL.v vs finite-diff : %.2e\n",
+         std::fabs(drev - dfd) / scale);
+  printf("  forward dL.v vs finite-diff : %.2e\n",
+         std::fabs(dfwd - dfd) / scale);
   printf("  forward / reverse agreement : %.2e\n",
          std::fabs(dfwd - drev) / (std::fabs(drev) + 1e-300));
 
@@ -160,7 +163,8 @@ int main() {
   for (int r = 0; r < reps; ++r) {
     volatile double q = __enzyme_fwddiff<double>(
         (void*)Loss, enzyme_dup, geom.data(), v.data(), enzyme_dup, work.data(),
-        dwork.data(), enzyme_dup, force.data(), dforce.data(), enzyme_const, &c);
+        dwork.data(), enzyme_dup, force.data(), dforce.data(), enzyme_const,
+        &c);
     (void)q;
   }
   auto t1 = std::chrono::steady_clock::now();
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h
index 6c0d69204..a5393da5f 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h
@@ -37,14 +37,12 @@ inline void ComputeEffectiveConstraintForce(
 
 // Add the bandpass-filtered constraint force gCon back into the MHD R/Z forces
 // (brmn, bzmn) and write the constraint-force outputs (frcon, fzcon).
-inline void AddConstraintForces(const double* rCon, const double* rCon0,
-                                const double* zCon, const double* zCon0,
-                                const double* ruFull, const double* zuFull,
-                                const double* gCon, const double* sqrtSF,
-                                int nZnT, int nsMinF, int nsMinF1, int nsMaxF,
-                                double* brmn_e, double* brmn_o, double* bzmn_e,
-                                double* bzmn_o, double* frcon_e, double* frcon_o,
-                                double* fzcon_e, double* fzcon_o) {
+inline void AddConstraintForces(
+    const double* rCon, const double* rCon0, const double* zCon,
+    const double* zCon0, const double* ruFull, const double* zuFull,
+    const double* gCon, const double* sqrtSF, int nZnT, int nsMinF, int nsMinF1,
+    int nsMaxF, double* brmn_e, double* brmn_o, double* bzmn_e, double* bzmn_o,
+    double* frcon_e, double* frcon_o, double* fzcon_e, double* fzcon_o) {
   for (int jF = nsMinF; jF < nsMaxF; ++jF) {
     for (int kl = 0; kl < nZnT; ++kl) {
       int idx_kl = (jF - nsMinF) * nZnT + kl;
@@ -66,13 +64,14 @@ inline void AddConstraintForces(const double* rCon, const double* rCon0,
 }
 
 // Fourier-space bandpass of the constraint force: forward transform gConEff to
-// the (m, n) coefficients gsc/gcs scaled by tcon, then inverse transform back to
-// real space scaled by faccon[m]. Bandpass keeps m in [1, mpol-1). The axis
+// the (m, n) coefficients gsc/gcs scaled by tcon, then inverse transform back
+// to real space scaled by faccon[m]. Bandpass keeps m in [1, mpol-1). The axis
 // surface has no poloidal angle and is skipped. Allocation-free over flat
 // buffers with explicit reductions (no Eigen temporaries), so it differentiates
-// under Enzyme; the free function vmecpp::deAliasConstraintForce wraps this with
-// the partition/basis structs. Basis layout: sinmui/cosmui/sinmu/cosmu indexed
-// m*nThetaReduced + l; cosnv/sinnv indexed k*(nnyq2+1) + n; gConEff/gCon indexed
+// under Enzyme; the free function vmecpp::deAliasConstraintForce wraps this
+// with the partition/basis structs. Basis layout: sinmui/cosmui/sinmu/cosmu
+// indexed m*nThetaReduced + l; cosnv/sinnv indexed k*(nnyq2+1) + n;
+// gConEff/gCon indexed
 // ((jF-nsMinF)*nZeta + k)*nThetaEff + l; gsc/gcs scratch of length ntor+1.
 inline void ComputeDeAliasConstraintForce(
     const double* gConEff, const double* faccon, const double* tcon,
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
index fb3a6e2db..5f384cd68 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
@@ -93,9 +93,10 @@ void ForcesToFourier3DSymmFastPoloidal(
         double zmksc_n = -czmn_seg.dot(sinmui_seg);
 
         // Assemble effective R and Z forces from MHD and spectral condensation
-        // contributions. Materialize to avoid re-evaluation in each dot product,
-        // reusing per-thread scratch so the inner loop allocates nothing (and
-        // stays Enzyme-differentiable). Same arithmetic as a fresh .eval().
+        // contributions. Materialize to avoid re-evaluation in each dot
+        // product, reusing per-thread scratch so the inner loop allocates
+        // nothing (and stays Enzyme-differentiable). Same arithmetic as a fresh
+        // .eval().
         thread_local Eigen::VectorXd tempR_seg, tempZ_seg;
         tempR_seg.noalias() = armn_seg + xmpq[m] * frcon_seg;
         tempZ_seg.noalias() = azmn_seg + xmpq[m] * fzcon_seg;
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
index e4b40a7d0..f7b5d5c31 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
@@ -10,8 +10,9 @@
 // for the single nonlinear pass and wraps it with the linear spectral
 // transforms.
 
-#include "vmecpp/common/enzyme/enzyme.h"
 #include "vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h"
+
+#include "vmecpp/common/enzyme/enzyme.h"
 #include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
 
 namespace vmecpp {
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 93398451f..8f538d101 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -25,8 +25,8 @@
 #include "vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h"
 #include "vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h"
-#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
 #include "vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h"
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
 #include "vmecpp/vmec/ideal_mhd_model/metric_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/mhdforce_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/pressure_kernel.h"
@@ -1549,8 +1549,8 @@ void IdealMhdModel::hybridLambdaForce() {
 #pragma omp barrier
 #endif  // _OPENMP
 
-  // Lambda force on the full grid via the shared kernel (lambda_force_kernel.h),
-  // also used by the Enzyme autodiff path.
+  // Lambda force on the full grid via the shared kernel
+  // (lambda_force_kernel.h), also used by the Enzyme autodiff path.
   ComputeHybridLambdaForce(
       bsubu.data(), bsubv.data(), gvv.data(), gsqrt.data(), guv.data(),
       bsupu.data(), lu_e.data(), lu_o.data(), m_p_.sqrtSH.data(),
@@ -2034,12 +2034,12 @@ void IdealMhdModel::assembleTotalForces() {
 
 #ifdef VMECPP_ENABLE_ENZYME
 void IdealMhdModel::packGeometry(FourierGeometry& m_decomposed,
-                                 FourierGeometry& m_physical_scratch, double* out,
-                                 int gS, bool primal) {
-  // Linear pre-chain decomposed -> real-space geometry, identical to the head of
-  // update(). Applied to a state it yields the geometry; applied to a tangent it
-  // yields the exact geometry tangent (the chain is linear), so no finite
-  // difference is needed.
+                                 FourierGeometry& m_physical_scratch,
+                                 double* out, int gS, bool primal) {
+  // Linear pre-chain decomposed -> real-space geometry, identical to the head
+  // of update(). Applied to a state it yields the geometry; applied to a
+  // tangent it yields the exact geometry tangent (the chain is linear), so no
+  // finite difference is needed.
   m_decomposed.decomposeInto(m_physical_scratch, m_p_.scalxc);
   m_physical_scratch.m1Constraint(1.0);
   m_physical_scratch.extrapolateTowardsAxis();
@@ -2087,7 +2087,8 @@ void IdealMhdModel::packGeometry(FourierGeometry& m_decomposed,
 }
 
 void IdealMhdModel::applyExactForceJacobian(const double* geomP,
-                                            const double* dgeom, int geom_stride,
+                                            const double* dgeom,
+                                            int geom_stride,
                                             FourierForces& m_physical_f,
                                             FourierForces& m_decomposed_hv) {
   LocalForceComposition comp;
@@ -2186,11 +2187,12 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   }
 }
 
-// Raw force-density tangent (20 blocks of (nsMaxFIncludingLcfs-nsMinF)*nZnT) from
-// one Enzyme forward pass, with no transform applied. For isolating the JVP from
-// the spectral-transform wrapping.
+// Raw force-density tangent (20 blocks of (nsMaxFIncludingLcfs-nsMinF)*nZnT)
+// from one Enzyme forward pass, with no transform applied. For isolating the
+// JVP from the spectral-transform wrapping.
 void IdealMhdModel::exactForceDensityTangent(const double* geomP,
-                                             const double* dgeom, int geom_stride,
+                                             const double* dgeom,
+                                             int geom_stride,
                                              double* dforce_out) {
   LocalForceComposition comp;
   comp.nZnT = s_.nZnT;
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index c4bbccff6..f02dccf0e 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -157,12 +157,12 @@ class IdealMhdModel {
 
   // Exact Hessian-vector product of the local force chain. Given the packed
   // real-space geometry primal geomP and a geometry tangent dgeom (each
-  // geom_stride doubles per block, see local_force_composition.h), differentiate
-  // the MHD-plus-lambda force density by one Enzyme forward pass, then apply the
-  // linear forward transform and preconditioner decomposition to obtain the
-  // decomposed force tangent in m_decomposed_hv. The constraint force is omitted
-  // here (it carries a linear bandpass handled separately). Used by the exact
-  // internal Newton-Krylov Hessian-vector product.
+  // geom_stride doubles per block, see local_force_composition.h),
+  // differentiate the MHD-plus-lambda force density by one Enzyme forward pass,
+  // then apply the linear forward transform and preconditioner decomposition to
+  // obtain the decomposed force tangent in m_decomposed_hv. The constraint
+  // force is omitted here (it carries a linear bandpass handled separately).
+  // Used by the exact internal Newton-Krylov Hessian-vector product.
   void applyExactForceJacobian(const double* geomP, const double* dgeom,
                                int geom_stride, FourierForces& m_physical_f,
                                FourierForces& m_decomposed_hv);
@@ -170,8 +170,8 @@ class IdealMhdModel {
   // Linear pre-chain decomposed -> real-space geometry (decomposeInto,
   // m1Constraint, extrapolate, geometryFromFourier) packed into the 20-block
   // layout of local_force_composition.h, with the computeBContra lambda
-  // normalization. Applied to a state (primal=true, adds phipF on lu_e) it gives
-  // the geometry; applied to a tangent (primal=false) it gives the exact
+  // normalization. Applied to a state (primal=true, adds phipF on lu_e) it
+  // gives the geometry; applied to a tangent (primal=false) it gives the exact
   // geometry tangent, no finite difference. Uses m_physical_scratch as scratch.
   void packGeometry(FourierGeometry& m_decomposed,
                     FourierGeometry& m_physical_scratch, double* out, int gS,
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h
index be7d54599..105df3730 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h
@@ -9,11 +9,11 @@ namespace vmecpp {
 
 // Hybrid lambda force on the full grid: blmn (and clmn in 3D). The covariant
 // field bsubv is interpolated from the half grid two ways (a plain average and
-// an alternative from gvv/gsqrt times lambda plus, in 3D, guv*bsupu) and blended
-// with the radialBlending profile. The radial sweep carries the inside half-grid
-// point in scratch (bsubu_i, bsubv_i, gvv_gsqrt_i, guv_bsupu_i, each nZnT) and
-// shifts it outward each surface. Allocation-free over flat buffers, shared
-// between IdealMhdModel::hybridLambdaForce and the Enzyme autodiff path.
+// an alternative from gvv/gsqrt times lambda plus, in 3D, guv*bsupu) and
+// blended with the radialBlending profile. The radial sweep carries the inside
+// half-grid point in scratch (bsubu_i, bsubv_i, gvv_gsqrt_i, guv_bsupu_i, each
+// nZnT) and shifts it outward each surface. Allocation-free over flat buffers,
+// shared between IdealMhdModel::hybridLambdaForce and the Enzyme autodiff path.
 //
 // Half-grid inputs (index (jH-nsMinH)*nZnT): bsubu, bsubv, gvv, gsqrt, guv,
 // bsupu. Full-grid lambda (index (jF-nsMinF1)*nZnT): lu_e, lu_o. Profiles:
@@ -72,8 +72,8 @@ inline void ComputeHybridLambdaForce(
         }
       }
 
-      double gvv_gsqrt_lu_e =
-          0.5 * (gvv_gsqrt_i[kl] + gvv_gsqrt_o) * lu_e[(jF - nsMinF1) * nZnT + kl];
+      double gvv_gsqrt_lu_e = 0.5 * (gvv_gsqrt_i[kl] + gvv_gsqrt_o) *
+                              lu_e[(jF - nsMinF1) * nZnT + kl];
       double gvv_gsqrt_lu_o =
           0.5 * (gvv_gsqrt_i[kl] * sqrtSHi + gvv_gsqrt_o * sqrtSHo) *
           lu_o[(jF - nsMinF1) * nZnT + kl];
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
index 37d6d557b..a0af2cf67 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -16,29 +16,33 @@
 
 namespace vmecpp {
 
-// Composition of the local force-density chain (MHD force + hybrid lambda force)
-// as a single allocation-free map g: real-space geometry -> real-space force
-// density. This is the nonlinear core of VMEC's force; the spectral transforms
-// around it are linear and applied separately. Shared between the Enzyme
-// autodiff validation and the exact Hessian-vector product. The spectral-
-// condensation constraint force is handled by the caller (it carries a linear
-// Fourier bandpass), so g here is the MHD-plus-lambda part of the force.
+// Composition of the local force-density chain (MHD force + hybrid lambda
+// force) as a single allocation-free map g: real-space geometry -> real-space
+// force density. This is the nonlinear core of VMEC's force; the spectral
+// transforms around it are linear and applied separately. Shared between the
+// Enzyme autodiff validation and the exact Hessian-vector product. The
+// spectral- condensation constraint force is handled by the caller (it carries
+// a linear Fourier bandpass), so g here is the MHD-plus-lambda part of the
+// force.
 //
 // Geometry layout (each block GeomStride doubles, index (jF-nsMinF1)*nZnT):
-//   r1_e r1_o z1_e z1_o ru_e ru_o zu_e zu_o rv_e rv_o zv_e zv_o lu_e lu_o lv_e lv_o
+//   r1_e r1_o z1_e z1_o ru_e ru_o zu_e zu_o rv_e rv_o zv_e zv_o lu_e lu_o lv_e
+//   lv_o
 // Force layout (each block ForceStride doubles): the 12 MHD densities then
 //   blmn_e blmn_o clmn_e clmn_o.
 struct LocalForceComposition {
   int nZnT;
   int geom_stride;   // doubles per geometry block (>= (nsMaxF1-nsMinF1)*nZnT)
-  int force_stride;  // doubles per force block (>= (nsMaxFIncludingLcfs-nsMinF)*nZnT)
+  int force_stride;  // doubles per force block (>=
+                     // (nsMaxFIncludingLcfs-nsMinF)*nZnT)
   int nsMinF, nsMinF1, nsMinH, nsMaxH;
-  int jMaxRZ;                 // MHD force surfaces: [nsMinF, jMaxRZ)
-  int nsMaxFIncludingLcfs;    // lambda force surfaces: [nsMinF, nsMaxFIncludingLcfs)
-  const double* sqrtSF;       // index jF-nsMinF1
-  const double* sqrtSH;       // index jH-nsMinH
-  const double* chipH;        // index jH-nsMinH (frozen for ncurr==0)
-  const double* presH;        // index jH-nsMinH
+  int jMaxRZ;                    // MHD force surfaces: [nsMinF, jMaxRZ)
+  int nsMaxFIncludingLcfs;       // lambda force surfaces: [nsMinF,
+                                 // nsMaxFIncludingLcfs)
+  const double* sqrtSF;          // index jF-nsMinF1
+  const double* sqrtSH;          // index jH-nsMinH
+  const double* chipH;           // index jH-nsMinH (frozen for ncurr==0)
+  const double* presH;           // index jH-nsMinH
   const double* radialBlending;  // index jF-nsMinF1
   double deltaS;
   double dSHalfDsInterp;
@@ -53,12 +57,12 @@ struct LocalForceComposition {
   const double* currH = nullptr;  // index jH-nsMinH
   const double* wInt = nullptr;   // index kl % nThetaEff
 
-  // Spectral-condensation constraint force. Enabled only when with_constraint is
-  // set; then geometry blocks 16-19 hold rCon, zCon, ruFull, zuFull and force
-  // blocks 16-19 receive frcon_e/o, fzcon_e/o. The bandpass uses the Fourier
-  // basis arrays and the tcon/faccon profiles; rCon0/zCon0 are frozen.
+  // Spectral-condensation constraint force. Enabled only when with_constraint
+  // is set; then geometry blocks 16-19 hold rCon, zCon, ruFull, zuFull and
+  // force blocks 16-19 receive frcon_e/o, fzcon_e/o. The bandpass uses the
+  // Fourier basis arrays and the tcon/faccon profiles; rCon0/zCon0 are frozen.
   bool with_constraint = false;
-  int nsMaxF = 0;       // constraint RZ range upper bound
+  int nsMaxF = 0;  // constraint RZ range upper bound
   int nZeta = 0, nThetaReduced = 0, mpol = 0, ntor = 0, nnyq2 = 0;
   const double* rCon0 = nullptr;
   const double* zCon0 = nullptr;
@@ -98,26 +102,41 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
   const double* lvo = geom + 15 * gS;
 
   double* p = work;
-  double* r12 = p; p += nH;
-  double* ru12 = p; p += nH;
-  double* zu12 = p; p += nH;
-  double* rs = p; p += nH;
-  double* zs = p; p += nH;
-  double* tau = p; p += nH;
-  double* gsqrt = p; p += nH;
-  double* guu = p; p += nH;
-  double* guv = p; p += nH;
-  double* gvv = p; p += nH;
-  double* bsupu = p; p += nH;
-  double* bsupv = p; p += nH;
-  double* bsubu = p; p += nH;
-  double* bsubv = p; p += nH;
-  double* tp = p; p += nH;
+  double* r12 = p;
+  p += nH;
+  double* ru12 = p;
+  p += nH;
+  double* zu12 = p;
+  p += nH;
+  double* rs = p;
+  p += nH;
+  double* zs = p;
+  p += nH;
+  double* tau = p;
+  p += nH;
+  double* gsqrt = p;
+  p += nH;
+  double* guu = p;
+  p += nH;
+  double* guv = p;
+  p += nH;
+  double* gvv = p;
+  p += nH;
+  double* bsupu = p;
+  p += nH;
+  double* bsupv = p;
+  p += nH;
+  double* bsubu = p;
+  p += nH;
+  double* bsubv = p;
+  p += nH;
+  double* tp = p;
+  p += nH;
   double* s = p;  // 30 * nZnT
 
   ComputeHalfGridJacobian(r1e, r1o, z1e, z1o, rue, ruo, zue, zuo, c->sqrtSH,
-                          c->deltaS, c->dSHalfDsInterp, nZnT, c->nsMinF1, c->nsMinH,
-                          c->nsMaxH, r12, ru12, zu12, rs, zs, tau);
+                          c->deltaS, c->dSHalfDsInterp, nZnT, c->nsMinF1,
+                          c->nsMinH, c->nsMaxH, r12, ru12, zu12, rs, zs, tau);
   ComputeMetricElements(r1e, r1o, rue, ruo, zue, zuo, rve, rvo, zve, zvo, tau,
                         r12, c->sqrtSF, c->sqrtSH, c->lthreed, nZnT, c->nsMinF1,
                         c->nsMinH, c->nsMaxH, gsqrt, guu, guv, gvv);
@@ -158,21 +177,66 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
       tp[(jH - c->nsMinH) * nZnT + kl] += c->presH[jH - c->nsMinH];
   }
 
-  double* P_i = s; s += nZnT; double* rup_i = s; s += nZnT;
-  double* zup_i = s; s += nZnT; double* rsp_i = s; s += nZnT;
-  double* zsp_i = s; s += nZnT; double* taup_i = s; s += nZnT;
-  double* gbubu_i = s; s += nZnT; double* gbubv_i = s; s += nZnT;
-  double* gbvbv_i = s; s += nZnT; double* P_o = s; s += nZnT;
-  double* rup_o = s; s += nZnT; double* zup_o = s; s += nZnT;
-  double* rsp_o = s; s += nZnT; double* zsp_o = s; s += nZnT;
-  double* taup_o = s; s += nZnT; double* gbubu_o = s; s += nZnT;
-  double* gbubv_o = s; s += nZnT; double* gbvbv_o = s; s += nZnT;
-  double* P_avg = s; s += nZnT; double* P_wavg = s; s += nZnT;
-  double* gbubu_avg = s; s += nZnT; double* gbubu_wavg = s; s += nZnT;
-  double* gbvbv_avg = s; s += nZnT; double* gbvbv_wavg = s; s += nZnT;
-  double* gbubv_avg = s; s += nZnT; double* gbubv_wavg = s; s += nZnT;
-  double* bsubu_i = s; s += nZnT; double* bsubv_i = s; s += nZnT;
-  double* gvv_gsqrt_i = s; s += nZnT; double* guv_bsupu_i = s; s += nZnT;
+  double* P_i = s;
+  s += nZnT;
+  double* rup_i = s;
+  s += nZnT;
+  double* zup_i = s;
+  s += nZnT;
+  double* rsp_i = s;
+  s += nZnT;
+  double* zsp_i = s;
+  s += nZnT;
+  double* taup_i = s;
+  s += nZnT;
+  double* gbubu_i = s;
+  s += nZnT;
+  double* gbubv_i = s;
+  s += nZnT;
+  double* gbvbv_i = s;
+  s += nZnT;
+  double* P_o = s;
+  s += nZnT;
+  double* rup_o = s;
+  s += nZnT;
+  double* zup_o = s;
+  s += nZnT;
+  double* rsp_o = s;
+  s += nZnT;
+  double* zsp_o = s;
+  s += nZnT;
+  double* taup_o = s;
+  s += nZnT;
+  double* gbubu_o = s;
+  s += nZnT;
+  double* gbubv_o = s;
+  s += nZnT;
+  double* gbvbv_o = s;
+  s += nZnT;
+  double* P_avg = s;
+  s += nZnT;
+  double* P_wavg = s;
+  s += nZnT;
+  double* gbubu_avg = s;
+  s += nZnT;
+  double* gbubu_wavg = s;
+  s += nZnT;
+  double* gbvbv_avg = s;
+  s += nZnT;
+  double* gbvbv_wavg = s;
+  s += nZnT;
+  double* gbubv_avg = s;
+  s += nZnT;
+  double* gbubv_wavg = s;
+  s += nZnT;
+  double* bsubu_i = s;
+  s += nZnT;
+  double* bsubv_i = s;
+  s += nZnT;
+  double* gvv_gsqrt_i = s;
+  s += nZnT;
+  double* guv_bsupu_i = s;
+  s += nZnT;
 
   double* armn_e = force + 0 * fS;
   double* armn_o = force + 1 * fS;
@@ -187,25 +251,24 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
   double* czmn_e = force + 10 * fS;
   double* czmn_o = force + 11 * fS;
   ComputeMHDForceDensity(
-      r1e, r1o, rue, ruo, zue, zuo, z1o, rve, rvo, zve, zvo, r12, ru12, zu12, rs,
-      zs, tau, tp, gsqrt, bsupu, bsupv, c->sqrtSF, c->sqrtSH, P_i, rup_i, zup_i,
-      rsp_i, zsp_i, taup_i, gbubu_i, gbubv_i, gbvbv_i, P_o, rup_o, zup_o, rsp_o,
-      zsp_o, taup_o, gbubu_o, gbubv_o, gbvbv_o, P_avg, P_wavg, gbubu_avg,
+      r1e, r1o, rue, ruo, zue, zuo, z1o, rve, rvo, zve, zvo, r12, ru12, zu12,
+      rs, zs, tau, tp, gsqrt, bsupu, bsupv, c->sqrtSF, c->sqrtSH, P_i, rup_i,
+      zup_i, rsp_i, zsp_i, taup_i, gbubu_i, gbubv_i, gbvbv_i, P_o, rup_o, zup_o,
+      rsp_o, zsp_o, taup_o, gbubu_o, gbubv_o, gbvbv_o, P_avg, P_wavg, gbubu_avg,
       gbubu_wavg, gbvbv_avg, gbvbv_wavg, gbubv_avg, gbubv_wavg, c->deltaS, nZnT,
-      c->nsMinF, c->nsMinF1, c->nsMinH, c->nsMaxH, c->jMaxRZ, c->lthreed, armn_e,
-      armn_o, azmn_e, azmn_o, brmn_e, brmn_o, bzmn_e, bzmn_o, crmn_e, crmn_o,
-      czmn_e, czmn_o);
+      c->nsMinF, c->nsMinF1, c->nsMinH, c->nsMaxH, c->jMaxRZ, c->lthreed,
+      armn_e, armn_o, azmn_e, azmn_o, brmn_e, brmn_o, bzmn_e, bzmn_o, crmn_e,
+      crmn_o, czmn_e, czmn_o);
 
   double* blmn_e = force + 12 * fS;
   double* blmn_o = force + 13 * fS;
   double* clmn_e = force + 14 * fS;
   double* clmn_o = force + 15 * fS;
-  ComputeHybridLambdaForce(bsubu, bsubv, gvv, gsqrt, guv, bsupu, lue, luo,
-                           c->sqrtSH, c->sqrtSF, c->radialBlending, c->lamscale,
-                           c->lthreed, nZnT, c->nsMinF, c->nsMinF1, c->nsMinH,
-                           c->nsMaxH, c->nsMaxFIncludingLcfs, bsubu_i, bsubv_i,
-                           gvv_gsqrt_i, guv_bsupu_i, blmn_e, blmn_o, clmn_e,
-                           clmn_o);
+  ComputeHybridLambdaForce(
+      bsubu, bsubv, gvv, gsqrt, guv, bsupu, lue, luo, c->sqrtSH, c->sqrtSF,
+      c->radialBlending, c->lamscale, c->lthreed, nZnT, c->nsMinF, c->nsMinF1,
+      c->nsMinH, c->nsMaxH, c->nsMaxFIncludingLcfs, bsubu_i, bsubv_i,
+      gvv_gsqrt_i, guv_bsupu_i, blmn_e, blmn_o, clmn_e, clmn_o);
 
   if (c->with_constraint) {
     // geometry blocks 16-19 carry the constraint coordinates and full-grid
@@ -243,11 +306,10 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
     ComputeEffectiveConstraintForce(rCon, rCon0, zCon, zCon0, ruFull, zuFull,
                                     nZnT, c->nsMinF, c->nsMaxFIncludingLcfs,
                                     gConEff);
-    ComputeDeAliasConstraintForce(gConEff, c->faccon, c->tcon, c->sinmui,
-                                  c->cosmui, c->cosnv, c->sinnv, c->sinmu,
-                                  c->cosmu, c->nsMinF, c->nsMaxF, c->nZeta,
-                                  c->nThetaEff, c->nThetaReduced, c->mpol,
-                                  c->ntor, c->nnyq2, gsc, gcs, gCon);
+    ComputeDeAliasConstraintForce(
+        gConEff, c->faccon, c->tcon, c->sinmui, c->cosmui, c->cosnv, c->sinnv,
+        c->sinmu, c->cosmu, c->nsMinF, c->nsMaxF, c->nZeta, c->nThetaEff,
+        c->nThetaReduced, c->mpol, c->ntor, c->nnyq2, gsc, gcs, gCon);
     double* frcon_e = force + 16 * fS;
     double* frcon_o = force + 17 * fS;
     double* fzcon_e = force + 18 * fS;
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index e84582c4e..dd8bc98bf 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -401,7 +401,8 @@ class VmecModel {
   // the directional step is finite-differenced. The current state is restored.
   Eigen::VectorXd HessianVectorProduct(const Eigen::VectorXd &v,
                                        double eps_rel = 1e-7) {
-    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    const Eigen::VectorXd x =
+        FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     const double vnorm = v.norm();
     if (vnorm == 0.0) {
       return Eigen::VectorXd::Zero(x.size());
@@ -409,22 +410,25 @@ class VmecModel {
     const double eps = eps_rel * (1.0 + x.norm()) / vnorm;
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + eps * v);
     Evaluate(2, 2, /*precondition=*/false);
-    const Eigen::VectorXd fp = FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
+    const Eigen::VectorXd fp =
+        FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x - eps * v);
     Evaluate(2, 2, /*precondition=*/false);
-    const Eigen::VectorXd fm = FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
+    const Eigen::VectorXd fm =
+        FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
     return (fp - fm) / (2.0 * eps);
   }
 
 #ifdef VMECPP_ENABLE_ENZYME
-  // Exact Hessian-vector product H v = T^T J_g T v of the augmented functional's
-  // MHD-plus-lambda part, computed with one Enzyme forward pass through the local
-  // force-density composition (J_g) wrapped by the linear spectral transforms.
-  // The geometry tangent T v is obtained exactly from the linearity of
-  // geometryFromFourier: T v = geom(x+v) - geom(x), so no finite-difference step
-  // enters. The constraint force (a linear bandpass over a nonlinear product) is
-  // omitted here. The model state is restored to x on return.
+  // Exact Hessian-vector product H v = T^T J_g T v of the augmented
+  // functional's MHD-plus-lambda part, computed with one Enzyme forward pass
+  // through the local force-density composition (J_g) wrapped by the linear
+  // spectral transforms. The geometry tangent T v is obtained exactly from the
+  // linearity of geometryFromFourier: T v = geom(x+v) - geom(x), so no
+  // finite-difference step enters. The constraint force (a linear bandpass over
+  // a nonlinear product) is omitted here. The model state is restored to x on
+  // return.
   Eigen::VectorXd ExactHessianVectorProduct(const Eigen::VectorXd &v) {
     vmecpp::IdealMhdModel &model = *vmec_->m_[0];
     const int gS = static_cast<int>(model.r1_e.size());
@@ -456,7 +460,8 @@ class VmecModel {
   // production force density; a nonzero value localizes a composition bug.
   double ComposedForceResidual() {
     vmecpp::IdealMhdModel &model = *vmec_->m_[0];
-    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    const Eigen::VectorXd x =
+        FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     const int gS = static_cast<int>(model.r1_e.size());
     Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
     auto blk = [&](int b, const Eigen::VectorXd &src) {
@@ -465,48 +470,91 @@ class VmecModel {
     };
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
     Evaluate(2, 2, false);
-    blk(0, model.r1_e); blk(1, model.r1_o); blk(2, model.z1_e);
-    blk(3, model.z1_o); blk(4, model.ru_e); blk(5, model.ru_o);
-    blk(6, model.zu_e); blk(7, model.zu_o); blk(8, model.rv_e);
-    blk(9, model.rv_o); blk(10, model.zv_e); blk(11, model.zv_o);
-    blk(12, model.lu_e); blk(13, model.lu_o); blk(14, model.lv_e);
-    blk(15, model.lv_o); blk(16, model.rCon); blk(17, model.zCon);
-    blk(18, model.ruFull); blk(19, model.zuFull);
+    blk(0, model.r1_e);
+    blk(1, model.r1_o);
+    blk(2, model.z1_e);
+    blk(3, model.z1_o);
+    blk(4, model.ru_e);
+    blk(5, model.ru_o);
+    blk(6, model.zu_e);
+    blk(7, model.zu_o);
+    blk(8, model.rv_e);
+    blk(9, model.rv_o);
+    blk(10, model.zv_e);
+    blk(11, model.zv_o);
+    blk(12, model.lu_e);
+    blk(13, model.lu_o);
+    blk(14, model.lv_e);
+    blk(15, model.lv_o);
+    blk(16, model.rCon);
+    blk(17, model.zCon);
+    blk(18, model.ruFull);
+    blk(19, model.zuFull);
     return model.composedForceResidual(geomP.data(), gS);
   }
 
-  // Diagnostic: relative difference between the Enzyme force-density tangent and
-  // a finite-difference force-density tangent (both in real space, before the
-  // spectral transform). Isolates the JVP + geometry tangent from the transform
-  // wrapping. Returns (per-block max-rel, overall-rel) so a nonzero block
-  // localizes the discrepancy.
+  // Diagnostic: relative difference between the Enzyme force-density tangent
+  // and a finite-difference force-density tangent (both in real space, before
+  // the spectral transform). Isolates the JVP + geometry tangent from the
+  // transform wrapping. Returns (per-block max-rel, overall-rel) so a nonzero
+  // block localizes the discrepancy.
   std::pair<std::vector<double>, double> ForceDensityJvpResidual(
       const Eigen::VectorXd &v) {
     vmecpp::IdealMhdModel &M = *vmec_->m_[0];
     const int nForce = static_cast<int>(M.blmn_e.size());
     const int gS = static_cast<int>(M.r1_e.size());
-    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    const Eigen::VectorXd x =
+        FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     auto packGeom = [&](Eigen::VectorXd &dst) {
       auto blk = [&](int b, const Eigen::VectorXd &s) {
         for (int i = 0; i < static_cast<int>(s.size()); ++i)
           dst[b * gS + i] = s[i];
       };
-      blk(0, M.r1_e); blk(1, M.r1_o); blk(2, M.z1_e); blk(3, M.z1_o);
-      blk(4, M.ru_e); blk(5, M.ru_o); blk(6, M.zu_e); blk(7, M.zu_o);
-      blk(8, M.rv_e); blk(9, M.rv_o); blk(10, M.zv_e); blk(11, M.zv_o);
-      blk(12, M.lu_e); blk(13, M.lu_o); blk(14, M.lv_e); blk(15, M.lv_o);
-      blk(16, M.rCon); blk(17, M.zCon); blk(18, M.ruFull); blk(19, M.zuFull);
+      blk(0, M.r1_e);
+      blk(1, M.r1_o);
+      blk(2, M.z1_e);
+      blk(3, M.z1_o);
+      blk(4, M.ru_e);
+      blk(5, M.ru_o);
+      blk(6, M.zu_e);
+      blk(7, M.zu_o);
+      blk(8, M.rv_e);
+      blk(9, M.rv_o);
+      blk(10, M.zv_e);
+      blk(11, M.zv_o);
+      blk(12, M.lu_e);
+      blk(13, M.lu_o);
+      blk(14, M.lv_e);
+      blk(15, M.lv_o);
+      blk(16, M.rCon);
+      blk(17, M.zCon);
+      blk(18, M.ruFull);
+      blk(19, M.zuFull);
     };
     auto packForce = [&](Eigen::VectorXd &dst) {
       auto blk = [&](int b, const Eigen::VectorXd &s) {
         for (int i = 0; i < static_cast<int>(s.size()); ++i)
           dst[b * nForce + i] = s[i];
       };
-      blk(0, M.armn_e); blk(1, M.armn_o); blk(2, M.azmn_e); blk(3, M.azmn_o);
-      blk(4, M.brmn_e); blk(5, M.brmn_o); blk(6, M.bzmn_e); blk(7, M.bzmn_o);
-      blk(8, M.crmn_e); blk(9, M.crmn_o); blk(10, M.czmn_e); blk(11, M.czmn_o);
-      blk(12, M.blmn_e); blk(13, M.blmn_o); blk(14, M.clmn_e); blk(15, M.clmn_o);
-      blk(16, M.frcon_e); blk(17, M.frcon_o); blk(18, M.fzcon_e);
+      blk(0, M.armn_e);
+      blk(1, M.armn_o);
+      blk(2, M.azmn_e);
+      blk(3, M.azmn_o);
+      blk(4, M.brmn_e);
+      blk(5, M.brmn_o);
+      blk(6, M.bzmn_e);
+      blk(7, M.bzmn_o);
+      blk(8, M.crmn_e);
+      blk(9, M.crmn_o);
+      blk(10, M.czmn_e);
+      blk(11, M.czmn_o);
+      blk(12, M.blmn_e);
+      blk(13, M.blmn_o);
+      blk(14, M.clmn_e);
+      blk(15, M.clmn_o);
+      blk(16, M.frcon_e);
+      blk(17, M.frcon_o);
+      blk(18, M.fzcon_e);
       blk(19, M.fzcon_o);
     };
     const double eps = 1e-6 * (1.0 + x.norm()) / v.norm();
@@ -554,8 +602,8 @@ class VmecModel {
 
   // Apply VMEC's preconditioner M^-1 to a vector in the decomposed internal
   // basis, mirroring the native apply sequence (m=1, radial, lambda). This is
-  // VMEC's hand-built approximate inverse Hessian; gradient-based solvers use it
-  // as the metric (preconditioned Krylov / quasi-Newton, and as the
+  // VMEC's hand-built approximate inverse Hessian; gradient-based solvers use
+  // it as the metric (preconditioned Krylov / quasi-Newton, and as the
   // preconditioner for the Hessian solve in adjoint sensitivities).
   //
   // Requires a prior evaluate(precondition=true) at the current state: the
@@ -640,8 +688,8 @@ class VmecModel {
 
   std::unique_ptr<vmecpp::Vmec> vmec_;
 
-  // Cached primal geometry for the exact Hessian-vector product: it depends only
-  // on the state, so it is reused across Krylov matvecs and invalidated by
+  // Cached primal geometry for the exact Hessian-vector product: it depends
+  // only on the state, so it is reused across Krylov matvecs and invalidated by
   // SetState. Mutable so SetState (const) can clear it.
   mutable Eigen::VectorXd exact_primal_;
   mutable bool exact_primal_valid_ = false;
diff --git a/tests/test_adjoint.py b/tests/test_adjoint.py
index 0d740d6be..c20c969ee 100644
--- a/tests/test_adjoint.py
+++ b/tests/test_adjoint.py
@@ -5,9 +5,8 @@
 """The adjoint boundary gradient matches brute-force finite differences.
 
 dJ/d(boundary) from one Hessian solve (implicit-function adjoint) agrees with the
-reference gradient obtained by re-converging the interior equilibrium for each
-perturbed boundary degree of freedom. J here is the MHD energy of the converged
-equilibrium.
+reference gradient obtained by re-converging the interior equilibrium for each perturbed
+boundary degree of freedom. J here is the MHD energy of the converged equilibrium.
 """
 
 import sys
diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
index 7f12ff51a..19bad3e06 100644
--- a/tests/test_external_optimizers.py
+++ b/tests/test_external_optimizers.py
@@ -4,10 +4,10 @@
 # SPDX-License-Identifier: MIT
 """External optimizers reach the same equilibrium as the native solver.
 
-The raw internal-basis force (gradient of VMEC's augmented functional) is the
-residual F(x); F(x) = 0 at equilibrium. Both a native-style preconditioned
-descent and a Jacobian-free Newton-Krylov solver drive it to zero and recover
-the native solver's converged state and energy.
+The raw internal-basis force (gradient of VMEC's augmented functional) is the residual
+F(x); F(x) = 0 at equilibrium. Both a native-style preconditioned descent and a
+Jacobian-free Newton-Krylov solver drive it to zero and recover the native solver's
+converged state and energy.
 """
 
 import sys
diff --git a/tests/test_internal_gradient.py b/tests/test_internal_gradient.py
index bc792d40a..022697143 100644
--- a/tests/test_internal_gradient.py
+++ b/tests/test_internal_gradient.py
@@ -4,15 +4,14 @@
 # SPDX-License-Identifier: MIT
 """Tests for the unpreconditioned internal-basis gradient.
 
-VmecModel.evaluate(precondition=False) returns at the INVARIANT_RESIDUALS
-checkpoint, so get_forces() yields the raw, unpreconditioned force: the gradient
-of VMEC's augmented functional (MHD energy plus the spectral-condensation and
-lambda constraints) with respect to the decomposed (internal-basis) state. This
-is the gradient an external optimizer working in the internal basis needs.
-
-The preconditioned force (precondition=True) is the native solver's search
-direction and is a different vector. The raw gradient must vanish at the
-converged equilibrium.
+VmecModel.evaluate(precondition=False) returns at the INVARIANT_RESIDUALS checkpoint, so
+get_forces() yields the raw, unpreconditioned force: the gradient of VMEC's augmented
+functional (MHD energy plus the spectral-condensation and lambda constraints) with
+respect to the decomposed (internal-basis) state. This is the gradient an external
+optimizer working in the internal basis needs.
+
+The preconditioned force (precondition=True) is the native solver's search direction and
+is a different vector. The raw gradient must vanish at the converged equilibrium.
 """
 
 from pathlib import Path
diff --git a/tests/test_preconditioner.py b/tests/test_preconditioner.py
index a0e1cd8b5..7009dee8a 100644
--- a/tests/test_preconditioner.py
+++ b/tests/test_preconditioner.py
@@ -4,12 +4,12 @@
 # SPDX-License-Identifier: MIT
 """VmecModel.apply_preconditioner exposes VMEC's preconditioner as an operator.
 
-The preconditioner M^-1 is VMEC's hand-built approximate inverse Hessian. The
-native solver applies it to the raw force to get its search direction, so
+The preconditioner M^-1 is VMEC's hand-built approximate inverse Hessian. The native
+solver applies it to the raw force to get its search direction, so
 apply_preconditioner(raw force) must equal the preconditioned force exactly. The
-operator is linear and, once assembled (via evaluate(precondition=True)), does
-not depend on the current state, so it can be reused as a frozen preconditioner
-for Krylov/quasi-Newton solvers.
+operator is linear and, once assembled (via evaluate(precondition=True)), does not
+depend on the current state, so it can be reused as a frozen preconditioner for
+Krylov/quasi-Newton solvers.
 """
 
 from pathlib import Path
diff --git a/tests/test_simsopt_gradient.py b/tests/test_simsopt_gradient.py
index 4080079fe..712f9eb20 100644
--- a/tests/test_simsopt_gradient.py
+++ b/tests/test_simsopt_gradient.py
@@ -4,10 +4,10 @@
 # SPDX-License-Identifier: MIT
 """VMEC++ exposes an analytic boundary gradient to SIMSOPT.
 
-The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches
-finite differences of its objective, and computing it is much cheaper than the
-conventional finite-difference boundary gradient (which re-solves the
-equilibrium per boundary degree of freedom).
+The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches finite
+differences of its objective, and computing it is much cheaper than the conventional
+finite-difference boundary gradient (which re-solves the equilibrium per boundary degree
+of freedom).
 """
 
 import sys

From c22a0c2f3b46c5bc93f57e5aeb273972f2c939b3 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 15:54:18 +0200
Subject: [PATCH 18/79] apply pre-commit formatting (ruff, docformatter,
 clang-format)

---
 examples/external_optimizers.py                 |  9 ++++-----
 examples/simsopt_vmec_gradient.py               |  6 +++---
 examples/vmecpp_adjoint.py                      |  6 +++---
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc     | 13 ++++++++-----
 tests/test_adjoint.py                           |  5 ++---
 tests/test_external_optimizers.py               |  8 ++++----
 tests/test_internal_gradient.py                 | 17 ++++++++---------
 tests/test_preconditioner.py                    | 10 +++++-----
 tests/test_simsopt_gradient.py                  |  8 ++++----
 9 files changed, 41 insertions(+), 41 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index d134b782b..f05cdf671 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -153,11 +153,10 @@ def solve_newton_hvp(
 ):
     """Globalized Newton-Krylov using VMEC++'s own Hessian-vector product.
 
-    Each Newton step solves H dx = -F with GMRES, where H v is
-    hessian_vector_product (the analytic force's directional derivative computed
-    inside VMEC++) and the inner solve is preconditioned by M^-1. A backtracking
-    line search on ||F|| globalizes the step, which is required on stiff 3D cases
-    where the full Newton step overshoots.
+    Each Newton step solves H dx = -F with GMRES, where H v is hessian_vector_product
+    (the analytic force's directional derivative computed inside VMEC++) and the inner
+    solve is preconditioned by M^-1. A backtracking line search on ||F|| globalizes the
+    step, which is required on stiff 3D cases where the full Newton step overshoots.
     """
     model = make_model(input_path, ns)
     F = residual(model)
diff --git a/examples/simsopt_vmec_gradient.py b/examples/simsopt_vmec_gradient.py
index 2240d2a66..4fad1c071 100644
--- a/examples/simsopt_vmec_gradient.py
+++ b/examples/simsopt_vmec_gradient.py
@@ -105,9 +105,9 @@ class GradResult:
 def gradient_cost(input_path: Path = DEFAULT_INPUT, ns: int = 11, analytic=True):
     """Cost of one full boundary gradient at the converged equilibrium.
 
-    This is what an external optimizer pays per iteration. The analytic adjoint
-    needs one Hessian solve regardless of the number of boundary DOFs; finite
-    differences re-converge the equilibrium twice per boundary DOF.
+    This is what an external optimizer pays per iteration. The analytic adjoint needs
+    one Hessian solve regardless of the number of boundary DOFs; finite differences re-
+    converge the equilibrium twice per boundary DOF.
     """
     problem = VmecBoundaryProblem(input_path, ns)
     x_star = problem._x_full.copy()
diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index a85ea2006..24838c5d0 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -93,9 +93,9 @@ def mii(bi):
 def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=80):
     """Converge the interior to force balance with the boundary held fixed.
 
-    Preconditioned Newton-Krylov on the interior residual with a backtracking
-    line search; the line search is required for stiff 3D equilibria, where the
-    full Newton step overshoots.
+    Preconditioned Newton-Krylov on the interior residual with a backtracking line
+    search; the line search is required for stiff 3D equilibria, where the full Newton
+    step overshoots.
     """
     x = np.asarray(x0, float).copy()
     x[boundary] = x_boundary
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index 766b9d8de..321a931b9 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -400,7 +400,8 @@ class VmecModel {
   // the directional step is finite-differenced. The current state is restored.
   Eigen::VectorXd HessianVectorProduct(const Eigen::VectorXd &v,
                                        double eps_rel = 1e-7) {
-    const Eigen::VectorXd x = FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    const Eigen::VectorXd x =
+        FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     const double vnorm = v.norm();
     if (vnorm == 0.0) {
       return Eigen::VectorXd::Zero(x.size());
@@ -408,18 +409,20 @@ class VmecModel {
     const double eps = eps_rel * (1.0 + x.norm()) / vnorm;
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + eps * v);
     Evaluate(2, 2, /*precondition=*/false);
-    const Eigen::VectorXd fp = FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
+    const Eigen::VectorXd fp =
+        FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x - eps * v);
     Evaluate(2, 2, /*precondition=*/false);
-    const Eigen::VectorXd fm = FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
+    const Eigen::VectorXd fm =
+        FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
     return (fp - fm) / (2.0 * eps);
   }
 
   // Apply VMEC's preconditioner M^-1 to a vector in the decomposed internal
   // basis, mirroring the native apply sequence (m=1, radial, lambda). This is
-  // VMEC's hand-built approximate inverse Hessian; gradient-based solvers use it
-  // as the metric (preconditioned Krylov / quasi-Newton, and as the
+  // VMEC's hand-built approximate inverse Hessian; gradient-based solvers use
+  // it as the metric (preconditioned Krylov / quasi-Newton, and as the
   // preconditioner for the Hessian solve in adjoint sensitivities).
   //
   // Requires a prior evaluate(precondition=true) at the current state: the
diff --git a/tests/test_adjoint.py b/tests/test_adjoint.py
index 0d740d6be..c20c969ee 100644
--- a/tests/test_adjoint.py
+++ b/tests/test_adjoint.py
@@ -5,9 +5,8 @@
 """The adjoint boundary gradient matches brute-force finite differences.
 
 dJ/d(boundary) from one Hessian solve (implicit-function adjoint) agrees with the
-reference gradient obtained by re-converging the interior equilibrium for each
-perturbed boundary degree of freedom. J here is the MHD energy of the converged
-equilibrium.
+reference gradient obtained by re-converging the interior equilibrium for each perturbed
+boundary degree of freedom. J here is the MHD energy of the converged equilibrium.
 """
 
 import sys
diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
index 7f12ff51a..19bad3e06 100644
--- a/tests/test_external_optimizers.py
+++ b/tests/test_external_optimizers.py
@@ -4,10 +4,10 @@
 # SPDX-License-Identifier: MIT
 """External optimizers reach the same equilibrium as the native solver.
 
-The raw internal-basis force (gradient of VMEC's augmented functional) is the
-residual F(x); F(x) = 0 at equilibrium. Both a native-style preconditioned
-descent and a Jacobian-free Newton-Krylov solver drive it to zero and recover
-the native solver's converged state and energy.
+The raw internal-basis force (gradient of VMEC's augmented functional) is the residual
+F(x); F(x) = 0 at equilibrium. Both a native-style preconditioned descent and a
+Jacobian-free Newton-Krylov solver drive it to zero and recover the native solver's
+converged state and energy.
 """
 
 import sys
diff --git a/tests/test_internal_gradient.py b/tests/test_internal_gradient.py
index bc792d40a..022697143 100644
--- a/tests/test_internal_gradient.py
+++ b/tests/test_internal_gradient.py
@@ -4,15 +4,14 @@
 # SPDX-License-Identifier: MIT
 """Tests for the unpreconditioned internal-basis gradient.
 
-VmecModel.evaluate(precondition=False) returns at the INVARIANT_RESIDUALS
-checkpoint, so get_forces() yields the raw, unpreconditioned force: the gradient
-of VMEC's augmented functional (MHD energy plus the spectral-condensation and
-lambda constraints) with respect to the decomposed (internal-basis) state. This
-is the gradient an external optimizer working in the internal basis needs.
-
-The preconditioned force (precondition=True) is the native solver's search
-direction and is a different vector. The raw gradient must vanish at the
-converged equilibrium.
+VmecModel.evaluate(precondition=False) returns at the INVARIANT_RESIDUALS checkpoint, so
+get_forces() yields the raw, unpreconditioned force: the gradient of VMEC's augmented
+functional (MHD energy plus the spectral-condensation and lambda constraints) with
+respect to the decomposed (internal-basis) state. This is the gradient an external
+optimizer working in the internal basis needs.
+
+The preconditioned force (precondition=True) is the native solver's search direction and
+is a different vector. The raw gradient must vanish at the converged equilibrium.
 """
 
 from pathlib import Path
diff --git a/tests/test_preconditioner.py b/tests/test_preconditioner.py
index a0e1cd8b5..7009dee8a 100644
--- a/tests/test_preconditioner.py
+++ b/tests/test_preconditioner.py
@@ -4,12 +4,12 @@
 # SPDX-License-Identifier: MIT
 """VmecModel.apply_preconditioner exposes VMEC's preconditioner as an operator.
 
-The preconditioner M^-1 is VMEC's hand-built approximate inverse Hessian. The
-native solver applies it to the raw force to get its search direction, so
+The preconditioner M^-1 is VMEC's hand-built approximate inverse Hessian. The native
+solver applies it to the raw force to get its search direction, so
 apply_preconditioner(raw force) must equal the preconditioned force exactly. The
-operator is linear and, once assembled (via evaluate(precondition=True)), does
-not depend on the current state, so it can be reused as a frozen preconditioner
-for Krylov/quasi-Newton solvers.
+operator is linear and, once assembled (via evaluate(precondition=True)), does not
+depend on the current state, so it can be reused as a frozen preconditioner for
+Krylov/quasi-Newton solvers.
 """
 
 from pathlib import Path
diff --git a/tests/test_simsopt_gradient.py b/tests/test_simsopt_gradient.py
index 4080079fe..712f9eb20 100644
--- a/tests/test_simsopt_gradient.py
+++ b/tests/test_simsopt_gradient.py
@@ -4,10 +4,10 @@
 # SPDX-License-Identifier: MIT
 """VMEC++ exposes an analytic boundary gradient to SIMSOPT.
 
-The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches
-finite differences of its objective, and computing it is much cheaper than the
-conventional finite-difference boundary gradient (which re-solves the
-equilibrium per boundary degree of freedom).
+The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches finite
+differences of its objective, and computing it is much cheaper than the conventional
+finite-difference boundary gradient (which re-solves the equilibrium per boundary degree
+of freedom).
 """
 
 import sys

From 5ad6d3768ac72699191d7758c1061ee4b33298ec Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 15:59:47 +0200
Subject: [PATCH 19/79] clang-tidy: drop dead store in composition; use
 reinterpret_cast in JVP wrapper

---
 .../cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc      | 6 +++---
 .../vmecpp/vmec/ideal_mhd_model/local_force_composition.h   | 3 +--
 2 files changed, 4 insertions(+), 5 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
index f7b5d5c31..2880969d1 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
@@ -20,9 +20,9 @@ namespace vmecpp {
 void ExactForceDensityJvp(const double* geom, const double* dgeom, double* work,
                           double* dwork, double* force, double* dforce,
                           const LocalForceComposition* c) {
-  __enzyme_fwddiff<void>((void*)ComputeLocalForceDensity, enzyme_dup, geom,
-                         dgeom, enzyme_dup, work, dwork, enzyme_dup, force,
-                         dforce, enzyme_const, c);
+  __enzyme_fwddiff<void>(reinterpret_cast<void*>(ComputeLocalForceDensity),
+                         enzyme_dup, geom, dgeom, enzyme_dup, work, dwork,
+                         enzyme_dup, force, dforce, enzyme_const, c);
 }
 
 }  // namespace vmecpp
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
index a0af2cf67..e0ae63a16 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -291,8 +291,7 @@ inline void ComputeLocalForceDensity(const double* geom, double* work,
     // exact HVP consistent with re-evaluating rzConIntoVolume each step.
     double* rCon0 = s;
     s += (c->nsMaxFIncludingLcfs - c->nsMinF) * nZnT;
-    double* zCon0 = s;
-    s += (c->nsMaxFIncludingLcfs - c->nsMinF) * nZnT;
+    double* zCon0 = s;  // last slice of the work buffer
     const int lcfs = (c->nsMaxFIncludingLcfs - 1 - c->nsMinF) * nZnT;
     for (int jF = (c->nsMinF > 1 ? c->nsMinF : 1); jF < c->nsMaxFIncludingLcfs;
          ++jF) {

From 8c7a59b64d203c755486fdfa6100bf38a012f357 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 17:01:03 +0200
Subject: [PATCH 20/79] bazel: declare shared force-chain kernel + autodiff
 headers in ideal_mhd_model

ideal_mhd_model.cc includes the header-only force kernels and the autodiff
composition/JVP header; the bazel sandbox needs them declared or the bazel
builds (opt/asan/ubsan/tsan, test_bazel) fail with 'jacobian_kernel.h: No such
file'. Add them via glob so each branch picks up whatever exists on it. The
Enzyme JVP .cc stays CMake-only (guarded by VMECPP_ENABLE_ENZYME).
---
 .../cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel       | 11 ++++++++++-
 1 file changed, 10 insertions(+), 1 deletion(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
index 5fca72477..bea93546e 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
@@ -82,7 +82,16 @@ cc_test(
 cc_library(
     name = "ideal_mhd_model",
     srcs = ["ideal_mhd_model.cc"],
-    hdrs = ["ideal_mhd_model.h", "dft_data.h"],
+    # The shared force-chain kernels and the autodiff composition/JVP header are
+    # header-only and included by ideal_mhd_model.cc; declare whatever exists on
+    # this branch so the bazel sandbox can see them. The Enzyme JVP entry point
+    # (exact_force_jvp.cc) is built only by the CMake VMECPP_ENABLE_ENZYME path,
+    # not by bazel; its use in ideal_mhd_model.cc is guarded by that define.
+    hdrs = ["ideal_mhd_model.h", "dft_data.h"] + glob([
+        "*_kernel.h",
+        "local_force_composition.h",
+        "exact_force_jvp.h",
+    ]),
     visibility = ["//visibility:public"],
     deps = [
         ":dft_toroidal",

From 6b6bf447bef7440d7220037d03b376515952a9d8 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 17:31:45 +0200
Subject: [PATCH 21/79] ci: skip benchmark result upload on fork PRs (token is
 read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.
---
 .github/workflows/benchmarks.yaml | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

diff --git a/.github/workflows/benchmarks.yaml b/.github/workflows/benchmarks.yaml
index c2f3fdf4e..54326ec6a 100644
--- a/.github/workflows/benchmarks.yaml
+++ b/.github/workflows/benchmarks.yaml
@@ -36,7 +36,11 @@ jobs:
           pytest benchmarks/test_benchmarks.py --benchmark-json=benchmark_results.json
 
       - name: Compare benchmark result
-        if: github.event_name == 'pull_request'
+        # Skip the result upload/compare for fork PRs: their GITHUB_TOKEN is
+        # read-only, so comment-on-alert/auto-push hit 'Resource not accessible
+        # by integration'. The benchmarks still run above; only the write-back
+        # is skipped for forks.
+        if: github.event_name == 'pull_request' && github.event.pull_request.head.repo.full_name == github.repository
         uses: benchmark-action/github-action-benchmark@v1.21.0
         with:
           tool: "pytest"

From 54ccecbb5a4df1f26a748cc50f71b82028273430 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 18:12:42 +0200
Subject: [PATCH 22/79] ci: build VMEC2000 from source so the compat test runs
 on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.
---
 .github/workflows/tests.yaml | 27 +++++++++++++++++++++------
 1 file changed, 21 insertions(+), 6 deletions(-)

diff --git a/.github/workflows/tests.yaml b/.github/workflows/tests.yaml
index a22e0cbdc..4a7964059 100644
--- a/.github/workflows/tests.yaml
+++ b/.github/workflows/tests.yaml
@@ -53,12 +53,27 @@ jobs:
       - name: Also install VMEC2000 (only on Ubuntu 22.04)
         if: ${{ matrix.os == 'ubuntu-22.04' && matrix.python-version == '3.10' }}
         run: |
-          # mpi4py is needed for VMEC2000
-          sudo apt-get install -y libopenmpi-dev
-          python -m pip install mpi4py
-          # custom wheel for VMEC2000, needed for some VMEC++/VMEC2000 compatibility tests
-          # NOTE: this wheel is only guaranteed to work on Ubuntu 22.04
-          python -m pip install vmec@https://anaconda.org/eguiraud-pf/vmec/0.0.6/download/vmec-0.0.6-cp310-cp310-linux_x86_64.whl
+          # Build VMEC2000 from source instead of the old prebuilt wheel. The
+          # pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x and
+          # fails to import under numpy 2 (f90wrap array-interface break), so the
+          # compatibility test could never run. Building from source with current
+          # f90wrap produces numpy-2-compatible bindings. Recipe mirrors SIMSOPT's
+          # own CI (hiddenSymmetries/VMEC2000 cmake/machines/ubuntu.json).
+          sudo apt-get install -y libopenmpi-dev libnetcdff-dev libscalapack-mpi-dev \
+            libhdf5-dev libhdf5-serial-dev libfftw3-dev libopenblas-dev ninja-build
+          # Build VMEC2000 with the SYSTEM cmake + ninja. Do not pip-install the
+          # cmake/ninja wheels: they would shadow the system cmake that
+          # scikit-build-core's editable vmecpp rebuild records, breaking its
+          # verify_globs step in the editable matrix job.
+          python -m pip install mpi4py numpy scikit-build f90wrap setuptools wheel
+          git clone --depth 1 https://github.com/hiddenSymmetries/VMEC2000.git /tmp/VMEC2000
+          cd /tmp/VMEC2000
+          cp cmake/machines/ubuntu.json cmake_config_file.json
+          LD_LIBRARY_PATH=/usr/lib/x86_64-linux-gnu \
+            python -m pip install -v --no-build-isolation .
+          cd -
+          # fail loudly here if the binding is still broken, not in the test step
+          python -c "import vmec; print('VMEC2000 import OK')"
       - name: Install package
         run: |
           # on Ubuntu we would not need this, but on MacOS we need to point CMake to gfortran-14 and gcc-14

From 3699230dbed10e631c74897a8c3f1f5d9e223154 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 18:20:09 +0200
Subject: [PATCH 23/79] test: skip vmecpp-only indata fields in the VMEC2000
 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.
---
 tests/test_simsopt_compat.py | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/tests/test_simsopt_compat.py b/tests/test_simsopt_compat.py
index 609dffea2..1443d8cbc 100644
--- a/tests/test_simsopt_compat.py
+++ b/tests/test_simsopt_compat.py
@@ -303,6 +303,11 @@ def test_ensure_vmec2000_input_from_vmecpp_input():
         if varname[1:-1] == "axis_":
             # these are called differently in VMEC2000, e.g. raxis_c -> raxis_cc
             varname_vmec2000 = f"{varname[:-1]}c{varname[-1]}"
+        if not hasattr(vmec2000.indata, varname_vmec2000):
+            # vmecpp-only field (e.g. free_boundary_method) with no counterpart
+            # in the legacy VMEC2000 INDATA namelist; not part of the common
+            # subset under test.
+            continue
         vmec2000_var = getattr(vmec2000.indata, varname_vmec2000)
 
         if vmecpp_var is None:

From 22ea7921443a80d9e43ebf3442797de98aff390d Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 17:31:45 +0200
Subject: [PATCH 24/79] ci: skip benchmark result upload on fork PRs (token is
 read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.
---
 .github/workflows/benchmarks.yaml | 6 +++++-
 1 file changed, 5 insertions(+), 1 deletion(-)

diff --git a/.github/workflows/benchmarks.yaml b/.github/workflows/benchmarks.yaml
index c2f3fdf4e..54326ec6a 100644
--- a/.github/workflows/benchmarks.yaml
+++ b/.github/workflows/benchmarks.yaml
@@ -36,7 +36,11 @@ jobs:
           pytest benchmarks/test_benchmarks.py --benchmark-json=benchmark_results.json
 
       - name: Compare benchmark result
-        if: github.event_name == 'pull_request'
+        # Skip the result upload/compare for fork PRs: their GITHUB_TOKEN is
+        # read-only, so comment-on-alert/auto-push hit 'Resource not accessible
+        # by integration'. The benchmarks still run above; only the write-back
+        # is skipped for forks.
+        if: github.event_name == 'pull_request' && github.event.pull_request.head.repo.full_name == github.repository
         uses: benchmark-action/github-action-benchmark@v1.21.0
         with:
           tool: "pytest"

From e10d2ec51ce6c1fbf713d6eb80c2c7962a4e81da Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 18:12:42 +0200
Subject: [PATCH 25/79] ci: build VMEC2000 from source so the compat test runs
 on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.
---
 .github/workflows/tests.yaml | 27 +++++++++++++++++++++------
 1 file changed, 21 insertions(+), 6 deletions(-)

diff --git a/.github/workflows/tests.yaml b/.github/workflows/tests.yaml
index a22e0cbdc..4a7964059 100644
--- a/.github/workflows/tests.yaml
+++ b/.github/workflows/tests.yaml
@@ -53,12 +53,27 @@ jobs:
       - name: Also install VMEC2000 (only on Ubuntu 22.04)
         if: ${{ matrix.os == 'ubuntu-22.04' && matrix.python-version == '3.10' }}
         run: |
-          # mpi4py is needed for VMEC2000
-          sudo apt-get install -y libopenmpi-dev
-          python -m pip install mpi4py
-          # custom wheel for VMEC2000, needed for some VMEC++/VMEC2000 compatibility tests
-          # NOTE: this wheel is only guaranteed to work on Ubuntu 22.04
-          python -m pip install vmec@https://anaconda.org/eguiraud-pf/vmec/0.0.6/download/vmec-0.0.6-cp310-cp310-linux_x86_64.whl
+          # Build VMEC2000 from source instead of the old prebuilt wheel. The
+          # pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x and
+          # fails to import under numpy 2 (f90wrap array-interface break), so the
+          # compatibility test could never run. Building from source with current
+          # f90wrap produces numpy-2-compatible bindings. Recipe mirrors SIMSOPT's
+          # own CI (hiddenSymmetries/VMEC2000 cmake/machines/ubuntu.json).
+          sudo apt-get install -y libopenmpi-dev libnetcdff-dev libscalapack-mpi-dev \
+            libhdf5-dev libhdf5-serial-dev libfftw3-dev libopenblas-dev ninja-build
+          # Build VMEC2000 with the SYSTEM cmake + ninja. Do not pip-install the
+          # cmake/ninja wheels: they would shadow the system cmake that
+          # scikit-build-core's editable vmecpp rebuild records, breaking its
+          # verify_globs step in the editable matrix job.
+          python -m pip install mpi4py numpy scikit-build f90wrap setuptools wheel
+          git clone --depth 1 https://github.com/hiddenSymmetries/VMEC2000.git /tmp/VMEC2000
+          cd /tmp/VMEC2000
+          cp cmake/machines/ubuntu.json cmake_config_file.json
+          LD_LIBRARY_PATH=/usr/lib/x86_64-linux-gnu \
+            python -m pip install -v --no-build-isolation .
+          cd -
+          # fail loudly here if the binding is still broken, not in the test step
+          python -c "import vmec; print('VMEC2000 import OK')"
       - name: Install package
         run: |
           # on Ubuntu we would not need this, but on MacOS we need to point CMake to gfortran-14 and gcc-14

From 441f1d16f996013d904fdda16f569de0d952b10c Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 18:20:09 +0200
Subject: [PATCH 26/79] test: skip vmecpp-only indata fields in the VMEC2000
 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.
---
 tests/test_simsopt_compat.py | 5 +++++
 1 file changed, 5 insertions(+)

diff --git a/tests/test_simsopt_compat.py b/tests/test_simsopt_compat.py
index 609dffea2..1443d8cbc 100644
--- a/tests/test_simsopt_compat.py
+++ b/tests/test_simsopt_compat.py
@@ -303,6 +303,11 @@ def test_ensure_vmec2000_input_from_vmecpp_input():
         if varname[1:-1] == "axis_":
             # these are called differently in VMEC2000, e.g. raxis_c -> raxis_cc
             varname_vmec2000 = f"{varname[:-1]}c{varname[-1]}"
+        if not hasattr(vmec2000.indata, varname_vmec2000):
+            # vmecpp-only field (e.g. free_boundary_method) with no counterpart
+            # in the legacy VMEC2000 INDATA namelist; not part of the common
+            # subset under test.
+            continue
         vmec2000_var = getattr(vmec2000.indata, varname_vmec2000)
 
         if vmecpp_var is None:

From 031e73969ee97e0851f10a5743039090338ea9ea Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Mon, 15 Jun 2026 07:23:11 +0200
Subject: [PATCH 27/79] ci: sync VMEC2000-from-source build, benchmark fork
 guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.
---
 .github/workflows/tests.yaml | 39 +++++++++++++++++++-----------------
 CMakeLists.txt               |  7 +++----
 2 files changed, 24 insertions(+), 22 deletions(-)

diff --git a/.github/workflows/tests.yaml b/.github/workflows/tests.yaml
index 4a7964059..0f68ab6e9 100644
--- a/.github/workflows/tests.yaml
+++ b/.github/workflows/tests.yaml
@@ -50,28 +50,31 @@ jobs:
         run: |
           # install VMEC++ deps as well as VMEC2000 deps (we need to import VMEC2000 in a test)
           sudo apt-get update && sudo apt-get install -y build-essential cmake libnetcdf-dev liblapack-dev libomp-dev
+      - name: Cache the VMEC2000 wheel
+        if: ${{ matrix.os == 'ubuntu-22.04' && matrix.python-version == '3.10' }}
+        uses: actions/cache@v4
+        with:
+          path: /tmp/vmec2000-wheel
+          # Keyed on the pinned VMEC2000 source commit: rebuild only when it changes.
+          key: vmec2000-728af8bd6c79-cp310-ubuntu22.04
       - name: Also install VMEC2000 (only on Ubuntu 22.04)
         if: ${{ matrix.os == 'ubuntu-22.04' && matrix.python-version == '3.10' }}
         run: |
-          # Build VMEC2000 from source instead of the old prebuilt wheel. The
-          # pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x and
-          # fails to import under numpy 2 (f90wrap array-interface break), so the
-          # compatibility test could never run. Building from source with current
-          # f90wrap produces numpy-2-compatible bindings. Recipe mirrors SIMSOPT's
-          # own CI (hiddenSymmetries/VMEC2000 cmake/machines/ubuntu.json).
           sudo apt-get install -y libopenmpi-dev libnetcdff-dev libscalapack-mpi-dev \
-            libhdf5-dev libhdf5-serial-dev libfftw3-dev libopenblas-dev ninja-build
-          # Build VMEC2000 with the SYSTEM cmake + ninja. Do not pip-install the
-          # cmake/ninja wheels: they would shadow the system cmake that
-          # scikit-build-core's editable vmecpp rebuild records, breaking its
-          # verify_globs step in the editable matrix job.
-          python -m pip install mpi4py numpy scikit-build f90wrap setuptools wheel
-          git clone --depth 1 https://github.com/hiddenSymmetries/VMEC2000.git /tmp/VMEC2000
-          cd /tmp/VMEC2000
-          cp cmake/machines/ubuntu.json cmake_config_file.json
-          LD_LIBRARY_PATH=/usr/lib/x86_64-linux-gnu \
-            python -m pip install -v --no-build-isolation .
-          cd -
+            libopenblas-dev ninja-build
+          python -m pip install mpi4py
+          if ! ls /tmp/vmec2000-wheel/vmec-*.whl >/dev/null 2>&1; then
+            # Build with the SYSTEM cmake + ninja; do NOT pip-install the
+            # cmake/ninja wheels (they shadow the system cmake that
+            # scikit-build-core's editable vmecpp rebuild records).
+            python -m pip install numpy scikit-build f90wrap setuptools wheel
+            git clone https://github.com/hiddenSymmetries/VMEC2000.git /tmp/VMEC2000
+            git -C /tmp/VMEC2000 checkout 728af8bd6c796b36a0aa85fe298e507791e57c6e
+            cp /tmp/VMEC2000/cmake/machines/ubuntu.json /tmp/VMEC2000/cmake_config_file.json
+            LD_LIBRARY_PATH=/usr/lib/x86_64-linux-gnu \
+              python -m pip wheel /tmp/VMEC2000 --no-build-isolation -w /tmp/vmec2000-wheel
+          fi
+          python -m pip install /tmp/vmec2000-wheel/vmec-*.whl
           # fail loudly here if the binding is still broken, not in the test step
           python -c "import vmec; print('VMEC2000 import OK')"
       - name: Install package
diff --git a/CMakeLists.txt b/CMakeLists.txt
index 9bfeb25df..be83e281e 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -66,10 +66,9 @@ find_package(LAPACK REQUIRED)
 FetchContent_Declare(
   abseil-cpp
   GIT_REPOSITORY https://github.com/abseil/abseil-cpp.git
-  # 20260107.1 LTS: required for Clang >= 21, where the 2024 pin fails to
-  # compile (absl::Nonnull SFINAE in absl/strings/ascii.cc). Clang is the
-  # compiler used for the Enzyme autodiff build.
-  GIT_TAG 20260107.1
+  # 20260107.1 LTS: older abseil fails to compile under Clang >= 21 (the
+  # Enzyme build) on absl::Nonnull SFINAE in absl/strings/ascii.cc.
+  GIT_TAG 255c84dadd029fd8ad25c5efb5933e47beaa00c7
   GIT_SHALLOW TRUE
 )
 FetchContent_Declare(

From cd9995573f33dc62eea519fd09641caba42576b8 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Mon, 15 Jun 2026 07:24:51 +0200
Subject: [PATCH 28/79] ci: sync VMEC2000-from-source build, benchmark fork
 guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash for Clang >= 21.
---
 .github/workflows/tests.yaml | 39 +++++++++++++++++++-----------------
 CMakeLists.txt               |  4 +++-
 2 files changed, 24 insertions(+), 19 deletions(-)

diff --git a/.github/workflows/tests.yaml b/.github/workflows/tests.yaml
index 4a7964059..0f68ab6e9 100644
--- a/.github/workflows/tests.yaml
+++ b/.github/workflows/tests.yaml
@@ -50,28 +50,31 @@ jobs:
         run: |
           # install VMEC++ deps as well as VMEC2000 deps (we need to import VMEC2000 in a test)
           sudo apt-get update && sudo apt-get install -y build-essential cmake libnetcdf-dev liblapack-dev libomp-dev
+      - name: Cache the VMEC2000 wheel
+        if: ${{ matrix.os == 'ubuntu-22.04' && matrix.python-version == '3.10' }}
+        uses: actions/cache@v4
+        with:
+          path: /tmp/vmec2000-wheel
+          # Keyed on the pinned VMEC2000 source commit: rebuild only when it changes.
+          key: vmec2000-728af8bd6c79-cp310-ubuntu22.04
       - name: Also install VMEC2000 (only on Ubuntu 22.04)
         if: ${{ matrix.os == 'ubuntu-22.04' && matrix.python-version == '3.10' }}
         run: |
-          # Build VMEC2000 from source instead of the old prebuilt wheel. The
-          # pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x and
-          # fails to import under numpy 2 (f90wrap array-interface break), so the
-          # compatibility test could never run. Building from source with current
-          # f90wrap produces numpy-2-compatible bindings. Recipe mirrors SIMSOPT's
-          # own CI (hiddenSymmetries/VMEC2000 cmake/machines/ubuntu.json).
           sudo apt-get install -y libopenmpi-dev libnetcdff-dev libscalapack-mpi-dev \
-            libhdf5-dev libhdf5-serial-dev libfftw3-dev libopenblas-dev ninja-build
-          # Build VMEC2000 with the SYSTEM cmake + ninja. Do not pip-install the
-          # cmake/ninja wheels: they would shadow the system cmake that
-          # scikit-build-core's editable vmecpp rebuild records, breaking its
-          # verify_globs step in the editable matrix job.
-          python -m pip install mpi4py numpy scikit-build f90wrap setuptools wheel
-          git clone --depth 1 https://github.com/hiddenSymmetries/VMEC2000.git /tmp/VMEC2000
-          cd /tmp/VMEC2000
-          cp cmake/machines/ubuntu.json cmake_config_file.json
-          LD_LIBRARY_PATH=/usr/lib/x86_64-linux-gnu \
-            python -m pip install -v --no-build-isolation .
-          cd -
+            libopenblas-dev ninja-build
+          python -m pip install mpi4py
+          if ! ls /tmp/vmec2000-wheel/vmec-*.whl >/dev/null 2>&1; then
+            # Build with the SYSTEM cmake + ninja; do NOT pip-install the
+            # cmake/ninja wheels (they shadow the system cmake that
+            # scikit-build-core's editable vmecpp rebuild records).
+            python -m pip install numpy scikit-build f90wrap setuptools wheel
+            git clone https://github.com/hiddenSymmetries/VMEC2000.git /tmp/VMEC2000
+            git -C /tmp/VMEC2000 checkout 728af8bd6c796b36a0aa85fe298e507791e57c6e
+            cp /tmp/VMEC2000/cmake/machines/ubuntu.json /tmp/VMEC2000/cmake_config_file.json
+            LD_LIBRARY_PATH=/usr/lib/x86_64-linux-gnu \
+              python -m pip wheel /tmp/VMEC2000 --no-build-isolation -w /tmp/vmec2000-wheel
+          fi
+          python -m pip install /tmp/vmec2000-wheel/vmec-*.whl
           # fail loudly here if the binding is still broken, not in the test step
           python -c "import vmec; print('VMEC2000 import OK')"
       - name: Install package
diff --git a/CMakeLists.txt b/CMakeLists.txt
index be3c3255b..8e07b1753 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -66,7 +66,9 @@ find_package(LAPACK REQUIRED)
 FetchContent_Declare(
   abseil-cpp
   GIT_REPOSITORY https://github.com/abseil/abseil-cpp.git
-  GIT_TAG 4447c7562e3bc702ade25105912dce503f0c4010
+  # 20260107.1 LTS: older abseil fails to compile under Clang >= 21 (the
+  # Enzyme build) on absl::Nonnull SFINAE in absl/strings/ascii.cc.
+  GIT_TAG 255c84dadd029fd8ad25c5efb5933e47beaa00c7
   GIT_SHALLOW TRUE
 )
 FetchContent_Declare(

From 161a26345a39b8dc44bd1bfa9d2815ed23bf8e5f Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Mon, 15 Jun 2026 21:57:07 +0200
Subject: [PATCH 29/79] enzyme: reverse-mode force VJP and transposed exact
 Hessian-vector product

Add the transpose H^T of the exact Hessian-vector product. VMEC's internal
force is a scaled gradient, so H = dF/dx is non-symmetric and the implicit-
function adjoint of a general objective needs H^T, not H.

- exact_force_vjp.{h,cc}: reverse-mode (Enzyme __enzyme_autodiff) VJP of the
  local force-density kernel, the transpose of exact_force_jvp.
- applyExactForceJacobianTranspose plus the 2D/3D transposed spectral
  transforms (dft_*Transpose_*): the linear transforms are reused as adjoints
  with the poloidal integration weight and constraint-coordinate parity handled
  exactly; the nonlinear kernel uses the reverse-mode VJP.
- extrapolateTowardsAxisTranspose; makeLocalForceComposition (dedup of the
  force-composition descriptor shared by the forward/reverse passes).
- pybind: exact_hessian_vector_product_transpose, plus kernel JVP/VJP
  diagnostics.

Validated against the assembled Hessian to machine precision: <J_g x, y> =
<x, J_g^T y> to 3e-15, and H^T w vs H.T w to 3e-16 (2D solovev) and 5e-16
(3D cth_like).
---
 CMakeLists.txt                                |  32 +-
 .../vmec/fourier_geometry/fourier_geometry.cc |  41 ++
 .../vmec/fourier_geometry/fourier_geometry.h  |  33 +-
 .../vmec/ideal_mhd_model/exact_force_vjp.cc   |  30 ++
 .../vmec/ideal_mhd_model/exact_force_vjp.h    |  26 +
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   | 495 ++++++++++++++++--
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    |  36 ++
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 187 ++++---
 8 files changed, 752 insertions(+), 128 deletions(-)
 create mode 100644 src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.cc
 create mode 100644 src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.h

diff --git a/CMakeLists.txt b/CMakeLists.txt
index 9bfeb25df..9b33c1e2e 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -184,16 +184,35 @@ install(TARGETS indata2json DESTINATION vmecpp/cpp/third_party/indata2json/)
 #   -DVMECPP_ENABLE_ENZYME=ON -DVMECPP_ENZYME_PLUGIN=/path/to/ClangEnzyme-NN.so
 option(VMECPP_ENABLE_ENZYME "Build Enzyme autodiff targets" OFF)
 if(VMECPP_ENABLE_ENZYME)
-  if(NOT CMAKE_CXX_COMPILER_ID MATCHES "Clang")
+  if(NOT CMAKE_CXX_COMPILER_ID MATCHES "Clang|IntelLLVM")
     message(FATAL_ERROR
-      "VMECPP_ENABLE_ENZYME requires a Clang compiler (got "
-      "${CMAKE_CXX_COMPILER_ID}); Enzyme attaches as a Clang plugin.")
+      "VMECPP_ENABLE_ENZYME requires a Clang-based compiler (got "
+      "${CMAKE_CXX_COMPILER_ID}); Enzyme attaches as a Clang plugin. "
+      "Intel oneAPI icx/icpx (IntelLLVM) is Clang-based and supported.")
   endif()
   set(VMECPP_ENZYME_PLUGIN "" CACHE FILEPATH "Path to ClangEnzyme-NN.so")
+  # The Enzyme pass is an LLVM-version-specific plugin. icx/icpx (IntelLLVM)
+  # cannot load a plugin built against a different upstream LLVM. Enzyme's output
+  # is an ordinary object file, though, so for such compilers the single Enzyme
+  # TU (exact_force_jvp.cc) can be precompiled with a matching ClangEnzyme and
+  # linked in via VMECPP_ENZYME_OBJECT, while the rest is built with icx.
+  set(VMECPP_ENZYME_OBJECT "" CACHE FILEPATH
+    "Prebuilt exact_force_jvp object (compiled elsewhere with a matching ClangEnzyme)")
+  if(VMECPP_ENZYME_OBJECT)
+    if(NOT EXISTS "${VMECPP_ENZYME_OBJECT}")
+      message(FATAL_ERROR "VMECPP_ENZYME_OBJECT not found: ${VMECPP_ENZYME_OBJECT}")
+    endif()
+    message(STATUS "Enzyme: linking prebuilt object ${VMECPP_ENZYME_OBJECT}")
+    set_source_files_properties("${VMECPP_ENZYME_OBJECT}" PROPERTIES
+      EXTERNAL_OBJECT TRUE GENERATED TRUE)
+    target_sources(vmecpp_core PRIVATE "${VMECPP_ENZYME_OBJECT}")
+    target_compile_definitions(vmecpp_core PUBLIC VMECPP_ENABLE_ENZYME)
+  else()
   if(NOT VMECPP_ENZYME_PLUGIN OR NOT EXISTS "${VMECPP_ENZYME_PLUGIN}")
     message(FATAL_ERROR
       "VMECPP_ENABLE_ENZYME=ON requires "
-      "-DVMECPP_ENZYME_PLUGIN=/path/to/ClangEnzyme-NN.so")
+      "-DVMECPP_ENZYME_PLUGIN=/path/to/ClangEnzyme-NN.so "
+      "(or -DVMECPP_ENZYME_OBJECT=/path/to/exact_force_jvp.o for icx)")
   endif()
   message(STATUS "Enzyme plugin: ${VMECPP_ENZYME_PLUGIN}")
   enable_testing()
@@ -229,9 +248,12 @@ if(VMECPP_ENABLE_ENZYME)
   # VMECPP_ENABLE_ENZYME guards the callers so the default (plugin-free) build is
   # unaffected.
   target_sources(vmecpp_core PRIVATE
-    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc)
+    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
+    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.cc)
   set_source_files_properties(
     ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_jvp.cc
+    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.cc
     PROPERTIES COMPILE_OPTIONS "-O2;-fplugin=${VMECPP_ENZYME_PLUGIN}")
   target_compile_definitions(vmecpp_core PUBLIC VMECPP_ENABLE_ENZYME)
+  endif()
 endif()
diff --git a/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.cc b/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.cc
index 02c8ebf7c..2a755ff5f 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.cc
@@ -351,6 +351,47 @@ void FourierGeometry::extrapolateTowardsAxis() {
   }  // n
 }
 
+void FourierGeometry::extrapolateTowardsAxisTranspose() {
+  if (nsMin_ > 0) {
+    return;
+  }
+  int axis = 0;
+  int firstSurface = 1;
+  for (int n = 0; n < s_.ntor + 1; ++n) {
+    int m0 = 0;
+    int m1 = 1;
+    int axis0 = (axis * s_.mpol + m0) * (s_.ntor + 1) + n;
+    int axis1 = (axis * s_.mpol + m1) * (s_.ntor + 1) + n;
+    int firstSurface0 = (firstSurface * s_.mpol + m0) * (s_.ntor + 1) + n;
+    int firstSurface1 = (firstSurface * s_.mpol + m1) * (s_.ntor + 1) + n;
+
+    auto fold = [](std::span<double> c, int axisIdx, int firstIdx) {
+      c[firstIdx] += c[axisIdx];
+      c[axisIdx] = 0.0;
+    };
+    fold(rmncc, axis1, firstSurface1);
+    fold(zmnsc, axis1, firstSurface1);
+    fold(lmnsc, axis1, firstSurface1);
+    if (s_.lthreed) {
+      fold(rmnss, axis1, firstSurface1);
+      fold(zmncs, axis1, firstSurface1);
+      fold(lmncs, axis1, firstSurface1);
+      fold(lmncs, axis0, firstSurface0);
+    }
+    if (s_.lasym) {
+      fold(rmnsc, axis1, firstSurface1);
+      fold(zmncc, axis1, firstSurface1);
+      fold(lmncc, axis1, firstSurface1);
+      fold(lmncc, axis0, firstSurface0);
+      if (s_.lthreed) {
+        fold(rmncs, axis1, firstSurface1);
+        fold(zmnss, axis1, firstSurface1);
+        fold(lmnss, axis1, firstSurface1);
+      }
+    }
+  }  // n
+}
+
 void FourierGeometry::ComputeSpectralWidth(
     const FourierBasisFastPoloidal& fourier_basis,
     RadialProfiles& m_radial_profiles, const int p, const int q) const {
diff --git a/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.h b/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.h
index fa65a020f..1e6059b4a 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/fourier_geometry/fourier_geometry.h
@@ -17,14 +17,14 @@ namespace vmecpp {
 
 class FourierGeometry : public FourierCoeffs {
  public:
-  FourierGeometry(const Sizes *s, const RadialPartitioning *r, int ns);
-  FourierGeometry(const FourierGeometry &other);
-  FourierGeometry &operator=(const FourierGeometry &other);
-  FourierGeometry(FourierGeometry &&other) noexcept;
-  FourierGeometry &operator=(FourierGeometry &&other) noexcept;
+  FourierGeometry(const Sizes* s, const RadialPartitioning* r, int ns);
+  FourierGeometry(const FourierGeometry& other);
+  FourierGeometry& operator=(const FourierGeometry& other);
+  FourierGeometry(FourierGeometry&& other) noexcept;
+  FourierGeometry& operator=(FourierGeometry&& other) noexcept;
 
-  void interpFromBoundaryAndAxis(const FourierBasisFastPoloidal &t,
-                                 const Boundaries &b, const RadialProfiles &p);
+  void interpFromBoundaryAndAxis(const FourierBasisFastPoloidal& t,
+                                 const Boundaries& b, const RadialProfiles& p);
 
   // Initialize the state of this FourierGeometry with the given Fourier
   // coefficients. If a Boundaries object is specified (defaults to nullptr; in
@@ -33,18 +33,23 @@ class FourierGeometry : public FourierCoeffs {
   // instead of from the Fourier coefficient matrices.
   // This latter use case applies to fixed-boundary hot-restart operation of
   // VMEC++.
-  void InitFromState(const FourierBasisFastPoloidal &fb,
-                     const RowMatrixXd &rmnc, const RowMatrixXd &zmns,
-                     const RowMatrixXd &lmns_full, const RadialProfiles &p,
-                     const VmecConstants &constants,
-                     const Boundaries *b = nullptr);
+  void InitFromState(const FourierBasisFastPoloidal& fb,
+                     const RowMatrixXd& rmnc, const RowMatrixXd& zmns,
+                     const RowMatrixXd& lmns_full, const RadialProfiles& p,
+                     const VmecConstants& constants,
+                     const Boundaries* b = nullptr);
 
   void extrapolateTowardsAxis();
 
+  // Transpose of extrapolateTowardsAxis: the forward copies surface-1 m=1 (and
+  // m=0 lambda) coefficients onto the axis, so the adjoint folds the axis
+  // cotangent back into surface 1 and zeroes the axis.
+  void extrapolateTowardsAxisTranspose();
+
   // Compute the spectral width of the R and Z Fourier coefficients
   // and write it into the spectral_width vector in the given RadialProfiles.
-  void ComputeSpectralWidth(const FourierBasisFastPoloidal &fourier_basis,
-                            RadialProfiles &m_radial_profiles, int p = 4,
+  void ComputeSpectralWidth(const FourierBasisFastPoloidal& fourier_basis,
+                            RadialProfiles& m_radial_profiles, int p = 4,
                             int q = 1) const;
 
   // N.B. all raw pointers below are non-owning pointers to data in
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.cc
new file mode 100644
index 000000000..8d217a26c
--- /dev/null
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.cc
@@ -0,0 +1,30 @@
+// SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+// <info@proximafusion.com>
+//
+// SPDX-License-Identifier: MIT
+
+// Reverse-mode vector-Jacobian product of the local force-density composition.
+// Compiled with the Clang/Enzyme plugin so that __enzyme_autodiff
+// differentiates ComputeLocalForceDensity in reverse. This is the transpose
+// J_g^T of the forward pass in exact_force_jvp.cc; the transposed exact
+// Hessian-vector product wraps it with the transposed linear spectral
+// transforms.
+
+#include "vmecpp/vmec/ideal_mhd_model/exact_force_vjp.h"
+
+#include "vmecpp/common/enzyme/enzyme.h"
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
+
+namespace vmecpp {
+
+void ExactForceDensityVjp(const double* geom, double* geom_bar, double* work,
+                          double* work_bar, double* force, double* force_bar,
+                          const LocalForceComposition* c) {
+  // force_bar carries the output cotangent (seed); geom_bar accumulates
+  // J_g^T force_bar. geom_bar and work_bar are zeroed by the caller.
+  __enzyme_autodiff(reinterpret_cast<void*>(ComputeLocalForceDensity),
+                    enzyme_dup, geom, geom_bar, enzyme_dup, work, work_bar,
+                    enzyme_dup, force, force_bar, enzyme_const, c);
+}
+
+}  // namespace vmecpp
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.h
new file mode 100644
index 000000000..97b07df47
--- /dev/null
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/exact_force_vjp.h
@@ -0,0 +1,26 @@
+// SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+// <info@proximafusion.com>
+//
+// SPDX-License-Identifier: MIT
+#ifndef VMECPP_VMEC_IDEAL_MHD_MODEL_EXACT_FORCE_VJP_H_
+#define VMECPP_VMEC_IDEAL_MHD_MODEL_EXACT_FORCE_VJP_H_
+
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
+
+namespace vmecpp {
+
+// Reverse-mode vector-Jacobian product of ComputeLocalForceDensity: given the
+// geometry primal and a force-density cotangent in force_bar, accumulates the
+// geometry cotangent J_g^T force_bar into geom_bar in one Enzyme reverse pass.
+// geom_bar and work_bar must be zeroed by the caller; work/work_bar/force are
+// caller-owned scratch sized as ComputeLocalForceDensity requires. This is the
+// transpose of ExactForceDensityJvp and is the nonlinear factor of the
+// transposed exact Hessian-vector product. Defined in exact_force_vjp.cc, which
+// is compiled with the Clang/Enzyme plugin.
+void ExactForceDensityVjp(const double* geom, double* geom_bar, double* work,
+                          double* work_bar, double* force, double* force_bar,
+                          const LocalForceComposition* c);
+
+}  // namespace vmecpp
+
+#endif  // VMECPP_VMEC_IDEAL_MHD_MODEL_EXACT_FORCE_VJP_H_
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 8f538d101..cd5bd904b 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -24,6 +24,7 @@
 #include "vmecpp/vmec/ideal_mhd_model/bcontra_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/constraint_force_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/exact_force_jvp.h"
+#include "vmecpp/vmec/ideal_mhd_model/exact_force_vjp.h"
 #include "vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/lambda_force_kernel.h"
 #include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
@@ -2086,16 +2087,12 @@ void IdealMhdModel::packGeometry(FourierGeometry& m_decomposed,
   blk(19, zuFull);
 }
 
-void IdealMhdModel::applyExactForceJacobian(const double* geomP,
-                                            const double* dgeom,
-                                            int geom_stride,
-                                            FourierForces& m_physical_f,
-                                            FourierForces& m_decomposed_hv) {
+LocalForceComposition IdealMhdModel::makeLocalForceComposition(
+    int geom_stride) {
   LocalForceComposition comp;
   comp.nZnT = s_.nZnT;
   comp.geom_stride = geom_stride;
-  const int nForce = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
-  comp.force_stride = nForce;
+  comp.force_stride = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
   comp.nsMinF = r_.nsMinF;
   comp.nsMinF1 = r_.nsMinF1;
   comp.nsMinH = r_.nsMinH;
@@ -2132,6 +2129,16 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   comp.sinnv = t_.sinnv.data();
   comp.sinmu = t_.sinmu.data();
   comp.cosmu = t_.cosmu.data();
+  return comp;
+}
+
+void IdealMhdModel::applyExactForceJacobian(const double* geomP,
+                                            const double* dgeom,
+                                            int geom_stride,
+                                            FourierForces& m_physical_f,
+                                            FourierForces& m_decomposed_hv) {
+  LocalForceComposition comp = makeLocalForceComposition(geom_stride);
+  const int nForce = comp.force_stride;
 
   const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
   // work holds the half-grid and per-point scratch plus the constraint scratch
@@ -2194,47 +2201,8 @@ void IdealMhdModel::exactForceDensityTangent(const double* geomP,
                                              const double* dgeom,
                                              int geom_stride,
                                              double* dforce_out) {
-  LocalForceComposition comp;
-  comp.nZnT = s_.nZnT;
-  comp.geom_stride = geom_stride;
-  const int nForce = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
-  comp.force_stride = nForce;
-  comp.nsMinF = r_.nsMinF;
-  comp.nsMinF1 = r_.nsMinF1;
-  comp.nsMinH = r_.nsMinH;
-  comp.nsMaxH = r_.nsMaxH;
-  comp.jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns - 1);
-  comp.nsMaxFIncludingLcfs = r_.nsMaxFIncludingLcfs;
-  comp.sqrtSF = m_p_.sqrtSF.data();
-  comp.sqrtSH = m_p_.sqrtSH.data();
-  comp.chipH = m_p_.chipH.data();
-  comp.presH = m_p_.presH.data();
-  comp.radialBlending = m_p_.radialBlending.data();
-  comp.deltaS = m_fc_.deltaS;
-  comp.dSHalfDsInterp = dSHalfDsInterp;
-  comp.lamscale = constants_.lamscale;
-  comp.lthreed = s_.lthreed;
-  comp.with_constraint = true;
-  comp.nsMaxF = r_.nsMaxF;
-  comp.nZeta = s_.nZeta;
-  comp.nThetaEff = s_.nThetaEff;
-  comp.ncurr = ncurr;
-  comp.currH = m_p_.currH.data();
-  comp.wInt = s_.wInt.data();
-  comp.nThetaReduced = s_.nThetaReduced;
-  comp.mpol = s_.mpol;
-  comp.ntor = s_.ntor;
-  comp.nnyq2 = s_.nnyq2;
-  comp.rCon0 = rCon0.data();
-  comp.zCon0 = zCon0.data();
-  comp.faccon = faccon.data();
-  comp.tcon = tcon.data();
-  comp.sinmui = t_.sinmui.data();
-  comp.cosmui = t_.cosmui.data();
-  comp.cosnv = t_.cosnv.data();
-  comp.sinnv = t_.sinnv.data();
-  comp.sinmu = t_.sinmu.data();
-  comp.cosmu = t_.cosmu.data();
+  LocalForceComposition comp = makeLocalForceComposition(geom_stride);
+  const int nForce = comp.force_stride;
   const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
   const int nWork = 15 * nH + 30 * s_.nZnT + 4 * nForce + 2 * (s_.ntor + 1);
   std::vector<double> work(nWork, 0.0);
@@ -2244,6 +2212,437 @@ void IdealMhdModel::exactForceDensityTangent(const double* geomP,
                        dforce_out, &comp);
 }
 
+void IdealMhdModel::exactForceDensityCotangent(const double* geomP,
+                                               const double* force_bar,
+                                               int geom_stride,
+                                               double* geom_bar_out) {
+  LocalForceComposition comp = makeLocalForceComposition(geom_stride);
+  const int nForce = comp.force_stride;
+  const int nH = (r_.nsMaxH - r_.nsMinH) * s_.nZnT;
+  const int nWork = 15 * nH + 30 * s_.nZnT + 4 * nForce + 2 * (s_.ntor + 1);
+  std::vector<double> work(nWork, 0.0);
+  std::vector<double> work_bar(nWork, 0.0);
+  std::vector<double> force(20 * nForce, 0.0);
+  // force_bar is the output cotangent seed; Enzyme consumes (and may clobber)
+  // the shadow, so pass a private copy. geom_bar_out is zeroed by the caller.
+  std::vector<double> fbar(force_bar, force_bar + 20 * nForce);
+  ExactForceDensityVjp(geomP, geom_bar_out, work.data(), work_bar.data(),
+                       force.data(), fbar.data(), &comp);
+}
+
+// (forcesToFourier)^T for the 2D case: scatter a decomposed-force Fourier
+// cotangent (frcc, fzsc, flsc) back to the real-space force-density members
+// through the same weighted basis the forward projection uses. Transpose of
+// dft_ForcesToFourier_2d_symm.
+void IdealMhdModel::dft_ForcesToFourierTranspose_2d_symm(
+    const FourierForces& m_coeff_bar) {
+  for (auto* v :
+       {&armn_e, &armn_o, &brmn_e, &brmn_o, &azmn_e, &azmn_o, &bzmn_e, &bzmn_o,
+        &frcon_e, &frcon_o, &fzcon_e, &fzcon_o, &blmn_e, &blmn_o}) {
+    v->setZero();
+  }
+  int jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns - 1);
+  if (m_fc_.lfreeb &&
+      (m_vacuum_pressure_state_ == VacuumPressureState::kInitialized ||
+       m_vacuum_pressure_state_ == VacuumPressureState::kActive)) {
+    jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns);
+  }
+  for (int jF = r_.nsMinF; jF < jMaxRZ; ++jF) {
+    const int num_m = (jF == 0) ? 1 : s_.mpol;
+    for (int m = 0; m < num_m; ++m) {
+      const bool m_even = m % 2 == 0;
+      auto& armn = m_even ? armn_e : armn_o;
+      auto& brmn = m_even ? brmn_e : brmn_o;
+      auto& azmn = m_even ? azmn_e : azmn_o;
+      auto& bzmn = m_even ? bzmn_e : bzmn_o;
+      auto& frcon = m_even ? frcon_e : frcon_o;
+      auto& fzcon = m_even ? fzcon_e : fzcon_o;
+      const int idx_jm = (jF - r_.nsMinF) * s_.mpol + m;
+      const double fr = m_coeff_bar.frcc[idx_jm];
+      const double fz = m_coeff_bar.fzsc[idx_jm];
+      for (int l = 0; l < s_.nThetaReduced; ++l) {
+        const int idx_jl = (jF - r_.nsMinF) * s_.nThetaEff + l;
+        const int idx_ml = m * s_.nThetaReduced + l;
+        const double cosmui = t_.cosmui[idx_ml];
+        const double sinmumi = t_.sinmumi[idx_ml];
+        const double sinmui = t_.sinmui[idx_ml];
+        const double cosmumi = t_.cosmumi[idx_ml];
+        armn[idx_jl] += fr * cosmui;
+        brmn[idx_jl] += fr * sinmumi;
+        frcon[idx_jl] += fr * xmpq[m] * cosmui;
+        azmn[idx_jl] += fz * sinmui;
+        bzmn[idx_jl] += fz * cosmumi;
+        fzcon[idx_jl] += fz * xmpq[m] * sinmui;
+      }  // l
+    }  // m
+  }  // jF
+  for (int jF = std::max(1, r_.nsMinF); jF < r_.nsMaxFIncludingLcfs; ++jF) {
+    for (int m = 0; m < s_.mpol; ++m) {
+      const bool m_even = m % 2 == 0;
+      auto& blmn = m_even ? blmn_e : blmn_o;
+      const int idx_jm = (jF - r_.nsMinF) * s_.mpol + m;
+      const double fl = m_coeff_bar.flsc[idx_jm];
+      for (int l = 0; l < s_.nThetaReduced; ++l) {
+        const int idx_jl = (jF - r_.nsMinF) * s_.nThetaEff + l;
+        const double cosmumi = t_.cosmumi[m * s_.nThetaReduced + l];
+        blmn[idx_jl] += fl * cosmumi;
+      }  // l
+    }  // m
+  }  // jF
+}
+
+// (geometryFromFourier)^T for the 2D case: project the real-space geometry
+// member cotangents (r1_e .. zCon) back to Fourier coefficient cotangents
+// through the unweighted basis. Transpose of dft_FourierToReal_2d_symm.
+void IdealMhdModel::dft_FourierToRealTranspose_2d_symm(
+    FourierGeometry& m_coeff_bar_out) {
+  m_coeff_bar_out.setZero();
+  for (int jF = r_.nsMinF1; jF < r_.nsMaxF1; ++jF) {
+    double* dst_rcc = &(m_coeff_bar_out.rmncc[(jF - r_.nsMinF1) * s_.mnsize]);
+    double* dst_zsc = &(m_coeff_bar_out.zmnsc[(jF - r_.nsMinF1) * s_.mnsize]);
+    double* dst_lsc = &(m_coeff_bar_out.lmnsc[(jF - r_.nsMinF1) * s_.mnsize]);
+    for (int l = 0; l < s_.nThetaReduced; ++l) {
+      const int idx_jl = (jF - r_.nsMinF1) * s_.nThetaEff + l;
+      const double r1eb = r1_e[idx_jl], rueb = ru_e[idx_jl],
+                   z1eb = z1_e[idx_jl], zueb = zu_e[idx_jl],
+                   lueb = lu_e[idx_jl];
+      const double r1ob = r1_o[idx_jl], ruob = ru_o[idx_jl],
+                   z1ob = z1_o[idx_jl], zuob = zu_o[idx_jl],
+                   luob = lu_o[idx_jl];
+      const int num_m = (jF == 0) ? 2 : s_.mpol;
+      for (int m = 0; m < num_m; ++m) {
+        const int p = m % 2;
+        const int idx_ml = m * s_.nThetaReduced + l;
+        const double cosmu = t_.cosmu[idx_ml];
+        const double sinmum = t_.sinmum[idx_ml];
+        const double sinmu = t_.sinmu[idx_ml];
+        const double cosmum = t_.cosmum[idx_ml];
+        dst_rcc[m] += (p ? r1ob : r1eb) * cosmu + (p ? ruob : rueb) * sinmum;
+        dst_zsc[m] += (p ? z1ob : z1eb) * sinmu + (p ? zuob : zueb) * cosmum;
+        dst_lsc[m] += (p ? luob : lueb) * cosmum;
+      }  // m
+    }  // l
+  }  // jF
+  for (int jF = r_.nsMinF; jF < r_.nsMaxFIncludingLcfs; ++jF) {
+    double* dst_rcc = &(m_coeff_bar_out.rmncc[(jF - r_.nsMinF1) * s_.mnsize]);
+    double* dst_zsc = &(m_coeff_bar_out.zmnsc[(jF - r_.nsMinF1) * s_.mnsize]);
+    const int num_m = (jF == 0) ? 2 : s_.mpol;
+    for (int m = 0; m < num_m; ++m) {
+      const int p = m % 2;
+      const double scale = xmpq[m] * (1 - p + p * m_p_.sqrtSF[jF - r_.nsMinF1]);
+      for (int l = 0; l < s_.nThetaReduced; ++l) {
+        const int idx_ml = m * s_.nThetaReduced + l;
+        const int idx_con = (jF - r_.nsMinF) * s_.nThetaEff + l;
+        dst_rcc[m] += rCon[idx_con] * t_.cosmu[idx_ml] * scale;
+        dst_zsc[m] += zCon[idx_con] * t_.sinmu[idx_ml] * scale;
+      }  // l
+    }  // m
+  }  // jF
+}
+
+// (forcesToFourier)^T for the 3D case. Transpose of
+// ForcesToFourier3DSymmFastPoloidal: undo the toroidal scatter, then the
+// poloidal projection, back onto the real-space force-density members.
+void IdealMhdModel::dft_ForcesToFourierTranspose_3d_symm(
+    const FourierForces& m_coeff_bar) {
+  for (auto* v :
+       {&armn_e, &armn_o, &azmn_e,  &azmn_o,  &blmn_e,  &blmn_o, &brmn_e,
+        &brmn_o, &bzmn_e, &bzmn_o,  &clmn_e,  &clmn_o,  &crmn_e, &crmn_o,
+        &czmn_e, &czmn_o, &frcon_e, &frcon_o, &fzcon_e, &fzcon_o}) {
+    v->setZero();
+  }
+  const int nThR = s_.nThetaReduced;
+  const int ntorp1 = s_.ntor + 1;
+  int jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns - 1);
+  if (m_fc_.lfreeb &&
+      (m_vacuum_pressure_state_ == VacuumPressureState::kInitialized ||
+       m_vacuum_pressure_state_ == VacuumPressureState::kActive)) {
+    jMaxRZ = std::min(r_.nsMaxF, m_fc_.ns);
+  }
+  const int jMinL = 1;
+  for (int jF = r_.nsMinF; jF < jMaxRZ; ++jF) {
+    const int mmax = (jF == 0) ? 1 : s_.mpol;
+    for (int m = 0; m < mmax; ++m) {
+      const bool m_even = m % 2 == 0;
+      auto& armn = m_even ? armn_e : armn_o;
+      auto& azmn = m_even ? azmn_e : azmn_o;
+      auto& blmn = m_even ? blmn_e : blmn_o;
+      auto& brmn = m_even ? brmn_e : brmn_o;
+      auto& bzmn = m_even ? bzmn_e : bzmn_o;
+      auto& clmn = m_even ? clmn_e : clmn_o;
+      auto& crmn = m_even ? crmn_e : crmn_o;
+      auto& czmn = m_even ? czmn_e : czmn_o;
+      auto& frcon = m_even ? frcon_e : frcon_o;
+      auto& fzcon = m_even ? fzcon_e : fzcon_o;
+      const int idx_ml_base = m * nThR;
+      for (int k = 0; k < s_.nZeta; ++k) {
+        const int idx_kl_base =
+            ((jF - r_.nsMinF) * s_.nZeta + k) * s_.nThetaEff;
+        const int idx_kn_base = k * (s_.nnyq2 + 1);
+        const int idx_mn_base = ((jF - r_.nsMinF) * s_.mpol + m) * ntorp1;
+        double rmkcc = 0, rmkss = 0, zmksc = 0, zmkcs = 0, rmkcc_n = 0,
+               zmkcs_n = 0, rmkss_n = 0, zmksc_n = 0, lmksc = 0, lmkcs = 0,
+               lmkcs_n = 0, lmksc_n = 0;
+        for (int nn = 0; nn < ntorp1; ++nn) {
+          const int kn = idx_kn_base + nn;
+          const int mn = idx_mn_base + nn;
+          const double cosnv = t_.cosnv[kn], sinnv = t_.sinnv[kn],
+                       cosnvn = t_.cosnvn[kn], sinnvn = t_.sinnvn[kn];
+          const double frcc = m_coeff_bar.frcc[mn], frss = m_coeff_bar.frss[mn],
+                       fzsc = m_coeff_bar.fzsc[mn], fzcs = m_coeff_bar.fzcs[mn];
+          rmkcc += frcc * cosnv;
+          rmkcc_n += frcc * sinnvn;
+          rmkss += frss * sinnv;
+          rmkss_n += frss * cosnvn;
+          zmksc += fzsc * cosnv;
+          zmksc_n += fzsc * sinnvn;
+          zmkcs += fzcs * sinnv;
+          zmkcs_n += fzcs * cosnvn;
+          if (jMinL <= jF) {
+            const double flsc = m_coeff_bar.flsc[mn],
+                         flcs = m_coeff_bar.flcs[mn];
+            lmksc += flsc * cosnv;
+            lmksc_n += flsc * sinnvn;
+            lmkcs += flcs * sinnv;
+            lmkcs_n += flcs * cosnvn;
+          }
+        }
+        for (int l = 0; l < nThR; ++l) {
+          const int im = idx_ml_base + l;
+          const int kl = idx_kl_base + l;
+          const double cosmui = t_.cosmui[im], sinmui = t_.sinmui[im],
+                       cosmumi = t_.cosmumi[im], sinmumi = t_.sinmumi[im];
+          const double tR = rmkcc * cosmui + rmkss * sinmui;
+          armn[kl] += tR;
+          frcon[kl] += xmpq[m] * tR;
+          brmn[kl] += rmkcc * sinmumi + rmkss * cosmumi;
+          const double tZ = zmksc * sinmui + zmkcs * cosmui;
+          azmn[kl] += tZ;
+          fzcon[kl] += xmpq[m] * tZ;
+          bzmn[kl] += zmksc * cosmumi + zmkcs * sinmumi;
+          crmn[kl] += -(rmkcc_n * cosmui + rmkss_n * sinmui);
+          czmn[kl] += -(zmkcs_n * cosmui + zmksc_n * sinmui);
+          blmn[kl] += lmksc * cosmumi + lmkcs * sinmumi;
+          clmn[kl] += -(lmkcs_n * cosmui + lmksc_n * sinmui);
+        }  // l
+      }  // k
+    }  // m
+  }  // jF
+  for (int jF = jMaxRZ; jF < r_.nsMaxFIncludingLcfs; ++jF) {
+    for (int m = 0; m < s_.mpol; ++m) {
+      const bool m_even = m % 2 == 0;
+      auto& blmn = m_even ? blmn_e : blmn_o;
+      auto& clmn = m_even ? clmn_e : clmn_o;
+      const int idx_ml_base = m * nThR;
+      for (int k = 0; k < s_.nZeta; ++k) {
+        const int idx_kl_base =
+            ((jF - r_.nsMinF) * s_.nZeta + k) * s_.nThetaEff;
+        const int idx_kn_base = k * (s_.nnyq2 + 1);
+        const int idx_mn_base = ((jF - r_.nsMinF) * s_.mpol + m) * ntorp1;
+        double lmksc = 0, lmkcs = 0, lmkcs_n = 0, lmksc_n = 0;
+        for (int nn = 0; nn < ntorp1; ++nn) {
+          const int kn = idx_kn_base + nn;
+          const int mn = idx_mn_base + nn;
+          lmksc += m_coeff_bar.flsc[mn] * t_.cosnv[kn];
+          lmksc_n += m_coeff_bar.flsc[mn] * t_.sinnvn[kn];
+          lmkcs += m_coeff_bar.flcs[mn] * t_.sinnv[kn];
+          lmkcs_n += m_coeff_bar.flcs[mn] * t_.cosnvn[kn];
+        }
+        for (int l = 0; l < nThR; ++l) {
+          const int im = idx_ml_base + l;
+          const int kl = idx_kl_base + l;
+          blmn[kl] += lmksc * t_.cosmumi[im] + lmkcs * t_.sinmumi[im];
+          clmn[kl] += -(lmkcs_n * t_.cosmui[im] + lmksc_n * t_.sinmui[im]);
+        }  // l
+      }  // k
+    }  // m
+  }  // jF
+}
+
+// (geometryFromFourier)^T for the 3D case. Transpose of
+// FourierToReal3DSymmFastPoloidal: undo the poloidal evaluation, then the
+// toroidal evaluation, back onto the Fourier coefficient cotangents.
+void IdealMhdModel::dft_FourierToRealTranspose_3d_symm(
+    FourierGeometry& m_coeff_bar_out) {
+  m_coeff_bar_out.setZero();
+  const int nThR = s_.nThetaReduced;
+  const int ntorp1 = s_.ntor + 1;
+  for (int jF = r_.nsMinF1; jF < r_.nsMaxF1; ++jF) {
+    for (int m = 0; m < s_.mpol; ++m) {
+      const bool m_even = m % 2 == 0;
+      const double con_factor =
+          m_even ? xmpq[m] : xmpq[m] * m_p_.sqrtSF[jF - r_.nsMinF1];
+      auto& r1 = m_even ? r1_e : r1_o;
+      auto& ru = m_even ? ru_e : ru_o;
+      auto& rv = m_even ? rv_e : rv_o;
+      auto& z1 = m_even ? z1_e : z1_o;
+      auto& zu = m_even ? zu_e : zu_o;
+      auto& zv = m_even ? zv_e : zv_o;
+      auto& lu = m_even ? lu_e : lu_o;
+      auto& lv = m_even ? lv_e : lv_o;
+      const int jMin = (m == 0 || m == 1) ? 0 : 1;
+      if (jF < jMin) {
+        continue;
+      }
+      const int idx_ml_base = m * nThR;
+      for (int k = 0; k < s_.nZeta; ++k) {
+        const int idx_kl_base =
+            ((jF - r_.nsMinF1) * s_.nZeta + k) * s_.nThetaEff;
+        const bool con_in_range =
+            (r_.nsMinF <= jF && jF < r_.nsMaxFIncludingLcfs);
+        const int idx_con_base =
+            ((jF - r_.nsMinF) * s_.nZeta + k) * s_.nThetaEff;
+        double rmkcc = 0, rmkss = 0, rmkcc_n = 0, rmkss_n = 0, zmksc = 0,
+               zmkcs = 0, zmksc_n = 0, zmkcs_n = 0, lmksc = 0, lmkcs = 0,
+               lmksc_n = 0, lmkcs_n = 0;
+        for (int l = 0; l < nThR; ++l) {
+          const int im = idx_ml_base + l;
+          const int kl = idx_kl_base + l;
+          const double cosmu = t_.cosmu[im], sinmu = t_.sinmu[im],
+                       sinmum = t_.sinmum[im], cosmum = t_.cosmum[im];
+          const double r1b = r1[kl], rub = ru[kl], rvb = rv[kl], z1b = z1[kl],
+                       zub = zu[kl], zvb = zv[kl], lub = lu[kl], lvb = lv[kl];
+          double rConb = 0, zConb = 0;
+          if (con_in_range) {
+            rConb = rCon[idx_con_base + l];
+            zConb = zCon[idx_con_base + l];
+          }
+          rmkcc += r1b * cosmu + rub * sinmum + rConb * cosmu * con_factor;
+          rmkss += r1b * sinmu + rub * cosmum + rConb * sinmu * con_factor;
+          rmkcc_n += rvb * cosmu;
+          rmkss_n += rvb * sinmu;
+          zmksc += z1b * sinmu + zub * cosmum + zConb * sinmu * con_factor;
+          zmkcs += z1b * cosmu + zub * sinmum + zConb * cosmu * con_factor;
+          zmksc_n += zvb * sinmu;
+          zmkcs_n += zvb * cosmu;
+          lmksc += lub * cosmum;
+          lmkcs += lub * sinmum;
+          lmksc_n += -lvb * sinmu;
+          lmkcs_n += -lvb * cosmu;
+        }  // l
+        const int idx_kn_base = k * (s_.nnyq2 + 1);
+        const int idx_mn_base = ((jF - r_.nsMinF1) * s_.mpol + m) * ntorp1;
+        for (int nn = 0; nn < ntorp1; ++nn) {
+          const int kn = idx_kn_base + nn;
+          const int mn = idx_mn_base + nn;
+          const double cosnv = t_.cosnv[kn], sinnv = t_.sinnv[kn],
+                       cosnvn = t_.cosnvn[kn], sinnvn = t_.sinnvn[kn];
+          m_coeff_bar_out.rmncc[mn] += rmkcc * cosnv + rmkcc_n * sinnvn;
+          m_coeff_bar_out.rmnss[mn] += rmkss * sinnv + rmkss_n * cosnvn;
+          m_coeff_bar_out.zmnsc[mn] += zmksc * cosnv + zmksc_n * sinnvn;
+          m_coeff_bar_out.zmncs[mn] += zmkcs * sinnv + zmkcs_n * cosnvn;
+          m_coeff_bar_out.lmnsc[mn] += lmksc * cosnv + lmksc_n * sinnvn;
+          m_coeff_bar_out.lmncs[mn] += lmkcs * sinnv + lmkcs_n * cosnvn;
+        }  // nn
+      }  // k
+    }  // m
+  }  // jF
+}
+
+void IdealMhdModel::applyExactForceJacobianTranspose(
+    const double* geomP, int geom_stride, FourierForces& m_decomposed_in,
+    FourierForces& m_physical_f, FourierGeometry& m_physical_scratch,
+    FourierGeometry& m_decomposed_out) {
+  const int gS = geom_stride;
+  const int nForce = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
+
+  // C^T: transpose of [scatter -> forcesToFourier -> decompose -> m1 -> zeroZ].
+  if (m_fc_.fsqz < 1.0e-6) {
+    m_decomposed_in.zeroZForceForM1();
+  }
+  m_decomposed_in.m1Constraint(1.0 / std::numbers::sqrt2);
+  m_decomposed_in.decomposeInto(m_physical_f, m_p_.scalxc);
+  if (s_.lthreed) {
+    dft_ForcesToFourierTranspose_3d_symm(m_physical_f);
+  } else {
+    dft_ForcesToFourierTranspose_2d_symm(m_physical_f);
+  }
+
+  // Gather the force-density member cotangents into the 20-block flat layout.
+  std::vector<double> force_bar(20 * nForce, 0.0);
+  auto gather = [&](int b, const Eigen::VectorXd& src) {
+    for (int i = 0; i < nForce; ++i) force_bar[b * nForce + i] = src[i];
+  };
+  gather(0, armn_e);
+  gather(1, armn_o);
+  gather(2, azmn_e);
+  gather(3, azmn_o);
+  gather(4, brmn_e);
+  gather(5, brmn_o);
+  gather(6, bzmn_e);
+  gather(7, bzmn_o);
+  gather(12, blmn_e);
+  gather(13, blmn_o);
+  gather(16, frcon_e);
+  gather(17, frcon_o);
+  gather(18, fzcon_e);
+  gather(19, fzcon_o);
+  if (s_.lthreed) {
+    gather(8, crmn_e);
+    gather(9, crmn_o);
+    gather(10, czmn_e);
+    gather(11, czmn_o);
+    gather(14, clmn_e);
+    gather(15, clmn_o);
+  }
+
+  // J_g^T: reverse-mode force-density kernel.
+  std::vector<double> geom_bar(20 * gS, 0.0);
+  exactForceDensityCotangent(geomP, force_bar.data(), gS, geom_bar.data());
+
+  // B^T: transpose of packGeometry [decompose -> m1 -> extrapolate ->
+  // geometryFromFourier -> block pack with lamscale + ruFull/zuFull].
+  auto scat = [&](int b, Eigen::VectorXd& dst) {
+    const int sz = std::min(gS, static_cast<int>(dst.size()));
+    for (int i = 0; i < sz; ++i) dst[i] = geom_bar[b * gS + i];
+  };
+  scat(0, r1_e);
+  scat(1, r1_o);
+  scat(2, z1_e);
+  scat(3, z1_o);
+  scat(4, ru_e);
+  scat(5, ru_o);
+  scat(6, zu_e);
+  scat(7, zu_o);
+  for (int i = 0; i < gS; ++i) {
+    lu_e[i] = constants_.lamscale * geom_bar[12 * gS + i];
+    lu_o[i] = constants_.lamscale * geom_bar[13 * gS + i];
+  }
+  if (s_.lthreed) {
+    scat(8, rv_e);
+    scat(9, rv_o);
+    scat(10, zv_e);
+    scat(11, zv_o);
+    for (int i = 0; i < gS; ++i) {
+      lv_e[i] = constants_.lamscale * geom_bar[14 * gS + i];
+      lv_o[i] = constants_.lamscale * geom_bar[15 * gS + i];
+    }
+  }
+  scat(16, rCon);
+  scat(17, zCon);
+  // ruFull/zuFull (blocks 18,19): forward ruFull = ru_e + sqrtSF*ru_o, so the
+  // adjoint folds the full-grid cotangent back into the even/odd ru, zu.
+  for (int jF = r_.nsMinF; jF < r_.nsMaxFIncludingLcfs; ++jF) {
+    const double sf = m_p_.sqrtSF[jF - r_.nsMinF1];
+    for (int kl = 0; kl < s_.nZnT; ++kl) {
+      const int idx_kl1 = (jF - r_.nsMinF1) * s_.nZnT + kl;
+      const int idx_kl = (jF - r_.nsMinF) * s_.nZnT + kl;
+      ru_e[idx_kl1] += geom_bar[18 * gS + idx_kl];
+      ru_o[idx_kl1] += sf * geom_bar[18 * gS + idx_kl];
+      zu_e[idx_kl1] += geom_bar[19 * gS + idx_kl];
+      zu_o[idx_kl1] += sf * geom_bar[19 * gS + idx_kl];
+    }
+  }
+  if (s_.lthreed) {
+    dft_FourierToRealTranspose_3d_symm(m_physical_scratch);
+  } else {
+    dft_FourierToRealTranspose_2d_symm(m_physical_scratch);
+  }
+  m_physical_scratch.extrapolateTowardsAxisTranspose();
+  m_physical_scratch.m1Constraint(1.0);
+  m_physical_scratch.decomposeInto(m_decomposed_out, m_p_.scalxc);
+}
+
 // Diagnostic: max |composed force density - production force density| at the
 // current state, to isolate composition bugs from the transform/tangent path.
 double IdealMhdModel::composedForceResidual(const double* geomP,
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index f02dccf0e..b98c7b17d 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -24,6 +24,7 @@
 #include "vmecpp/vmec/fourier_geometry/fourier_geometry.h"
 #include "vmecpp/vmec/handover_storage/handover_storage.h"
 #include "vmecpp/vmec/ideal_mhd_model/dft_toroidal.h"
+#include "vmecpp/vmec/ideal_mhd_model/local_force_composition.h"
 #ifdef VMECPP_USE_FFTX
 #include "vmecpp/vmec/ideal_mhd_model/fft_toroidal.h"
 #endif
@@ -187,6 +188,41 @@ class IdealMhdModel {
   void exactForceDensityTangent(const double* geomP, const double* dgeom,
                                 int geom_stride, double* dforce_out);
 
+  // Reverse-mode force-density cotangent: J_g^T applied to the force-density
+  // cotangent force_bar (20 blocks of (nsMaxFIncludingLcfs-nsMinF)*nZnT),
+  // accumulated into geom_bar_out (20 blocks of geom_stride, zeroed by caller),
+  // by one Enzyme reverse pass. The transpose of exactForceDensityTangent.
+  void exactForceDensityCotangent(const double* geomP, const double* force_bar,
+                                  int geom_stride, double* geom_bar_out);
+
+  // Fill the local force-density composition descriptor shared by the exact
+  // forward/reverse force-density passes (geom_stride sized blocks).
+  LocalForceComposition makeLocalForceComposition(int geom_stride);
+
+  // Transpose of applyExactForceJacobian: H^T w. Given a decomposed-force
+  // cotangent (the space applyExactForceJacobian writes), apply C^T (transpose
+  // of the output transform), the reverse-mode force-density kernel J_g^T, and
+  // B^T (transpose of packGeometry's pre-chain), yielding the state cotangent
+  // in m_decomposed_out. The two spectral transforms are reused as each other's
+  // adjoint with the poloidal integration weight; the rest of the linear chain
+  // (decomposeInto, m1Constraint, zeroZForceForM1, extrapolateTowardsAxis,
+  // ruFull/zuFull, lamscale) is transposed analytically.
+  void applyExactForceJacobianTranspose(const double* geomP, int geom_stride,
+                                        FourierForces& m_decomposed_in,
+                                        FourierForces& m_physical_f,
+                                        FourierGeometry& m_physical_scratch,
+                                        FourierGeometry& m_decomposed_out);
+
+  // Transposes of the spectral transforms, for the transposed exact Hessian.
+  // dft_ForcesToFourierTranspose: (forcesToFourier)^T, decomposed-force coeff
+  // cotangent -> real-space force-density member cotangents (armn_e ..
+  // fzcon_o). dft_FourierToRealTranspose: (geometryFromFourier)^T, real-space
+  // geometry member cotangents (r1_e .. zCon) -> Fourier coeff cotangent.
+  void dft_ForcesToFourierTranspose_2d_symm(const FourierForces& m_coeff_bar);
+  void dft_FourierToRealTranspose_2d_symm(FourierGeometry& m_coeff_bar_out);
+  void dft_ForcesToFourierTranspose_3d_symm(const FourierForces& m_coeff_bar);
+  void dft_FourierToRealTranspose_3d_symm(FourierGeometry& m_coeff_bar_out);
+
   // Computes the forward-DFT of forces for the 3D (Stellarator) case.
   // Dispatching dft_ForcesToFourier_3d_symm
   void dft_ForcesToFourier_3d_symm(FourierForces& m_physical_f);
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index dd8bc98bf..c1eedc511 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -41,24 +41,24 @@ namespace {
 //
 // Use as: def_eigen_property(pywrapperclass, "rmnc", &WOutFileContents::rmnc);
 template <typename PywrapperClass, typename EigenMatrix, typename Class>
-void DefEigenProperty(PywrapperClass &pywrapper, const std::string &name,
-                      EigenMatrix Class::*member_ptr) {
+void DefEigenProperty(PywrapperClass& pywrapper, const std::string& name,
+                      EigenMatrix Class::* member_ptr) {
   static_assert(std::is_same_v<EigenMatrix, vmecpp::RowMatrixXd> ||
                 std::is_same_v<EigenMatrix, Eigen::VectorXd> ||
                 std::is_same_v<EigenMatrix, Eigen::VectorXi>);
   // similar to what pybind11's def_readwrite does, but returning a non-const
   // value from the getter
-  auto getter = [member_ptr](Class &obj) -> EigenMatrix & {
+  auto getter = [member_ptr](Class& obj) -> EigenMatrix& {
     return obj.*member_ptr;
   };
-  auto setter = [member_ptr](Class &obj, const EigenMatrix &val) {
+  auto setter = [member_ptr](Class& obj, const EigenMatrix& val) {
     obj.*member_ptr = val;
   };
   pywrapper.def_property(name.c_str(), getter, setter);
 }
 
 template <typename T>
-T &GetValueOrThrow(absl::StatusOr<T> &s) {
+T& GetValueOrThrow(absl::StatusOr<T>& s) {
   if (!s.ok()) {
     // Could handle more exceptions, but only some are translated to meaningful
     // python exception types.
@@ -73,7 +73,7 @@ T &GetValueOrThrow(absl::StatusOr<T> &s) {
 }
 
 vmecpp::HotRestartState MakeHotRestartState(vmecpp::WOutFileContents wout,
-                                            const vmecpp::VmecINDATA &indata) {
+                                            const vmecpp::VmecINDATA& indata) {
   return vmecpp::HotRestartState(std::move(wout), indata);
 }
 
@@ -81,8 +81,8 @@ vmecpp::HotRestartState MakeHotRestartState(vmecpp::WOutFileContents wout,
 // fixed canonical order set by (lthreed, lasym). Used to (un)flatten the VMEC++
 // decision vector for Python. FourierForces shares the same underlying
 // FourierCoeffs storage layout, so the same ordering applies.
-std::vector<std::span<double>> ActiveSpans(vmecpp::FourierGeometry &x,
-                                           const vmecpp::Sizes &s) {
+std::vector<std::span<double>> ActiveSpans(vmecpp::FourierGeometry& x,
+                                           const vmecpp::Sizes& s) {
   std::vector<std::span<double>> out = {x.rmncc};
   if (s.lthreed) out.push_back(x.rmnss);
   if (s.lasym) out.push_back(x.rmnsc);
@@ -98,8 +98,8 @@ std::vector<std::span<double>> ActiveSpans(vmecpp::FourierGeometry &x,
   return out;
 }
 
-std::vector<std::span<double>> ActiveSpans(vmecpp::FourierForces &x,
-                                           const vmecpp::Sizes &s) {
+std::vector<std::span<double>> ActiveSpans(vmecpp::FourierForces& x,
+                                           const vmecpp::Sizes& s) {
   std::vector<std::span<double>> out = {x.frcc};
   if (s.lthreed) out.push_back(x.frss);
   if (s.lasym) out.push_back(x.frsc);
@@ -116,13 +116,13 @@ std::vector<std::span<double>> ActiveSpans(vmecpp::FourierForces &x,
 }
 
 template <typename FourierObject>
-Eigen::VectorXd FlattenActive(FourierObject &x, const vmecpp::Sizes &s) {
+Eigen::VectorXd FlattenActive(FourierObject& x, const vmecpp::Sizes& s) {
   const std::vector<std::span<double>> spans = ActiveSpans(x, s);
   Eigen::Index total = 0;
-  for (const auto &sp : spans) total += static_cast<Eigen::Index>(sp.size());
+  for (const auto& sp : spans) total += static_cast<Eigen::Index>(sp.size());
   Eigen::VectorXd out(total);
   Eigen::Index offset = 0;
-  for (const auto &sp : spans) {
+  for (const auto& sp : spans) {
     const Eigen::Index n = static_cast<Eigen::Index>(sp.size());
     out.segment(offset, n) = Eigen::Map<const Eigen::VectorXd>(sp.data(), n);
     offset += n;
@@ -131,11 +131,11 @@ Eigen::VectorXd FlattenActive(FourierObject &x, const vmecpp::Sizes &s) {
 }
 
 template <typename FourierObject>
-void UnflattenActive(FourierObject &m_x, const vmecpp::Sizes &s,
-                     const Eigen::VectorXd &flat) {
+void UnflattenActive(FourierObject& m_x, const vmecpp::Sizes& s,
+                     const Eigen::VectorXd& flat) {
   const std::vector<std::span<double>> spans = ActiveSpans(m_x, s);
   Eigen::Index total = 0;
-  for (const auto &sp : spans) total += static_cast<Eigen::Index>(sp.size());
+  for (const auto& sp : spans) total += static_cast<Eigen::Index>(sp.size());
   if (flat.size() != total) {
     throw std::runtime_error(
         "VmecModel.set_state: state vector has wrong length (got " +
@@ -143,7 +143,7 @@ void UnflattenActive(FourierObject &m_x, const vmecpp::Sizes &s,
         ")");
   }
   Eigen::Index offset = 0;
-  for (auto &sp : spans) {
+  for (auto& sp : spans) {
     const Eigen::Index n = static_cast<Eigen::Index>(sp.size());
     Eigen::Map<Eigen::VectorXd>(sp.data(), n) = flat.segment(offset, n);
     offset += n;
@@ -167,8 +167,8 @@ class VmecModel {
   // (the inner solve VMEC++ performs at one multi-grid step). The Python loop
   // owns the multi-grid sequencing.
   static std::unique_ptr<VmecModel> Create(
-      const VmecINDATA &indata, int ns,
-      const std::optional<vmecpp::HotRestartState> &initial_state) {
+      const VmecINDATA& indata, int ns,
+      const std::optional<vmecpp::HotRestartState>& initial_state) {
     auto vmec_or = vmecpp::Vmec::FromIndata(
         indata, /*magnetic_response_table=*/nullptr, /*max_threads=*/1,
         vmecpp::OutputMode::kSilent);
@@ -176,7 +176,7 @@ class VmecModel {
       throw std::runtime_error(std::string(vmec_or.status().message()));
     }
     auto model = std::make_unique<VmecModel>(std::move(vmec_or.value()));
-    vmecpp::Vmec &v = *model->vmec_;
+    vmecpp::Vmec& v = *model->vmec_;
 
     // Mirror the per-multi-grid-step setup that Vmec::run performs before
     // SolveEquilibrium (vmec.cc), for a single ns value.
@@ -187,7 +187,7 @@ class VmecModel {
 
     // ftol/niter for this resolution: use the ns_array entry matching ns, else
     // the last entry of the arrays.
-    const Eigen::VectorXi &ns_array = v.indata_.ns_array;
+    const Eigen::VectorXi& ns_array = v.indata_.ns_array;
     int idx = static_cast<int>(ns_array.size()) - 1;
     for (int i = 0; i < ns_array.size(); ++i) {
       if (ns_array[i] == ns) {
@@ -333,7 +333,7 @@ class VmecModel {
   // the coarse->fine ramp from Python. `new_ns` must be finer than the current
   // ns (multi-grid only refines).
   void RefineTo(int new_ns) {
-    vmecpp::Vmec &v = *vmec_;
+    vmecpp::Vmec& v = *vmec_;
     if (new_ns <= v.fc_.ns) {
       throw std::runtime_error("VmecModel.refine_to: new_ns (" +
                                std::to_string(new_ns) +
@@ -349,7 +349,7 @@ class VmecModel {
 
     // ftol/niter for the new resolution: the ns_array entry matching new_ns,
     // else the last entry (mirrors Create()).
-    const Eigen::VectorXi &ns_array = v.indata_.ns_array;
+    const Eigen::VectorXi& ns_array = v.indata_.ns_array;
     int idx = static_cast<int>(ns_array.size()) - 1;
     for (int i = 0; i < ns_array.size(); ++i) {
       if (ns_array[i] == new_ns) {
@@ -384,7 +384,7 @@ class VmecModel {
   Eigen::VectorXd GetState() const {
     return FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
   }
-  void SetState(const Eigen::VectorXd &flat) const {
+  void SetState(const Eigen::VectorXd& flat) const {
     UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, flat);
     exact_primal_valid_ = false;  // primal geometry cache is stale
   }
@@ -399,7 +399,7 @@ class VmecModel {
   // decomposed internal basis. This is the matrix-free Hessian information an
   // internal or external Newton-Krylov solver needs; F itself is exact, so only
   // the directional step is finite-differenced. The current state is restored.
-  Eigen::VectorXd HessianVectorProduct(const Eigen::VectorXd &v,
+  Eigen::VectorXd HessianVectorProduct(const Eigen::VectorXd& v,
                                        double eps_rel = 1e-7) {
     const Eigen::VectorXd x =
         FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
@@ -429,8 +429,8 @@ class VmecModel {
   // finite-difference step enters. The constraint force (a linear bandpass over
   // a nonlinear product) is omitted here. The model state is restored to x on
   // return.
-  Eigen::VectorXd ExactHessianVectorProduct(const Eigen::VectorXd &v) {
-    vmecpp::IdealMhdModel &model = *vmec_->m_[0];
+  Eigen::VectorXd ExactHessianVectorProduct(const Eigen::VectorXd& v) {
+    vmecpp::IdealMhdModel& model = *vmec_->m_[0];
     const int gS = static_cast<int>(model.r1_e.size());
     Eigen::VectorXd dgeom = Eigen::VectorXd::Zero(20 * gS);
     // The primal geometry depends only on the current state, not on v, so cache
@@ -455,16 +455,40 @@ class VmecModel {
     return FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
   }
 
+  // Transpose of the exact Hessian-vector product: H^T w, in the decomposed
+  // internal basis. The force Jacobian is non-symmetric (VMEC's force is a
+  // scaled gradient), so the adjoint boundary gradient needs H^T, not H. Uses
+  // the same cached primal geometry as ExactHessianVectorProduct.
+  Eigen::VectorXd ExactHessianVectorProductTranspose(const Eigen::VectorXd& w) {
+    vmecpp::IdealMhdModel& model = *vmec_->m_[0];
+    const int gS = static_cast<int>(model.r1_e.size());
+    if (!exact_primal_valid_ ||
+        exact_primal_.size() != static_cast<Eigen::Index>(20 * gS)) {
+      exact_primal_.setZero(20 * gS);
+      model.packGeometry(*vmec_->decomposed_x_[0], *vmec_->physical_x_[0],
+                         exact_primal_.data(), gS, /*primal=*/true);
+      exact_primal_valid_ = true;
+    }
+    vmec_->decomposed_f_[0]->setZero();
+    UnflattenActive(*vmec_->decomposed_f_[0], vmec_->s_, w);
+    vmec_->physical_x_backup_[0]->setZero();
+    model.applyExactForceJacobianTranspose(
+        exact_primal_.data(), gS, *vmec_->decomposed_f_[0],
+        *vmec_->physical_f_[0], *vmec_->physical_x_[0],
+        *vmec_->physical_x_backup_[0]);
+    return FlattenActive(*vmec_->physical_x_backup_[0], vmec_->s_);
+  }
+
   // Diagnostic: max |composed force density - production force density| at the
   // current state. Zero confirms the flat-buffer composition reproduces the
   // production force density; a nonzero value localizes a composition bug.
   double ComposedForceResidual() {
-    vmecpp::IdealMhdModel &model = *vmec_->m_[0];
+    vmecpp::IdealMhdModel& model = *vmec_->m_[0];
     const Eigen::VectorXd x =
         FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
     const int gS = static_cast<int>(model.r1_e.size());
     Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
-    auto blk = [&](int b, const Eigen::VectorXd &src) {
+    auto blk = [&](int b, const Eigen::VectorXd& src) {
       for (int i = 0; i < static_cast<int>(src.size()); ++i)
         geomP[b * gS + i] = src[i];
     };
@@ -499,14 +523,14 @@ class VmecModel {
   // transform wrapping. Returns (per-block max-rel, overall-rel) so a nonzero
   // block localizes the discrepancy.
   std::pair<std::vector<double>, double> ForceDensityJvpResidual(
-      const Eigen::VectorXd &v) {
-    vmecpp::IdealMhdModel &M = *vmec_->m_[0];
+      const Eigen::VectorXd& v) {
+    vmecpp::IdealMhdModel& M = *vmec_->m_[0];
     const int nForce = static_cast<int>(M.blmn_e.size());
     const int gS = static_cast<int>(M.r1_e.size());
     const Eigen::VectorXd x =
         FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
-    auto packGeom = [&](Eigen::VectorXd &dst) {
-      auto blk = [&](int b, const Eigen::VectorXd &s) {
+    auto packGeom = [&](Eigen::VectorXd& dst) {
+      auto blk = [&](int b, const Eigen::VectorXd& s) {
         for (int i = 0; i < static_cast<int>(s.size()); ++i)
           dst[b * gS + i] = s[i];
       };
@@ -531,8 +555,8 @@ class VmecModel {
       blk(18, M.ruFull);
       blk(19, M.zuFull);
     };
-    auto packForce = [&](Eigen::VectorXd &dst) {
-      auto blk = [&](int b, const Eigen::VectorXd &s) {
+    auto packForce = [&](Eigen::VectorXd& dst) {
+      auto blk = [&](int b, const Eigen::VectorXd& s) {
         for (int i = 0; i < static_cast<int>(s.size()); ++i)
           dst[b * nForce + i] = s[i];
       };
@@ -598,6 +622,42 @@ class VmecModel {
     const double overall = (jvp - fd).norm() / (fd.norm() + 1e-300);
     return {per_block, overall};
   }
+
+  // Primal geometry (20 blocks of gS) at the current state, the linearization
+  // point for the exact force-density kernel.
+  Eigen::VectorXd packExactGeometryPrimal(int gS) {
+    Eigen::VectorXd geomP = Eigen::VectorXd::Zero(20 * gS);
+    vmec_->m_[0]->packGeometry(*vmec_->decomposed_x_[0], *vmec_->physical_x_[0],
+                               geomP.data(), gS, /*primal=*/true);
+    return geomP;
+  }
+
+  // Force-density kernel J_g (20 geometry-block tangent -> 20 force-block
+  // tangent) and its transpose J_g^T, at the current state. For validating the
+  // Enzyme reverse pass via the bilinear identity <J_g x, y> == <x, J_g^T y>.
+  Eigen::VectorXd KernelJvp(const Eigen::VectorXd& dgeom) {
+    vmecpp::IdealMhdModel& M = *vmec_->m_[0];
+    const int nForce = static_cast<int>(M.blmn_e.size());
+    const int gS = static_cast<int>(M.r1_e.size());
+    Eigen::VectorXd geomP = packExactGeometryPrimal(gS);
+    Eigen::VectorXd jvp = Eigen::VectorXd::Zero(20 * nForce);
+    M.exactForceDensityTangent(geomP.data(), dgeom.data(), gS, jvp.data());
+    return jvp;
+  }
+  Eigen::VectorXd KernelVjp(const Eigen::VectorXd& force_bar) {
+    vmecpp::IdealMhdModel& M = *vmec_->m_[0];
+    const int gS = static_cast<int>(M.r1_e.size());
+    Eigen::VectorXd geomP = packExactGeometryPrimal(gS);
+    Eigen::VectorXd vjp = Eigen::VectorXd::Zero(20 * gS);
+    M.exactForceDensityCotangent(geomP.data(), force_bar.data(), gS,
+                                 vjp.data());
+    return vjp;
+  }
+  // (geom-block size gS, force-block size nForce) for the kernel diagnostics.
+  std::pair<int, int> KernelDims() {
+    vmecpp::IdealMhdModel& M = *vmec_->m_[0];
+    return {static_cast<int>(M.r1_e.size()), static_cast<int>(M.blmn_e.size())};
+  }
 #endif  // VMECPP_ENABLE_ENZYME
 
   // Apply VMEC's preconditioner M^-1 to a vector in the decomposed internal
@@ -608,11 +668,11 @@ class VmecModel {
   //
   // Requires a prior evaluate(precondition=true) at the current state: the
   // radial preconditioner is assembled inside that forward-model call.
-  Eigen::VectorXd ApplyPreconditioner(const Eigen::VectorXd &v) const {
+  Eigen::VectorXd ApplyPreconditioner(const Eigen::VectorXd& v) const {
     vmecpp::FourierForces tmp(&vmec_->s_, vmec_->r_[0].get(), vmec_->fc_.ns);
     tmp.setZero();
     UnflattenActive(tmp, vmec_->s_, v);
-    vmecpp::IdealMhdModel &model = *vmec_->m_[0];
+    vmecpp::IdealMhdModel& model = *vmec_->m_[0];
     model.applyM1Preconditioner(tmp);
     const absl::Status status = model.applyRZPreconditioner(tmp);
     if (!status.ok()) {
@@ -670,7 +730,7 @@ class VmecModel {
   std::vector<int> restart_reasons() const {
     std::vector<int> out;
     out.reserve(vmec_->fc_.restart_reasons.size());
-    for (const auto &r : vmec_->fc_.restart_reasons) {
+    for (const auto& r : vmec_->fc_.restart_reasons) {
       out.push_back(static_cast<int>(r));
     }
     return out;
@@ -794,30 +854,30 @@ PYBIND11_MODULE(_vmecpp, m) {
       // disallow re-assignment of the whole vector (to preserve sizes
       // consistent with mpol/ntor) but allow changing the individual elements
       .def_property_readonly(
-          "raxis_c", [](VmecINDATA &w) -> VectorXd & { return w.raxis_c; })
+          "raxis_c", [](VmecINDATA& w) -> VectorXd& { return w.raxis_c; })
       .def_property_readonly(
-          "zaxis_s", [](VmecINDATA &w) -> VectorXd & { return w.zaxis_s; })
+          "zaxis_s", [](VmecINDATA& w) -> VectorXd& { return w.zaxis_s; })
       .def_property_readonly(
           "raxis_s",
-          [](VmecINDATA &w) -> std::optional<VectorXd> & { return w.raxis_s; })
+          [](VmecINDATA& w) -> std::optional<VectorXd>& { return w.raxis_s; })
       .def_property_readonly(
           "zaxis_c",
-          [](VmecINDATA &w) -> std::optional<VectorXd> & { return w.zaxis_c; })
+          [](VmecINDATA& w) -> std::optional<VectorXd>& { return w.zaxis_c; })
 
       // (initial guess for) boundary shape
       // disallow re-assignment of the whole matrix (to preserve shapes
       // consistent with mpol/ntor) but allow changing the individual elements
       .def_property_readonly(
-          "rbc", [](VmecINDATA &w) -> vmecpp::RowMatrixXd & { return w.rbc; })
+          "rbc", [](VmecINDATA& w) -> vmecpp::RowMatrixXd& { return w.rbc; })
       .def_property_readonly(
-          "zbs", [](VmecINDATA &w) -> vmecpp::RowMatrixXd & { return w.zbs; })
+          "zbs", [](VmecINDATA& w) -> vmecpp::RowMatrixXd& { return w.zbs; })
       .def_property_readonly(
           "rbs",
-          [](VmecINDATA &w) -> std::optional<vmecpp::RowMatrixXd> & {
+          [](VmecINDATA& w) -> std::optional<vmecpp::RowMatrixXd>& {
             return w.rbs;
           })
       .def_property_readonly(
-          "zbc", [](VmecINDATA &w) -> std::optional<vmecpp::RowMatrixXd> & {
+          "zbc", [](VmecINDATA& w) -> std::optional<vmecpp::RowMatrixXd>& {
             return w.zbc;
           });
 
@@ -1070,7 +1130,7 @@ PYBIND11_MODULE(_vmecpp, m) {
       .def_readonly("delta3", &vmecpp::Threed1ShafranovIntegrals::delta3);
 
   py::class_<vmecpp::WOutFileContents>(m, "WOutFileContents")
-      .def(py::init<const vmecpp::WOutFileContents &>(), py::arg("wout"))
+      .def(py::init<const vmecpp::WOutFileContents&>(), py::arg("wout"))
       .def(py::init())
       .def_readwrite("version_", &vmecpp::WOutFileContents::version_)
       .def_readwrite("input_extension",
@@ -1260,8 +1320,8 @@ PYBIND11_MODULE(_vmecpp, m) {
       .def_readonly("indata", &vmecpp::OutputQuantities::indata)
       .def(
           "save",
-          [](const vmecpp::OutputQuantities &oq,
-             const std::filesystem::path &path) {
+          [](const vmecpp::OutputQuantities& oq,
+             const std::filesystem::path& path) {
             absl::Status s = oq.Save(path);
 
             if (!s.ok()) {
@@ -1272,7 +1332,7 @@ PYBIND11_MODULE(_vmecpp, m) {
             }
           },
           py::arg("path"))
-      .def_static("load", [](const std::filesystem::path &path) {
+      .def_static("load", [](const std::filesystem::path& path) {
         auto maybe_oq = vmecpp::OutputQuantities::Load(path);
         if (!maybe_oq.ok()) {
           const std::string msg =
@@ -1291,7 +1351,7 @@ PYBIND11_MODULE(_vmecpp, m) {
 
   m.def(
       "run",
-      [](const VmecINDATA &indata,
+      [](const VmecINDATA& indata,
          std::optional<vmecpp::HotRestartState> initial_state,
          std::optional<int> max_threads,
          vmecpp::OutputMode verbose) -> vmecpp::OutputQuantities {
@@ -1331,7 +1391,7 @@ PYBIND11_MODULE(_vmecpp, m) {
            "number_of_z_grid_points"_a, "number_of_phi_grid_points"_a)
       .def_static(
           "from_file",
-          [](const std::filesystem::path &file) {
+          [](const std::filesystem::path& file) {
             auto maybe_params =
                 makegrid::ImportMakegridParametersFromFile(file);
             return GetValueOrThrow(maybe_params);
@@ -1361,7 +1421,7 @@ PYBIND11_MODULE(_vmecpp, m) {
   py::class_<magnetics::MagneticConfiguration>(m, "MagneticConfiguration")
       .def_static(
           "from_file",
-          [](const std::filesystem::path &file) {
+          [](const std::filesystem::path& file) {
             auto maybe_config =
                 magnetics::ImportMagneticConfigurationFromCoilsFile(file);
             return GetValueOrThrow(maybe_config);
@@ -1370,10 +1430,10 @@ PYBIND11_MODULE(_vmecpp, m) {
   auto response_table =
       py::class_<makegrid::MagneticFieldResponseTable>(
           m, "MagneticFieldResponseTable")
-          .def(py::init<const makegrid::MakegridParameters &,
-                        const makegrid::RowMatrixXd &,
-                        const makegrid::RowMatrixXd &,
-                        const makegrid::RowMatrixXd &>(),
+          .def(py::init<const makegrid::MakegridParameters&,
+                        const makegrid::RowMatrixXd&,
+                        const makegrid::RowMatrixXd&,
+                        const makegrid::RowMatrixXd&>(),
                py::arg("parameters"), py::arg("b_r"), py::arg("b_p"),
                py::arg("b_z"))
           .def_readonly("parameters",
@@ -1387,8 +1447,8 @@ PYBIND11_MODULE(_vmecpp, m) {
 
   m.def(
       "compute_magnetic_field_response_table",
-      [](const makegrid::MakegridParameters &mgrid_params,
-         const magnetics::MagneticConfiguration &magnetic_configuration) {
+      [](const makegrid::MakegridParameters& mgrid_params,
+         const magnetics::MagneticConfiguration& magnetic_configuration) {
         auto ret = makegrid::ComputeMagneticFieldResponseTable(
             mgrid_params, magnetic_configuration);
         return GetValueOrThrow(ret);
@@ -1397,8 +1457,8 @@ PYBIND11_MODULE(_vmecpp, m) {
 
   m.def(
       "run",
-      [](const VmecINDATA &indata,
-         const makegrid::MagneticFieldResponseTable &magnetic_response_table,
+      [](const VmecINDATA& indata,
+         const makegrid::MagneticFieldResponseTable& magnetic_response_table,
          std::optional<vmecpp::HotRestartState> initial_state,
          std::optional<int> max_threads, vmecpp::OutputMode verbose) {
         bool was_interrupted = false;
@@ -1458,9 +1518,14 @@ PYBIND11_MODULE(_vmecpp, m) {
 #ifdef VMECPP_ENABLE_ENZYME
       .def("exact_hessian_vector_product",
            &VmecModel::ExactHessianVectorProduct, py::arg("v"))
+      .def("exact_hessian_vector_product_transpose",
+           &VmecModel::ExactHessianVectorProductTranspose, py::arg("w"))
       .def("composed_force_residual", &VmecModel::ComposedForceResidual)
       .def("force_density_jvp_residual", &VmecModel::ForceDensityJvpResidual,
            py::arg("v"))
+      .def("_kernel_jvp", &VmecModel::KernelJvp, py::arg("dgeom"))
+      .def("_kernel_vjp", &VmecModel::KernelVjp, py::arg("force_bar"))
+      .def("_kernel_dims", &VmecModel::KernelDims)
 #endif  // VMECPP_ENABLE_ENZYME
       .def_property_readonly("force_eval_count", &VmecModel::force_eval_count)
       .def("reset_force_eval_count", &VmecModel::reset_force_eval_count)

From 9aae482836b373121432c401f66c9d2a8dd9c9fb Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Mon, 15 Jun 2026 21:57:07 +0200
Subject: [PATCH 30/79] examples: exact boundary gradient via forward
 sensitivities and O(1) reverse adjoint

The previous adjoint solved H_II lambda = dJ/dx_I assuming H symmetric, which is
wrong for a general objective (the energy only passed because dJ/dx_I ~ 0 at
equilibrium). Replace it with two finite-difference-free, exact forms driven by
the autodiff Hessian-vector product:

- forward_boundary_gradient: one Hessian solve per boundary DOF.
- adjoint_boundary_gradient: one solve total via the transpose
  exact_hessian_vector_product_transpose, with state-independent structural-null
  deflation (structural_nullfree_interior) and the symmetric preconditioner.

boundary_gradient now routes through the forward sensitivities. Validated vs the
re-solve finite-difference reference (solovev 5e-6) and cross-checked
forward vs reverse (cth_like 9e-6).
---
 examples/vmecpp_adjoint.py | 170 ++++++++++++++++++++++++++++++-------
 1 file changed, 140 insertions(+), 30 deletions(-)

diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index bd34ec056..d4471b8cf 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -2,23 +2,32 @@
 # <info@proximafusion.com>
 #
 # SPDX-License-Identifier: MIT
-"""Adjoint sensitivity of a converged VMEC++ equilibrium to its boundary.
+"""Sensitivity of a converged VMEC++ equilibrium to its boundary.
 
 A fixed-boundary equilibrium satisfies the interior force balance F_I(x) = 0,
-where x is the decomposed internal-basis state and F is the gradient of VMEC's
-augmented functional. The outermost flux surface (the boundary) is the last
-radial block of the state and is held fixed during the solve. For a scalar
-objective J(x), the sensitivity to the boundary degrees of freedom follows from
-the implicit function theorem:
-
-    dJ/dx_B = dJ/dx_B|_x - (dF_I/dx_B)^T lambda,   H_II lambda = dJ/dx_I,
-
-with H = dF/dx the (symmetric) Hessian of the augmented functional. Every
-operator is matrix-free and already exposed by VmecModel: the Hessian-vector
-product (``hessian_vector_product``) and the preconditioner
-(``apply_preconditioner``), used to solve the adjoint system. Only one Hessian
-solve is needed for the full boundary gradient, versus one equilibrium re-solve
-per boundary degree of freedom for finite differences.
+where x is the decomposed internal-basis state and F is VMEC's internal force.
+The outermost flux surface (the boundary) is the last radial block of the state
+and is held fixed during the solve. For a scalar objective J(x), the implicit
+function theorem gives the boundary sensitivity from the equilibrium response
+dx_I/dx_B = -H_II^{-1} H_IB, H = dF/dx:
+
+    dJ/dx_Bj = dJ/dx_Bj|_x + dJ/dx_I . dx_I,   H_II dx_I = -H_IB e_j.
+
+Two finite-difference-free forms are provided, both driven by the exact autodiff
+Hessian-vector product and preconditioned by VMEC's approximate inverse Hessian
+(``apply_preconditioner``):
+
+* ``forward_boundary_gradient``: one Hessian solve per boundary DOF
+  (``H_II dx_I = -H_IB e_j``). Works on any Enzyme build.
+* ``adjoint_boundary_gradient``: one Hessian solve total, independent of the
+  boundary DOF count. VMEC's force is a *scaled* gradient, so H = dF/dx is
+  non-symmetric and the adjoint needs the transpose H^T, exposed by
+  ``exact_hessian_vector_product_transpose``; the structural null space is
+  deflated once (state-independent) and the symmetric preconditioner serves H^T
+  as well as H.
+
+Both are far cheaper than the finite-difference reference, which re-solves the
+equilibrium once per boundary DOF.
 """
 
 from __future__ import annotations
@@ -142,24 +151,17 @@ def objective_state_gradient(model, x, objective, h=1e-6):
 def boundary_gradient(
     model, x_star, interior, boundary, objective, h=1e-6, exact=False
 ):
-    """Adjoint gradient dJ/dx_B at the converged equilibrium x_star.
+    """dJ/dx_B at the converged equilibrium for a scalar objective.
 
-    With exact=True the analytic autodiff Hessian-vector product is used (no force
-    evaluation per matvec); otherwise the finite-difference HVP.
+    The equilibrium response is captured by forward sensitivities driven by the
+    exact autodiff Hessian-vector product (no nonlinear re-solves). The state
+    cotangent dJ/dx is taken by finite differences over the state here; pass an
+    analytic one to ``forward_boundary_gradient`` to remove that step too (the QS
+    objective does, via ``qs_boundary_gradient``).
     """
-    n = x_star.size
     dj = objective_state_gradient(model, x_star, objective, h)
-    model.set_state(np.ascontiguousarray(x_star))
-    model.evaluate(2, 2, True)  # assemble preconditioner + set base state to x_star
-    h_op, m_op = _interior_operators(model, x_star, interior, exact)
-    lam, _ = gmres(h_op, dj[interior], M=m_op, rtol=1e-6, restart=100, maxiter=30)
-    embedded = np.zeros(n)
-    embedded[interior] = lam
-    hvp = model.exact_hessian_vector_product if exact else model.hessian_vector_product
-    model.set_state(np.ascontiguousarray(x_star))
-    model.evaluate(2, 2, False)
-    coupling = np.asarray(hvp(np.ascontiguousarray(embedded)), float)[boundary]
-    return dj[boundary] - coupling
+    grad, _ = forward_boundary_gradient(model, x_star, interior, boundary, dj, exact)
+    return grad
 
 
 def finite_difference_boundary_gradient(
@@ -186,3 +188,111 @@ def finite_difference_boundary_gradient(
 
 def mhd_energy(model):
     return model.mhd_energy
+
+
+def forward_boundary_gradient(
+    model, x_star, interior, boundary, dj, exact=True, rtol=1e-8, maxiter=60
+):
+    """dJ/dx_B from forward equilibrium sensitivities, finite-difference-free.
+
+    For each boundary DOF j the interior responds along the equilibrium manifold
+    by dx_I = -H_II^{-1} H_IB e_j (implicit function theorem, F_I = 0), and
+    dJ/dx_Bj = dJ/dx_Bj|_x + dJ/dx_I . dx_I. Every operator is the exact autodiff
+    Hessian-vector product.
+
+    The reverse (adjoint) form would need one solve total instead of one per
+    boundary DOF, but it requires the transpose H^T: VMEC's force is a *scaled*
+    gradient, so H = dF/dx is genuinely non-symmetric (H_BI != H_IB^T) and the
+    naive reverse solve is wrong. The scaling cancels in the forward sensitivity
+    (H_II dx = -H_IB e_j has the same solution as the symmetric system), so this
+    form is exact. An exact O(1) adjoint needs a reverse-mode force VJP.
+    """
+    hvp = model.exact_hessian_vector_product if exact else model.hessian_vector_product
+    n = x_star.size
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, True)  # assemble preconditioner at x_star
+    h_op, m_op = _interior_operators(model, x_star, interior, exact)
+    dj_i = dj[interior]
+    grad = np.empty(boundary.size)
+    failures = 0
+    for col, j in enumerate(boundary):
+        ej = np.zeros(n)
+        ej[j] = 1.0
+        hib = np.asarray(hvp(np.ascontiguousarray(ej)), float)[interior]
+        dxi, info = gmres(h_op, -hib, M=m_op, rtol=rtol, restart=100, maxiter=maxiter)
+        failures += int(info != 0)
+        grad[col] = dj[j] + dj_i @ dxi
+    return grad, failures
+
+
+def structural_nullfree_interior(model, interior, tol=1e-9):
+    """Interior DOFs that actually enter the force, i.e. not in the augmented
+    Hessian's structural null space (state-independent gauge/parity modes). A DOF
+    is kept when both its Hessian column and row are nonzero. The set is
+    state-independent, so detect it once and reuse it across adjoint solves."""
+    n = int(np.asarray(model.get_state()).size)
+    col = np.zeros(n)
+    row = np.zeros(n)
+    for i in interior:
+        e = np.zeros(n)
+        e[i] = 1.0
+        col[i] = np.linalg.norm(
+            np.asarray(
+                model.exact_hessian_vector_product(np.ascontiguousarray(e)), float
+            )
+        )
+        row[i] = np.linalg.norm(
+            np.asarray(
+                model.exact_hessian_vector_product_transpose(np.ascontiguousarray(e)),
+                float,
+            )
+        )
+    thr = tol * max(col.max(), 1.0)
+    return np.array([i for i in interior if col[i] > thr and row[i] > thr])
+
+
+def adjoint_boundary_gradient(
+    model, x_star, interior, boundary, dj, keep=None, rtol=1e-9, maxiter=400
+):
+    """dJ/dx_B from the reverse adjoint: one Hessian solve, O(1) in boundary DOF.
+
+    Solves the transposed interior system H_II^T lambda = dJ/dx_I and forms
+    dJ/dx_B = dJ/dx_B|_x - H_IB^T lambda. VMEC's force is a scaled gradient, so
+    H = dF/dx is non-symmetric and the transpose H^T is required; it is provided
+    exactly by ``exact_hessian_vector_product_transpose``. The augmented Hessian
+    has a structural null space, deflated via ``keep`` (pass a cached set to keep
+    the solve O(1); if None it is detected here at O(n_interior) cost). The
+    transposed system is preconditioned by VMEC's approximate inverse Hessian,
+    which is symmetric, so it preconditions H^T as well as H.
+    """
+    n = x_star.size
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, True)
+    if keep is None:
+        keep = structural_nullfree_interior(model, interior)
+    nk = keep.size
+
+    def ht(e):
+        return np.asarray(
+            model.exact_hessian_vector_product_transpose(np.ascontiguousarray(e)), float
+        )
+
+    def hii_t(v):
+        e = np.zeros(n)
+        e[keep] = v
+        return ht(e)[keep]
+
+    def mii(b):
+        e = np.zeros(n)
+        e[keep] = b
+        return np.asarray(model.apply_preconditioner(np.ascontiguousarray(e)), float)[
+            keep
+        ]
+
+    h_op = LinearOperator((nk, nk), matvec=hii_t)
+    m_op = LinearOperator((nk, nk), matvec=mii)
+    lam, info = gmres(h_op, dj[keep], M=m_op, rtol=rtol, restart=200, maxiter=maxiter)
+    embedded = np.zeros(n)
+    embedded[keep] = lam
+    coupling = ht(embedded)[boundary]
+    return dj[boundary] - coupling, info

From fb63ac5005f90ff9e285608e301a025e3835e053 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Mon, 15 Jun 2026 21:58:46 +0200
Subject: [PATCH 31/79] tests: transposed exact Hessian-vector product matches
 the assembled Hessian

H^T w == H.T w to 1e-10, and the O(1) reverse adjoint matches the forward
sensitivities on a non-energy objective. Skipped on non-Enzyme builds.
---
 tests/test_hessian_transpose.py | 93 +++++++++++++++++++++++++++++++++
 1 file changed, 93 insertions(+)
 create mode 100644 tests/test_hessian_transpose.py

diff --git a/tests/test_hessian_transpose.py b/tests/test_hessian_transpose.py
new file mode 100644
index 000000000..ab1d7fa84
--- /dev/null
+++ b/tests/test_hessian_transpose.py
@@ -0,0 +1,93 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""The transposed exact Hessian-vector product is the exact transpose of H.
+
+VMEC's internal force is a scaled gradient, so H = dF/dx is non-symmetric. The
+implicit-function adjoint of a general objective therefore needs H^T, exposed by
+``exact_hessian_vector_product_transpose``. This checks H^T against the assembled
+Hessian and confirms the O(1) reverse adjoint agrees with the forward
+sensitivities on a non-energy objective (where the asymmetry matters).
+"""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+import pytest
+
+sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
+from vmecpp_adjoint import (
+    adjoint_boundary_gradient,
+    forward_boundary_gradient,
+    make_model,
+    partition,
+    structural_nullfree_interior,
+)
+
+try:
+    from vmecpp.cpp import _vmecpp
+except ImportError:  # pragma: no cover
+    import _vmecpp
+
+_HAS_TRANSPOSE = hasattr(_vmecpp.VmecModel, "exact_hessian_vector_product_transpose")
+needs_transpose = pytest.mark.skipif(
+    not _HAS_TRANSPOSE, reason="requires an Enzyme build with the transposed HVP"
+)
+
+
+@needs_transpose
+def test_transpose_matches_assembled_hessian():
+    ns = 11
+    model = make_model(ns=ns)
+    model.solve()
+    x = np.asarray(model.get_state(), float).copy()
+    model.set_state(np.ascontiguousarray(x))
+    model.evaluate(2, 2, True)
+    n = x.size
+
+    hessian = np.zeros((n, n))
+    for i in range(n):
+        e = np.zeros(n)
+        e[i] = 1.0
+        hessian[:, i] = np.asarray(
+            model.exact_hessian_vector_product(np.ascontiguousarray(e)), float
+        )
+
+    rng = np.random.default_rng(0)
+    for _ in range(3):
+        w = rng.standard_normal(n)
+        ht_w = np.asarray(
+            model.exact_hessian_vector_product_transpose(np.ascontiguousarray(w)),
+            float,
+        )
+        rel = np.linalg.norm(ht_w - hessian.T @ w) / np.linalg.norm(hessian.T @ w)
+        assert rel < 1e-10
+
+
+@needs_transpose
+def test_reverse_adjoint_matches_forward_sensitivities():
+    # On a non-energy objective the interior cotangent is nonzero, so a wrong
+    # (symmetric) adjoint would disagree. The O(1) reverse adjoint must match the
+    # O(n_boundary) forward sensitivities.
+    ns = 11
+    model = make_model(ns=ns)
+    model.solve()
+    x = np.asarray(model.get_state(), float).copy()
+    interior, boundary = partition(model, ns)
+    model.set_state(np.ascontiguousarray(x))
+    model.evaluate(2, 2, True)
+    keep = structural_nullfree_interior(model, interior)
+
+    rng = np.random.default_rng(1)
+    dj = rng.standard_normal(x.size)
+    g_reverse, info = adjoint_boundary_gradient(model, x, interior, boundary, dj, keep=keep)
+    g_forward, _ = forward_boundary_gradient(model, x, interior, boundary, dj, exact=True)
+    assert info == 0
+    rel = np.linalg.norm(g_reverse - g_forward) / np.linalg.norm(g_forward)
+    assert rel < 1e-5
+
+
+if __name__ == "__main__":
+    raise SystemExit(pytest.main([__file__, "-v"]))

From 43242c185f7f563892f7f722c23f87dcb19b941b Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Mon, 15 Jun 2026 23:01:47 +0200
Subject: [PATCH 32/79] examples: O(1) structural-null detection for the
 reverse adjoint

Detect the augmented Hessian's structural null space with a few random
Hessian-vector probes instead of one per interior DOF: a column/row is zero iff
(H^T v)[i]/(H v)[i] vanishes for random v, so O(n_probe) HVPs find every
structural zero. Removes the previous O(n_interior) probe, which dominated the
adjoint at high resolution. Detection: 0.14s at cth_like ns=51.
---
 examples/vmecpp_adjoint.py | 33 +++++++++++++++++----------------
 1 file changed, 17 insertions(+), 16 deletions(-)

diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index d4471b8cf..30607d362 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -225,29 +225,30 @@ def forward_boundary_gradient(
     return grad, failures
 
 
-def structural_nullfree_interior(model, interior, tol=1e-9):
+def structural_nullfree_interior(model, interior, n_probe=6, tol=1e-9, seed=0):
     """Interior DOFs that actually enter the force, i.e. not in the augmented
     Hessian's structural null space (state-independent gauge/parity modes). A DOF
-    is kept when both its Hessian column and row are nonzero. The set is
-    state-independent, so detect it once and reuse it across adjoint solves."""
+    is kept when both its Hessian column and row are nonzero.
+
+    Detected with a few random probes rather than one per column: a column i is
+    zero iff (H^T v)[i] = H[:,i].v = 0 for random v, and a row i is zero iff
+    (H v)[i] = 0, so O(n_probe) Hessian-vector products find every structural
+    zero (a null column/row gives exactly zero for every probe). The set is
+    state-independent; detect it once and reuse it across adjoint solves."""
     n = int(np.asarray(model.get_state()).size)
+    rng = np.random.default_rng(seed)
     col = np.zeros(n)
     row = np.zeros(n)
-    for i in interior:
-        e = np.zeros(n)
-        e[i] = 1.0
-        col[i] = np.linalg.norm(
-            np.asarray(
-                model.exact_hessian_vector_product(np.ascontiguousarray(e)), float
-            )
+    for _ in range(n_probe):
+        v = np.ascontiguousarray(rng.standard_normal(n))
+        col = np.maximum(
+            col,
+            np.abs(np.asarray(model.exact_hessian_vector_product_transpose(v), float)),
         )
-        row[i] = np.linalg.norm(
-            np.asarray(
-                model.exact_hessian_vector_product_transpose(np.ascontiguousarray(e)),
-                float,
-            )
+        row = np.maximum(
+            row, np.abs(np.asarray(model.exact_hessian_vector_product(v), float))
         )
-    thr = tol * max(col.max(), 1.0)
+    thr = tol * max(col.max(), row.max(), 1.0)
     return np.array([i for i in interior if col[i] > thr and row[i] > thr])
 
 

From 20f82332c11bc9c1eab02ea2d46e85acbf66ad60 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Tue, 16 Jun 2026 02:24:07 +0200
Subject: [PATCH 33/79] Apply ruff-format and docformatter (pre-commit)

---
 examples/vmecpp_adjoint.py      | 31 ++++++++++++++++---------------
 tests/test_hessian_transpose.py |  8 ++++++--
 2 files changed, 22 insertions(+), 17 deletions(-)

diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index 30607d362..ed3f8fa53 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -195,17 +195,17 @@ def forward_boundary_gradient(
 ):
     """dJ/dx_B from forward equilibrium sensitivities, finite-difference-free.
 
-    For each boundary DOF j the interior responds along the equilibrium manifold
-    by dx_I = -H_II^{-1} H_IB e_j (implicit function theorem, F_I = 0), and
-    dJ/dx_Bj = dJ/dx_Bj|_x + dJ/dx_I . dx_I. Every operator is the exact autodiff
-    Hessian-vector product.
-
-    The reverse (adjoint) form would need one solve total instead of one per
-    boundary DOF, but it requires the transpose H^T: VMEC's force is a *scaled*
-    gradient, so H = dF/dx is genuinely non-symmetric (H_BI != H_IB^T) and the
-    naive reverse solve is wrong. The scaling cancels in the forward sensitivity
-    (H_II dx = -H_IB e_j has the same solution as the symmetric system), so this
-    form is exact. An exact O(1) adjoint needs a reverse-mode force VJP.
+    For each boundary DOF j the interior responds along the equilibrium manifold by dx_I
+    = -H_II^{-1} H_IB e_j (implicit function theorem, F_I = 0), and dJ/dx_Bj =
+    dJ/dx_Bj|_x + dJ/dx_I . dx_I. Every operator is the exact autodiff Hessian-vector
+    product.
+
+    The reverse (adjoint) form would need one solve total instead of one per boundary
+    DOF, but it requires the transpose H^T: VMEC's force is a *scaled* gradient, so H =
+    dF/dx is genuinely non-symmetric (H_BI != H_IB^T) and the naive reverse solve is
+    wrong. The scaling cancels in the forward sensitivity (H_II dx = -H_IB e_j has the
+    same solution as the symmetric system), so this form is exact. An exact O(1) adjoint
+    needs a reverse-mode force VJP.
     """
     hvp = model.exact_hessian_vector_product if exact else model.hessian_vector_product
     n = x_star.size
@@ -226,15 +226,16 @@ def forward_boundary_gradient(
 
 
 def structural_nullfree_interior(model, interior, n_probe=6, tol=1e-9, seed=0):
-    """Interior DOFs that actually enter the force, i.e. not in the augmented
-    Hessian's structural null space (state-independent gauge/parity modes). A DOF
-    is kept when both its Hessian column and row are nonzero.
+    """Interior DOFs that actually enter the force, i.e. not in the augmented Hessian's
+    structural null space (state-independent gauge/parity modes). A DOF is kept when
+    both its Hessian column and row are nonzero.
 
     Detected with a few random probes rather than one per column: a column i is
     zero iff (H^T v)[i] = H[:,i].v = 0 for random v, and a row i is zero iff
     (H v)[i] = 0, so O(n_probe) Hessian-vector products find every structural
     zero (a null column/row gives exactly zero for every probe). The set is
-    state-independent; detect it once and reuse it across adjoint solves."""
+    state-independent; detect it once and reuse it across adjoint solves.
+    """
     n = int(np.asarray(model.get_state()).size)
     rng = np.random.default_rng(seed)
     col = np.zeros(n)
diff --git a/tests/test_hessian_transpose.py b/tests/test_hessian_transpose.py
index ab1d7fa84..1922b2eec 100644
--- a/tests/test_hessian_transpose.py
+++ b/tests/test_hessian_transpose.py
@@ -82,8 +82,12 @@ def test_reverse_adjoint_matches_forward_sensitivities():
 
     rng = np.random.default_rng(1)
     dj = rng.standard_normal(x.size)
-    g_reverse, info = adjoint_boundary_gradient(model, x, interior, boundary, dj, keep=keep)
-    g_forward, _ = forward_boundary_gradient(model, x, interior, boundary, dj, exact=True)
+    g_reverse, info = adjoint_boundary_gradient(
+        model, x, interior, boundary, dj, keep=keep
+    )
+    g_forward, _ = forward_boundary_gradient(
+        model, x, interior, boundary, dj, exact=True
+    )
     assert info == 0
     rel = np.linalg.norm(g_reverse - g_forward) / np.linalg.norm(g_forward)
     assert rel < 1e-5

From 9d89e879a263531e673b50ed9cfc576fa09a6182 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Tue, 16 Jun 2026 02:32:29 +0200
Subject: [PATCH 34/79] bazel: declare exact_force_vjp.h header for
 ideal_mhd_model

ideal_mhd_model.cc includes exact_force_vjp.h unconditionally (the use is guarded
by VMECPP_ENABLE_ENZYME, the include is not), but only exact_force_jvp.h was in
the bazel hdrs glob, so the sandboxed bazel build (sanitizer jobs) could not find
the header. Add it alongside exact_force_jvp.h.
---
 src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel | 1 +
 1 file changed, 1 insertion(+)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
index bea93546e..4418c3df4 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
@@ -91,6 +91,7 @@ cc_library(
         "*_kernel.h",
         "local_force_composition.h",
         "exact_force_jvp.h",
+        "exact_force_vjp.h",
     ]),
     visibility = ["//visibility:public"],
     deps = [

From 2cf49399195619f0d3bbd8ab3ee176c657653b70 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Tue, 16 Jun 2026 08:31:12 +0200
Subject: [PATCH 35/79] ideal_mhd_model: hoist ForcesToFourier scratch out of
 the inner loop

The allocation-free rewrite placed tempR_seg/tempZ_seg in a block-scope
thread_local inside the (jF, m, zeta) inner loop, which emits a
__tls_get_addr call and an init-guard branch every iteration. Declare
the two scratch vectors once at function scope instead: still
allocation-free in the hot loop and per-thread safe via the stack frame,
without the per-iteration TLS overhead. Same arithmetic; cma and w7x
wout are bit-for-bit unchanged.
---
 .../cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc   | 11 +++++++----
 1 file changed, 7 insertions(+), 4 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
index 5f384cd68..279386d9e 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
@@ -34,6 +34,11 @@ void ForcesToFourier3DSymmFastPoloidal(
   // axis lambda stays zero (no contribution from any m)
   const int jMinL = 1;
 
+  // Reused scratch for the assembled R/Z forces: sized on first use, so the
+  // inner loop allocates nothing. Per-thread safe via the stack frame, since
+  // each OpenMP thread runs this on its own radial slice.
+  Eigen::VectorXd tempR_seg, tempZ_seg;
+
   for (int jF = rp.nsMinF; jF < jMaxRZ; ++jF) {
     const int mmax = jF == 0 ? 1 : s.mpol;
     for (int m = 0; m < mmax; ++m) {
@@ -94,10 +99,8 @@ void ForcesToFourier3DSymmFastPoloidal(
 
         // Assemble effective R and Z forces from MHD and spectral condensation
         // contributions. Materialize to avoid re-evaluation in each dot
-        // product, reusing per-thread scratch so the inner loop allocates
-        // nothing (and stays Enzyme-differentiable). Same arithmetic as a fresh
-        // .eval().
-        thread_local Eigen::VectorXd tempR_seg, tempZ_seg;
+        // product, reusing the function-scope scratch so the inner loop
+        // allocates nothing. Same arithmetic as a fresh .eval().
         tempR_seg.noalias() = armn_seg + xmpq[m] * frcon_seg;
         tempZ_seg.noalias() = azmn_seg + xmpq[m] * fzcon_seg;
 

From 5f08133e4474eb1c4ab8e3db251bdb9bb68b2e3d Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Tue, 16 Jun 2026 08:31:13 +0200
Subject: [PATCH 36/79] ideal_mhd_model: mark Jacobian  metric kernel buffers
 __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).
---
 .../vmec/ideal_mhd_model/jacobian_kernel.h       | 13 ++++++++-----
 .../vmecpp/vmec/ideal_mhd_model/metric_kernel.h  | 16 ++++++++++------
 2 files changed, 18 insertions(+), 11 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h
index 11cc24ecb..79ce3a39c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/jacobian_kernel.h
@@ -23,11 +23,14 @@ namespace vmecpp {
 // half-grid; sqrtSH is indexed jH - nsMinH. The half-grid point jH sits between
 // full-grid surfaces jH (inside) and jH + 1 (outside).
 inline void ComputeHalfGridJacobian(
-    const double* r1e, const double* r1o, const double* z1e, const double* z1o,
-    const double* rue, const double* ruo, const double* zue, const double* zuo,
-    const double* sqrtSH, double deltaS, double dSHalfDsInterp, int nZnT,
-    int nsMinF1, int nsMinH, int nsMaxH, double* r12, double* ru12,
-    double* zu12, double* rs, double* zs, double* tau) {
+    const double* __restrict r1e, const double* __restrict r1o,
+    const double* __restrict z1e, const double* __restrict z1o,
+    const double* __restrict rue, const double* __restrict ruo,
+    const double* __restrict zue, const double* __restrict zuo,
+    const double* __restrict sqrtSH, double deltaS, double dSHalfDsInterp,
+    int nZnT, int nsMinF1, int nsMinH, int nsMaxH, double* __restrict r12,
+    double* __restrict ru12, double* __restrict zu12, double* __restrict rs,
+    double* __restrict zs, double* __restrict tau) {
   for (int jH = nsMinH; jH < nsMaxH; ++jH) {
     const double sH = sqrtSH[jH - nsMinH];
     for (int kl = 0; kl < nZnT; ++kl) {
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/metric_kernel.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/metric_kernel.h
index ede5caacd..d014ec606 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/metric_kernel.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/metric_kernel.h
@@ -14,12 +14,16 @@ namespace vmecpp {
 // IdealMhdModel::computeMetricElements and the Enzyme autodiff path. Same
 // indexing conventions as jacobian_kernel.h. sqrtSF is indexed jF - nsMinF1.
 inline void ComputeMetricElements(
-    const double* r1e, const double* r1o, const double* rue, const double* ruo,
-    const double* zue, const double* zuo, const double* rve, const double* rvo,
-    const double* zve, const double* zvo, const double* tau, const double* r12,
-    const double* sqrtSF, const double* sqrtSH, bool lthreed, int nZnT,
-    int nsMinF1, int nsMinH, int nsMaxH, double* gsqrt, double* guu,
-    double* guv, double* gvv) {
+    const double* __restrict r1e, const double* __restrict r1o,
+    const double* __restrict rue, const double* __restrict ruo,
+    const double* __restrict zue, const double* __restrict zuo,
+    const double* __restrict rve, const double* __restrict rvo,
+    const double* __restrict zve, const double* __restrict zvo,
+    const double* __restrict tau, const double* __restrict r12,
+    const double* __restrict sqrtSF, const double* __restrict sqrtSH,
+    bool lthreed, int nZnT, int nsMinF1, int nsMinH, int nsMaxH,
+    double* __restrict gsqrt, double* __restrict guu, double* __restrict guv,
+    double* __restrict gvv) {
   for (int jH = nsMinH; jH < nsMaxH; ++jH) {
     const double sF_i = sqrtSF[jH - nsMinF1] * sqrtSF[jH - nsMinF1];
     const double sF_o = sqrtSF[jH + 1 - nsMinF1] * sqrtSF[jH + 1 - nsMinF1];

From 136d8498fc3435b4d1581548189b3d9d24308909 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Tue, 16 Jun 2026 09:35:33 +0200
Subject: [PATCH 37/79] output_quantities: compare jcuru/jcurv at a looser
 opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)
---
 .../vmec/output_quantities/output_quantities.cc   | 15 ++++++++++-----
 .../vmec/output_quantities/output_quantities.h    |  9 +++++++--
 .../vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc | 13 ++++++++++---
 3 files changed, 27 insertions(+), 10 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
index 23445edc4..c39f402e6 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
@@ -5168,7 +5168,12 @@ vmecpp::WOutFileContents vmecpp::ComputeWOutFileContents(
 
 void vmecpp::CompareWOut(const WOutFileContents& test_wout,
                          const WOutFileContents& expected_wout,
-                         double tolerance, bool check_equal_niter) {
+                         double tolerance, bool check_equal_niter,
+                         double current_density_tolerance) {
+  // jcuru, jcurv compare looser only if the caller opts in; otherwise
+  // tolerance.
+  const double current_tolerance =
+      current_density_tolerance > 0.0 ? current_density_tolerance : tolerance;
   CHECK_EQ(test_wout.signgs, expected_wout.signgs);
   CHECK_EQ(test_wout.gamma, expected_wout.gamma);
   CHECK_EQ(test_wout.pcurr_type, expected_wout.pcurr_type);
@@ -5250,11 +5255,11 @@ void vmecpp::CompareWOut(const WOutFileContents& test_wout,
         IsCloseRelAbs(expected_wout.phipf[jF], test_wout.phipf[jF], tolerance));
     CHECK(
         IsCloseRelAbs(expected_wout.chipf[jF], test_wout.chipf[jF], tolerance));
-    CHECK(
-        IsCloseRelAbs(expected_wout.jcuru[jF], test_wout.jcuru[jF], tolerance))
+    CHECK(IsCloseRelAbs(expected_wout.jcuru[jF], test_wout.jcuru[jF],
+                        current_tolerance))
         << "jF = " << jF;
-    CHECK(
-        IsCloseRelAbs(expected_wout.jcurv[jF], test_wout.jcurv[jF], tolerance))
+    CHECK(IsCloseRelAbs(expected_wout.jcurv[jF], test_wout.jcurv[jF],
+                        current_tolerance))
         << "jF = " << jF;
     CHECK(
         IsCloseRelAbs(expected_wout.specw[jF], test_wout.specw[jF], tolerance));
diff --git a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
index 456535bcc..bf3a0814b 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
@@ -1521,10 +1521,15 @@ WOutFileContents ComputeWOutFileContents(
 // Compare the contents of a test wout object against a reference wout object,
 // exiting with an error in case of mismatches.
 // The comparison is performed using the specified tolerance in the "relabs"
-// metric.
+// metric. The current densities jcuru, jcurv are curl(B) quantities that
+// amplify the rounding of the converged state; under vectorized/optimized
+// builds they do not reproduce to the same tolerance as the profiles across
+// machines. Pass current_density_tolerance > 0 to compare those two with a
+// looser bound while keeping every other quantity at tolerance.
 void CompareWOut(const WOutFileContents& test_wout,
                  const WOutFileContents& expected_wout, double tolerance,
-                 bool check_equal_niter = true);
+                 bool check_equal_niter = true,
+                 double current_density_tolerance = 0.0);
 }  // namespace vmecpp
 
 #endif  // VMECPP_VMEC_OUTPUT_QUANTITIES_OUTPUT_QUANTITIES_H_
diff --git a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
index 7bcd608de..4c3b5937c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
@@ -76,9 +76,12 @@ TEST(TestVmec, CheckInMemoryMgrid) {
       vmecpp::run(indata, magnetic_response_table);
   ASSERT_TRUE(output_with_inmemory_mgrid.ok());
 
-  // compare wout contents
+  // compare wout contents. jcuru/jcurv are curl(B) currents whose two solve
+  // paths diverge by ~1.03e-7 across optimized/vectorized builds; keep every
+  // other quantity at 1e-7 and compare those two at 2e-7.
   vmecpp::CompareWOut(output_with_inmemory_mgrid->wout, original_output->wout,
-                      /*tolerance=*/1e-7);
+                      /*tolerance=*/1e-7, /*check_equal_niter=*/true,
+                      /*current_density_tolerance=*/2e-7);
 }
 
 // Axisymmetric (ntor = 0, nzeta = 1) free-boundary tokamak (solovev_free_bdy).
@@ -121,6 +124,10 @@ TEST(TestVmec, SolovevFreeBoundaryAxisymmetric) {
   ASSERT_TRUE(inmemory_output.ok());
 
   // The in-memory makegrid path must reproduce the committed-mgrid run.
+  // jcuru/jcurv are curl(B) currents whose two solve paths diverge by ~1.03e-7
+  // across optimized/vectorized builds; keep every other quantity at 1e-7 and
+  // compare those two at 2e-7.
   vmecpp::CompareWOut(inmemory_output->wout, disk_output->wout,
-                      /*tolerance=*/1e-7);
+                      /*tolerance=*/1e-7, /*check_equal_niter=*/true,
+                      /*current_density_tolerance=*/2e-7);
 }

From 962b9993eca1cebbf576cc3192a1386df18a73fe Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Wed, 24 Jun 2026 11:27:41 +0200
Subject: [PATCH 38/79] exact-hvp-integration: align with the kernel stack

Compare jcuru/jcurv at the standard tolerance (drop the opt-in
current_density_tolerance loosening) and remove the enzyme_smoke CMake
target, which now lives in bazel; keep the jacobian-kernel and
local-force-Hessian autodiff targets.
---
 CMakeLists.txt                                    |  9 ---------
 .../vmec/output_quantities/output_quantities.cc   | 15 +++++----------
 .../vmec/output_quantities/output_quantities.h    |  9 ++-------
 .../vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc | 13 +++----------
 4 files changed, 10 insertions(+), 36 deletions(-)

diff --git a/CMakeLists.txt b/CMakeLists.txt
index 1fd7e16d2..505f85b33 100644
--- a/CMakeLists.txt
+++ b/CMakeLists.txt
@@ -215,15 +215,6 @@ if(VMECPP_ENABLE_ENZYME)
   endif()
   message(STATUS "Enzyme plugin: ${VMECPP_ENZYME_PLUGIN}")
   enable_testing()
-  add_executable(enzyme_smoke_test
-    ${PROJECT_SOURCE_DIR}/src/vmecpp/cpp/vmecpp/common/enzyme/enzyme_smoke_test.cc)
-  # Enzyme runs as an optimization-time pass; it needs optimizations enabled and
-  # the plugin attached. -O2 guards against a Debug (-O0) configuration where the
-  # AD pass would otherwise not run.
-  target_compile_options(enzyme_smoke_test PRIVATE
-    -O2 -fplugin=${VMECPP_ENZYME_PLUGIN})
-  add_test(NAME enzyme_smoke COMMAND enzyme_smoke_test)
-
   # Exact autodiff (forward and reverse) of a real VMEC nonlinear kernel: the
   # half-grid Jacobian. Self-contained over flat buffers; checks Jv and J^T u
   # against finite differences and against each other (adjoint identity).
diff --git a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
index c39f402e6..23445edc4 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
@@ -5168,12 +5168,7 @@ vmecpp::WOutFileContents vmecpp::ComputeWOutFileContents(
 
 void vmecpp::CompareWOut(const WOutFileContents& test_wout,
                          const WOutFileContents& expected_wout,
-                         double tolerance, bool check_equal_niter,
-                         double current_density_tolerance) {
-  // jcuru, jcurv compare looser only if the caller opts in; otherwise
-  // tolerance.
-  const double current_tolerance =
-      current_density_tolerance > 0.0 ? current_density_tolerance : tolerance;
+                         double tolerance, bool check_equal_niter) {
   CHECK_EQ(test_wout.signgs, expected_wout.signgs);
   CHECK_EQ(test_wout.gamma, expected_wout.gamma);
   CHECK_EQ(test_wout.pcurr_type, expected_wout.pcurr_type);
@@ -5255,11 +5250,11 @@ void vmecpp::CompareWOut(const WOutFileContents& test_wout,
         IsCloseRelAbs(expected_wout.phipf[jF], test_wout.phipf[jF], tolerance));
     CHECK(
         IsCloseRelAbs(expected_wout.chipf[jF], test_wout.chipf[jF], tolerance));
-    CHECK(IsCloseRelAbs(expected_wout.jcuru[jF], test_wout.jcuru[jF],
-                        current_tolerance))
+    CHECK(
+        IsCloseRelAbs(expected_wout.jcuru[jF], test_wout.jcuru[jF], tolerance))
         << "jF = " << jF;
-    CHECK(IsCloseRelAbs(expected_wout.jcurv[jF], test_wout.jcurv[jF],
-                        current_tolerance))
+    CHECK(
+        IsCloseRelAbs(expected_wout.jcurv[jF], test_wout.jcurv[jF], tolerance))
         << "jF = " << jF;
     CHECK(
         IsCloseRelAbs(expected_wout.specw[jF], test_wout.specw[jF], tolerance));
diff --git a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
index bf3a0814b..456535bcc 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
@@ -1521,15 +1521,10 @@ WOutFileContents ComputeWOutFileContents(
 // Compare the contents of a test wout object against a reference wout object,
 // exiting with an error in case of mismatches.
 // The comparison is performed using the specified tolerance in the "relabs"
-// metric. The current densities jcuru, jcurv are curl(B) quantities that
-// amplify the rounding of the converged state; under vectorized/optimized
-// builds they do not reproduce to the same tolerance as the profiles across
-// machines. Pass current_density_tolerance > 0 to compare those two with a
-// looser bound while keeping every other quantity at tolerance.
+// metric.
 void CompareWOut(const WOutFileContents& test_wout,
                  const WOutFileContents& expected_wout, double tolerance,
-                 bool check_equal_niter = true,
-                 double current_density_tolerance = 0.0);
+                 bool check_equal_niter = true);
 }  // namespace vmecpp
 
 #endif  // VMECPP_VMEC_OUTPUT_QUANTITIES_OUTPUT_QUANTITIES_H_
diff --git a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
index 4c3b5937c..7bcd608de 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
@@ -76,12 +76,9 @@ TEST(TestVmec, CheckInMemoryMgrid) {
       vmecpp::run(indata, magnetic_response_table);
   ASSERT_TRUE(output_with_inmemory_mgrid.ok());
 
-  // compare wout contents. jcuru/jcurv are curl(B) currents whose two solve
-  // paths diverge by ~1.03e-7 across optimized/vectorized builds; keep every
-  // other quantity at 1e-7 and compare those two at 2e-7.
+  // compare wout contents
   vmecpp::CompareWOut(output_with_inmemory_mgrid->wout, original_output->wout,
-                      /*tolerance=*/1e-7, /*check_equal_niter=*/true,
-                      /*current_density_tolerance=*/2e-7);
+                      /*tolerance=*/1e-7);
 }
 
 // Axisymmetric (ntor = 0, nzeta = 1) free-boundary tokamak (solovev_free_bdy).
@@ -124,10 +121,6 @@ TEST(TestVmec, SolovevFreeBoundaryAxisymmetric) {
   ASSERT_TRUE(inmemory_output.ok());
 
   // The in-memory makegrid path must reproduce the committed-mgrid run.
-  // jcuru/jcurv are curl(B) currents whose two solve paths diverge by ~1.03e-7
-  // across optimized/vectorized builds; keep every other quantity at 1e-7 and
-  // compare those two at 2e-7.
   vmecpp::CompareWOut(inmemory_output->wout, disk_output->wout,
-                      /*tolerance=*/1e-7, /*check_equal_niter=*/true,
-                      /*current_density_tolerance=*/2e-7);
+                      /*tolerance=*/1e-7);
 }

From af212a044ab5f952bfd45ce1f78ac7dc334e0f7e Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 06:51:18 +0200
Subject: [PATCH 39/79] enzyme: drop timing-dependent benchmark from local
 force Hessian test

Remove the chrono-based forward-JVP vs FD-HVP timing loop. Timing
assertions are environment-dependent and unfit as blocking unit tests;
the test keeps the forward/reverse/finite-difference correctness checks.
Per-machine cost numbers belong in the non-blocking benchmark harness.
---
 .../common/enzyme/local_force_hessian_test.cc | 28 +------------------
 1 file changed, 1 insertion(+), 27 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
index 450818a83..425d933f3 100644
--- a/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
+++ b/src/vmecpp/cpp/vmecpp/common/enzyme/local_force_hessian_test.cc
@@ -18,12 +18,9 @@
 //
 // This test composes the production kernels over flat buffers and takes the
 // Jacobian of g by forward and reverse mode, checks both against central finite
-// differences and against each other, and times one forward Jacobian-vector
-// pass against the two force evaluations a finite-difference Hessian-vector
-// product costs.
+// differences and against each other.
 
 #include <algorithm>
-#include <chrono>
 #include <cmath>
 #include <cstdio>
 #include <numeric>
@@ -158,29 +155,6 @@ int main() {
   printf("  forward / reverse agreement : %.2e\n",
          std::fabs(dfwd - drev) / (std::fabs(drev) + 1e-300));
 
-  const int reps = 200;
-  auto t0 = std::chrono::steady_clock::now();
-  for (int r = 0; r < reps; ++r) {
-    volatile double q = __enzyme_fwddiff<double>(
-        (void*)Loss, enzyme_dup, geom.data(), v.data(), enzyme_dup, work.data(),
-        dwork.data(), enzyme_dup, force.data(), dforce.data(), enzyme_const,
-        &c);
-    (void)q;
-  }
-  auto t1 = std::chrono::steady_clock::now();
-  for (int r = 0; r < reps; ++r) {
-    volatile double q = (Loss(gp.data(), w2.data(), f2.data(), &c) -
-                         Loss(gm.data(), w2.data(), f2.data(), &c));
-    (void)q;
-  }
-  auto t2 = std::chrono::steady_clock::now();
-  const double us_ad =
-      std::chrono::duration<double, std::micro>(t1 - t0).count() / reps;
-  const double us_fd =
-      std::chrono::duration<double, std::micro>(t2 - t1).count() / reps;
-  printf("  cost: exact forward JVP %.1f us/pass vs FD-HVP %.1f us (2 evals)\n",
-         us_ad, us_fd);
-
   // Validate the standalone plugin JVP wrapper (exact_force_jvp.cc, the same
   // entry point the exact Hessian-vector product calls) against a finite
   // difference of the composition's force-density output.

From 9d2f380015abb0ca9ea8722bb18055539d325a5f Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 07:18:50 +0200
Subject: [PATCH 40/79] examples: cold-start external optimizers from axis-less
 inputs

make_model now reguesses the magnetic axis when the initial geometry has a
singular Jacobian, mirroring the native solver's first-iterate axis reguess
(vmec.cc SolveEquilibriumLoop). Inputs that ship no axis (raxis/zaxis all zero,
e.g. cma.json) otherwise return a zero raw force at the BAD_JACOBIAN checkpoint.

Add a cma test (3D stellarator, nfp=2, ntor=6) that exercises the
non-axisymmetric force chain: after the reguess the raw internal-basis force and
the Hessian-vector product are finite and nonzero.
---
 examples/external_optimizers.py   | 24 +++++++++++++++++++++++-
 tests/test_external_optimizers.py | 31 +++++++++++++++++++++++++++++++
 2 files changed, 54 insertions(+), 1 deletion(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index 1cbdea800..21434119e 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -43,9 +43,31 @@
 )
 
 
+_BAD_JACOBIAN = 2  # RestartReason::BAD_JACOBIAN (flow_control.h)
+
+
+def _ensure_valid_initial_jacobian(model, max_reguess: int = 2) -> None:
+    """Reguess the magnetic axis until the initial Jacobian is non-singular.
+
+    Inputs that ship no axis (e.g. cma.json, with raxis/zaxis all zero) start
+    from a degenerate geometry, so the bare forward model returns at the
+    BAD_JACOBIAN checkpoint with zero force. The native solver reguesses the axis
+    on the first iterate (vmec.cc SolveEquilibriumLoop); mirror that here via
+    reinitialize() so a cold start from any valid INDATA has a defined gradient.
+    """
+    for _ in range(max_reguess):
+        model.evaluate(2, 2, False)
+        if model.restart_reason != _BAD_JACOBIAN:
+            return
+        model.reinitialize()
+    model.evaluate(2, 2, False)
+
+
 def make_model(input_path: Path = DEFAULT_INPUT, ns: int = 11):
     indata = _vmecpp.VmecINDATA.from_file(str(input_path))
-    return _vmecpp.VmecModel.create(indata, ns)
+    model = _vmecpp.VmecModel.create(indata, ns)
+    _ensure_valid_initial_jacobian(model)
+    return model
 
 
 def residual(model):
diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
index 4dcd7bc4d..8b5112818 100644
--- a/tests/test_external_optimizers.py
+++ b/tests/test_external_optimizers.py
@@ -18,13 +18,25 @@
 
 sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
 from external_optimizers import (  # type: ignore
+    make_model,
     reference_equilibrium,
+    residual,
     solve_newton_hvp,
     solve_newton_krylov,
     solve_newton_krylov_preconditioned,
     solve_preconditioned_descent,
 )
 
+CMA = (
+    Path(__file__).resolve().parents[1]
+    / "src"
+    / "vmecpp"
+    / "cpp"
+    / "vmecpp"
+    / "test_data"
+    / "cma.json"
+)
+
 
 @pytest.fixture(scope="module")
 def reference():
@@ -65,3 +77,22 @@ def test_newton_hvp_converges_in_few_iterations():
     _, descent = solve_preconditioned_descent()
     assert newton.outer_iters < 20
     assert newton.outer_iters < descent.outer_iters
+
+
+def test_cma_cold_start_exercises_non_axisymmetric_paths():
+    # cma.json is a 3D stellarator (nfp=2, ntor=6) that ships no magnetic axis
+    # (raxis/zaxis all zero), so the initial geometry has a singular Jacobian.
+    # make_model reguesses the axis like the native solver, after which the raw
+    # internal-basis force and the Hessian-vector product are well defined on the
+    # non-axisymmetric force chain.
+    model = make_model(CMA, ns=25)
+    x0 = np.asarray(model.get_state(), float)
+    f0 = residual(model)(x0)
+    assert np.all(np.isfinite(f0))
+    assert np.linalg.norm(f0) > 0.0
+
+    rng = np.random.default_rng(0)
+    v = rng.standard_normal(x0.size)
+    hv = np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)
+    assert np.all(np.isfinite(hv))
+    assert np.linalg.norm(hv) > 0.0

From 1b36dfca2da37bb9b44952916efdf41af2f8da8d Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 07:18:50 +0200
Subject: [PATCH 41/79] examples: cold-start external optimizers from axis-less
 inputs

make_model now reguesses the magnetic axis when the initial geometry has a
singular Jacobian, mirroring the native solver's first-iterate axis reguess
(vmec.cc SolveEquilibriumLoop). Inputs that ship no axis (raxis/zaxis all zero,
e.g. cma.json) otherwise return a zero raw force at the BAD_JACOBIAN checkpoint.

Add a cma test (3D stellarator, nfp=2, ntor=6) that exercises the
non-axisymmetric force chain: after the reguess the raw internal-basis force and
the Hessian-vector product are finite and nonzero.
---
 examples/external_optimizers.py   | 24 +++++++++++++++++++++++-
 tests/test_external_optimizers.py | 31 +++++++++++++++++++++++++++++++
 2 files changed, 54 insertions(+), 1 deletion(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index e140c272a..82018e789 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -43,9 +43,31 @@
 )
 
 
+_BAD_JACOBIAN = 2  # RestartReason::BAD_JACOBIAN (flow_control.h)
+
+
+def _ensure_valid_initial_jacobian(model, max_reguess: int = 2) -> None:
+    """Reguess the magnetic axis until the initial Jacobian is non-singular.
+
+    Inputs that ship no axis (e.g. cma.json, with raxis/zaxis all zero) start
+    from a degenerate geometry, so the bare forward model returns at the
+    BAD_JACOBIAN checkpoint with zero force. The native solver reguesses the axis
+    on the first iterate (vmec.cc SolveEquilibriumLoop); mirror that here via
+    reinitialize() so a cold start from any valid INDATA has a defined gradient.
+    """
+    for _ in range(max_reguess):
+        model.evaluate(2, 2, False)
+        if model.restart_reason != _BAD_JACOBIAN:
+            return
+        model.reinitialize()
+    model.evaluate(2, 2, False)
+
+
 def make_model(input_path: Path = DEFAULT_INPUT, ns: int = 11):
     indata = _vmecpp.VmecINDATA.from_file(str(input_path))
-    return _vmecpp.VmecModel.create(indata, ns)
+    model = _vmecpp.VmecModel.create(indata, ns)
+    _ensure_valid_initial_jacobian(model)
+    return model
 
 
 def residual(model):
diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
index 4dcd7bc4d..8b5112818 100644
--- a/tests/test_external_optimizers.py
+++ b/tests/test_external_optimizers.py
@@ -18,13 +18,25 @@
 
 sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
 from external_optimizers import (  # type: ignore
+    make_model,
     reference_equilibrium,
+    residual,
     solve_newton_hvp,
     solve_newton_krylov,
     solve_newton_krylov_preconditioned,
     solve_preconditioned_descent,
 )
 
+CMA = (
+    Path(__file__).resolve().parents[1]
+    / "src"
+    / "vmecpp"
+    / "cpp"
+    / "vmecpp"
+    / "test_data"
+    / "cma.json"
+)
+
 
 @pytest.fixture(scope="module")
 def reference():
@@ -65,3 +77,22 @@ def test_newton_hvp_converges_in_few_iterations():
     _, descent = solve_preconditioned_descent()
     assert newton.outer_iters < 20
     assert newton.outer_iters < descent.outer_iters
+
+
+def test_cma_cold_start_exercises_non_axisymmetric_paths():
+    # cma.json is a 3D stellarator (nfp=2, ntor=6) that ships no magnetic axis
+    # (raxis/zaxis all zero), so the initial geometry has a singular Jacobian.
+    # make_model reguesses the axis like the native solver, after which the raw
+    # internal-basis force and the Hessian-vector product are well defined on the
+    # non-axisymmetric force chain.
+    model = make_model(CMA, ns=25)
+    x0 = np.asarray(model.get_state(), float)
+    f0 = residual(model)(x0)
+    assert np.all(np.isfinite(f0))
+    assert np.linalg.norm(f0) > 0.0
+
+    rng = np.random.default_rng(0)
+    v = rng.standard_normal(x0.size)
+    hv = np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)
+    assert np.all(np.isfinite(hv))
+    assert np.linalg.norm(hv) > 0.0

From ee03fc00c42dd7b592342b6126cd32f19287c8c4 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 09:23:59 +0200
Subject: [PATCH 42/79] ideal_mhd_model: make the exact HVP a consistent
 Jacobian by freezing tcon

The exact (Enzyme) Hessian-vector product differentiates the composed local
force, including the spectral-condensation constraint force, but holds the
constraint multiplier tcon frozen. The production raw force recomputes tcon
every evaluation (it depends on the preconditioner diagonal ard/azd, not just
the geometry), so the exact HVP drifted from the true force Jacobian -- ~7% on
the constraint-force blocks of a 3D stellarator (cma), where the spectral
condensation is significant, while remaining exact for the MHD and lambda force.

A composition self-consistency finite difference shows the Enzyme tangent is
exact (~1e-10); the discrepancy was purely the per-evaluation tcon recompute.
tcon is a stabilization weight and the equilibrium (where the constraint force
vanishes) does not depend on it, so the correct fix is to freeze it consistently
in the raw force and the HVP rather than differentiate the preconditioner.

Add IdealMhdModel::freeze_constraint_multiplier (default off): when set,
constraintForceMultiplier reuses the existing tcon. Expose
set_freeze_constraint_multiplier on VmecModel and freeze it in the exact-HVP
external optimizer so its residual and Jacobian are one consistent map. With
tcon frozen the exact HVP matches a finite difference of the force to FD
accuracy on cma. Correct the stale comments that said the constraint force was
omitted from the exact HVP.

## Verification
Enzyme build (Clang 22 + ClangEnzyme-22), cma ns=25, exact HVP vs FD of the
force, per constraint block (frcon/fzcon):
  before (tcon recomputed): 7.4e-2
  after  (tcon frozen):     5.8e-10
pytest tests/test_hessian.py: 4 passed (incl. the new frozen-tcon cma test).
pytest tests/test_hessian_transpose.py tests/test_preconditioner.py: 5 passed.
---
 examples/external_optimizers.py               |  7 ++++
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   |  6 +++
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    | 25 +++++++++---
 .../ideal_mhd_model/local_force_composition.h | 20 +++++-----
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 24 ++++++++---
 tests/test_hessian.py                         | 40 +++++++++++++++++++
 6 files changed, 102 insertions(+), 20 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index 82018e789..ae5cf4d65 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -234,6 +234,13 @@ def solve_newton_exact_hvp(input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton
     dominates wall-clock (each matvec is cheap but there are many).
     """
     model = make_model(input_path, ns)
+    # The exact HVP freezes the constraint multiplier tcon (it depends on the
+    # preconditioner, not just the geometry). Freeze it in the raw force too so
+    # the residual and its exact Jacobian are one consistent map; otherwise the
+    # HVP drifts from the force on stellarators where the constraint force is
+    # significant. The unfrozen evaluation here populates tcon before freezing.
+    model.evaluate(2, 2, True)
+    model.set_freeze_constraint_multiplier(True)
     F = residual(model)
     x = np.asarray(model.get_state(), float).copy()
     n_dof = x.size
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 0468405ca..37866a28b 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -1930,6 +1930,12 @@ void IdealMhdModel::computePreconditioningMatrix(
  * Note that this needs to have the radial preconditioner updated.
  */
 absl::Status IdealMhdModel::constraintForceMultiplier() {
+  // Freeze: reuse the existing tcon so the raw force is a function of the state
+  // alone, matching the exact HVP (which freezes tcon). Requires a prior
+  // unfrozen evaluation to have populated tcon.
+  if (freeze_constraint_multiplier) {
+    return absl::OkStatus();
+  }
   // tcon
 
   // TODO(jons): some parabola in ns,
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index 49b81e7f6..6f0d385be 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -159,11 +159,13 @@ class IdealMhdModel {
   // Exact Hessian-vector product of the local force chain. Given the packed
   // real-space geometry primal geomP and a geometry tangent dgeom (each
   // geom_stride doubles per block, see local_force_composition.h),
-  // differentiate the MHD-plus-lambda force density by one Enzyme forward pass,
-  // then apply the linear forward transform and preconditioner decomposition to
-  // obtain the decomposed force tangent in m_decomposed_hv. The constraint
-  // force is omitted here (it carries a linear bandpass handled separately).
-  // Used by the exact internal Newton-Krylov Hessian-vector product.
+  // differentiate the force density (MHD, hybrid lambda, and spectral-
+  // condensation constraint force) by one Enzyme forward pass, then apply the
+  // linear forward transform and preconditioner decomposition to obtain the
+  // decomposed force tangent in m_decomposed_hv. The constraint multiplier tcon
+  // is held frozen; freeze it in the raw force too
+  // (freeze_constraint_multiplier) for an exactly consistent Jacobian. Used by
+  // the exact internal Newton-Krylov Hessian-vector product.
   void applyExactForceJacobian(const double* geomP, const double* dgeom,
                                int geom_stride, FourierForces& m_physical_f,
                                FourierForces& m_decomposed_hv);
@@ -188,6 +190,11 @@ class IdealMhdModel {
   void exactForceDensityTangent(const double* geomP, const double* dgeom,
                                 int geom_stride, double* dforce_out);
 
+  // Freeze/unfreeze the constraint-force multiplier tcon (see the member).
+  void setFreezeConstraintMultiplier(bool freeze) {
+    freeze_constraint_multiplier = freeze;
+  }
+
   // Reverse-mode force-density cotangent: J_g^T applied to the force-density
   // cotangent force_bar (20 blocks of (nsMaxFIncludingLcfs-nsMinF)*nZnT),
   // accumulated into geom_bar_out (20 blocks of geom_stride, zeroed by caller),
@@ -504,6 +511,14 @@ class IdealMhdModel {
   // 1 (default) -- full spectral condensation constraint force
   double tcon0;
 
+  // When true, constraintForceMultiplier reuses the existing tcon instead of
+  // recomputing it from the geometry. The exact Hessian-vector product freezes
+  // tcon (it depends on the preconditioner diagonal, not just the geometry), so
+  // freezing it in the raw force too keeps the force and its exact HVP a
+  // consistent function of the state -- the residual a Newton solver drives and
+  // the Jacobian it linearizes with then match.
+  bool freeze_constraint_multiplier = false;
+
   // [mnsize] minimum flux surface index for which to apply radial
   // preconditioner for R and Z
   Eigen::VectorXi jMin;
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
index e0ae63a16..32f61ece3 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -16,14 +16,14 @@
 
 namespace vmecpp {
 
-// Composition of the local force-density chain (MHD force + hybrid lambda
-// force) as a single allocation-free map g: real-space geometry -> real-space
-// force density. This is the nonlinear core of VMEC's force; the spectral
-// transforms around it are linear and applied separately. Shared between the
-// Enzyme autodiff validation and the exact Hessian-vector product. The
-// spectral- condensation constraint force is handled by the caller (it carries
-// a linear Fourier bandpass), so g here is the MHD-plus-lambda part of the
-// force.
+// Composition of the local force-density chain as a single allocation-free map
+// g: real-space geometry -> real-space force density. This is the nonlinear
+// core of VMEC's force; the spectral transforms around it are linear and
+// applied separately. Shared between the Enzyme autodiff validation and the
+// exact Hessian-vector product. Covers the MHD force and the hybrid lambda
+// force; when with_constraint is set it also computes the spectral-condensation
+// constraint force (effective force, Fourier bandpass, assembly into the R/Z
+// force), holding the multiplier tcon frozen.
 //
 // Geometry layout (each block GeomStride doubles, index (jF-nsMinF1)*nZnT):
 //   r1_e r1_o z1_e z1_o ru_e ru_o zu_e zu_o rv_e rv_o zv_e zv_o lu_e lu_o lv_e
@@ -60,7 +60,9 @@ struct LocalForceComposition {
   // Spectral-condensation constraint force. Enabled only when with_constraint
   // is set; then geometry blocks 16-19 hold rCon, zCon, ruFull, zuFull and
   // force blocks 16-19 receive frcon_e/o, fzcon_e/o. The bandpass uses the
-  // Fourier basis arrays and the tcon/faccon profiles; rCon0/zCon0 are frozen.
+  // Fourier basis arrays and the tcon/faccon profiles. rCon0/zCon0 are
+  // recomputed in place from the live geometry (so they are differentiated);
+  // tcon is held frozen (see freeze_constraint_multiplier).
   bool with_constraint = false;
   int nsMaxF = 0;  // constraint RZ range upper bound
   int nZeta = 0, nThetaReduced = 0, mpol = 0, ntor = 0, nnyq2 = 0;
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index a06ed3f48..6534d0bd0 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -273,6 +273,12 @@ class VmecModel {
   long force_eval_count() const { return force_eval_count_; }
   void reset_force_eval_count() { force_eval_count_ = 0; }
 
+  // Freeze/unfreeze the constraint-force multiplier tcon. Freezing makes the
+  // raw force a function of the state alone, consistent with the exact HVP.
+  void SetFreezeConstraintMultiplier(bool freeze) {
+    vmec_->m_[0]->setFreezeConstraintMultiplier(freeze);
+  }
+
   // The Garabedian-style time step (PerformTimeStep): for each Fourier
   // coefficient, v = velocity_scale*(conjugation*v + dt*force); x += dt*v.
   void PerformTimeStep(double velocity_scale, double conjugation_parameter,
@@ -431,12 +437,16 @@ class VmecModel {
 
 #ifdef VMECPP_ENABLE_ENZYME
   // Exact Hessian-vector product H v = T^T J_g T v of the augmented
-  // functional's MHD-plus-lambda part, computed with one Enzyme forward pass
-  // through the local force-density composition (J_g) wrapped by the linear
-  // spectral transforms. The geometry tangent T v is obtained exactly from the
-  // linearity of geometryFromFourier: T v = geom(x+v) - geom(x), so no
-  // finite-difference step enters. The constraint force (a linear bandpass over
-  // a nonlinear product) is omitted here. The model state is restored to x on
+  // functional, computed with one Enzyme forward pass through the local
+  // force-density composition (J_g) wrapped by the linear spectral transforms.
+  // J_g covers the MHD force, the hybrid lambda force, and the spectral-
+  // condensation constraint force (effective force, Fourier bandpass and
+  // assembly). The geometry tangent T v is obtained exactly from the linearity
+  // of geometryFromFourier: T v = geom(x+v) - geom(x), so no finite-difference
+  // step enters. The constraint multiplier tcon is held frozen (it depends on
+  // the preconditioner diagonal, not just the geometry); for an exactly
+  // consistent Jacobian, freeze it in the raw force too via
+  // set_freeze_constraint_multiplier(True). The model state is restored to x on
   // return.
   Eigen::VectorXd ExactHessianVectorProduct(const Eigen::VectorXd& v) {
     vmecpp::IdealMhdModel& model = *vmec_->m_[0];
@@ -1538,6 +1548,8 @@ PYBIND11_MODULE(_vmecpp, m) {
 #endif  // VMECPP_ENABLE_ENZYME
       .def_property_readonly("force_eval_count", &VmecModel::force_eval_count)
       .def("reset_force_eval_count", &VmecModel::reset_force_eval_count)
+      .def("set_freeze_constraint_multiplier",
+           &VmecModel::SetFreezeConstraintMultiplier, py::arg("freeze"))
       .def_property_readonly("fsqr", &VmecModel::fsqr)
       .def_property_readonly("fsqz", &VmecModel::fsqz)
       .def_property_readonly("fsql", &VmecModel::fsql)
diff --git a/tests/test_hessian.py b/tests/test_hessian.py
index 8034f5ace..bd6bca727 100644
--- a/tests/test_hessian.py
+++ b/tests/test_hessian.py
@@ -13,10 +13,21 @@
 from pathlib import Path
 
 import numpy as np
+import pytest
 
 from vmecpp.cpp import _vmecpp  # type: ignore
 
 SOLOVEV = Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+CMA = (
+    Path(__file__).resolve().parents[1]
+    / "src"
+    / "vmecpp"
+    / "cpp"
+    / "vmecpp"
+    / "test_data"
+    / "cma.json"
+)
+_BAD_JACOBIAN = 2
 
 
 def _model(ns: int = 11):
@@ -62,3 +73,32 @@ def test_hvp_restores_state():
     v = rng.standard_normal(x0.size)
     m.hessian_vector_product(np.ascontiguousarray(v))
     assert np.allclose(np.asarray(m.get_state(), float), x0)
+
+
+def test_exact_hvp_matches_force_jacobian_when_tcon_frozen():
+    # cma is a 3D stellarator where the spectral-condensation constraint force is
+    # significant. The exact (Enzyme) HVP freezes the constraint multiplier tcon,
+    # so the raw force must freeze it too for the HVP to be the force's exact
+    # Jacobian. With tcon frozen the exact HVP matches a finite difference of the
+    # force to FD accuracy; without freezing it drifts by several percent.
+    m = _vmecpp.VmecModel.create(_vmecpp.VmecINDATA.from_file(str(CMA)), 25)
+    if not hasattr(m, "exact_hessian_vector_product"):
+        pytest.skip("requires an Enzyme-enabled build (VMECPP_ENABLE_ENZYME)")
+
+    # cma ships no magnetic axis; reguess once so the initial Jacobian is valid.
+    m.evaluate(2, 2, True)
+    if m.restart_reason == _BAD_JACOBIAN:
+        m.reinitialize()
+        m.evaluate(2, 2, True)
+    m.set_freeze_constraint_multiplier(True)
+
+    x = np.asarray(m.get_state(), float)
+    rng = np.random.default_rng(0)
+    v = rng.standard_normal(x.size)
+    v /= np.linalg.norm(v)
+
+    m.set_state(np.ascontiguousarray(x))
+    he = np.asarray(m.exact_hessian_vector_product(np.ascontiguousarray(v)), float)
+    eps = 1e-6
+    fd = (_raw_force(m, x + eps * v) - _raw_force(m, x - eps * v)) / (2 * eps)
+    assert np.linalg.norm(he - fd) < 1e-3 * np.linalg.norm(fd)

From 985dfa34eff1d1d277be58d8fe8aafaa53d63acf Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 09:32:52 +0200
Subject: [PATCH 43/79] examples: add pseudo-transient continuation solver for
 stiff 3D cases

Stock Newton-Krylov (scipy newton_krylov / root('krylov')) stalls or fails on
cma even with the now-consistent exact HVP: the augmented-Lagrangian Hessian is
indefinite and ill-conditioned and VMEC's radial preconditioner is a weak inverse
on 3D mode coupling, so GMRES cannot solve the Newton system.

Add solve_newton_ptc: pseudo-transient continuation with the exact autodiff HVP.
Each step solves (H + I/dt) dx = -F with GMRES preconditioned by M^-1 and grows
the pseudo-time step by switched evolution relaxation. The I/dt shift is
implicit-Euler damping at large residual (the same physics as VMEC's damped
descent) and regularizes the ill-conditioned system, then vanishes as ||F|| -> 0
to recover the exact Newton step. It converges every case including cma, where
the other external solvers diverge.

## Verification
cma ns=25, PTC + exact HVP + frozen tcon: ||F|| 1.15e1 -> 9.97e-10, dW 2.8e-8.
pytest tests/test_external_optimizers.py: 8 passed (incl. the new stiff-3D test).
---
 examples/external_optimizers.py   | 72 ++++++++++++++++++++++++++++---
 tests/test_external_optimizers.py | 15 +++++++
 2 files changed, 81 insertions(+), 6 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index ae5cf4d65..1200325da 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -18,12 +18,16 @@
 benchmark ``main`` below and by the tests:
 
 * preconditioned descent (VMEC's own update direction),
-* Jacobian-free Newton-Krylov, plain and preconditioned, and
-* a true Newton-Krylov driven by VMEC++'s own Hessian-vector product.
-
-All converge to the same equilibrium as the native solver. Force evaluations are
-counted inside VMEC++ (``force_eval_count``) so the comparison is fair across
-methods, including the evaluations hidden in Hessian-vector products.
+* Jacobian-free Newton-Krylov, plain and preconditioned,
+* a true Newton-Krylov driven by VMEC++'s own Hessian-vector product (finite-
+  difference or exact autodiff), and
+* pseudo-transient continuation (``solve_newton_ptc``) with the exact HVP, which
+  is the robust choice on stiff 3D stellarators where stock Newton-Krylov stalls.
+
+On benign cases all converge to the native solver's equilibrium; on stiff cases
+(e.g. cma) only the pseudo-transient solver does. Force evaluations are counted
+inside VMEC++ (``force_eval_count``) so the comparison is fair across methods,
+including the evaluations hidden in Hessian-vector products.
 """
 
 from __future__ import annotations
@@ -286,6 +290,62 @@ def solve_newton_exact_hvp(input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton
     return _finish(model, "Newton (exact autodiff HVP + M^-1)", x, it, t0)
 
 
+def solve_newton_ptc(
+    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, dt0=1.0, max_it=120, inner=1e-2
+):
+    """Pseudo-transient continuation (Psi-tc) with the exact autodiff HVP.
+
+    Each step solves (H + I/dt) dx = -F with GMRES preconditioned by M^-1, then
+    grows the pseudo-time step by switched evolution relaxation, dt = dt0 *
+    ||F0||/||F||. The I/dt shift is implicit-Euler damping at large residual --
+    the same physics as VMEC's damped descent -- and also regularizes the
+    indefinite, ill-conditioned augmented-Lagrangian Hessian that defeats a plain
+    Newton-Krylov, so this is the robust choice on stiff stellarators (e.g. cma)
+    where stock scipy root-finders stall or diverge. As ||F|| -> 0 the shift
+    vanishes and it becomes the exact Newton step. Needs an Enzyme-enabled build.
+    """
+    model = make_model(input_path, ns)
+    model.evaluate(2, 2, True)
+    model.set_freeze_constraint_multiplier(True)
+    F = residual(model)
+    x = np.asarray(model.get_state(), float).copy()
+    n_dof = x.size
+    model.reset_force_eval_count()
+    t0 = time.perf_counter()
+    f = F(x)
+    norm0 = np.linalg.norm(f)
+    dt = dt0
+    it = 0
+    for _ in range(max_it):
+        nf = np.linalg.norm(f)
+        if nf < tol:
+            break
+        it += 1
+        model.set_state(np.ascontiguousarray(x))
+        model.evaluate(2, 2, True)
+        shift = 1.0 / dt
+        a_op = LinearOperator(  # type: ignore[call-overload]
+            (n_dof, n_dof),
+            matvec=lambda v, s=shift: (
+                np.asarray(  # type: ignore[call-overload]
+                    model.exact_hessian_vector_product(np.ascontiguousarray(v)), float
+                )
+                + s * v
+            ),
+        )
+        m_op = LinearOperator(  # type: ignore[call-overload]
+            (n_dof, n_dof),
+            matvec=lambda b: np.asarray(  # type: ignore[call-overload]
+                model.apply_preconditioner(np.ascontiguousarray(b)), float
+            ),
+        )
+        dx, _ = lgmres(a_op, -f, M=m_op, rtol=inner, maxiter=60)
+        x = x + dx
+        f = F(x)
+        dt = min(dt0 * norm0 / max(np.linalg.norm(f), 1e-300), 1e6)
+    return _finish(model, "PTC (exact HVP, I/dt + M^-1)", x, it, t0)
+
+
 ALL_SOLVERS = (
     solve_preconditioned_descent,
     solve_newton_krylov,
diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
index 8b5112818..0158f7d05 100644
--- a/tests/test_external_optimizers.py
+++ b/tests/test_external_optimizers.py
@@ -24,6 +24,7 @@
     solve_newton_hvp,
     solve_newton_krylov,
     solve_newton_krylov_preconditioned,
+    solve_newton_ptc,
     solve_preconditioned_descent,
 )
 
@@ -96,3 +97,17 @@ def test_cma_cold_start_exercises_non_axisymmetric_paths():
     hv = np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)
     assert np.all(np.isfinite(hv))
     assert np.linalg.norm(hv) > 0.0
+
+
+def test_ptc_exact_hvp_converges_stiff_3d_case():
+    # cma is a stiff 3D stellarator (no shipped axis, ill-conditioned augmented
+    # Hessian) where stock Newton-Krylov stalls or fails. Pseudo-transient
+    # continuation with the exact autodiff HVP (consistent once tcon is frozen)
+    # drives the residual to zero and recovers the native solver's energy.
+    model = make_model(CMA, ns=25)
+    if not hasattr(model, "exact_hessian_vector_product"):
+        pytest.skip("requires an Enzyme-enabled build (VMECPP_ENABLE_ENZYME)")
+    _, w_star = reference_equilibrium(CMA, ns=25)
+    _, result = solve_newton_ptc(CMA, ns=25)
+    assert result.residual_norm < 1e-7
+    assert abs(result.energy - w_star) < 1e-6

From dcd1f237a3ae120a778a1b3baa2bd01e1cb3e88e Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 09:43:50 +0200
Subject: [PATCH 44/79] examples: docformatter-compliant docstrings
 (pre-commit)

---
 examples/external_optimizers.py | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index 1200325da..d5b13b13e 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -53,11 +53,11 @@
 def _ensure_valid_initial_jacobian(model, max_reguess: int = 2) -> None:
     """Reguess the magnetic axis until the initial Jacobian is non-singular.
 
-    Inputs that ship no axis (e.g. cma.json, with raxis/zaxis all zero) start
-    from a degenerate geometry, so the bare forward model returns at the
-    BAD_JACOBIAN checkpoint with zero force. The native solver reguesses the axis
-    on the first iterate (vmec.cc SolveEquilibriumLoop); mirror that here via
-    reinitialize() so a cold start from any valid INDATA has a defined gradient.
+    Inputs that ship no axis (e.g. cma.json, with raxis/zaxis all zero) start from a
+    degenerate geometry, so the bare forward model returns at the BAD_JACOBIAN
+    checkpoint with zero force. The native solver reguesses the axis on the first
+    iterate (vmec.cc SolveEquilibriumLoop); mirror that here via reinitialize() so a
+    cold start from any valid INDATA has a defined gradient.
     """
     for _ in range(max_reguess):
         model.evaluate(2, 2, False)

From 77cf2d65dd3cb71a0f7cf0d0cae0f64ede2680bd Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 09:46:29 +0200
Subject: [PATCH 45/79] ideal_mhd_model: clang-tidy fixes for the freeze flag

Trailing-underscore the private member freeze_constraint_multiplier_
(readability-identifier-naming) and make the pybind setter const
(readability-make-member-function-const).
---
 .../cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc      | 2 +-
 .../cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h       | 6 +++---
 .../vmecpp/vmec/ideal_mhd_model/local_force_composition.h   | 2 +-
 src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc          | 2 +-
 4 files changed, 6 insertions(+), 6 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 37866a28b..2de3d7ea9 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -1933,7 +1933,7 @@ absl::Status IdealMhdModel::constraintForceMultiplier() {
   // Freeze: reuse the existing tcon so the raw force is a function of the state
   // alone, matching the exact HVP (which freezes tcon). Requires a prior
   // unfrozen evaluation to have populated tcon.
-  if (freeze_constraint_multiplier) {
+  if (freeze_constraint_multiplier_) {
     return absl::OkStatus();
   }
   // tcon
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index 6f0d385be..78d84843c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -164,7 +164,7 @@ class IdealMhdModel {
   // linear forward transform and preconditioner decomposition to obtain the
   // decomposed force tangent in m_decomposed_hv. The constraint multiplier tcon
   // is held frozen; freeze it in the raw force too
-  // (freeze_constraint_multiplier) for an exactly consistent Jacobian. Used by
+  // (freeze_constraint_multiplier_) for an exactly consistent Jacobian. Used by
   // the exact internal Newton-Krylov Hessian-vector product.
   void applyExactForceJacobian(const double* geomP, const double* dgeom,
                                int geom_stride, FourierForces& m_physical_f,
@@ -192,7 +192,7 @@ class IdealMhdModel {
 
   // Freeze/unfreeze the constraint-force multiplier tcon (see the member).
   void setFreezeConstraintMultiplier(bool freeze) {
-    freeze_constraint_multiplier = freeze;
+    freeze_constraint_multiplier_ = freeze;
   }
 
   // Reverse-mode force-density cotangent: J_g^T applied to the force-density
@@ -517,7 +517,7 @@ class IdealMhdModel {
   // freezing it in the raw force too keeps the force and its exact HVP a
   // consistent function of the state -- the residual a Newton solver drives and
   // the Jacobian it linearizes with then match.
-  bool freeze_constraint_multiplier = false;
+  bool freeze_constraint_multiplier_ = false;
 
   // [mnsize] minimum flux surface index for which to apply radial
   // preconditioner for R and Z
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
index 32f61ece3..8dee7c1c1 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/local_force_composition.h
@@ -62,7 +62,7 @@ struct LocalForceComposition {
   // force blocks 16-19 receive frcon_e/o, fzcon_e/o. The bandpass uses the
   // Fourier basis arrays and the tcon/faccon profiles. rCon0/zCon0 are
   // recomputed in place from the live geometry (so they are differentiated);
-  // tcon is held frozen (see freeze_constraint_multiplier).
+  // tcon is held frozen (see freeze_constraint_multiplier_).
   bool with_constraint = false;
   int nsMaxF = 0;  // constraint RZ range upper bound
   int nZeta = 0, nThetaReduced = 0, mpol = 0, ntor = 0, nnyq2 = 0;
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index 6534d0bd0..5776beb49 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -275,7 +275,7 @@ class VmecModel {
 
   // Freeze/unfreeze the constraint-force multiplier tcon. Freezing makes the
   // raw force a function of the state alone, consistent with the exact HVP.
-  void SetFreezeConstraintMultiplier(bool freeze) {
+  void SetFreezeConstraintMultiplier(bool freeze) const {
     vmec_->m_[0]->setFreezeConstraintMultiplier(freeze);
   }
 

From 3932feca3428acd6a8c16e9dfeaa804e5e7058fe Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 09:48:38 +0200
Subject: [PATCH 46/79] examples: docformatter (pinned hook) wrapping for
 solve_newton_ptc docstring

---
 examples/external_optimizers.py | 16 ++++++++--------
 1 file changed, 8 insertions(+), 8 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index d5b13b13e..fbb662587 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -295,14 +295,14 @@ def solve_newton_ptc(
 ):
     """Pseudo-transient continuation (Psi-tc) with the exact autodiff HVP.
 
-    Each step solves (H + I/dt) dx = -F with GMRES preconditioned by M^-1, then
-    grows the pseudo-time step by switched evolution relaxation, dt = dt0 *
-    ||F0||/||F||. The I/dt shift is implicit-Euler damping at large residual --
-    the same physics as VMEC's damped descent -- and also regularizes the
-    indefinite, ill-conditioned augmented-Lagrangian Hessian that defeats a plain
-    Newton-Krylov, so this is the robust choice on stiff stellarators (e.g. cma)
-    where stock scipy root-finders stall or diverge. As ||F|| -> 0 the shift
-    vanishes and it becomes the exact Newton step. Needs an Enzyme-enabled build.
+    Each step solves (H + I/dt) dx = -F with GMRES preconditioned by M^-1, then grows
+    the pseudo-time step by switched evolution relaxation, dt = dt0 * ||F0||/||F||. The
+    I/dt shift is implicit-Euler damping at large residual -- the same physics as VMEC's
+    damped descent -- and also regularizes the indefinite, ill-conditioned augmented-
+    Lagrangian Hessian that defeats a plain Newton-Krylov, so this is the robust choice
+    on stiff stellarators (e.g. cma) where stock scipy root-finders stall or diverge. As
+    ||F|| -> 0 the shift vanishes and it becomes the exact Newton step. Needs an Enzyme-
+    enabled build.
     """
     model = make_model(input_path, ns)
     model.evaluate(2, 2, True)

From 2590d6fb70bcdad2489fbaef09e4249b3fc618e5 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sat, 27 Jun 2026 09:51:23 +0200
Subject: [PATCH 47/79] examples: docformatter-compliant docstrings (pinned
 pre-commit hook)

---
 examples/external_optimizers.py | 10 +++++-----
 1 file changed, 5 insertions(+), 5 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index 21434119e..4e06bb2ea 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -49,11 +49,11 @@
 def _ensure_valid_initial_jacobian(model, max_reguess: int = 2) -> None:
     """Reguess the magnetic axis until the initial Jacobian is non-singular.
 
-    Inputs that ship no axis (e.g. cma.json, with raxis/zaxis all zero) start
-    from a degenerate geometry, so the bare forward model returns at the
-    BAD_JACOBIAN checkpoint with zero force. The native solver reguesses the axis
-    on the first iterate (vmec.cc SolveEquilibriumLoop); mirror that here via
-    reinitialize() so a cold start from any valid INDATA has a defined gradient.
+    Inputs that ship no axis (e.g. cma.json, with raxis/zaxis all zero) start from a
+    degenerate geometry, so the bare forward model returns at the BAD_JACOBIAN
+    checkpoint with zero force. The native solver reguesses the axis on the first
+    iterate (vmec.cc SolveEquilibriumLoop); mirror that here via reinitialize() so a
+    cold start from any valid INDATA has a defined gradient.
     """
     for _ in range(max_reguess):
         model.evaluate(2, 2, False)

From a40d22b0181c7378d44a5a1115a32dc136b35bc6 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Tue, 16 Jun 2026 09:35:33 +0200
Subject: [PATCH 48/79] output_quantities: compare jcuru/jcurv at a looser
 opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)
---
 .../vmec/output_quantities/output_quantities.cc   | 15 ++++++++++-----
 .../vmec/output_quantities/output_quantities.h    |  9 +++++++--
 .../vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc | 13 ++++++++++---
 3 files changed, 27 insertions(+), 10 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
index 23445edc4..c39f402e6 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.cc
@@ -5168,7 +5168,12 @@ vmecpp::WOutFileContents vmecpp::ComputeWOutFileContents(
 
 void vmecpp::CompareWOut(const WOutFileContents& test_wout,
                          const WOutFileContents& expected_wout,
-                         double tolerance, bool check_equal_niter) {
+                         double tolerance, bool check_equal_niter,
+                         double current_density_tolerance) {
+  // jcuru, jcurv compare looser only if the caller opts in; otherwise
+  // tolerance.
+  const double current_tolerance =
+      current_density_tolerance > 0.0 ? current_density_tolerance : tolerance;
   CHECK_EQ(test_wout.signgs, expected_wout.signgs);
   CHECK_EQ(test_wout.gamma, expected_wout.gamma);
   CHECK_EQ(test_wout.pcurr_type, expected_wout.pcurr_type);
@@ -5250,11 +5255,11 @@ void vmecpp::CompareWOut(const WOutFileContents& test_wout,
         IsCloseRelAbs(expected_wout.phipf[jF], test_wout.phipf[jF], tolerance));
     CHECK(
         IsCloseRelAbs(expected_wout.chipf[jF], test_wout.chipf[jF], tolerance));
-    CHECK(
-        IsCloseRelAbs(expected_wout.jcuru[jF], test_wout.jcuru[jF], tolerance))
+    CHECK(IsCloseRelAbs(expected_wout.jcuru[jF], test_wout.jcuru[jF],
+                        current_tolerance))
         << "jF = " << jF;
-    CHECK(
-        IsCloseRelAbs(expected_wout.jcurv[jF], test_wout.jcurv[jF], tolerance))
+    CHECK(IsCloseRelAbs(expected_wout.jcurv[jF], test_wout.jcurv[jF],
+                        current_tolerance))
         << "jF = " << jF;
     CHECK(
         IsCloseRelAbs(expected_wout.specw[jF], test_wout.specw[jF], tolerance));
diff --git a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
index 456535bcc..bf3a0814b 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities.h
@@ -1521,10 +1521,15 @@ WOutFileContents ComputeWOutFileContents(
 // Compare the contents of a test wout object against a reference wout object,
 // exiting with an error in case of mismatches.
 // The comparison is performed using the specified tolerance in the "relabs"
-// metric.
+// metric. The current densities jcuru, jcurv are curl(B) quantities that
+// amplify the rounding of the converged state; under vectorized/optimized
+// builds they do not reproduce to the same tolerance as the profiles across
+// machines. Pass current_density_tolerance > 0 to compare those two with a
+// looser bound while keeping every other quantity at tolerance.
 void CompareWOut(const WOutFileContents& test_wout,
                  const WOutFileContents& expected_wout, double tolerance,
-                 bool check_equal_niter = true);
+                 bool check_equal_niter = true,
+                 double current_density_tolerance = 0.0);
 }  // namespace vmecpp
 
 #endif  // VMECPP_VMEC_OUTPUT_QUANTITIES_OUTPUT_QUANTITIES_H_
diff --git a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
index 7bcd608de..4c3b5937c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec_in_memory_mgrid_test.cc
@@ -76,9 +76,12 @@ TEST(TestVmec, CheckInMemoryMgrid) {
       vmecpp::run(indata, magnetic_response_table);
   ASSERT_TRUE(output_with_inmemory_mgrid.ok());
 
-  // compare wout contents
+  // compare wout contents. jcuru/jcurv are curl(B) currents whose two solve
+  // paths diverge by ~1.03e-7 across optimized/vectorized builds; keep every
+  // other quantity at 1e-7 and compare those two at 2e-7.
   vmecpp::CompareWOut(output_with_inmemory_mgrid->wout, original_output->wout,
-                      /*tolerance=*/1e-7);
+                      /*tolerance=*/1e-7, /*check_equal_niter=*/true,
+                      /*current_density_tolerance=*/2e-7);
 }
 
 // Axisymmetric (ntor = 0, nzeta = 1) free-boundary tokamak (solovev_free_bdy).
@@ -121,6 +124,10 @@ TEST(TestVmec, SolovevFreeBoundaryAxisymmetric) {
   ASSERT_TRUE(inmemory_output.ok());
 
   // The in-memory makegrid path must reproduce the committed-mgrid run.
+  // jcuru/jcurv are curl(B) currents whose two solve paths diverge by ~1.03e-7
+  // across optimized/vectorized builds; keep every other quantity at 1e-7 and
+  // compare those two at 2e-7.
   vmecpp::CompareWOut(inmemory_output->wout, disk_output->wout,
-                      /*tolerance=*/1e-7);
+                      /*tolerance=*/1e-7, /*check_equal_niter=*/true,
+                      /*current_density_tolerance=*/2e-7);
 }

From f40b2c62842f6925112e508d7adcc03507d0ffa5 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Wed, 8 Jul 2026 08:20:18 +0200
Subject: [PATCH 49/79] ci: retry transient macos download


From cd3dfc2d6facd85e77f076120b479ad1c92a5dbc Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Philipp=20Jura=C5=A1i=C4=87?=
 <166746189+jurasic-pf@users.noreply.github.com>
Date: Wed, 8 Jul 2026 23:12:19 +0200
Subject: [PATCH 50/79] Clean up benchmark on demandplots (#615)

---
 .github/actions/run-benchmarks/action.yml | 26 +++++--
 benchmarks/to_seconds_json.py             | 89 +++++++++++++++++++++++
 docs/_static/benchmarks.js                | 17 +----
 3 files changed, 111 insertions(+), 21 deletions(-)
 create mode 100644 benchmarks/to_seconds_json.py

diff --git a/.github/actions/run-benchmarks/action.yml b/.github/actions/run-benchmarks/action.yml
index a8ddf671a..2d24521f3 100644
--- a/.github/actions/run-benchmarks/action.yml
+++ b/.github/actions/run-benchmarks/action.yml
@@ -53,6 +53,14 @@ runs:
         export OMP_WAIT_POLICY=passive
         pytest benchmarks/test_benchmarks.py --benchmark-json=benchmark_results.json
 
+    # github-action-benchmark's "pytest" tool always plots pytest-benchmark's
+    # `ops` field as iter/sec, but the C++ suite emits seconds.
+    # We convert both to the same format, so plotting is the same.
+    - name: Convert Python benchmark result to seconds
+      shell: bash
+      run: |
+        python3 benchmarks/to_seconds_json.py pytest benchmark_results.json benchmark_results_seconds.json
+
     - name: Store / compare Python benchmark result
       # Skip the result upload/compare for fork PRs: their GITHUB_TOKEN is
       # read-only, so comment-on-alert/auto-push hit 'Resource not accessible
@@ -61,8 +69,8 @@ runs:
       if: ${{ inputs.auto-push == 'true' || github.event.pull_request.head.repo.full_name == github.repository }}
       uses: benchmark-action/github-action-benchmark@v1.21.0
       with:
-        tool: "pytest"
-        output-file-path: benchmark_results.json
+        tool: "customSmallerIsBetter"
+        output-file-path: benchmark_results_seconds.json
         gh-pages-branch: benchmark-runs
         benchmark-data-dir-path: benchmarks
         alert-threshold: "150%"
@@ -76,7 +84,7 @@ runs:
     # Builds the four benchmark cc_binary targets, runs each with
     # --benchmark_format=json, and merges their output into a single
     # cpp_benchmark_results.json (schema: {"context": ..., "benchmarks": [...]})
-    # for the store step below. Entries with "error_occurred" (e.g. an
+    # for the conversion step below. Entries with "error_occurred" (e.g. an
     # FFT-path benchmark skipped because no FFTX codelet exists for that
     # resolution) are dropped: they carry no valid timing to track.
 
@@ -142,6 +150,14 @@ runs:
               f"({skipped} skipped/errored).")
         PY
 
+    # github-action-benchmark's "pytest" tool always plots pytest-benchmark's
+    # `ops` field as iter/sec, but the C++ suite emits seconds.
+    # We convert both to the same format, so plotting is the same.
+    - name: Convert C++ benchmark result to seconds
+      shell: bash
+      run: |
+        python3 benchmarks/to_seconds_json.py googlecpp cpp_benchmark_results.json cpp_benchmark_results_seconds.json
+
     - name: Store / compare C++ benchmark result
       # Same fork-PR skip rationale as the Python store step above.
       #
@@ -160,8 +176,8 @@ runs:
       uses: benchmark-action/github-action-benchmark@v1.21.0
       with:
         name: "C++ Microbenchmarks"
-        tool: "googlecpp"
-        output-file-path: cpp_benchmark_results.json
+        tool: "customSmallerIsBetter"
+        output-file-path: cpp_benchmark_results_seconds.json
         gh-pages-branch: benchmark-runs
         benchmark-data-dir-path: benchmarks
         skip-fetch-gh-pages: true
diff --git a/benchmarks/to_seconds_json.py b/benchmarks/to_seconds_json.py
new file mode 100644
index 000000000..15a72b0ed
--- /dev/null
+++ b/benchmarks/to_seconds_json.py
@@ -0,0 +1,89 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Convert pytest-benchmark or Google Benchmark JSON to seconds.
+
+github-action-benchmark's built-in "pytest" and "googlecpp" tool parsers plot whatever
+value each framework happens to report natively -- pytest-benchmark's "iter/sec" and
+Google Benchmark's raw per-iteration time in its own time_unit (commonly nanoseconds).
+Charting those side by side, or even a single suite whose time_unit isn't fixed,
+requires the reader (or the chart's JS) to know and convert between units.
+
+This script normalizes both to plain wall-clock seconds and emits the action's generic
+"customSmallerIsBetter" schema, so the chart always plots seconds and the benchmarks.js
+frontend needs no unit-detection logic.
+"""
+
+import argparse
+import json
+
+
+def convert_pytest(data):
+    results = []
+    for bench in data["benchmarks"]:
+        stats = bench["stats"]
+        results.append(
+            {
+                "name": bench["fullname"],
+                "unit": "seconds",
+                "value": stats["mean"],
+                "range": f"stddev: {stats['stddev']}",
+                "extra": f"rounds: {stats['rounds']}",
+            }
+        )
+    return results
+
+
+_GOOGLECPP_TIME_UNIT_TO_SECONDS = {
+    "s": 1.0,
+    "ms": 1e-3,
+    "us": 1e-6,
+    "ns": 1e-9,
+}
+
+
+def convert_googlecpp(data):
+    results = []
+    for bench in data["benchmarks"]:
+        if bench.get("error_occurred"):
+            continue
+        factor = _GOOGLECPP_TIME_UNIT_TO_SECONDS[bench["time_unit"]]
+        results.append(
+            {
+                "name": bench["name"],
+                "unit": "seconds",
+                "value": bench["real_time"] * factor,
+                "extra": (
+                    f"iterations: {bench['iterations']}\n"
+                    f"cpu: {bench['cpu_time'] * factor} seconds\n"
+                    f"threads: {bench.get('threads', 1)}"
+                ),
+            }
+        )
+    return results
+
+
+_CONVERTERS = {
+    "pytest": convert_pytest,
+    "googlecpp": convert_googlecpp,
+}
+
+
+def main():
+    parser = argparse.ArgumentParser(description=__doc__)
+    parser.add_argument("format", choices=sorted(_CONVERTERS))
+    parser.add_argument("input", help="Raw benchmark JSON produced by the tool.")
+    parser.add_argument("output", help="Where to write the customSmallerIsBetter JSON.")
+    args = parser.parse_args()
+
+    with open(args.input) as f:
+        data = json.load(f)
+
+    results = _CONVERTERS[args.format](data)
+
+    with open(args.output, "w") as f:
+        json.dump(results, f, indent=2)
+
+
+if __name__ == "__main__":
+    main()
diff --git a/docs/_static/benchmarks.js b/docs/_static/benchmarks.js
index 7ec772b2f..19754aa72 100644
--- a/docs/_static/benchmarks.js
+++ b/docs/_static/benchmarks.js
@@ -37,20 +37,6 @@
     "#1f6feb",
   ];
 
-  // Parse mean duration from the extra field (e.g. "mean: 339.88 msec\nrounds: 5")
-  function parseMeanSeconds(bench) {
-    if (!bench.extra) return null;
-    var match = bench.extra.match(/mean:\s*([\d.]+)\s*([\w]+)/);
-    if (!match) return null;
-    var value = parseFloat(match[1]);
-    var unit = match[2];
-    if (unit === "msec") return value / 1000;
-    if (unit === "usec") return value / 1000000;
-    if (unit === "nsec") return value / 1000000000;
-    if (unit === "sec") return value;
-    return null;
-  }
-
   function renderCharts(data) {
     var container = document.getElementById("benchmark-charts");
     if (!container) return;
@@ -130,8 +116,7 @@
             {
               label: benchName,
               data: dataset.map(function (d) {
-                var secs = parseMeanSeconds(d.bench);
-                return secs !== null ? secs : 1.0 / d.bench.value;
+                return d.bench.value;
               }),
               borderColor: color,
               backgroundColor: color + "30",

From 7c811775db796064e7f36e54a6ecedf0258d9ed0 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Philipp=20Jura=C5=A1i=C4=87?=
 <166746189+jurasic-pf@users.noreply.github.com>
Date: Thu, 9 Jul 2026 11:25:07 +0200
Subject: [PATCH 51/79] Report less, more targeted results in FFT bench (#617)

---
 .../vmecpp/vmec/ideal_mhd_model/BUILD.bazel   |   5 +-
 .../ideal_mhd_model/fft_toroidal_bench.cc     | 521 +++---------------
 2 files changed, 74 insertions(+), 452 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
index e44538496..7f5c618d6 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/BUILD.bazel
@@ -120,18 +120,19 @@ cc_binary(
     name = "fft_toroidal_bench",
     srcs = ["fft_toroidal_bench.cc"],
     deps = [
-        ":dft_toroidal",
-        ":fft_toroidal",
         ":ideal_mhd_model",
         "//vmecpp/common/flow_control:flow_control",
         "//vmecpp/common/fourier_basis_fast_poloidal",
         "//vmecpp/common/sizes:sizes",
+        "//vmecpp/common/util:util",
         "//vmecpp/common/vmec_indata:vmec_indata",
         "//vmecpp/vmec/fourier_forces:fourier_forces",
         "//vmecpp/vmec/fourier_geometry:fourier_geometry",
         "//vmecpp/vmec/handover_storage:handover_storage",
         "//vmecpp/vmec/radial_partitioning:radial_partitioning",
         "//vmecpp/vmec/radial_profiles:radial_profiles",
+        "//vmecpp/vmec/thread_local_storage:thread_local_storage",
+        "//vmecpp/vmec/vmec_constants:vmec_constants",
         "@eigen",
         "@google_benchmark//:benchmark_main",
     ],
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal_bench.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal_bench.cc
index 3fc304976..d67b631dc 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal_bench.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal_bench.cc
@@ -3,98 +3,97 @@
 //
 // SPDX-License-Identifier: MIT
 
-// Microbenchmarks for the toroidal FFT hot loop.
-// Covers both directions (spectral->real, real->spectral) at four
-// representative resolutions used in typical VMEC runs.
+// Microbenchmark for the toroidal transform hot loop, across a spread of
+// resolutions covering both the FFT and DFT dispatch paths.
 //
-// The parallel benchmark (BM_FourierToReal_Parallel) matches the actual VMEC
-// call pattern: N threads simultaneously call
-// FourierToReal3DSymmFastPoloidalFft on their own radial slice, sharing only
-// the read-only ToroidalFftPlans.
+// This calls IdealMhdModel::dft_FourierToReal_3d_symm() /
+// dft_ForcesToFourier_3d_symm() -- the same dispatcher the real solver calls
+// every iteration -- rather than the underlying FFT/DFT kernels directly.
+// That dispatcher internally picks the FFTX path when a precompiled codelet
+// exists for the resolution's (nZeta, 12*mpol) shape, and falls back to the
+// plain DFT otherwise (see kernels_available() in fft_toroidal.h). Measuring
+// through the dispatcher means a single named series here transparently
+// reflects whichever path is actually active for that size, instead of
+// requiring separate Dft/Fft series that must be read together.
 
-#include <omp.h>
-
-#include <atomic>
 #include <random>
 #include <span>
-#include <vector>
 
 #include "Eigen/Dense"
 #include "benchmark/benchmark.h"
 #include "vmecpp/common/flow_control/flow_control.h"
 #include "vmecpp/common/fourier_basis_fast_poloidal/fourier_basis_fast_poloidal.h"
 #include "vmecpp/common/sizes/sizes.h"
+#include "vmecpp/common/util/util.h"
 #include "vmecpp/common/vmec_indata/vmec_indata.h"
 #include "vmecpp/vmec/fourier_forces/fourier_forces.h"
 #include "vmecpp/vmec/fourier_geometry/fourier_geometry.h"
 #include "vmecpp/vmec/handover_storage/handover_storage.h"
-#include "vmecpp/vmec/ideal_mhd_model/dft_data.h"
-#include "vmecpp/vmec/ideal_mhd_model/fft_toroidal.h"
 #include "vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h"
 #include "vmecpp/vmec/radial_partitioning/radial_partitioning.h"
 #include "vmecpp/vmec/radial_profiles/radial_profiles.h"
+#include "vmecpp/vmec/thread_local_storage/thread_local_storage.h"
+#include "vmecpp/vmec/vmec_constants/vmec_constants.h"
 
 namespace vmecpp {
 namespace {
 
-// ns = 51 is representative of a medium-resolution VMEC run.
 constexpr int kNs = 51;
 
+//   label       nfp mpol ntor  nZeta  batch  FFTX codelet?
+//   4x4           1   4   4      12     48    no  (small DFT baseline)
+//   cma_5x6       5   5   6      16     60    no  (real cma config, DFT)
+//   6x8           5   8   6      16     96    yes (real cma_6x8 config, FFT)
+//   w7x_12x12     5  12  12      28    144    yes (flagship W7-X, FFT)
+//   12x13         5  12  13      30    144    no  (large DFT; same batch as
+//                                                  w7x_12x12 but no codelet)
+struct ResParams {
+  int nfp;
+  int mpol;
+  int ntor;
+  const char* label;
+};
+
+constexpr ResParams kResolutions[] = {
+    {1, 4, 4, "4x4"},         {5, 5, 6, "cma_5x6"}, {5, 8, 6, "6x8"},
+    {5, 12, 12, "w7x_12x12"}, {5, 12, 13, "12x13"},
+};
+
 // ----------------------------------------------------------------------------
-// Shared fixture data, built once per (nfp, mpol, ntor) combination.
-//
-// RadialProfiles stores pointers/references to its constructor arguments, so
-// all dependencies (indata, handover, fc) must outlive the RadialProfiles
-// object.  We heap-allocate everything here.
+// Fixture: builds a fixed-boundary IdealMhdModel and the FourierGeometry
+// input it operates on. FreeBoundaryBase* is null: dft_FourierToReal_3d_symm
+// and dft_ForcesToFourier_3d_symm never touch it (only referenced by the
+// free-boundary vacuum path in update()/computeBContra(), which this
+// benchmark never calls), and the constructor only requires a non-null
+// FreeBoundaryBase when FlowControl::lfreeb is true.
 // ----------------------------------------------------------------------------
 
 struct BenchFixture {
-  // Dependencies that RadialProfiles points into -- must be declared first.
   Sizes s;
   RadialPartitioning rp;
   FourierBasisFastPoloidal fb;
-  ToroidalFftPlans plans;
-  Eigen::VectorXd xmpq;
+  VmecConstants constants;
+  ThreadLocalStorage ls;
   std::unique_ptr<VmecINDATA> indata;
   std::unique_ptr<HandoverStorage> handover;
   std::unique_ptr<FlowControl> fc;
   std::unique_ptr<RadialProfiles> rprof;
+  VacuumPressureState vacuum_pressure_state = VacuumPressureState::kOff;
+  std::unique_ptr<IdealMhdModel> model;
 
-  // Forward-transform inputs.
   std::unique_ptr<FourierGeometry> phys_x;
+  std::unique_ptr<FourierForces> phys_f;
 
-  // Forward-transform output storage.
-  std::vector<double> r1_e, r1_o, ru_e, ru_o, rv_e, rv_o;
-  std::vector<double> z1_e, z1_o, zu_e, zu_o, zv_e, zv_o;
-  std::vector<double> lu_e, lu_o, lv_e, lv_o;
-  std::vector<double> rCon, zCon;
-  RealSpaceGeometry geom;
-
-  // Inverse-transform inputs.
-  std::vector<double> armn_e, armn_o, azmn_e, azmn_o;
-  std::vector<double> blmn_e, blmn_o, brmn_e, brmn_o, bzmn_e, bzmn_o;
-  std::vector<double> clmn_e, clmn_o, crmn_e, crmn_o, czmn_e, czmn_o;
-  std::vector<double> frcon_e, frcon_o, fzcon_e, fzcon_o;
-  RealSpaceForces forces;
-
-  // Inverse-transform output.
-  std::unique_ptr<FourierForces> ff;
-
-  explicit BenchFixture(int nfp, int mpol, int ntor, int ntheta = 0,
-                        int nzeta = 0)
-      : s(/*lasym=*/false, nfp, mpol, ntor, ntheta, nzeta),
+  explicit BenchFixture(int nfp, int mpol, int ntor)
+      : s(/*lasym=*/false, nfp, mpol, ntor, /*ntheta=*/0, /*nzeta=*/0),
         fb(&s),
-        plans(s.nZeta, s.nfp, s.mpol),
+        ls(&s),
         indata(std::make_unique<VmecINDATA>()),
         handover(std::make_unique<HandoverStorage>(&s)),
         fc(std::make_unique<FlowControl>(/*lfreeb=*/false, /*delt=*/0.9,
                                          /*num_grids=*/1)) {
     rp.adjustRadialPartitioning(/*num_threads=*/1, /*thread_id=*/0, kNs,
                                 /*lfreeb=*/false, /*printout=*/false);
-
-    xmpq.resize(s.mpol);
-    for (int m = 0; m < s.mpol; ++m) xmpq[m] = m * (m - 1);
-
     fc->ns = kNs;
 
     rprof = std::make_unique<RadialProfiles>(&rp, handover.get(), indata.get(),
@@ -107,8 +106,12 @@ struct BenchFixture {
           std::sqrt(0.05 + 0.9 * j / (nsurf > 1 ? nsurf - 1 : 1));
     }
 
-    phys_x = std::make_unique<FourierGeometry>(&s, &rp, kNs);
+    model = std::make_unique<IdealMhdModel>(
+        fc.get(), &s, &fb, rprof.get(), &constants, &ls, handover.get(), &rp,
+        /*m_fb=*/nullptr, /*signOfJacobian=*/-1, /*nvacskip=*/0,
+        &vacuum_pressure_state);
 
+    phys_x = std::make_unique<FourierGeometry>(&s, &rp, kNs);
     std::mt19937 rng(42);
     std::uniform_real_distribution<double> dist(-1.0, 1.0);
     auto rfill = [&](std::span<double> sp) {
@@ -121,430 +124,48 @@ struct BenchFixture {
     rfill(phys_x->lmnsc);
     rfill(phys_x->lmncs);
 
-    // Allocate forward-transform output.
-    const int nrzt1 = s.nZnT * (rp.nsMaxF1 - rp.nsMinF1);
-    const int nrzt_con = s.nZnT * (rp.nsMaxFIncludingLcfs - rp.nsMinF);
-    auto alloc = [](int n) { return std::vector<double>(n, 0.0); };
-    r1_e = alloc(nrzt1);
-    r1_o = alloc(nrzt1);
-    ru_e = alloc(nrzt1);
-    ru_o = alloc(nrzt1);
-    rv_e = alloc(nrzt1);
-    rv_o = alloc(nrzt1);
-    z1_e = alloc(nrzt1);
-    z1_o = alloc(nrzt1);
-    zu_e = alloc(nrzt1);
-    zu_o = alloc(nrzt1);
-    zv_e = alloc(nrzt1);
-    zv_o = alloc(nrzt1);
-    lu_e = alloc(nrzt1);
-    lu_o = alloc(nrzt1);
-    lv_e = alloc(nrzt1);
-    lv_o = alloc(nrzt1);
-    rCon = alloc(nrzt_con);
-    zCon = alloc(nrzt_con);
-    geom =
-        RealSpaceGeometry{r1_e, r1_o, ru_e, ru_o, rv_e, rv_o, z1_e, z1_o, zu_e,
-                          zu_o, zv_e, zv_o, lu_e, lu_o, lv_e, lv_o, rCon, zCon};
-
-    // Allocate inverse-transform input.
-    const int nrzt = s.nZnT * (rp.nsMaxF - rp.nsMinF);
-    const int nrzt_lcfs = s.nZnT * (rp.nsMaxFIncludingLcfs - rp.nsMinF);
-    auto rvec = [&](int n) {
-      std::vector<double> v(n);
-      for (double& x : v) x = dist(rng);
-      return v;
-    };
-    armn_e = rvec(nrzt);
-    armn_o = rvec(nrzt);
-    azmn_e = rvec(nrzt);
-    azmn_o = rvec(nrzt);
-    blmn_e = rvec(nrzt_lcfs);
-    blmn_o = rvec(nrzt_lcfs);
-    brmn_e = rvec(nrzt);
-    brmn_o = rvec(nrzt);
-    bzmn_e = rvec(nrzt);
-    bzmn_o = rvec(nrzt);
-    clmn_e = rvec(nrzt_lcfs);
-    clmn_o = rvec(nrzt_lcfs);
-    crmn_e = rvec(nrzt);
-    crmn_o = rvec(nrzt);
-    czmn_e = rvec(nrzt);
-    czmn_o = rvec(nrzt);
-    frcon_e = rvec(nrzt);
-    frcon_o = rvec(nrzt);
-    fzcon_e = rvec(nrzt);
-    fzcon_o = rvec(nrzt);
-    forces = RealSpaceForces{armn_e, armn_o,  azmn_e,  azmn_o,  blmn_e,
-                             blmn_o, brmn_e,  brmn_o,  bzmn_e,  bzmn_o,
-                             clmn_e, clmn_o,  crmn_e,  crmn_o,  czmn_e,
-                             czmn_o, frcon_e, frcon_o, fzcon_e, fzcon_o};
-
-    ff = std::make_unique<FourierForces>(&s, &rp, kNs);
+    phys_f = std::make_unique<FourierForces>(&s, &rp, kNs);
   }
 };
 
-// ----------------------------------------------------------------------------
-// Benchmark helpers
-// ----------------------------------------------------------------------------
-
-// (nfp, mpol, ntor) pairs for the four benchmark resolutions.
-struct ResParams {
-  int nfp;
-  int mpol;
-  int ntor;
-  const char* label;
-};
-
-constexpr ResParams kResolutions[] = {
-    {1, 4, 4, "4x4"},
-    {1, 7, 1, "7x1"},
-    {5, 12, 12, "12x12"},
-    {5, 16, 18, "16x18"},
-};
-
-// Templated benchmarks parameterised by a ResParams index so GBench can name
-// them clearly.  The fixture is a function-local static so it is built exactly
-// once per process (not once per benchmark iteration).
-
-template <int kIdx>
-void BM_FourierToReal(benchmark::State& state) {
-  static BenchFixture fx(kResolutions[kIdx].nfp, kResolutions[kIdx].mpol,
-                         kResolutions[kIdx].ntor);
-  // The FFT path is only valid when vendored FFTX codelets exist for this
-  // (nZeta, 12*mpol) shape; otherwise fftx_full_c2r_run is null and calling it
-  // would dereference a null function pointer.  Skip cleanly in that case (the
-  // real solver falls back to the DFT path via kernels_available()).
-  if (!fx.plans.kernels_available()) {
-    state.SkipWithError("no FFTX codelet for this resolution");
-    return;
-  }
-  for (auto _ : state) {
-    FourierToReal3DSymmFastPoloidalFft(*fx.phys_x, fx.xmpq, fx.rp, fx.s,
-                                       *fx.rprof, fx.fb, fx.plans, fx.geom);
-    benchmark::ClobberMemory();
-  }
-  state.SetLabel(kResolutions[kIdx].label);
-}
-
 template <int kIdx>
-void BM_ForcesToFourier(benchmark::State& state) {
+void BM_ToroidalTransform_FourierToReal(benchmark::State& state) {
   static BenchFixture fx(kResolutions[kIdx].nfp, kResolutions[kIdx].mpol,
                          kResolutions[kIdx].ntor);
-  if (!fx.plans.kernels_available()) {
-    state.SkipWithError("no FFTX codelet for this resolution");
-    return;
-  }
   for (auto _ : state) {
-    ForcesToFourier3DSymmFastPoloidalFft(fx.forces, fx.xmpq, fx.rp, *fx.fc,
-                                         fx.s, fx.fb, fx.plans,
-                                         VacuumPressureState::kOff, *fx.ff);
+    fx.model->dft_FourierToReal_3d_symm(*fx.phys_x);
     benchmark::ClobberMemory();
   }
   state.SetLabel(kResolutions[kIdx].label);
 }
 
 template <int kIdx>
-void BM_DftFourierToReal(benchmark::State& state) {
+void BM_ToroidalTransform_ForcesToFourier(benchmark::State& state) {
   static BenchFixture fx(kResolutions[kIdx].nfp, kResolutions[kIdx].mpol,
                          kResolutions[kIdx].ntor);
+  // dft_ForcesToFourier_3d_symm reads IdealMhdModel's private real-space
+  // force members (armn_e, blmn_e, ...), which this fixture never populates
+  // -- they stay zero-initialized. That's fine for a timing benchmark: the
+  // transform cost doesn't depend on the input values, only its size.
   for (auto _ : state) {
-    FourierToReal3DSymmFastPoloidal(*fx.phys_x, fx.xmpq, fx.rp, fx.s, *fx.rprof,
-                                    fx.fb, fx.geom);
+    fx.model->dft_ForcesToFourier_3d_symm(*fx.phys_f);
     benchmark::ClobberMemory();
   }
   state.SetLabel(kResolutions[kIdx].label);
 }
 
-BENCHMARK_TEMPLATE(BM_DftFourierToReal, 0)->Name("DftFourierToReal/4x4");
-BENCHMARK_TEMPLATE(BM_FourierToReal, 0)->Name("FftFourierToReal/4x4");
-BENCHMARK_TEMPLATE(BM_DftFourierToReal, 1)->Name("DftFourierToReal/7x1");
-BENCHMARK_TEMPLATE(BM_FourierToReal, 1)->Name("FftFourierToReal/7x1");
-BENCHMARK_TEMPLATE(BM_DftFourierToReal, 2)->Name("DftFourierToReal/12x12");
-BENCHMARK_TEMPLATE(BM_FourierToReal, 2)->Name("FftFourierToReal/12x12");
-BENCHMARK_TEMPLATE(BM_DftFourierToReal, 3)->Name("DftFourierToReal/16x18");
-BENCHMARK_TEMPLATE(BM_FourierToReal, 3)->Name("FftFourierToReal/16x18");
-
-// ----------------------------------------------------------------------------
-// Real-space resolution sweep at fixed spectral resolution (mpol=12, ntor=12,
-// nfp=5).  Default real-space grid is (ntheta=2*mpol+6=30, nzeta=2*ntor+4=28
-// when nfp=1; for nfp=5 the toroidal grid extends accordingly).  We vary
-// ntheta and nzeta independently to isolate poloidal-AXPY cost (ntheta) from
-// toroidal-FFT cost (nzeta).
-// ----------------------------------------------------------------------------
-
-struct RealSpaceParams {
-  int nfp;
-  int mpol;
-  int ntor;
-  int ntheta;  // 0 = use Sizes default
-  int nzeta;   // 0 = use Sizes default
-  const char* label;
-};
-
-constexpr RealSpaceParams kRealSpaceSweep[] = {
-    // Vary nzeta (toroidal real-space grid) at fixed ntheta default.
-    {5, 12, 12, 0, 28, "12x12_ntheta-default_nzeta-28"},    // baseline default
-    {5, 12, 12, 0, 56, "12x12_ntheta-default_nzeta-56"},    // 2x toroidal
-    {5, 12, 12, 0, 84, "12x12_ntheta-default_nzeta-84"},    // 3x toroidal
-    {5, 12, 12, 0, 112, "12x12_ntheta-default_nzeta-112"},  // 4x toroidal
-    // Vary ntheta (poloidal real-space grid) at fixed nzeta default.
-    {5, 12, 12, 30, 0, "12x12_ntheta-30_nzeta-default"},    // baseline default
-    {5, 12, 12, 60, 0, "12x12_ntheta-60_nzeta-default"},    // 2x poloidal
-    {5, 12, 12, 90, 0, "12x12_ntheta-90_nzeta-default"},    // 3x poloidal
-    {5, 12, 12, 120, 0, "12x12_ntheta-120_nzeta-default"},  // 4x poloidal
-};
-
-template <int kIdx>
-void BM_FftFourierToReal_RealSpace(benchmark::State& state) {
-  static BenchFixture fx(kRealSpaceSweep[kIdx].nfp, kRealSpaceSweep[kIdx].mpol,
-                         kRealSpaceSweep[kIdx].ntor,
-                         kRealSpaceSweep[kIdx].ntheta,
-                         kRealSpaceSweep[kIdx].nzeta);
-  if (!fx.plans.kernels_available()) {
-    state.SkipWithError("no FFTX codelet for this resolution");
-    return;
-  }
-  for (auto _ : state) {
-    FourierToReal3DSymmFastPoloidalFft(*fx.phys_x, fx.xmpq, fx.rp, fx.s,
-                                       *fx.rprof, fx.fb, fx.plans, fx.geom);
-    benchmark::ClobberMemory();
-  }
-  state.SetLabel(kRealSpaceSweep[kIdx].label);
-}
-
-template <int kIdx>
-void BM_DftFourierToReal_RealSpace(benchmark::State& state) {
-  static BenchFixture fx(kRealSpaceSweep[kIdx].nfp, kRealSpaceSweep[kIdx].mpol,
-                         kRealSpaceSweep[kIdx].ntor,
-                         kRealSpaceSweep[kIdx].ntheta,
-                         kRealSpaceSweep[kIdx].nzeta);
-  for (auto _ : state) {
-    FourierToReal3DSymmFastPoloidal(*fx.phys_x, fx.xmpq, fx.rp, fx.s, *fx.rprof,
-                                    fx.fb, fx.geom);
-    benchmark::ClobberMemory();
-  }
-  state.SetLabel(kRealSpaceSweep[kIdx].label);
-}
+#define REGISTER_RES(IDX, LABEL)                                \
+  BENCHMARK_TEMPLATE(BM_ToroidalTransform_FourierToReal, IDX)   \
+      ->Name("ToroidalFourierToReal/" LABEL);                   \
+  BENCHMARK_TEMPLATE(BM_ToroidalTransform_ForcesToFourier, IDX) \
+      ->Name("ToroidalForcesToFourier/" LABEL)
 
-#define REGISTER_RS(I, NAME)                                               \
-  BENCHMARK_TEMPLATE(BM_DftFourierToReal_RealSpace, I)->Name("Dft/" NAME); \
-  BENCHMARK_TEMPLATE(BM_FftFourierToReal_RealSpace, I)->Name("Fft/" NAME)
+REGISTER_RES(0, "4x4");
+REGISTER_RES(1, "6x8");
+REGISTER_RES(2, "12x12");
+REGISTER_RES(3, "12x13");
 
-REGISTER_RS(0, "12x12_nzeta-28");
-REGISTER_RS(1, "12x12_nzeta-56");
-REGISTER_RS(2, "12x12_nzeta-84");
-REGISTER_RS(3, "12x12_nzeta-112");
-REGISTER_RS(4, "12x12_ntheta-30");
-REGISTER_RS(5, "12x12_ntheta-60");
-REGISTER_RS(6, "12x12_ntheta-90");
-REGISTER_RS(7, "12x12_ntheta-120");
-
-#undef REGISTER_RS
-
-// ----------------------------------------------------------------------------
-// Parallel fixture: one BenchFixture per thread, each covering its own radial
-// slice, sharing the same ToroidalFftPlans (read-only during the hot loop).
-// This matches the real VMEC calling pattern exactly:
-//   #pragma omp parallel
-//   { models[omp_get_thread_num()].geometryFromFourier(phys_x); }
-// ----------------------------------------------------------------------------
-
-struct ParallelBenchFixture {
-  // Shared across threads (read-only during hot loop).
-  Sizes s;
-  FourierBasisFastPoloidal fb;
-  ToroidalFftPlans plans;
-  Eigen::VectorXd xmpq;
-
-  // Per-thread state: each thread gets its own radial slice.
-  struct ThreadSlice {
-    RadialPartitioning rp;
-    std::unique_ptr<VmecINDATA> indata;
-    std::unique_ptr<HandoverStorage> handover;
-    std::unique_ptr<FlowControl> fc;
-    std::unique_ptr<RadialProfiles> rprof;
-    std::unique_ptr<FourierGeometry> phys_x;
-    // Output buffers (each thread writes only to its own slice).
-    std::vector<double> r1_e, r1_o, ru_e, ru_o, rv_e, rv_o;
-    std::vector<double> z1_e, z1_o, zu_e, zu_o, zv_e, zv_o;
-    std::vector<double> lu_e, lu_o, lv_e, lv_o;
-    std::vector<double> rCon, zCon;
-    RealSpaceGeometry geom;
-  };
-
-  std::vector<ThreadSlice> threads;
-
-  explicit ParallelBenchFixture(int nfp, int mpol, int ntor, int num_threads)
-      : s(/*lasym=*/false, nfp, mpol, ntor, /*ntheta=*/0, /*nzeta=*/0),
-        fb(&s),
-        plans(s.nZeta, s.nfp, s.mpol),
-        threads(num_threads) {
-    xmpq.resize(s.mpol);
-    for (int m = 0; m < s.mpol; ++m) xmpq[m] = m * (m - 1);
-
-    std::mt19937 rng(42);
-    std::uniform_real_distribution<double> dist(-1.0, 1.0);
-
-    for (int t = 0; t < num_threads; ++t) {
-      ThreadSlice& sl = threads[t];
-      sl.rp.adjustRadialPartitioning(num_threads, t, kNs,
-                                     /*lfreeb=*/false, /*printout=*/false);
-      sl.indata = std::make_unique<VmecINDATA>();
-      sl.handover = std::make_unique<HandoverStorage>(&s);
-      sl.fc = std::make_unique<FlowControl>(/*lfreeb=*/false, /*delt=*/0.9,
-                                            /*num_grids=*/1);
-      sl.fc->ns = kNs;
-      sl.rprof = std::make_unique<RadialProfiles>(
-          &sl.rp, sl.handover.get(), sl.indata.get(), sl.fc.get(),
-          /*signOfJacobian=*/-1, /*pDamp=*/0.05);
-      const int nsurf = sl.rp.nsMaxF1 - sl.rp.nsMinF1;
-      sl.rprof->sqrtSF.resize(nsurf);
-      for (int j = 0; j < nsurf; ++j) {
-        sl.rprof->sqrtSF[j] =
-            std::sqrt(0.05 + 0.9 * j / (nsurf > 1 ? nsurf - 1 : 1));
-      }
-
-      sl.phys_x = std::make_unique<FourierGeometry>(&s, &sl.rp, kNs);
-      auto rfill = [&](std::span<double> sp) {
-        for (double& x : sp) x = dist(rng);
-      };
-      rfill(sl.phys_x->rmncc);
-      rfill(sl.phys_x->rmnss);
-      rfill(sl.phys_x->zmnsc);
-      rfill(sl.phys_x->zmncs);
-      rfill(sl.phys_x->lmnsc);
-      rfill(sl.phys_x->lmncs);
-
-      const int nrzt1 = s.nZnT * (sl.rp.nsMaxF1 - sl.rp.nsMinF1);
-      const int nrzt_con = s.nZnT * (sl.rp.nsMaxFIncludingLcfs - sl.rp.nsMinF);
-      auto alloc = [](int n) { return std::vector<double>(n, 0.0); };
-      sl.r1_e = alloc(nrzt1);
-      sl.r1_o = alloc(nrzt1);
-      sl.ru_e = alloc(nrzt1);
-      sl.ru_o = alloc(nrzt1);
-      sl.rv_e = alloc(nrzt1);
-      sl.rv_o = alloc(nrzt1);
-      sl.z1_e = alloc(nrzt1);
-      sl.z1_o = alloc(nrzt1);
-      sl.zu_e = alloc(nrzt1);
-      sl.zu_o = alloc(nrzt1);
-      sl.zv_e = alloc(nrzt1);
-      sl.zv_o = alloc(nrzt1);
-      sl.lu_e = alloc(nrzt1);
-      sl.lu_o = alloc(nrzt1);
-      sl.lv_e = alloc(nrzt1);
-      sl.lv_o = alloc(nrzt1);
-      sl.rCon = alloc(nrzt_con);
-      sl.zCon = alloc(nrzt_con);
-      sl.geom = RealSpaceGeometry{sl.r1_e, sl.r1_o, sl.ru_e, sl.ru_o, sl.rv_e,
-                                  sl.rv_o, sl.z1_e, sl.z1_o, sl.zu_e, sl.zu_o,
-                                  sl.zv_e, sl.zv_o, sl.lu_e, sl.lu_o, sl.lv_e,
-                                  sl.lv_o, sl.rCon, sl.zCon};
-    }
-  }
-};
-
-// w7x at 4 threads: matches the real VMEC calling pattern exactly.
-//
-// VMEC keeps a single #pragma omp parallel team alive for the entire solver
-// run.  Each thread owns its own IdealMhdModel (with its own RadialPartitioning
-// slice) and calls dft_FourierToReal_3d_symm directly from within that
-// persistent team -- there is no nested fork/join per iteration, only
-// #pragma omp barrier between phases.
-//
-// We replicate that by opening one persistent team and looping inside it.
-// Thread 0 drives the Google Benchmark state machine; the others mirror it
-// via a shared atomic flag.  Each call is bracketed by barriers so all threads
-// start and finish together, and the wall-clock time reflects the slowest.
-void BM_FourierToReal_Parallel_W7x_4t(benchmark::State& state) {
-  constexpr int kNumThreads = 6;
-  static ParallelBenchFixture fx(/*nfp=*/5, /*mpol=*/12, /*ntor=*/12,
-                                 kNumThreads);
-  if (!fx.plans.kernels_available()) {
-    state.SkipWithError("no FFTX codelet for this resolution");
-    return;
-  }
-
-  std::atomic<bool> keep_going{true};
-
-#pragma omp parallel num_threads(kNumThreads)
-  {
-    const int tid = omp_get_thread_num();
-    ParallelBenchFixture::ThreadSlice& sl = fx.threads[tid];
-
-    // Warmup: each thread faults in its own plan's twiddle pages and sizes
-    // the thread_local scratch buffer before timing starts.
-    FourierToReal3DSymmFastPoloidalFft(*sl.phys_x, fx.xmpq, sl.rp, fx.s,
-                                       *sl.rprof, fx.fb, fx.plans, sl.geom);
-#pragma omp barrier
-
-    if (tid == 0) {
-      for (auto _ : state) {
-#pragma omp barrier
-        FourierToReal3DSymmFastPoloidalFft(*sl.phys_x, fx.xmpq, sl.rp, fx.s,
-                                           *sl.rprof, fx.fb, fx.plans, sl.geom);
-#pragma omp barrier
-      }
-      keep_going.store(false, std::memory_order_release);
-      // Final barrier so workers see the flag and exit cleanly.
-#pragma omp barrier
-    } else {
-      while (true) {
-#pragma omp barrier
-        if (!keep_going.load(std::memory_order_acquire)) break;
-        FourierToReal3DSymmFastPoloidalFft(*sl.phys_x, fx.xmpq, sl.rp, fx.s,
-                                           *sl.rprof, fx.fb, fx.plans, sl.geom);
-#pragma omp barrier
-      }
-    }
-  }
-
-  state.SetLabel("12x12 6-thread parallel fft");
-}
-BENCHMARK(BM_FourierToReal_Parallel_W7x_4t);
-
-// DFT parallel: same structure, no FFT plans needed.
-void BM_DftFourierToReal_Parallel_W7x_4t(benchmark::State& state) {
-  constexpr int kNumThreads = 6;
-  static ParallelBenchFixture fx(/*nfp=*/5, /*mpol=*/12, /*ntor=*/12,
-                                 kNumThreads);
-
-  std::atomic<bool> keep_going{true};
-
-#pragma omp parallel num_threads(kNumThreads)
-  {
-    const int tid = omp_get_thread_num();
-    ParallelBenchFixture::ThreadSlice& sl = fx.threads[tid];
-
-    // Warmup: fault in thread_local memory / cache lines before timing.
-    FourierToReal3DSymmFastPoloidal(*sl.phys_x, fx.xmpq, sl.rp, fx.s, *sl.rprof,
-                                    fx.fb, sl.geom);
-#pragma omp barrier
-
-    if (tid == 0) {
-      for (auto _ : state) {
-#pragma omp barrier
-        FourierToReal3DSymmFastPoloidal(*sl.phys_x, fx.xmpq, sl.rp, fx.s,
-                                        *sl.rprof, fx.fb, sl.geom);
-#pragma omp barrier
-      }
-      keep_going.store(false, std::memory_order_release);
-#pragma omp barrier
-    } else {
-      while (true) {
-#pragma omp barrier
-        if (!keep_going.load(std::memory_order_acquire)) break;
-        FourierToReal3DSymmFastPoloidal(*sl.phys_x, fx.xmpq, sl.rp, fx.s,
-                                        *sl.rprof, fx.fb, sl.geom);
-#pragma omp barrier
-      }
-    }
-  }
-
-  state.SetLabel("12x12 6-thread parallel dft");
-}
-BENCHMARK(BM_DftFourierToReal_Parallel_W7x_4t);
+#undef REGISTER_RES
 
 }  // namespace
 }  // namespace vmecpp

From 686bd269a67c9e0ab0bf5e8b30c23ed97a38486d Mon Sep 17 00:00:00 2001
From: CharlesCNorton <machineelv@gmail.com>
Date: Thu, 9 Jul 2026 08:15:48 -0400
Subject: [PATCH 52/79] Honor iteration_style=parvmec in the native solver
 (#612)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* Honor iteration_style=parvmec in the native solver

The PARVMEC time-step control (dual preconditioned/invariant residual minima,
a permissive 1e4 revert leash, gentle non-escalating revert) was only reachable
through the Python iteration driver. Implement it natively in
Vmec::SolveEquilibriumLoop gated on indata.iteration_style, plumb iteration_style
through the VmecInput model, and lift the run() guard, so vmecpp.run() honors the
input-file flag. The default vmec_8_52 path is unchanged.

* Drop the obsolete iteration_style skip in test_vmec_input_validation

VmecInput now carries iteration_style, so the field is present on both sides of
the INDATA/VmecInput serialization round-trip and no longer needs to be deleted
before the comparison.

* Restrict PARVMEC residual tracking to the PARVMEC branch

Compute the invariant residual minimum res1 and its inputs only when the PARVMEC
control is active, so the default vmec_8_52 time-step control adds no work to its
path and stays byte-for-byte unchanged, including under multithreading.

* Inline the PARVMEC iteration-style check at its use sites

* Strengthen the iteration-style physics check beyond volume

Compare geometry (volume, aspect), beta, pressure energy, and magnetic energy
between the vmec_8_52 and parvmec convergence controls, which all match to
machine precision; local profiles like iota are path-sensitive at finite ftol.

* Tighten the native/Python PARVMEC trace tolerance from 1e-3 to 1e-8

The two loops make identical control decisions, so the force-residual traces
agree to ~4e-9 (floating-point accumulation of the control arithmetic); the old
1e-3 relative tolerance was loose enough to mask real divergence.

* Add a PARVMEC-reference match test for the parvmec iteration style

Assert vmecpp's parvmec style reproduces the committed reference wouts, which
were verified against fresh ORNL-Fusion/PARVMEC output (bulk quantities to ~1e-15,
geometry and iota to ~1e-7 for cth_like, machine precision for solovev).

* Pin the parvmec iteration style to ORNL PARVMEC's force-residual trace

Adds a per-iteration force-residual reference from ORNL-Fusion/PARVMEC for
cth_like_fixed_bdy and a test that the native parvmec control reproduces it
step-for-step: machine precision for the first steps, a bounded ~1e-4 relative
drift over the full solve, and the same step count. A companion test asserts the
vmec_8_52 and parvmec controls take measurably different paths on the
restart-triggering cma ns=72 case, which is chaotic and so cannot be matched to
PARVMEC trace-for-trace.

---------

Co-authored-by: Philipp Jurašić <166746189+jurasic-pf@users.noreply.github.com>
---
 src/vmecpp/__init__.py                        |  36 +++-
 .../common/flow_control/flow_control.cc       |   1 +
 .../vmecpp/common/flow_control/flow_control.h |   4 +
 .../vmecpp/common/vmec_indata/vmec_indata.cc  |  15 +-
 ...parvmec_cth_like_fixed_bdy_force_trace.csv | 125 +++++++++++++
 src/vmecpp/cpp/vmecpp/vmec/vmec/vmec.cc       |  28 ++-
 tests/test_init.py                            |   2 -
 tests/test_iteration.py                       | 166 ++++++++++++++++++
 8 files changed, 366 insertions(+), 11 deletions(-)
 create mode 100644 src/vmecpp/cpp/vmecpp/test_data/parvmec_cth_like_fixed_bdy_force_trace.csv

diff --git a/src/vmecpp/__init__.py b/src/vmecpp/__init__.py
index 54ffb4077..9883157d5 100644
--- a/src/vmecpp/__init__.py
+++ b/src/vmecpp/__init__.py
@@ -113,6 +113,16 @@ class FreeBoundaryMethod(str, enum.Enum):
     """Boundary Integral Equation Solver for Toroidal systems."""
 
 
+class IterationStyle(str, enum.Enum):
+    """Time-step / restart control scheme for the equilibrium iteration."""
+
+    VMEC_8_52 = "vmec_8_52"
+    """The Fortran VMEC 8.52 control (the default)."""
+
+    PARVMEC = "parvmec"
+    """The PARVMEC / VMEC2000 9.0 control."""
+
+
 class OutputMode(enum.Enum):
     """Controls the output format of iteration logging.."""
 
@@ -138,6 +148,15 @@ def _validate_free_boundary_method(
     return FreeBoundaryMethod(str(value))
 
 
+def _validate_iteration_style(
+    value: _vmecpp.IterationStyle | str | IterationStyle,
+) -> IterationStyle:
+    """Convert various representations to IterationStyle."""
+    if isinstance(value, _vmecpp.IterationStyle):
+        return IterationStyle(value.name.lower())  # pyright: ignore[reportAttributeAccessIssue]
+    return IterationStyle(str(value))
+
+
 # This is a pure Python equivalent of VmecINDATAPyWrapper.
 # In the future VmecINDATAPyWrapper and the C++ VmecINDATA will merge into one type,
 # and this will become a Python wrapper around the one C++ VmecINDATA type.
@@ -347,6 +366,14 @@ class VmecInput(BaseModelWithNumpy):
     ] = FreeBoundaryMethod.NESTOR
     """Method for handling free-boundary conditions."""
 
+    iteration_style: typing.Annotated[
+        IterationStyle,
+        pydantic.BeforeValidator(_validate_iteration_style),
+        pydantic.Field(),
+    ] = IterationStyle.VMEC_8_52
+    """Time-step / restart control scheme for the equilibrium iteration (``"vmec_8_52"``
+    or ``"parvmec"``)."""
+
     nstep: int = 10
     """Printout interval at which convergence progress is logged."""
 
@@ -591,7 +618,10 @@ def _to_cpp_vmecindata(self) -> _vmecpp.VmecINDATA:
         }
 
         for attr in VmecInput.model_fields:
-            if attr in readonly_attrs or attr == "free_boundary_method":
+            if attr in readonly_attrs or attr in (
+                "free_boundary_method",
+                "iteration_style",
+            ):
                 continue  # these must be set separately
             setattr(cpp_indata, attr, getattr(self, attr))
 
@@ -599,6 +629,9 @@ def _to_cpp_vmecindata(self) -> _vmecpp.VmecINDATA:
         cpp_indata.free_boundary_method = getattr(
             _vmecpp.FreeBoundaryMethod, self.free_boundary_method.upper()
         )
+        cpp_indata.iteration_style = getattr(
+            _vmecpp.IterationStyle, self.iteration_style.upper()
+        )
 
         # this also resizes the readonly_attrs
         cpp_indata._set_mpol_ntor(self.mpol, self.ntor)
@@ -2468,6 +2501,7 @@ def set_profile(
     "MakegridParameters",
     "MagneticFieldResponseTable",
     "FreeBoundaryMethod",
+    "IterationStyle",
     "set_profile",
     "iterate",
     "solve_equilibrium",
diff --git a/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.cc b/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.cc
index ae577452c..70a013022 100644
--- a/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.cc
+++ b/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.cc
@@ -60,6 +60,7 @@ FlowControl::FlowControl(bool lfreeb, double delt, int num_grids,
   ijacob = 0;
   restart_reason = RestartReason::NO_RESTART;
   res0 = -1;
+  res1 = -1;
   delt0r = delt;
   multi_ns_grid = num_grids;
   neqs_old = 0;
diff --git a/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.h b/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.h
index dd4900572..ab7e2cb79 100644
--- a/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.h
+++ b/src/vmecpp/cpp/vmecpp/common/flow_control/flow_control.h
@@ -107,7 +107,11 @@ class FlowControl {
   // occurred)
   std::vector<RestartReason> restart_reasons;
 
+  // Running minimum of the preconditioned residual sum (fsq).
   double res0;
+  // Running minimum of the invariant residual sum (fsqr + fsqz + fsql); used
+  // only by the PARVMEC time-step control.
+  double res1;
 
   Eigen::Vector3d fResInvar;
   Eigen::Vector3d fResPrecd;
diff --git a/src/vmecpp/cpp/vmecpp/common/vmec_indata/vmec_indata.cc b/src/vmecpp/cpp/vmecpp/common/vmec_indata/vmec_indata.cc
index 86d15b98e..22fef164f 100644
--- a/src/vmecpp/cpp/vmecpp/common/vmec_indata/vmec_indata.cc
+++ b/src/vmecpp/cpp/vmecpp/common/vmec_indata/vmec_indata.cc
@@ -1460,13 +1460,14 @@ absl::Status IsConsistent(const VmecINDATA& vmec_indata,
   // nothing to check here: lforbal can be true or false and both are valid...
 
   // iteration_style
-  // For the current state of the code, we only accept VMEC_8_52,
-  // but in the future [TODO(jons)] also all other (implemented) enum values
-  // are valid.
-  if (vmec_indata.iteration_style != IterationStyle::VMEC_8_52) {
-    return absl::InvalidArgumentError(absl::StrFormat(
-        "input variable 'iteration_style' must be 'vmec_8_52', but is %s\n",
-        ToString(vmec_indata.iteration_style)));
+  // VMEC_8_52 and PARVMEC are both implemented in Vmec::SolveEquilibriumLoop.
+  if (vmec_indata.iteration_style != IterationStyle::VMEC_8_52 &&
+      vmec_indata.iteration_style != IterationStyle::PARVMEC) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("input variable 'iteration_style' must be 'vmec_8_52' "
+                        "or 'parvmec', but "
+                        "is %s\n",
+                        ToString(vmec_indata.iteration_style)));
   }
 
   // return_outputs_even_if_not_converged
diff --git a/src/vmecpp/cpp/vmecpp/test_data/parvmec_cth_like_fixed_bdy_force_trace.csv b/src/vmecpp/cpp/vmecpp/test_data/parvmec_cth_like_fixed_bdy_force_trace.csv
new file mode 100644
index 000000000..57c74bc5a
--- /dev/null
+++ b/src/vmecpp/cpp/vmecpp/test_data/parvmec_cth_like_fixed_bdy_force_trace.csv
@@ -0,0 +1,125 @@
+# Per-iteration force residuals (fsqr, fsqz, fsql) from ORNL-Fusion/PARVMEC
+# (PARVMEC / VMEC2000, version 10.0) for input.cth_like_fixed_bdy at ns=25,
+# ftol=1e-6, single radial grid. Pins the native parvmec iteration style to the
+# independent Fortran implementation. See test_parvmec_follows_ornl_parvmec_trace.
+iteration,fsqr,fsqz,fsql
+1,2.4089006554949474e-01,1.8189242874036761e-02,5.3233806831922206e-02
+2,3.6681393963863773e-02,2.9257078637605996e-03,2.0822882493332223e-02
+3,2.7779199334269252e-02,3.1344672791890939e-03,7.8400861928866594e-03
+4,4.5783030023920118e-02,4.9600076740968823e-03,1.1774119147540077e-02
+5,5.1533978999608258e-02,4.2652972219809670e-03,1.7993385241339890e-02
+6,5.7018736996709410e-02,4.6757668824585806e-03,1.7574015932580466e-02
+7,4.3109330291012618e-02,3.7895229757328045e-03,1.4163129388007296e-02
+8,3.5282867818118714e-02,9.3090538363044869e-03,1.4740903073900581e-02
+9,6.7512273795342706e-02,1.0963609530785181e-02,1.2272875086431585e-02
+10,2.7731565647694276e-02,5.4971626708317858e-03,7.0968082879577458e-03
+11,1.8717883628984681e-02,7.9409877631612287e-03,7.7416656727391040e-03
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+83,3.4571481241582132e-05,1.3222284564468306e-05,8.9849791597011068e-06
+84,3.6445289666553020e-05,1.3490211875571353e-05,7.6715832688247774e-06
+85,2.6836598884880545e-05,1.2228820709752092e-05,7.1047069567287085e-06
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+87,2.3045574529612850e-05,8.2783972746246295e-06,6.6274790604370499e-06
+88,2.0370542800839981e-05,8.7002943805473195e-06,5.2252480315205973e-06
+89,1.7977669721492016e-05,7.8672988854320640e-06,3.7936399705513430e-06
+90,1.3430524155009589e-05,6.9087201322425244e-06,3.2787527308475006e-06
+91,1.1181120201723126e-05,5.4025043317155100e-06,3.0809032723935404e-06
+92,1.3559887209373415e-05,4.2014661938091536e-06,3.4608729407792537e-06
+93,1.2746061880701487e-05,5.1789278661230076e-06,3.5352969772966303e-06
+94,9.4217905731992719e-06,4.9835523918641098e-06,2.2751170648794482e-06
+95,7.7366127289888493e-06,3.0759568914052026e-06,1.5196010009991481e-06
+96,8.3603327348265684e-06,2.9096250885540066e-06,2.1524084232659252e-06
+97,8.6955511799238599e-06,3.2835936714182820e-06,2.1679155065792652e-06
+98,6.9502126484870322e-06,2.5593521364206751e-06,1.3457562878104268e-06
+99,5.9463989880255614e-06,2.2659281857306051e-06,1.2728569415488959e-06
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+101,5.0693593578870755e-06,1.7912387403060682e-06,1.0948523289948467e-06
+102,4.8241421692710616e-06,1.3960215997118064e-06,8.2136369354832465e-07
+103,4.5679347253786716e-06,1.3256231291206880e-06,1.0512473810209935e-06
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+105,3.1489417069699166e-06,1.1025622870406821e-06,6.5276467896386802e-07
+106,3.2098909735179557e-06,9.3601268365983292e-07,5.3290041464301652e-07
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+113,1.4921799483519888e-06,5.1292691554744418e-07,3.1973232345980027e-07
+114,1.4023514921744768e-06,4.8103063561005090e-07,2.6861565212357621e-07
+115,1.3156479553749446e-06,4.8034511015759358e-07,2.7239727527235290e-07
+116,1.4063099029231428e-06,4.6454601027385580e-07,2.6220737517779700e-07
+117,1.1147425009542418e-06,4.2478177802809816e-07,1.9162536410246975e-07
+118,1.0646051378747073e-06,3.6844629041079497e-07,1.8411084320916765e-07
+119,1.0231548922946661e-06,3.3310841632637473e-07,2.5467984006468637e-07
+120,9.3467676899647807e-07,3.1138879132238200e-07,2.6141971225702011e-07
diff --git a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec.cc
index 9f6c1ea96..4915f753f 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/vmec/vmec.cc
@@ -429,6 +429,7 @@ bool Vmec::InitializeRadial(
   fc_.ijacob = 0;
   fc_.restart_reason = RestartReason::NO_RESTART;
   fc_.res0 = -1;
+  fc_.res1 = -1;
   m_delt0 = indata_.delt;
 
   // INITIALIZE MESH-DEPENDENT SCALARS
@@ -963,9 +964,34 @@ absl::StatusOr<Vmec::SolveEqLoopStatus> Vmec::SolveEquilibriumLoop(
 
       // res0 is the best force residual we got so far
       fc_.res0 = std::min(fc_.res0, fc_.fsq);
+
+      // PARVMEC additionally tracks the invariant residual minimum res1. Keep
+      // it (and its inputs) off the vmec_8_52 path so the default control stays
+      // byte-for-byte unchanged.
+      if (indata_.iteration_style == IterationStyle::PARVMEC) {
+        const double fsq_invariant = fc_.fsqr + fc_.fsqz + fc_.fsql;
+        if (iter2 == iter1_ || fc_.res1 == -1) {
+          fc_.res1 = fsq_invariant;
+        }
+        fc_.res1 = std::min(fc_.res1, fsq_invariant);
+      }
     }
 
-    if (fc_.fsq <= fc_.res0 && (iter2 - iter1_) > 10) {
+    if (indata_.iteration_style == IterationStyle::PARVMEC) {
+      // PARVMEC control: store when both residual minima improve; revert via
+      // BAD_PROGRESS (delt0r /= 1.03, no ijacob) when either exceeds 1e4 * its
+      // minimum after 10 steps.
+      const double fsq_invariant = fc_.fsqr + fc_.fsqz + fc_.fsql;
+      if (fc_.fsq <= fc_.res0 && fsq_invariant <= fc_.res1) {
+        RestartIteration(fc_.delt0r, thread_id);
+      } else if ((iter2 - iter1_) > 10 && (fc_.fsq > 1.0e4 * fc_.res0 ||
+                                           fsq_invariant > 1.0e4 * fc_.res1)) {
+#ifdef _OPENMP
+#pragma omp single
+#endif  // _OPENMP
+        fc_.restart_reason = RestartReason::BAD_PROGRESS;
+      }
+    } else if (fc_.fsq <= fc_.res0 && (iter2 - iter1_) > 10) {
       // Store current state (restart_reason=NO_RESTART)
       // --> was able to reduce force consistenly over at least 10 iterations
       RestartIteration(fc_.delt0r, thread_id);
diff --git a/tests/test_init.py b/tests/test_init.py
index 0d223c8ec..7a4779a1e 100644
--- a/tests/test_init.py
+++ b/tests/test_init.py
@@ -615,8 +615,6 @@ def test_vmec_input_validation():
     # The test_file json may exclude fields that have default values,
     # while the parsed versions should have all fields populated.
     indata_dict_from_json = json.loads(vmec_input._to_cpp_vmecindata().to_json())
-    # TODO(jurasic): iteration_style is not yet present in VmecInput, since there's only one option atm.
-    del indata_dict_from_json["iteration_style"]
     vmec_input_dict_from_json = json.loads(vmec_input.model_dump_json())
 
     if not vmec_input.lasym:
diff --git a/tests/test_iteration.py b/tests/test_iteration.py
index 8932ab092..f8699daca 100644
--- a/tests/test_iteration.py
+++ b/tests/test_iteration.py
@@ -208,6 +208,172 @@ def test_alternative_styles_converge_cma():
     assert results["parvmec"].restarts == 0
 
 
+def test_native_parvmec_matches_python_parvmec():
+    """The native C++ PARVMEC control reproduces the ported Python PARVMEC control.
+
+    Vmec::SolveEquilibriumLoop runs the PARVMEC time-step control when
+    indata.iteration_style == PARVMEC; it must match the Python parvmec loop on the
+    same forward model step for step, the analog of
+    test_python_iteration_matches_cpp_restart_path for the default style.
+    """
+    cpp_indata = _single_resolution_indata("cma", 72, ftol=1.0e-16, niter=200)
+    cpp_indata.iteration_style = _vmecpp.IterationStyle.PARVMEC
+
+    reference = _vmecpp.VmecModel.create(cpp_indata, 72)
+    reference.solve()
+
+    model = _vmecpp.VmecModel.create(cpp_indata, 72)
+    result = vmecpp.solve_equilibrium(model, style="parvmec")
+
+    assert not result.failed
+    np.testing.assert_array_equal(
+        np.asarray(result.restart_reasons), np.asarray(reference.restart_reasons)
+    )
+    cpp_r = np.asarray(reference.force_residual_r)
+    py_r = np.asarray(result.force_residual_r)
+    # The two loops make identical control decisions (restart_reasons match
+    # exactly above), so the force-residual traces agree up to floating-point
+    # accumulation of the control arithmetic: ~1e-9 relative early, growing to
+    # only a few 1e-9 by the deep-convergence tail (max abs ~6e-13).
+    np.testing.assert_allclose(py_r[:50], cpp_r[:50], rtol=1.0e-9, atol=1e-15)
+    np.testing.assert_allclose(py_r, cpp_r, rtol=1.0e-8, atol=1e-12)
+
+
+def test_run_honors_iteration_style_flag():
+    """vmecpp.run() honors the iteration_style input flag through the native solver.
+
+    The two styles take different iteration paths to the same equilibrium, so the flag
+    must survive the VmecInput -> C++ round-trip and reach Vmec::SolveEquilibriumLoop.
+    """
+    base = vmecpp.VmecInput.from_file(TEST_DATA / "cma.json").model_copy(
+        update={
+            "ns_array": np.array([51], dtype=np.int64),
+            "ftol_array": np.array([1.0e-12]),
+            "niter_array": np.array([3000], dtype=np.int64),
+        }
+    )
+    outputs = {}
+    for style in ("vmec_8_52", "parvmec"):
+        inp = base.model_copy(update={"iteration_style": style})
+        assert inp.iteration_style == style
+        assert inp._to_cpp_vmecindata().iteration_style == getattr(
+            _vmecpp.IterationStyle, style.upper()
+        )
+        outputs[style] = vmecpp.run(inp, max_threads=1, verbose=False)
+    # Same physics regardless of the iteration scheme: the two styles take
+    # different paths to the same equilibrium, so global geometry, pressure, and
+    # magnetic-field quantities must agree, not just the volume. (Local profiles
+    # such as the iota profile are path-sensitive at finite ftol, so they are not
+    # asserted here; the exact-reference check against PARVMEC covers those.)
+    ref = outputs["vmec_8_52"].wout
+    par = outputs["parvmec"].wout
+    assert par.volume_p == pytest.approx(ref.volume_p, rel=1.0e-9)  # geometry
+    assert par.aspect == pytest.approx(ref.aspect, rel=1.0e-9)  # geometry
+    assert par.betatotal == pytest.approx(ref.betatotal, rel=1.0e-9)  # beta
+    assert par.wp == pytest.approx(ref.wp, rel=1.0e-9)  # pressure energy
+    assert par.wb == pytest.approx(ref.wb, rel=1.0e-5)  # magnetic energy
+
+
+@pytest.mark.parametrize("case", ["cth_like_fixed_bdy", "solovev"])
+def test_parvmec_matches_parvmec_reference(case):
+    """The PARVMEC iteration style reproduces the ORNL-Fusion/PARVMEC wout.
+
+    The committed reference wouts match fresh output from the Fortran ORNL-
+    Fusion/PARVMEC to machine precision (volume/aspect ~1e-15, geometry and iota ~1e-7
+    for cth_like, ~0 for solovev), so this pins the new iteration style to the
+    independent parallel implementation, not only to the vmec_8_52 control.
+    """
+    reference = vmecpp.VmecWOut.from_wout_file(TEST_DATA / f"wout_{case}.nc")
+    base = vmecpp.VmecInput.from_file(TEST_DATA / f"{case}.json")
+    result = vmecpp.run(
+        base.model_copy(update={"iteration_style": "parvmec"}),
+        max_threads=1,
+        verbose=False,
+    )
+    w = result.wout
+
+    assert w.volume_p == pytest.approx(reference.volume_p, rel=1.0e-9)
+    assert w.aspect == pytest.approx(reference.aspect, rel=1.0e-9)
+    np.testing.assert_allclose(w.iotaf, reference.iotaf, rtol=1.0e-5, atol=1.0e-6)
+    np.testing.assert_allclose(w.rmnc, reference.rmnc, rtol=1.0e-5, atol=1.0e-6)
+    np.testing.assert_allclose(w.zmns, reference.zmns, rtol=1.0e-5, atol=1.0e-6)
+    np.testing.assert_allclose(w.bmnc, reference.bmnc, rtol=1.0e-5, atol=1.0e-6)
+
+
+def test_parvmec_follows_ornl_parvmec_trace():
+    """The native parvmec control follows ORNL PARVMEC's per-iteration residual trace.
+
+    cth_like_fixed_bdy has a well-posed linear guess (no cold-start axis reguess) and
+    converges without a restart, so the iteration is numerically stable: the native
+    parvmec time-step control reproduces the committed ORNL-Fusion/PARVMEC force
+    residuals step-for-step. The first several steps agree to machine precision; the
+    difference then settles into a bounded ~1e-4 relative drift (floating-point
+    accumulation between two independent implementations) that does not grow, and both
+    reach force balance in the same number of steps. This pins the control to the
+    Fortran PARVMEC itself, not only to the vmec_8_52 baseline or the Python port.
+
+    A step-for-step match is only meaningful on a non-chaotic case. Inputs that trip the
+    parvmec-specific restart / reguess logic have violent transients (fsqr ~ 1e4) on
+    which the ~1e-13 floating-point difference between any two implementations amplifies
+    to order unity within a couple of steps; those converge to the same equilibrium but
+    cannot be compared trace-for-trace. The divergence between the two styles on such a
+    case is covered by test_iteration_styles_diverge_on_stiff_case.
+    """
+    ref = np.loadtxt(
+        TEST_DATA / "parvmec_cth_like_fixed_bdy_force_trace.csv",
+        delimiter=",",
+        skiprows=5,
+    )
+    ref_fsqr, ref_fsqz, ref_fsql = ref[:, 1], ref[:, 2], ref[:, 3]
+
+    cpp_indata = _single_resolution_indata("cth_like_fixed_bdy", 25, 1.0e-6, 25000)
+    cpp_indata.iteration_style = _vmecpp.IterationStyle.PARVMEC
+    model = _vmecpp.VmecModel.create(cpp_indata, 25)
+    model.solve()
+    fsqr = np.asarray(model.force_residual_r)
+    fsqz = np.asarray(model.force_residual_z)
+    fsql = np.asarray(model.force_residual_lambda)
+
+    # Same number of steps to force balance (up to the last step at finite ftol).
+    assert abs(len(fsqr) - len(ref_fsqr)) <= 2
+    n = min(len(fsqr), len(ref_fsqr))
+
+    # Bit-faithful for the first several steps.
+    np.testing.assert_allclose(fsqr[:6], ref_fsqr[:6], rtol=1.0e-9, atol=1.0e-14)
+    np.testing.assert_allclose(fsqz[:6], ref_fsqz[:6], rtol=1.0e-9, atol=1.0e-14)
+    np.testing.assert_allclose(fsql[:6], ref_fsql[:6], rtol=1.0e-9, atol=1.0e-14)
+
+    # Tracks ORNL PARVMEC over the whole solve; the bounded drift never turns into a
+    # flow-control divergence (which would show up as a discrete jump, not a drift).
+    np.testing.assert_allclose(fsqr[:n], ref_fsqr[:n], rtol=3.0e-3, atol=1.0e-9)
+    np.testing.assert_allclose(fsqz[:n], ref_fsqz[:n], rtol=3.0e-3, atol=1.0e-9)
+    np.testing.assert_allclose(fsql[:n], ref_fsql[:n], rtol=3.0e-3, atol=1.0e-9)
+
+
+def test_iteration_styles_diverge_on_stiff_case():
+    """vmec_8_52 and parvmec take measurably different paths on a restart-triggering
+    case.
+
+    cma at ns=72 has a cold-start axis reguess and a violent initial transient that
+    trips the time-step-control restart logic. The two styles' restart bookkeeping and
+    force-residual progressions differ -- the reason the parvmec control exists -- even
+    though both converge to the same equilibrium. This is the flow-control difference
+    that the trace test cannot pin to ORNL PARVMEC directly (the case is chaotic).
+    """
+    traces = {}
+    for style in ("vmec_8_52", "parvmec"):
+        cpp_indata = _single_resolution_indata("cma", 72, 1.0e-11, 200)
+        cpp_indata.iteration_style = getattr(_vmecpp.IterationStyle, style.upper())
+        model = _vmecpp.VmecModel.create(cpp_indata, 72)
+        model.solve()
+        traces[style] = np.asarray(model.force_residual_r)
+
+    r852, rpar = traces["vmec_8_52"], traces["parvmec"]
+    n = min(len(r852), len(rpar))
+    # The two controls share only a prefix, then take genuinely different paths.
+    assert not np.allclose(r852[:n], rpar[:n], rtol=1.0e-6, atol=1.0e-12)
+
+
 def test_callback_records_iteration_state():
     """The per-iteration callback fires once per recorded iteration with a consistent
     IterationState snapshot of the convergence / flow-control state.

From e0f1985e4ff668a5c80b92561bb610d93b59f861 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Philipp=20Jura=C5=A1i=C4=87?=
 <166746189+jurasic-pf@users.noreply.github.com>
Date: Thu, 9 Jul 2026 14:23:00 +0200
Subject: [PATCH 53/79] Abseil status handling for mgrid errors (#613)

* Abseil status handling for mgrid errors

* Add full validation to CI (#614)

* Guard mgrid field reads against shape mismatches

Co-authored-by: jurasic-pf <166746189+jurasic-pf@users.noreply.github.com>

* Use safe move extraction for mgrid status values

Co-authored-by: jurasic-pf <166746189+jurasic-pf@users.noreply.github.com>

---------

Co-authored-by: copilot-swe-agent[bot] <198982749+Copilot@users.noreply.github.com>
---
 .github/workflows/full_validation.yaml        |  70 +++
 README.md                                     |   5 +
 src/vmecpp/cpp/util/netcdf_io/BUILD.bazel     |   5 +-
 src/vmecpp/cpp/util/netcdf_io/netcdf_io.cc    | 421 ++++++++++--------
 src/vmecpp/cpp/util/netcdf_io/netcdf_io.h     |  24 +-
 .../cpp/util/netcdf_io/netcdf_io_test.cc      |  65 ++-
 .../common/makegrid_lib/makegrid_lib_test.cc  |  58 ++-
 .../mgrid_provider/mgrid_provider.cc          | 137 +++++-
 .../output_quantities_test.cc                 | 333 +++++++-------
 .../mgrid_provider/mgrid_provider_test.cc     |  16 +-
 .../output_quantities_test.cc                 |   4 +-
 tests/test_init.py                            |   3 +-
 12 files changed, 720 insertions(+), 421 deletions(-)
 create mode 100644 .github/workflows/full_validation.yaml

diff --git a/.github/workflows/full_validation.yaml b/.github/workflows/full_validation.yaml
new file mode 100644
index 000000000..c378dc018
--- /dev/null
+++ b/.github/workflows/full_validation.yaml
@@ -0,0 +1,70 @@
+name: Full V&V against reference VMEC
+
+# Runs the full Verification & Validation suite from proximafusion/vmecpp-validation
+# (all ~219 input configurations against reference Fortran VMEC2000), using the
+# vmecpp version built from this commit rather than the version pinned on PyPI/GitHub.
+# This is a lot more exhaustive (and slower) than the "short" validation that
+# vmecpp-validation itself runs on its own PRs, so it is not run on every push here.
+on:
+  workflow_dispatch:
+  push:
+    branches:
+      - main
+
+concurrency:
+  group: ${{ github.workflow }}-${{ github.ref || github.run_id }}
+  cancel-in-progress: ${{ github.ref_name != 'main' }}
+
+jobs:
+  full-validation:
+    name: Run full VMEC++ validation
+    runs-on: ubuntu-22.04
+    # The full parameter scan runs ~219 configurations, each computing a reference
+    # wout with Fortran VMEC2000 in Docker plus one with VMEC++: budget generously.
+    timeout-minutes: 360
+    steps:
+      - name: Check out VMEC++
+        uses: actions/checkout@v4
+        with:
+          path: vmecpp
+          lfs: true
+
+      - name: Check out vmecpp-validation
+        uses: actions/checkout@v4
+        with:
+          repository: proximafusion/vmecpp-validation
+          path: vmecpp-validation
+          lfs: true
+
+      - uses: actions/setup-python@v5
+        with:
+          python-version: "3.10"
+
+      - name: Install required system packages for Ubuntu
+        run: |
+          sudo apt-get update && sudo apt-get install -y \
+            build-essential cmake libnetcdf-dev liblapack-dev liblapacke-dev libopenmpi-dev \
+            libomp-dev libeigen3-dev nlohmann-json3-dev libhdf5-dev
+
+      - name: Install vmecpp-validation's Python requirements
+        run: |
+          cd vmecpp-validation
+          # Install everything except vmecpp itself: we build and install vmecpp
+          # from this commit below instead of the version pinned in requirements.txt.
+          grep -v '^vmecpp@' requirements.txt > requirements.no-vmecpp.txt
+          python -m pip install -r requirements.no-vmecpp.txt
+
+      - name: Install VMEC++ from this commit
+        run: python -m pip install ./vmecpp
+
+      - name: Run full validation
+        working-directory: vmecpp-validation
+        run: python validate_vmec.py
+
+      - name: Upload V&V results
+        if: always()
+        uses: actions/upload-artifact@v4
+        with:
+          name: vnvresults
+          path: vmecpp-validation/vnvresults_*/
+          retention-days: 30
diff --git a/README.md b/README.md
index fa0800d5a..18cc7eefe 100644
--- a/README.md
+++ b/README.md
@@ -19,6 +19,7 @@
 
 [![CI](https://github.com/proximafusion/vmecpp/actions/workflows/tests.yaml/badge.svg)](https://github.com/proximafusion/vmecpp/actions/workflows/tests.yaml)
 [![C++ core tests](https://github.com/proximafusion/vmecpp/actions/workflows/test_bazel.yaml/badge.svg)](https://github.com/proximafusion/vmecpp/actions/workflows/test_bazel.yaml)
+[![Full V&V against reference VMEC](https://github.com/proximafusion/vmecpp/actions/workflows/full_validation.yaml/badge.svg)](https://github.com/proximafusion/vmecpp/actions/workflows/full_validation.yaml)
 [![Publish wheels to PyPI](https://github.com/proximafusion/vmecpp/actions/workflows/pypi_publish.yml/badge.svg)](https://github.com/proximafusion/vmecpp/actions/workflows/pypi_publish.yml)
 
 VMEC++ is a Python-friendly, from-scratch reimplementation in C++ of the Variational Moments Equilibrium Code (VMEC),
@@ -315,6 +316,10 @@ The single-thread runtimes as well as the contents of the "wout" file produced b
 The full validation test can be found at https://github.com/proximafusion/vmecpp-validation, including a set of sensible input configurations,
 parameter scan values and tolerances that make the comparison pass. See that repo for more information.
 
+This full validation (~219 input configurations) is run against every commit to `main` and can also be triggered
+on demand from the [Full V&V against reference VMEC](https://github.com/proximafusion/vmecpp/actions/workflows/full_validation.yaml) workflow
+(click "Run workflow"). It builds `vmecpp` from the corresponding commit rather than using the version pinned by `vmecpp-validation`.
+
 ## Differences with respect to PARVMEC/VMEC2000
 
 VMEC++:
diff --git a/src/vmecpp/cpp/util/netcdf_io/BUILD.bazel b/src/vmecpp/cpp/util/netcdf_io/BUILD.bazel
index 2d02e19d4..5452bd619 100644
--- a/src/vmecpp/cpp/util/netcdf_io/BUILD.bazel
+++ b/src/vmecpp/cpp/util/netcdf_io/BUILD.bazel
@@ -7,8 +7,8 @@ cc_library(
     hdrs = ["netcdf_io.h"],
     visibility = ["//visibility:public"],
     deps = [
-        "@abseil-cpp//absl/log:check",
-        "@abseil-cpp//absl/log:log",
+        "@abseil-cpp//absl/status",
+        "@abseil-cpp//absl/status:statusor",
         "@abseil-cpp//absl/strings",
         "@abseil-cpp//absl/strings:str_format",
         "//third_party/netcdf4",
@@ -31,6 +31,7 @@ cc_test(
     ],
     deps = [
         ":netcdf_io",
+        "@abseil-cpp//absl/status",
         "@googletest//:gtest_main",
     ],
 )
diff --git a/src/vmecpp/cpp/util/netcdf_io/netcdf_io.cc b/src/vmecpp/cpp/util/netcdf_io/netcdf_io.cc
index 54c9340b0..eef621527 100644
--- a/src/vmecpp/cpp/util/netcdf_io/netcdf_io.cc
+++ b/src/vmecpp/cpp/util/netcdf_io/netcdf_io.cc
@@ -7,156 +7,213 @@
 #include <string>
 #include <vector>
 
-#include "absl/log/check.h"
-#include "absl/log/log.h"
+#include "absl/status/status.h"
+#include "absl/status/statusor.h"
 #include "absl/strings/ascii.h"
 #include "absl/strings/str_format.h"
 #include "netcdf.h"
 
 namespace netcdf_io {
 
-bool NetcdfReadBool(int ncid, const std::string& variable_name) {
-  // VMEC uses `int` to store booleans: 0 means false, otherwise true.
-  // Also, the actual variable name is `<variable_name>__logical__`.
-  // AFAIK this is because NetCDF3 did not have a `boolean` data type.
+namespace {
 
-  // find variable ID for given variable name
+absl::StatusOr<int> FindVariableId(int ncid, const std::string& variable_name) {
   int variable_id = 0;
-  CHECK_EQ(
-      nc_inq_varid(ncid, (variable_name + "__logical__").c_str(), &variable_id),
-      NC_NOERR)
-      << "variable '" << variable_name << "' not found";
+  if (nc_inq_varid(ncid, variable_name.c_str(), &variable_id) != NC_NOERR) {
+    return absl::NotFoundError(
+        absl::StrFormat("variable '%s' not found", variable_name));
+  }
+  return variable_id;
+}
 
-  // figure out rank of data, i.e., how many dimensions does it have
+absl::StatusOr<int> GetVariableRank(int ncid, int variable_id,
+                                    const std::string& variable_name) {
   int rank = 0;
-  CHECK_EQ(nc_inq_varndims(ncid, variable_id, &rank), NC_NOERR);
+  if (nc_inq_varndims(ncid, variable_id, &rank) != NC_NOERR) {
+    return absl::InternalError(absl::StrFormat(
+        "could not determine rank of variable '%s'", variable_name));
+  }
+  return rank;
+}
 
-  // only accept zero-dimensional array for scalar
-  CHECK_EQ(rank, 0) << "Not a rank-0 array: " << variable_name;
+absl::StatusOr<std::vector<size_t> > GetVariableDimensions(
+    int ncid, int variable_id, int rank, const std::string& variable_name) {
+  std::vector<int> dimension_ids(rank, 0);
+  if (nc_inq_vardimid(ncid, variable_id, dimension_ids.data()) != NC_NOERR) {
+    return absl::InternalError(absl::StrFormat(
+        "could not determine dimension ids of variable '%s'", variable_name));
+  }
 
-  // actually read data
-  int variable_data = 0;
-  CHECK_EQ(nc_get_var_int(ncid, variable_id, &variable_data), NC_NOERR);
+  std::vector<size_t> dimensions(rank, 0);
+  for (int i = 0; i < rank; ++i) {
+    size_t dimension = 0;
+    if (nc_inq_dimlen(ncid, dimension_ids[i], &dimension) != NC_NOERR) {
+      return absl::InternalError(
+          absl::StrFormat("could not determine dimension %d of variable '%s'",
+                          i, variable_name));
+    }
+    dimensions[i] = dimension;
+  }
+  return dimensions;
+}
 
-  return (variable_data != 0);
-}  // NetcdfReadBool
+}  // namespace
 
-char NetcdfReadChar(int ncid, const std::string& variable_name) {
-  // find variable ID for given variable name
-  int variable_id = 0;
-  CHECK_EQ(nc_inq_varid(ncid, variable_name.c_str(), &variable_id), NC_NOERR)
-      << "variable '" << variable_name << "' not found";
+absl::StatusOr<bool> NetcdfReadBool(int ncid,
+                                    const std::string& variable_name) {
+  // VMEC uses `int` to store booleans: 0 means false, otherwise true.
+  // Also, the actual variable name is `<variable_name>__logical__`.
+  // AFAIK this is because NetCDF3 did not have a `boolean` data type.
+  const std::string logical_variable_name = variable_name + "__logical__";
 
-  // figure out rank of data, i.e., how many dimensions does it have
-  int rank = 1;
-  CHECK_EQ(nc_inq_varndims(ncid, variable_id, &rank), NC_NOERR);
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, logical_variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
 
-  // only accept zero-dimensional array for scalar
-  CHECK_EQ(rank, 1) << "Not a rank-1 array: " << variable_name;
+  absl::StatusOr<int> rank =
+      GetVariableRank(ncid, *variable_id, logical_variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
+  if (*rank != 0) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-0 array: %s", logical_variable_name));
+  }
 
-  // figure out the dimension IDs
-  std::vector<int> dimension_ids(rank, 0);
-  CHECK_EQ(nc_inq_vardimid(ncid, variable_id, dimension_ids.data()), NC_NOERR);
+  int variable_data = 0;
+  if (nc_get_var_int(ncid, *variable_id, &variable_data) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", logical_variable_name));
+  }
 
-  // figure out dimension of data, i.e., length of string
-  std::vector<size_t> dimensions(rank, 0);
-  size_t total_element_count = 1;
-  for (int i = 0; i < rank; ++i) {
-    size_t dimension = 0;
-    CHECK_EQ(nc_inq_dimlen(ncid, dimension_ids[i], &dimension), NC_NOERR);
-    dimensions[i] = dimension;
-    total_element_count *= dimension;
+  return variable_data != 0;
+}  // NetcdfReadBool
+
+absl::StatusOr<char> NetcdfReadChar(int ncid,
+                                    const std::string& variable_name) {
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
+
+  absl::StatusOr<int> rank = GetVariableRank(ncid, *variable_id, variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
+  if (*rank != 1) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-1 array: %s", variable_name));
+  }
+
+  absl::StatusOr<std::vector<size_t> > dimensions =
+      GetVariableDimensions(ncid, *variable_id, *rank, variable_name);
+  if (!dimensions.ok()) {
+    return dimensions.status();
   }
 
   // for a single char, make sure that the array dimension is 1
-  CHECK_EQ(dimensions[0], (size_t)1)
-      << "Not a length-1 array: " << variable_name;
+  if ((*dimensions)[0] != 1) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a length-1 array: %s", variable_name));
+  }
 
   // actually read data
-  std::vector<size_t> read_start_indices(rank, 0);
-  std::vector<char> variable_data(total_element_count, 0);
-  CHECK_EQ(nc_get_vara(ncid, variable_id, read_start_indices.data(),
-                       dimensions.data(), variable_data.data()),
-           NC_NOERR);
+  std::vector<size_t> read_start_indices(*rank, 0);
+  std::vector<char> variable_data(1, 0);
+  if (nc_get_vara(ncid, *variable_id, read_start_indices.data(),
+                  dimensions->data(), variable_data.data()) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", variable_name));
+  }
 
   return variable_data[0];
 }  // NetcdfReadChar
 
-int NetcdfReadInt(int ncid, const std::string& variable_name) {
-  // find variable ID for given variable name
-  int variable_id = 0;
-  CHECK_EQ(nc_inq_varid(ncid, variable_name.c_str(), &variable_id), NC_NOERR)
-      << "variable '" << variable_name << "' not found";
-
-  // figure out rank of data, i.e., how many dimensions does it have
-  int rank = 0;
-  CHECK_EQ(nc_inq_varndims(ncid, variable_id, &rank), NC_NOERR);
+absl::StatusOr<int> NetcdfReadInt(int ncid, const std::string& variable_name) {
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
 
-  // only accept zero-dimensional array for scalar
-  CHECK_EQ(rank, 0) << "Not a rank-0 array: " << variable_name;
+  absl::StatusOr<int> rank = GetVariableRank(ncid, *variable_id, variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
+  if (*rank != 0) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-0 array: %s", variable_name));
+  }
 
-  // actually read data
   int variable_data = 0;
-  CHECK_EQ(nc_get_var_int(ncid, variable_id, &variable_data), NC_NOERR);
+  if (nc_get_var_int(ncid, *variable_id, &variable_data) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", variable_name));
+  }
 
   return variable_data;
 }  // NetcdfReadInt
 
-double NetcdfReadDouble(int ncid, const std::string& variable_name) {
-  // find variable ID for given variable name
-  int variable_id = 0;
-  CHECK_EQ(nc_inq_varid(ncid, variable_name.c_str(), &variable_id), NC_NOERR)
-      << "variable '" << variable_name << "' not found";
-
-  // figure out rank of data, i.e., how many dimensions does it have
-  int rank = 0;
-  CHECK_EQ(nc_inq_varndims(ncid, variable_id, &rank), NC_NOERR);
+absl::StatusOr<double> NetcdfReadDouble(int ncid,
+                                        const std::string& variable_name) {
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
 
-  // only accept zero-dimensional array for scalar
-  CHECK_EQ(rank, 0) << "Not a rank-0 array: " << variable_name;
+  absl::StatusOr<int> rank = GetVariableRank(ncid, *variable_id, variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
+  if (*rank != 0) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-0 array: %s", variable_name));
+  }
 
-  // actually read data
   double variable_data = 0;
-  CHECK_EQ(nc_get_var_double(ncid, variable_id, &variable_data), NC_NOERR);
+  if (nc_get_var_double(ncid, *variable_id, &variable_data) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", variable_name));
+  }
 
   return variable_data;
 }  // NetcdfReadDouble
 
-std::string NetcdfReadString(int ncid, const std::string& variable_name) {
-  // find variable ID for given variable name
-  int varid = 0;
-  CHECK_EQ(nc_inq_varid(ncid, variable_name.c_str(), &varid), NC_NOERR)
-      << "variable '" << variable_name << "' not found";
-
-  // figure out rank of data, i.e., how many dimensions does it have
-  int rank = 0;
-  CHECK_EQ(nc_inq_varndims(ncid, varid, &rank), NC_NOERR);
+absl::StatusOr<std::string> NetcdfReadString(int ncid,
+                                             const std::string& variable_name) {
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
 
+  absl::StatusOr<int> rank = GetVariableRank(ncid, *variable_id, variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
   // only accept one-dimensional array of CHAR for strings
-  CHECK_EQ(rank, 1) << "Not a rank-1 array: " << variable_name;
-
-  // figure out the dimension IDs
-  std::vector<int> dimension_ids(rank, 0);
-  CHECK_EQ(nc_inq_vardimid(ncid, varid, dimension_ids.data()), NC_NOERR);
+  if (*rank != 1) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-1 array: %s", variable_name));
+  }
 
-  // figure out dimension of data, i.e., length of string
-  std::vector<size_t> dimensions(rank, 0);
-  size_t total_element_count = 1;
-  for (int i = 0; i < rank; ++i) {
-    size_t dimension = 0;
-    CHECK_EQ(nc_inq_dimlen(ncid, dimension_ids[i], &dimension), NC_NOERR);
-    dimensions[i] = dimension;
-    total_element_count *= dimension;
+  absl::StatusOr<std::vector<size_t> > dimensions =
+      GetVariableDimensions(ncid, *variable_id, *rank, variable_name);
+  if (!dimensions.ok()) {
+    return dimensions.status();
   }
 
+  size_t total_element_count = (*dimensions)[0];
+
   // actually read data
-  std::vector<size_t> read_start_indices(rank, 0);
+  std::vector<size_t> read_start_indices(*rank, 0);
   // one extra element that stays at 0 in order to properly zero-terminate the
   // string
   std::vector<char> variable_data(total_element_count + 1, 0);
-  CHECK_EQ(nc_get_vara(ncid, varid, read_start_indices.data(),
-                       dimensions.data(), variable_data.data()),
-           NC_NOERR);
+  if (nc_get_vara(ncid, *variable_id, read_start_indices.data(),
+                  dimensions->data(), variable_data.data()) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", variable_name));
+  }
   std::string string_from_char_array = std::string(variable_data.data());
 
   // Strings are usually whitespace-padded when coming from Fortran
@@ -164,78 +221,73 @@ std::string NetcdfReadString(int ncid, const std::string& variable_name) {
   return std::string(absl::StripAsciiWhitespace(string_from_char_array));
 }  // NetcdfReadString
 
-std::vector<double> NetcdfReadArray1D(int ncid,
-                                      const std::string& variable_name) {
-  // find variable ID for given variable name
-  int variable_id = 0;
-  CHECK_EQ(nc_inq_varid(ncid, variable_name.c_str(), &variable_id), NC_NOERR)
-      << "variable '" << variable_name << "' not found";
-
-  // figure out rank of data, i.e., how many dimensions does it have
-  int rank = 1;
-  CHECK_EQ(nc_inq_varndims(ncid, variable_id, &rank), NC_NOERR);
-
-  // only accept zero-dimensional array for scalar
-  CHECK_EQ(rank, 1) << "Not a rank-1 array: " << variable_name;
+absl::StatusOr<std::vector<double> > NetcdfReadArray1D(
+    int ncid, const std::string& variable_name) {
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
 
-  // figure out the dimension IDs
-  std::vector<int> dimension_ids(rank, 0);
-  CHECK_EQ(nc_inq_vardimid(ncid, variable_id, dimension_ids.data()), NC_NOERR);
+  absl::StatusOr<int> rank = GetVariableRank(ncid, *variable_id, variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
+  if (*rank != 1) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-1 array: %s", variable_name));
+  }
 
-  // figure out dimension of data, i.e., length of string
-  std::vector<size_t> dimensions(rank, 0);
-  size_t total_element_count = 1;
-  for (int i = 0; i < rank; ++i) {
-    size_t dimension = 0;
-    CHECK_EQ(nc_inq_dimlen(ncid, dimension_ids[i], &dimension), NC_NOERR);
-    dimensions[i] = dimension;
-    total_element_count *= dimension;
+  absl::StatusOr<std::vector<size_t> > dimensions =
+      GetVariableDimensions(ncid, *variable_id, *rank, variable_name);
+  if (!dimensions.ok()) {
+    return dimensions.status();
   }
 
-  // actually read data
-  std::vector<size_t> read_start_indices(rank, 0);
+  size_t total_element_count = (*dimensions)[0];
+
+  std::vector<size_t> read_start_indices(*rank, 0);
   std::vector<double> variable_data(total_element_count, 0.0);
-  CHECK_EQ(nc_get_vara(ncid, variable_id, read_start_indices.data(),
-                       dimensions.data(), variable_data.data()),
-           NC_NOERR);
+  if (nc_get_vara(ncid, *variable_id, read_start_indices.data(),
+                  dimensions->data(), variable_data.data()) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", variable_name));
+  }
 
   return variable_data;
 }  // NetcdfReadArray1D
 
-std::vector<std::vector<double> > NetcdfReadArray2D(
+absl::StatusOr<std::vector<std::vector<double> > > NetcdfReadArray2D(
     int ncid, const std::string& variable_name) {
-  // find variable ID for given variable name
-  int variable_id = 0;
-  CHECK_EQ(nc_inq_varid(ncid, variable_name.c_str(), &variable_id), NC_NOERR)
-      << "variable '" << variable_name << "' not found";
-
-  // figure out rank of data, i.e., how many dimensions does it have
-  int rank = 1;
-  CHECK_EQ(nc_inq_varndims(ncid, variable_id, &rank), NC_NOERR);
-
-  // only accept zero-dimensional array for scalar
-  CHECK_EQ(rank, 2) << "Not a rank-2 array: " << variable_name;
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
 
-  // figure out the dimension IDs
-  std::vector<int> dimension_ids(rank, 0);
-  CHECK_EQ(nc_inq_vardimid(ncid, variable_id, dimension_ids.data()), NC_NOERR);
+  absl::StatusOr<int> rank = GetVariableRank(ncid, *variable_id, variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
+  if (*rank != 2) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-2 array: %s", variable_name));
+  }
 
-  // figure out dimension of data, i.e., length of string
-  std::vector<size_t> dimensions(rank, 0);
-  size_t total_element_count = 1;
-  for (int i = 0; i < rank; ++i) {
-    size_t dimension = 0;
-    CHECK_EQ(nc_inq_dimlen(ncid, dimension_ids[i], &dimension), NC_NOERR);
-    dimensions[i] = dimension;
-    total_element_count *= dimension;
+  absl::StatusOr<std::vector<size_t> > dimensions_or =
+      GetVariableDimensions(ncid, *variable_id, *rank, variable_name);
+  if (!dimensions_or.ok()) {
+    return dimensions_or.status();
   }
+  auto dimensions = dimensions_or.value();
 
-  // actually read data
-  std::vector<size_t> read_start_indices(rank, 0);
+  size_t total_element_count = dimensions[0] * dimensions[1];
+
+  std::vector<size_t> read_start_indices(*rank, 0);
   std::vector<double> variable_data(total_element_count, 0.0);
-  CHECK_EQ(nc_get_vara(ncid, variable_id, read_start_indices.data(),
-                       dimensions.data(), variable_data.data()),
-           NC_NOERR);
+  if (nc_get_vara(ncid, *variable_id, read_start_indices.data(),
+                  dimensions.data(), variable_data.data()) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", variable_name));
+  }
 
   // copy from flattened vector into two-dimensional vector of vectors
   std::vector<std::vector<double> > two_dimensional_data(dimensions[0]);
@@ -249,40 +301,37 @@ std::vector<std::vector<double> > NetcdfReadArray2D(
   return two_dimensional_data;
 }  // NetcdfReadArray2D
 
-std::vector<std::vector<std::vector<double> > > NetcdfReadArray3D(
-    int ncid, const std::string& variable_name) {
-  // find variable ID for given variable name
-  int variable_id = 0;
-  CHECK_EQ(nc_inq_varid(ncid, variable_name.c_str(), &variable_id), NC_NOERR)
-      << "variable '" << variable_name << "' not found";
-
-  // figure out rank of data, i.e., how many dimensions does it have
-  int rank = 1;
-  CHECK_EQ(nc_inq_varndims(ncid, variable_id, &rank), NC_NOERR);
-
-  // only accept zero-dimensional array for scalar
-  CHECK_EQ(rank, 3) << "Not a rank-3 array: " << variable_name;
+absl::StatusOr<std::vector<std::vector<std::vector<double> > > >
+NetcdfReadArray3D(int ncid, const std::string& variable_name) {
+  absl::StatusOr<int> variable_id = FindVariableId(ncid, variable_name);
+  if (!variable_id.ok()) {
+    return variable_id.status();
+  }
 
-  // figure out the dimension IDs
-  std::vector<int> dimension_ids(rank, 0);
-  CHECK_EQ(nc_inq_vardimid(ncid, variable_id, dimension_ids.data()), NC_NOERR);
+  absl::StatusOr<int> rank = GetVariableRank(ncid, *variable_id, variable_name);
+  if (!rank.ok()) {
+    return rank.status();
+  }
+  if (*rank != 3) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Not a rank-3 array: %s", variable_name));
+  }
 
-  // figure out dimension of data, i.e., length of string
-  std::vector<size_t> dimensions(rank, 0);
-  size_t total_element_count = 1;
-  for (int i = 0; i < rank; ++i) {
-    size_t dimension = 0;
-    CHECK_EQ(nc_inq_dimlen(ncid, dimension_ids[i], &dimension), NC_NOERR);
-    dimensions[i] = dimension;
-    total_element_count *= dimension;
+  absl::StatusOr<std::vector<size_t> > dimensions_or =
+      GetVariableDimensions(ncid, *variable_id, *rank, variable_name);
+  if (!dimensions_or.ok()) {
+    return dimensions_or.status();
   }
+  auto dimensions = dimensions_or.value();
 
-  // actually read data
-  std::vector<size_t> read_start_indices(rank, 0);
+  size_t total_element_count = dimensions[0] * dimensions[1] * dimensions[2];
+  std::vector<size_t> read_start_indices(*rank, 0);
   std::vector<double> variable_data(total_element_count, 0.0);
-  CHECK_EQ(nc_get_vara(ncid, variable_id, read_start_indices.data(),
-                       dimensions.data(), variable_data.data()),
-           NC_NOERR);
+  if (nc_get_vara(ncid, *variable_id, read_start_indices.data(),
+                  dimensions.data(), variable_data.data()) != NC_NOERR) {
+    return absl::InternalError(
+        absl::StrFormat("could not read variable '%s'", variable_name));
+  }
 
   // copy from flattened vector into three-dimensional vector of vectors
   std::vector<std::vector<std::vector<double> > > three_dimensional_data(
diff --git a/src/vmecpp/cpp/util/netcdf_io/netcdf_io.h b/src/vmecpp/cpp/util/netcdf_io/netcdf_io.h
index fc2fc2f67..87d189823 100644
--- a/src/vmecpp/cpp/util/netcdf_io/netcdf_io.h
+++ b/src/vmecpp/cpp/util/netcdf_io/netcdf_io.h
@@ -8,44 +8,48 @@
 #include <string>
 #include <vector>
 
+#include "absl/status/statusor.h"
+
 namespace netcdf_io {
 
 // Read a scalar `bool` variable in Fortran VMEC style from the (opened) NetCDF
 // file identified by `ncid`. It is expected that the value is stored in a
 // scalar `int` variable named `<variable_name>__logical__`.
-bool NetcdfReadBool(int ncid, const std::string& variable_name);
+absl::StatusOr<bool> NetcdfReadBool(int ncid, const std::string& variable_name);
 
 // Read a scalar `char` variable from the (opened) NetCDF file identified by
 // `ncid`. It is expected that the value is stored in a length-1 `char` array.
-char NetcdfReadChar(int ncid, const std::string& variable_name);
+absl::StatusOr<char> NetcdfReadChar(int ncid, const std::string& variable_name);
 
 // Read a scalar `int` variable  from the (opened) NetCDF file identified by
 // `ncid`.
-int NetcdfReadInt(int ncid, const std::string& variable_name);
+absl::StatusOr<int> NetcdfReadInt(int ncid, const std::string& variable_name);
 
 // Read a scalar `double` variable  from the (opened) NetCDF file identified by
 // `ncid`.
-double NetcdfReadDouble(int ncid, const std::string& variable_name);
+absl::StatusOr<double> NetcdfReadDouble(int ncid,
+                                        const std::string& variable_name);
 
 // Read a string from the (opened) NetCDF file identified by `ncid`.
 // It is expected that the data is stored as a rank-1 `char` array.
 // Whitespace at the start and end of the `char` array is stripped.
-std::string NetcdfReadString(int ncid, const std::string& variable_name);
+absl::StatusOr<std::string> NetcdfReadString(int ncid,
+                                             const std::string& variable_name);
 
 // Read a rank-1 `double` array from the (opened) NetCDF file identified by
 // `ncid`.
-std::vector<double> NetcdfReadArray1D(int ncid,
-                                      const std::string& variable_name);
+absl::StatusOr<std::vector<double> > NetcdfReadArray1D(
+    int ncid, const std::string& variable_name);
 
 // Read a rank-2 `double` array from the (opened) NetCDF file identified by
 // `ncid`.
-std::vector<std::vector<double> > NetcdfReadArray2D(
+absl::StatusOr<std::vector<std::vector<double> > > NetcdfReadArray2D(
     int ncid, const std::string& variable_name);
 
 // Read a rank-3 `double` array from the (opened) NetCDF file identified by
 // `ncid`.
-std::vector<std::vector<std::vector<double> > > NetcdfReadArray3D(
-    int ncid, const std::string& variable_name);
+absl::StatusOr<std::vector<std::vector<std::vector<double> > > >
+NetcdfReadArray3D(int ncid, const std::string& variable_name);
 
 }  // namespace netcdf_io
 
diff --git a/src/vmecpp/cpp/util/netcdf_io/netcdf_io_test.cc b/src/vmecpp/cpp/util/netcdf_io/netcdf_io_test.cc
index 8afab1824..d23391702 100644
--- a/src/vmecpp/cpp/util/netcdf_io/netcdf_io_test.cc
+++ b/src/vmecpp/cpp/util/netcdf_io/netcdf_io_test.cc
@@ -9,6 +9,7 @@
 #include <string>
 #include <vector>
 
+#include "absl/status/status.h"
 #include "gmock/gmock.h"
 #include "gtest/gtest.h"
 
@@ -22,9 +23,10 @@ TEST(TestNetcdfIO, CheckReadBool) {
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  const bool lasym = NetcdfReadBool(ncid, "lasym");
+  absl::StatusOr<bool> lasym = NetcdfReadBool(ncid, "lasym");
 
-  EXPECT_FALSE(lasym);
+  ASSERT_TRUE(lasym.ok());
+  EXPECT_FALSE(*lasym);
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
 }  // CheckReadBool
@@ -35,9 +37,10 @@ TEST(TestNetcdfIO, CheckReadChar) {
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  const char mgrid_mode = NetcdfReadChar(ncid, "mgrid_mode");
+  absl::StatusOr<char> mgrid_mode = NetcdfReadChar(ncid, "mgrid_mode");
 
-  EXPECT_EQ(mgrid_mode, 'R');
+  ASSERT_TRUE(mgrid_mode.ok());
+  EXPECT_EQ(*mgrid_mode, 'R');
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
 }  // CheckReadChar
@@ -48,22 +51,38 @@ TEST(TestNetcdfIO, CheckReadInt) {
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  const int nfp = NetcdfReadInt(ncid, "nfp");
+  absl::StatusOr<int> nfp = NetcdfReadInt(ncid, "nfp");
 
-  EXPECT_EQ(nfp, 5);
+  ASSERT_TRUE(nfp.ok());
+  EXPECT_EQ(*nfp, 5);
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
 }  // CheckReadInt
 
+TEST(TestNetcdfIO, CheckReadIntMissingVariable) {
+  const std::string example_netcdf = "util/netcdf_io/example_netcdf.nc";
+
+  int ncid = 0;
+  ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
+
+  absl::StatusOr<int> missing = NetcdfReadInt(ncid, "does_not_exist");
+
+  EXPECT_FALSE(missing.ok());
+  EXPECT_EQ(missing.status().code(), absl::StatusCode::kNotFound);
+
+  ASSERT_EQ(nc_close(ncid), NC_NOERR);
+}  // CheckReadIntMissingVariable
+
 TEST(TestNetcdfIO, CheckReadDouble) {
   const std::string example_netcdf = "util/netcdf_io/example_netcdf.nc";
 
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  const double ftolv = NetcdfReadDouble(ncid, "ftolv");
+  absl::StatusOr<double> ftolv = NetcdfReadDouble(ncid, "ftolv");
 
-  EXPECT_EQ(ftolv, 1.0e-10);
+  ASSERT_TRUE(ftolv.ok());
+  EXPECT_EQ(*ftolv, 1.0e-10);
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
 }  // CheckReadDouble
@@ -74,9 +93,10 @@ TEST(TestNetcdfIO, CheckReadString) {
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  const std::string mgrid_file = NetcdfReadString(ncid, "mgrid_file");
+  absl::StatusOr<std::string> mgrid_file = NetcdfReadString(ncid, "mgrid_file");
 
-  EXPECT_EQ(mgrid_file, "mgrid_cth_like.nc");
+  ASSERT_TRUE(mgrid_file.ok());
+  EXPECT_EQ(*mgrid_file, "mgrid_cth_like.nc");
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
 }  // CheckReadString
@@ -87,7 +107,9 @@ TEST(TestNetcdfIO, CheckReadArray1D) {
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  std::vector<double> am = NetcdfReadArray1D(ncid, "am");
+  absl::StatusOr<std::vector<double> > am = NetcdfReadArray1D(ncid, "am");
+
+  ASSERT_TRUE(am.ok());
 
   // `am` is stored in the wout file with its default (maximum) length
   // and only the first few (relevant) entries are actually populated.
@@ -96,7 +118,7 @@ TEST(TestNetcdfIO, CheckReadArray1D) {
   reference_am[1] = 5.0;
   reference_am[2] = 10.0;
 
-  EXPECT_THAT(am, ElementsAreArray(reference_am));
+  EXPECT_THAT(*am, ElementsAreArray(reference_am));
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
 }  // CheckReadArray1D
@@ -107,14 +129,17 @@ TEST(TestNetcdfIO, CheckReadArray2D) {
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  std::vector<std::vector<double> > rmnc = NetcdfReadArray2D(ncid, "rmnc");
+  absl::StatusOr<std::vector<std::vector<double> > > rmnc =
+      NetcdfReadArray2D(ncid, "rmnc");
+
+  ASSERT_TRUE(rmnc.ok());
 
   std::vector<std::vector<double> > reference_rmnc = {{0.0, 1.0, 2.0},
                                                       {0.1, 1.1, 2.1}};
 
-  ASSERT_EQ(rmnc.size(), reference_rmnc.size());
+  ASSERT_EQ(rmnc->size(), reference_rmnc.size());
   for (size_t i = 0; i < reference_rmnc.size(); ++i) {
-    EXPECT_THAT(rmnc[i], ElementsAreArray(reference_rmnc[i]));
+    EXPECT_THAT((*rmnc)[i], ElementsAreArray(reference_rmnc[i]));
   }
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
@@ -126,9 +151,11 @@ TEST(TestNetcdfIO, CheckReadArray3D) {
   int ncid = 0;
   ASSERT_EQ(nc_open(example_netcdf.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  std::vector<std::vector<std::vector<double> > > br_001 =
+  absl::StatusOr<std::vector<std::vector<std::vector<double> > > > br_001 =
       NetcdfReadArray3D(ncid, "br_001");
 
+  ASSERT_TRUE(br_001.ok());
+
   std::vector<std::vector<std::vector<double> > > reference_br_001 = {
       {{0.00, 0.01, 0.02, 0.03},
        {0.10, 0.11, 0.12, 0.13},
@@ -137,11 +164,11 @@ TEST(TestNetcdfIO, CheckReadArray3D) {
        {1.10, 1.11, 1.12, 1.13},
        {1.20, 1.21, 1.22, 1.23}}};
 
-  ASSERT_EQ(br_001.size(), reference_br_001.size());
+  ASSERT_EQ(br_001->size(), reference_br_001.size());
   for (size_t i = 0; i < reference_br_001.size(); ++i) {
-    ASSERT_EQ(br_001[i].size(), reference_br_001[i].size());
+    ASSERT_EQ((*br_001)[i].size(), reference_br_001[i].size());
     for (size_t j = 0; j < reference_br_001[i].size(); ++j) {
-      EXPECT_THAT(br_001[i][j], ElementsAreArray(reference_br_001[i][j]));
+      EXPECT_THAT((*br_001)[i][j], ElementsAreArray(reference_br_001[i][j]));
     }
   }
 
diff --git a/src/vmecpp/cpp/vmecpp/common/makegrid_lib/makegrid_lib_test.cc b/src/vmecpp/cpp/vmecpp/common/makegrid_lib/makegrid_lib_test.cc
index ed2952b5f..d328ee5fa 100644
--- a/src/vmecpp/cpp/vmecpp/common/makegrid_lib/makegrid_lib_test.cc
+++ b/src/vmecpp/cpp/vmecpp/common/makegrid_lib/makegrid_lib_test.cc
@@ -509,19 +509,23 @@ TEST_P(CheckComputeMagneticFieldResponseTable, MatchesFortranReference) {
   int ncid = 0;
   ASSERT_EQ(nc_open(p.reference_nc_file.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  EXPECT_EQ(NetcdfReadInt(ncid, "nfp"),
+  EXPECT_EQ(NetcdfReadInt(ncid, "nfp").value(),
             makegrid_parameters.number_of_field_periods);
-  EXPECT_EQ(NetcdfReadInt(ncid, "ir"),
+  EXPECT_EQ(NetcdfReadInt(ncid, "ir").value(),
             makegrid_parameters.number_of_r_grid_points);
-  EXPECT_EQ(NetcdfReadDouble(ncid, "rmin"), makegrid_parameters.r_grid_minimum);
-  EXPECT_EQ(NetcdfReadDouble(ncid, "rmax"), makegrid_parameters.r_grid_maximum);
-  EXPECT_EQ(NetcdfReadInt(ncid, "jz"),
+  EXPECT_EQ(NetcdfReadDouble(ncid, "rmin").value(),
+            makegrid_parameters.r_grid_minimum);
+  EXPECT_EQ(NetcdfReadDouble(ncid, "rmax").value(),
+            makegrid_parameters.r_grid_maximum);
+  EXPECT_EQ(NetcdfReadInt(ncid, "jz").value(),
             makegrid_parameters.number_of_z_grid_points);
-  EXPECT_EQ(NetcdfReadDouble(ncid, "zmin"), makegrid_parameters.z_grid_minimum);
-  EXPECT_EQ(NetcdfReadDouble(ncid, "zmax"), makegrid_parameters.z_grid_maximum);
-  EXPECT_EQ(NetcdfReadInt(ncid, "kp"),
+  EXPECT_EQ(NetcdfReadDouble(ncid, "zmin").value(),
+            makegrid_parameters.z_grid_minimum);
+  EXPECT_EQ(NetcdfReadDouble(ncid, "zmax").value(),
+            makegrid_parameters.z_grid_maximum);
+  EXPECT_EQ(NetcdfReadInt(ncid, "kp").value(),
             makegrid_parameters.number_of_phi_grid_points);
-  EXPECT_EQ(NetcdfReadInt(ncid, "nextcur"), number_of_serial_circuits);
+  EXPECT_EQ(NetcdfReadInt(ncid, "nextcur").value(), number_of_serial_circuits);
 
   for (int circuit_index = 0; circuit_index < number_of_serial_circuits;
        ++circuit_index) {
@@ -537,11 +541,14 @@ TEST_P(CheckComputeMagneticFieldResponseTable, MatchesFortranReference) {
 
     // load mgrid data from NetCDF file
     std::vector<std::vector<std::vector<double>>> b_r_contribution =
-        NetcdfReadArray3D(ncid, absl::StrFormat("br_%03d", circuit_index + 1));
+        NetcdfReadArray3D(ncid, absl::StrFormat("br_%03d", circuit_index + 1))
+            .value();
     std::vector<std::vector<std::vector<double>>> b_p_contribution =
-        NetcdfReadArray3D(ncid, absl::StrFormat("bp_%03d", circuit_index + 1));
+        NetcdfReadArray3D(ncid, absl::StrFormat("bp_%03d", circuit_index + 1))
+            .value();
     std::vector<std::vector<std::vector<double>>> b_z_contribution =
-        NetcdfReadArray3D(ncid, absl::StrFormat("bz_%03d", circuit_index + 1));
+        NetcdfReadArray3D(ncid, absl::StrFormat("bz_%03d", circuit_index + 1))
+            .value();
 
     // perform comparison of points that are not explicitly excluded from the
     // comparison
@@ -647,31 +654,31 @@ TEST_P(CheckComputeVectorPotentialCache, MatchesFortranReference) {
   int ncid = 0;
   ASSERT_EQ(nc_open(p.reference_nc_file.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  const int nfp = NetcdfReadInt(ncid, "nfp");
+  const int nfp = NetcdfReadInt(ncid, "nfp").value();
   EXPECT_EQ(nfp, makegrid_parameters.number_of_field_periods);
 
-  const int numR = NetcdfReadInt(ncid, "ir");
+  const int numR = NetcdfReadInt(ncid, "ir").value();
   EXPECT_EQ(numR, makegrid_parameters.number_of_r_grid_points);
 
-  const double minR = NetcdfReadDouble(ncid, "rmin");
+  const double minR = NetcdfReadDouble(ncid, "rmin").value();
   EXPECT_EQ(minR, makegrid_parameters.r_grid_minimum);
 
-  const double maxR = NetcdfReadDouble(ncid, "rmax");
+  const double maxR = NetcdfReadDouble(ncid, "rmax").value();
   EXPECT_EQ(maxR, makegrid_parameters.r_grid_maximum);
 
-  const int numZ = NetcdfReadInt(ncid, "jz");
+  const int numZ = NetcdfReadInt(ncid, "jz").value();
   EXPECT_EQ(numZ, makegrid_parameters.number_of_z_grid_points);
 
-  const double minZ = NetcdfReadDouble(ncid, "zmin");
+  const double minZ = NetcdfReadDouble(ncid, "zmin").value();
   EXPECT_EQ(minZ, makegrid_parameters.z_grid_minimum);
 
-  const double maxZ = NetcdfReadDouble(ncid, "zmax");
+  const double maxZ = NetcdfReadDouble(ncid, "zmax").value();
   EXPECT_EQ(maxZ, makegrid_parameters.z_grid_maximum);
 
-  const int numPhi = NetcdfReadInt(ncid, "kp");
+  const int numPhi = NetcdfReadInt(ncid, "kp").value();
   EXPECT_EQ(numPhi, makegrid_parameters.number_of_phi_grid_points);
 
-  const int nextcur = NetcdfReadInt(ncid, "nextcur");
+  const int nextcur = NetcdfReadInt(ncid, "nextcur").value();
   EXPECT_EQ(nextcur, number_of_serial_circuits);
 
   for (int circuit_index = 0; circuit_index < number_of_serial_circuits;
@@ -688,11 +695,14 @@ TEST_P(CheckComputeVectorPotentialCache, MatchesFortranReference) {
 
     // load mgrid data from NetCDF file
     std::vector<std::vector<std::vector<double>>> a_r_contribution =
-        NetcdfReadArray3D(ncid, absl::StrFormat("ar_%03d", circuit_index + 1));
+        NetcdfReadArray3D(ncid, absl::StrFormat("ar_%03d", circuit_index + 1))
+            .value();
     std::vector<std::vector<std::vector<double>>> a_p_contribution =
-        NetcdfReadArray3D(ncid, absl::StrFormat("ap_%03d", circuit_index + 1));
+        NetcdfReadArray3D(ncid, absl::StrFormat("ap_%03d", circuit_index + 1))
+            .value();
     std::vector<std::vector<std::vector<double>>> a_z_contribution =
-        NetcdfReadArray3D(ncid, absl::StrFormat("az_%03d", circuit_index + 1));
+        NetcdfReadArray3D(ncid, absl::StrFormat("az_%03d", circuit_index + 1))
+            .value();
 
     // perform comparison of points that are not explicitly excluded from the
     // comparison
diff --git a/src/vmecpp/cpp/vmecpp/free_boundary/mgrid_provider/mgrid_provider.cc b/src/vmecpp/cpp/vmecpp/free_boundary/mgrid_provider/mgrid_provider.cc
index e6a02b5a2..b1291d81e 100644
--- a/src/vmecpp/cpp/vmecpp/free_boundary/mgrid_provider/mgrid_provider.cc
+++ b/src/vmecpp/cpp/vmecpp/free_boundary/mgrid_provider/mgrid_provider.cc
@@ -26,6 +26,40 @@ using netcdf_io::NetcdfReadDouble;
 using netcdf_io::NetcdfReadInt;
 using netcdf_io::NetcdfReadString;
 
+namespace {
+
+absl::Status ValidateFieldContributionShape(
+    const std::vector<std::vector<std::vector<double> > >& field_contribution,
+    const std::string& variable_name, int num_phi, int num_z, int num_r) {
+  if (field_contribution.size() != static_cast<size_t>(num_phi)) {
+    return absl::InvalidArgumentError(
+        absl::StrFormat("Variable '%s' has %d phi slices, expected %d.",
+                        variable_name, field_contribution.size(), num_phi));
+  }
+
+  for (int index_phi = 0; index_phi < num_phi; ++index_phi) {
+    if (field_contribution[index_phi].size() != static_cast<size_t>(num_z)) {
+      return absl::InvalidArgumentError(absl::StrFormat(
+          "Variable '%s' has %d z slices at phi index %d, expected %d.",
+          variable_name, field_contribution[index_phi].size(), index_phi,
+          num_z));
+    }
+    for (int index_z = 0; index_z < num_z; ++index_z) {
+      if (field_contribution[index_phi][index_z].size() !=
+          static_cast<size_t>(num_r)) {
+        return absl::InvalidArgumentError(absl::StrFormat(
+            "Variable '%s' has %d r points at phi index %d and z index %d, "
+            "expected %d.",
+            variable_name, field_contribution[index_phi][index_z].size(),
+            index_phi, index_z, num_r));
+      }
+    }
+  }
+  return absl::OkStatus();
+}
+
+}  // namespace
+
 MGridProvider::MGridProvider() {
   nfp = -1;
 
@@ -73,30 +107,71 @@ absl::Status MGridProvider::LoadFile(const std::filesystem::path& filename,
                         filename.string()));
   }
 
-  // TODO(jurasic) All of these should be handled with abseil status, but
-  // terminate on error with absl::CHECK instead.
-  nfp = NetcdfReadInt(ncid, "nfp");
+  // Reads below return absl::Status on failure (e.g. a missing variable),
+  // which we propagate to the caller instead of aborting the process.
+  auto with_context = [&filename](const absl::Status& s) {
+    return absl::Status(s.code(),
+                        absl::StrFormat("While reading mgrid file '%s': %s",
+                                        filename.string(), s.message()));
+  };
+
+  absl::StatusOr<int> nfp_or = NetcdfReadInt(ncid, "nfp");
+
+  absl::StatusOr<int> num_r_or = NetcdfReadInt(ncid, "ir");
+  absl::StatusOr<double> min_r_or = NetcdfReadDouble(ncid, "rmin");
+  absl::StatusOr<double> max_r_or = NetcdfReadDouble(ncid, "rmax");
+
+  absl::StatusOr<int> num_z_or = NetcdfReadInt(ncid, "jz");
+  absl::StatusOr<double> min_z_or = NetcdfReadDouble(ncid, "zmin");
+  absl::StatusOr<double> max_z_or = NetcdfReadDouble(ncid, "zmax");
+
+  absl::StatusOr<int> num_phi_or = NetcdfReadInt(ncid, "kp");
+
+  absl::StatusOr<int> nextcur_or = NetcdfReadInt(ncid, "nextcur");
+
+  absl::StatusOr<std::string> mgrid_mode_or =
+      NetcdfReadString(ncid, "mgrid_mode");
+
+  absl::Status read_status;
+  read_status.Update(nfp_or.status());
+  read_status.Update(num_r_or.status());
+  read_status.Update(min_r_or.status());
+  read_status.Update(max_r_or.status());
+  read_status.Update(num_z_or.status());
+  read_status.Update(min_z_or.status());
+  read_status.Update(max_z_or.status());
+  read_status.Update(num_phi_or.status());
+  read_status.Update(nextcur_or.status());
+  read_status.Update(mgrid_mode_or.status());
+  if (!read_status.ok()) {
+    nc_close(ncid);
+    return with_context(read_status);
+  }
+
+  nfp = *nfp_or;
 
-  numR = NetcdfReadInt(ncid, "ir");
-  minR = NetcdfReadDouble(ncid, "rmin");
-  maxR = NetcdfReadDouble(ncid, "rmax");
+  numR = *num_r_or;
+  minR = *min_r_or;
+  maxR = *max_r_or;
   deltaR = (maxR - minR) / (numR - 1.0);
 
-  numZ = NetcdfReadInt(ncid, "jz");
-  minZ = NetcdfReadDouble(ncid, "zmin");
-  maxZ = NetcdfReadDouble(ncid, "zmax");
+  numZ = *num_z_or;
+  minZ = *min_z_or;
+  maxZ = *max_z_or;
   deltaZ = (maxZ - minZ) / (numZ - 1.0);
 
-  numPhi = NetcdfReadInt(ncid, "kp");
+  numPhi = *num_phi_or;
 
-  nextcur = NetcdfReadInt(ncid, "nextcur");
+  nextcur = *nextcur_or;
   if (coil_currents.size() != nextcur) {
+    nc_close(ncid);
     return absl::InvalidArgumentError(
         absl::StrFormat("Number of currents %d does not match number of mgrid "
                         "coil fields nextcur=%d.",
                         coil_currents.size(), nextcur));
   }
-  mgrid_mode = NetcdfReadString(ncid, "mgrid_mode");
+
+  mgrid_mode = *mgrid_mode_or;
 
   // Resize and make sure that the accumulation arrays are reset to zeros
   // if they contained previous contents from an earlier call to this routine.
@@ -111,16 +186,44 @@ absl::Status MGridProvider::LoadFile(const std::filesystem::path& filename,
     // from i=1, 2, ..., nextcur
 
     std::string br_variable = absl::StrFormat("br_%03d", i + 1);
-    std::vector<std::vector<std::vector<double> > > b_r_contribution =
-        NetcdfReadArray3D(ncid, br_variable);
+    absl::StatusOr<std::vector<std::vector<std::vector<double> > > >
+        b_r_contribution_or = NetcdfReadArray3D(ncid, br_variable);
 
     std::string bp_variable = absl::StrFormat("bp_%03d", i + 1);
-    std::vector<std::vector<std::vector<double> > > b_p_contribution =
-        NetcdfReadArray3D(ncid, bp_variable);
+    absl::StatusOr<std::vector<std::vector<std::vector<double> > > >
+        b_p_contribution_or = NetcdfReadArray3D(ncid, bp_variable);
 
     std::string bz_variable = absl::StrFormat("bz_%03d", i + 1);
+    absl::StatusOr<std::vector<std::vector<std::vector<double> > > >
+        b_z_contribution_or = NetcdfReadArray3D(ncid, bz_variable);
+
+    absl::Status contribution_status;
+    contribution_status.Update(b_r_contribution_or.status());
+    contribution_status.Update(b_p_contribution_or.status());
+    contribution_status.Update(b_z_contribution_or.status());
+    if (!contribution_status.ok()) {
+      nc_close(ncid);
+      return with_context(contribution_status);
+    }
+
+    std::vector<std::vector<std::vector<double> > > b_r_contribution =
+        std::move(*b_r_contribution_or);
+    std::vector<std::vector<std::vector<double> > > b_p_contribution =
+        std::move(*b_p_contribution_or);
     std::vector<std::vector<std::vector<double> > > b_z_contribution =
-        NetcdfReadArray3D(ncid, bz_variable);
+        std::move(*b_z_contribution_or);
+
+    absl::Status shape_status;
+    shape_status.Update(ValidateFieldContributionShape(
+        b_r_contribution, br_variable, numPhi, numZ, numR));
+    shape_status.Update(ValidateFieldContributionShape(
+        b_p_contribution, bp_variable, numPhi, numZ, numR));
+    shape_status.Update(ValidateFieldContributionShape(
+        b_z_contribution, bz_variable, numPhi, numZ, numR));
+    if (!shape_status.ok()) {
+      nc_close(ncid);
+      return with_context(shape_status);
+    }
 
     for (int index_phi = 0; index_phi < numPhi; ++index_phi) {
       for (int index_z = 0; index_z < numZ; ++index_z) {
diff --git a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities_test.cc b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities_test.cc
index e696b52e4..44ecb7f86 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities_test.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/output_quantities/output_quantities_test.cc
@@ -88,15 +88,15 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
   int ncid;
   ASSERT_EQ(nc_open(filename.c_str(), NC_NOWRITE, &ncid), NC_NOERR);
 
-  EXPECT_EQ(wout.signgs, NetcdfReadInt(ncid, "signgs"));
+  EXPECT_EQ(wout.signgs, NetcdfReadInt(ncid, "signgs").value());
 
-  EXPECT_EQ(wout.gamma, NetcdfReadDouble(ncid, "gamma"));
+  EXPECT_EQ(wout.gamma, NetcdfReadDouble(ncid, "gamma").value());
 
-  EXPECT_EQ(wout.pcurr_type, NetcdfReadString(ncid, "pcurr_type"));
-  EXPECT_EQ(wout.pmass_type, NetcdfReadString(ncid, "pmass_type"));
-  EXPECT_EQ(wout.piota_type, NetcdfReadString(ncid, "piota_type"));
+  EXPECT_EQ(wout.pcurr_type, NetcdfReadString(ncid, "pcurr_type").value());
+  EXPECT_EQ(wout.pmass_type, NetcdfReadString(ncid, "pmass_type").value());
+  EXPECT_EQ(wout.piota_type, NetcdfReadString(ncid, "piota_type").value());
 
-  std::vector<double> reference_am = NetcdfReadArray1D(ncid, "am");
+  std::vector<double> reference_am = NetcdfReadArray1D(ncid, "am").value();
   // remove zero-padding at end
   reference_am.resize(wout.am.size());
   EXPECT_THAT(wout.am, ElementsAreArray(reference_am));
@@ -105,37 +105,37 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
   if (vmec_indata->ncurr == 0) {
     // constrained-iota; ignore current profile coefficients
     // TODO(jons): check for spline profiles -> need to check ai_aux_*
-    std::vector<double> reference_ai = NetcdfReadArray1D(ncid, "ai");
+    std::vector<double> reference_ai = NetcdfReadArray1D(ncid, "ai").value();
     // remove zero-padding at end
     reference_ai.resize(wout.ai.size());
     EXPECT_THAT(wout.ai, ElementsAreArray(reference_ai));
   } else {
     // constrained-current
     // TODO(jons): check for spline profiles -> need to check ac_aux_*
-    std::vector<double> reference_ac = NetcdfReadArray1D(ncid, "ac");
+    std::vector<double> reference_ac = NetcdfReadArray1D(ncid, "ac").value();
     reference_ac.resize(wout.ac.size());
     EXPECT_THAT(wout.ac, ElementsAreArray(reference_ac));
 
     if (wout.ai.size() > 0) {
       // iota profile (if present) taken as initial guess for first iteration
       // TODO(jons): check for spline profiles -> need to check ai_aux_*
-      std::vector<double> reference_ai = NetcdfReadArray1D(ncid, "ai");
+      std::vector<double> reference_ai = NetcdfReadArray1D(ncid, "ai").value();
       // remove zero-padding at end
       reference_ai.resize(wout.ai.size());
       EXPECT_THAT(wout.ai, ElementsAreArray(reference_ai));
     }
   }
 
-  EXPECT_EQ(wout.nfp, NetcdfReadInt(ncid, "nfp"));
-  EXPECT_EQ(wout.mpol, NetcdfReadInt(ncid, "mpol"));
-  EXPECT_EQ(wout.ntor, NetcdfReadInt(ncid, "ntor"));
-  EXPECT_EQ(wout.lasym, NetcdfReadBool(ncid, "lasym"));
+  EXPECT_EQ(wout.nfp, NetcdfReadInt(ncid, "nfp").value());
+  EXPECT_EQ(wout.mpol, NetcdfReadInt(ncid, "mpol").value());
+  EXPECT_EQ(wout.ntor, NetcdfReadInt(ncid, "ntor").value());
+  EXPECT_EQ(wout.lasym, NetcdfReadBool(ncid, "lasym").value());
 
-  EXPECT_EQ(wout.ns, NetcdfReadInt(ncid, "ns"));
-  EXPECT_EQ(wout.ftolv, NetcdfReadDouble(ncid, "ftolv"));
-  EXPECT_EQ(wout.niter, NetcdfReadInt(ncid, "niter"));
+  EXPECT_EQ(wout.ns, NetcdfReadInt(ncid, "ns").value());
+  EXPECT_EQ(wout.ftolv, NetcdfReadDouble(ncid, "ftolv").value());
+  EXPECT_EQ(wout.niter, NetcdfReadInt(ncid, "niter").value());
 
-  EXPECT_EQ(wout.lfreeb, NetcdfReadBool(ncid, "lfreeb"));
+  EXPECT_EQ(wout.lfreeb, NetcdfReadBool(ncid, "lfreeb").value());
   if (wout.lfreeb) {
     // The reference data is generated using educational_VMEC,
     // which is run from within //vmecpp/test_data.
@@ -147,88 +147,99 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
     // //vmecpp/test_data/regenerate_test_data.sh.
     EXPECT_EQ(wout.mgrid_file,
               absl::StrCat("vmecpp/test_data/",
-                           NetcdfReadString(ncid, "mgrid_file")));
-    std::vector<double> reference_extcur = NetcdfReadArray1D(ncid, "extcur");
+                           NetcdfReadString(ncid, "mgrid_file").value()));
+    std::vector<double> reference_extcur =
+        NetcdfReadArray1D(ncid, "extcur").value();
     EXPECT_THAT(wout.extcur, ElementsAreArray(reference_extcur));
     EXPECT_EQ(wout.nextcur, static_cast<int>(reference_extcur.size()));
   }
-  EXPECT_EQ(wout.mgrid_mode, NetcdfReadString(ncid, "mgrid_mode"));
+  EXPECT_EQ(wout.mgrid_mode, NetcdfReadString(ncid, "mgrid_mode").value());
 
   // -------------------
   // scalar quantities
 
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "wb"), wout.wb, tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "wp"), wout.wp, tolerance));
+  EXPECT_TRUE(
+      IsCloseRelAbs(NetcdfReadDouble(ncid, "wb").value(), wout.wb, tolerance));
+  EXPECT_TRUE(
+      IsCloseRelAbs(NetcdfReadDouble(ncid, "wp").value(), wout.wp, tolerance));
 
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "rmax_surf"), wout.rmax_surf,
-                            tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "rmin_surf"), wout.rmin_surf,
-                            tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "zmax_surf"), wout.zmax_surf,
-                            tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "rmax_surf").value(),
+                            wout.rmax_surf, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "rmin_surf").value(),
+                            wout.rmin_surf, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "zmax_surf").value(),
+                            wout.zmax_surf, tolerance));
 
-  EXPECT_EQ(wout.mnmax, NetcdfReadInt(ncid, "mnmax"));
-  EXPECT_EQ(wout.mnmax_nyq, NetcdfReadInt(ncid, "mnmax_nyq"));
+  EXPECT_EQ(wout.mnmax, NetcdfReadInt(ncid, "mnmax").value());
+  EXPECT_EQ(wout.mnmax_nyq, NetcdfReadInt(ncid, "mnmax_nyq").value());
 
-  EXPECT_EQ(wout.ier_flag, NetcdfReadInt(ncid, "ier_flag"));
+  EXPECT_EQ(wout.ier_flag, NetcdfReadInt(ncid, "ier_flag").value());
 
-  EXPECT_TRUE(
-      IsCloseRelAbs(NetcdfReadDouble(ncid, "aspect"), wout.aspect, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "aspect").value(),
+                            wout.aspect, tolerance));
 
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betatotal"), wout.betatotal,
-                            tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betapol"), wout.betapol,
-                            tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betator"), wout.betator,
-                            tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betaxis"), wout.betaxis,
-                            tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betatotal").value(),
+                            wout.betatotal, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betapol").value(),
+                            wout.betapol, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betator").value(),
+                            wout.betator, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "betaxis").value(),
+                            wout.betaxis, tolerance));
 
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "b0"), wout.b0, tolerance));
-
-  EXPECT_TRUE(
-      IsCloseRelAbs(NetcdfReadDouble(ncid, "rbtor0"), wout.rbtor0, tolerance));
   EXPECT_TRUE(
-      IsCloseRelAbs(NetcdfReadDouble(ncid, "rbtor"), wout.rbtor, tolerance));
+      IsCloseRelAbs(NetcdfReadDouble(ncid, "b0").value(), wout.b0, tolerance));
 
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "IonLarmor"), wout.IonLarmor,
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "rbtor0").value(),
+                            wout.rbtor0, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "rbtor").value(), wout.rbtor,
                             tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "volavgB"), wout.volavgB,
+
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "IonLarmor").value(),
+                            wout.IonLarmor, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "volavgB").value(),
+                            wout.volavgB, tolerance));
+
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "ctor").value(), wout.ctor,
                             tolerance));
 
-  EXPECT_TRUE(
-      IsCloseRelAbs(NetcdfReadDouble(ncid, "ctor"), wout.ctor, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "Aminor_p").value(),
+                            wout.Aminor_p, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "Rmajor_p").value(),
+                            wout.Rmajor_p, tolerance));
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "volume_p").value(),
+                            wout.volume, tolerance));
 
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "Aminor_p"), wout.Aminor_p,
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "fsqr").value(), wout.fsqr,
                             tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "Rmajor_p"), wout.Rmajor_p,
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "fsqz").value(), wout.fsqz,
                             tolerance));
-  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "volume_p"), wout.volume,
+  EXPECT_TRUE(IsCloseRelAbs(NetcdfReadDouble(ncid, "fsql").value(), wout.fsql,
                             tolerance));
 
-  EXPECT_TRUE(
-      IsCloseRelAbs(NetcdfReadDouble(ncid, "fsqr"), wout.fsqr, tolerance));
-  EXPECT_TRUE(
-      IsCloseRelAbs(NetcdfReadDouble(ncid, "fsqz"), wout.fsqz, tolerance));
-  EXPECT_TRUE(
-      IsCloseRelAbs(NetcdfReadDouble(ncid, "fsql"), wout.fsql, tolerance));
-
   // -------------------
   // one-dimensional array quantities
 
-  std::vector<double> reference_iota_full = NetcdfReadArray1D(ncid, "iotaf");
+  std::vector<double> reference_iota_full =
+      NetcdfReadArray1D(ncid, "iotaf").value();
   std::vector<double> reference_safety_factor =
-      NetcdfReadArray1D(ncid, "q_factor");
+      NetcdfReadArray1D(ncid, "q_factor").value();
   std::vector<double> reference_pressure_full =
-      NetcdfReadArray1D(ncid, "presf");
-  std::vector<double> reference_toroidal_flux = NetcdfReadArray1D(ncid, "phi");
-  std::vector<double> reference_poloidal_flux = NetcdfReadArray1D(ncid, "chi");
-  std::vector<double> reference_phipf = NetcdfReadArray1D(ncid, "phipf");
-  std::vector<double> reference_chipf = NetcdfReadArray1D(ncid, "chipf");
-  std::vector<double> reference_jcuru = NetcdfReadArray1D(ncid, "jcuru");
-  std::vector<double> reference_jcurv = NetcdfReadArray1D(ncid, "jcurv");
+      NetcdfReadArray1D(ncid, "presf").value();
+  std::vector<double> reference_toroidal_flux =
+      NetcdfReadArray1D(ncid, "phi").value();
+  std::vector<double> reference_poloidal_flux =
+      NetcdfReadArray1D(ncid, "chi").value();
+  std::vector<double> reference_phipf =
+      NetcdfReadArray1D(ncid, "phipf").value();
+  std::vector<double> reference_chipf =
+      NetcdfReadArray1D(ncid, "chipf").value();
+  std::vector<double> reference_jcuru =
+      NetcdfReadArray1D(ncid, "jcuru").value();
+  std::vector<double> reference_jcurv =
+      NetcdfReadArray1D(ncid, "jcurv").value();
   std::vector<double> reference_spectral_width =
-      NetcdfReadArray1D(ncid, "specw");
+      NetcdfReadArray1D(ncid, "specw").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     EXPECT_TRUE(
         IsCloseRelAbs(reference_iota_full[jF], wout.iotaf[jF], tolerance));
@@ -248,15 +259,21 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
         IsCloseRelAbs(reference_spectral_width[jF], wout.specw[jF], tolerance));
   }  // jF
 
-  std::vector<double> reference_iota_half = NetcdfReadArray1D(ncid, "iotas");
-  std::vector<double> reference_mass_half = NetcdfReadArray1D(ncid, "mass");
-  std::vector<double> reference_pressure_half = NetcdfReadArray1D(ncid, "pres");
-  std::vector<double> reference_beta = NetcdfReadArray1D(ncid, "beta_vol");
-  std::vector<double> reference_buco = NetcdfReadArray1D(ncid, "buco");
-  std::vector<double> reference_bvco = NetcdfReadArray1D(ncid, "bvco");
-  std::vector<double> reference_dVds = NetcdfReadArray1D(ncid, "vp");
-  std::vector<double> reference_phips = NetcdfReadArray1D(ncid, "phips");
-  std::vector<double> reference_overr = NetcdfReadArray1D(ncid, "over_r");
+  std::vector<double> reference_iota_half =
+      NetcdfReadArray1D(ncid, "iotas").value();
+  std::vector<double> reference_mass_half =
+      NetcdfReadArray1D(ncid, "mass").value();
+  std::vector<double> reference_pressure_half =
+      NetcdfReadArray1D(ncid, "pres").value();
+  std::vector<double> reference_beta =
+      NetcdfReadArray1D(ncid, "beta_vol").value();
+  std::vector<double> reference_buco = NetcdfReadArray1D(ncid, "buco").value();
+  std::vector<double> reference_bvco = NetcdfReadArray1D(ncid, "bvco").value();
+  std::vector<double> reference_dVds = NetcdfReadArray1D(ncid, "vp").value();
+  std::vector<double> reference_phips =
+      NetcdfReadArray1D(ncid, "phips").value();
+  std::vector<double> reference_overr =
+      NetcdfReadArray1D(ncid, "over_r").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     EXPECT_TRUE(
         IsCloseRelAbs(reference_iota_half[jF], wout.iotas[jF], tolerance));
@@ -273,10 +290,12 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
     EXPECT_TRUE(IsCloseRelAbs(reference_overr[jF], wout.over_r[jF], tolerance));
   }  // jF
 
-  std::vector<double> reference_jdotb = NetcdfReadArray1D(ncid, "jdotb");
-  std::vector<double> reference_bdotb = NetcdfReadArray1D(ncid, "bdotb");
+  std::vector<double> reference_jdotb =
+      NetcdfReadArray1D(ncid, "jdotb").value();
+  std::vector<double> reference_bdotb =
+      NetcdfReadArray1D(ncid, "bdotb").value();
   std::vector<double> reference_bdotgradv =
-      NetcdfReadArray1D(ncid, "bdotgradv");
+      NetcdfReadArray1D(ncid, "bdotgradv").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     EXPECT_TRUE(
         IsCloseRelAbs(reference_jdotb[jF], wout.jdotb[jF], 10 * tolerance));
@@ -285,11 +304,16 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
         IsCloseRelAbs(reference_bdotgradv[jF], wout.bdotgradv[jF], tolerance));
   }  // jF
 
-  std::vector<double> reference_DMerc = NetcdfReadArray1D(ncid, "DMerc");
-  std::vector<double> reference_Dshear = NetcdfReadArray1D(ncid, "DShear");
-  std::vector<double> reference_Dwell = NetcdfReadArray1D(ncid, "DWell");
-  std::vector<double> reference_Dcurr = NetcdfReadArray1D(ncid, "DCurr");
-  std::vector<double> reference_Dgeod = NetcdfReadArray1D(ncid, "DGeod");
+  std::vector<double> reference_DMerc =
+      NetcdfReadArray1D(ncid, "DMerc").value();
+  std::vector<double> reference_Dshear =
+      NetcdfReadArray1D(ncid, "DShear").value();
+  std::vector<double> reference_Dwell =
+      NetcdfReadArray1D(ncid, "DWell").value();
+  std::vector<double> reference_Dcurr =
+      NetcdfReadArray1D(ncid, "DCurr").value();
+  std::vector<double> reference_Dgeod =
+      NetcdfReadArray1D(ncid, "DGeod").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     EXPECT_TRUE(IsCloseRelAbs(reference_DMerc[jF], wout.DMerc[jF], tolerance));
     EXPECT_TRUE(
@@ -299,7 +323,8 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
     EXPECT_TRUE(IsCloseRelAbs(reference_Dgeod[jF], wout.DGeod[jF], tolerance));
   }  // jF
 
-  std::vector<double> reference_equif = NetcdfReadArray1D(ncid, "equif");
+  std::vector<double> reference_equif =
+      NetcdfReadArray1D(ncid, "equif").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     EXPECT_TRUE(IsCloseRelAbs(reference_equif[jF], wout.equif[jF], tolerance));
   }
@@ -309,15 +334,17 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
   // -------------------
   // mode numbers for Fourier coefficient arrays below
 
-  std::vector<double> reference_xm = NetcdfReadArray1D(ncid, "xm");
-  std::vector<double> reference_xn = NetcdfReadArray1D(ncid, "xn");
+  std::vector<double> reference_xm = NetcdfReadArray1D(ncid, "xm").value();
+  std::vector<double> reference_xn = NetcdfReadArray1D(ncid, "xn").value();
   for (int mn = 0; mn < wout.mnmax; ++mn) {
     EXPECT_EQ(wout.xm[mn], reference_xm[mn]);
     EXPECT_EQ(wout.xn[mn], reference_xn[mn]);
   }  // mn
 
-  std::vector<double> reference_xm_nyq = NetcdfReadArray1D(ncid, "xm_nyq");
-  std::vector<double> reference_xn_nyq = NetcdfReadArray1D(ncid, "xn_nyq");
+  std::vector<double> reference_xm_nyq =
+      NetcdfReadArray1D(ncid, "xm_nyq").value();
+  std::vector<double> reference_xn_nyq =
+      NetcdfReadArray1D(ncid, "xn_nyq").value();
   for (int mn_nyq = 0; mn_nyq < wout.mnmax_nyq; ++mn_nyq) {
     EXPECT_EQ(wout.xm_nyq[mn_nyq], reference_xm_nyq[mn_nyq]);
     EXPECT_EQ(wout.xn_nyq[mn_nyq], reference_xn_nyq[mn_nyq]);
@@ -326,8 +353,10 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
   // -------------------
   // stellarator-symmetric Fourier coefficients
 
-  std::vector<double> reference_raxis_cc = NetcdfReadArray1D(ncid, "raxis_cc");
-  std::vector<double> reference_zaxis_cs = NetcdfReadArray1D(ncid, "zaxis_cs");
+  std::vector<double> reference_raxis_cc =
+      NetcdfReadArray1D(ncid, "raxis_cc").value();
+  std::vector<double> reference_zaxis_cs =
+      NetcdfReadArray1D(ncid, "zaxis_cs").value();
   for (int n = 0; n <= wout.ntor; ++n) {
     EXPECT_TRUE(
         IsCloseRelAbs(reference_raxis_cc[n], wout.raxis_cc[n], tolerance));
@@ -336,9 +365,9 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
   }  // n
 
   std::vector<std::vector<double>> reference_rmnc =
-      NetcdfReadArray2D(ncid, "rmnc");
+      NetcdfReadArray2D(ncid, "rmnc").value();
   std::vector<std::vector<double>> reference_zmns =
-      NetcdfReadArray2D(ncid, "zmns");
+      NetcdfReadArray2D(ncid, "zmns").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     for (int mn = 0; mn < s.mnmax; ++mn) {
       EXPECT_TRUE(
@@ -349,7 +378,7 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
   }  // jF
 
   std::vector<std::vector<double>> reference_lmns =
-      NetcdfReadArray2D(ncid, "lmns");
+      NetcdfReadArray2D(ncid, "lmns").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     for (int mn = 0; mn < s.mnmax; ++mn) {
       EXPECT_TRUE(
@@ -358,19 +387,19 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
   }  // jF
 
   std::vector<std::vector<double>> reference_gmnc =
-      NetcdfReadArray2D(ncid, "gmnc");
+      NetcdfReadArray2D(ncid, "gmnc").value();
   std::vector<std::vector<double>> reference_bmnc =
-      NetcdfReadArray2D(ncid, "bmnc");
+      NetcdfReadArray2D(ncid, "bmnc").value();
   std::vector<std::vector<double>> reference_bsubumnc =
-      NetcdfReadArray2D(ncid, "bsubumnc");
+      NetcdfReadArray2D(ncid, "bsubumnc").value();
   std::vector<std::vector<double>> reference_bsubvmnc =
-      NetcdfReadArray2D(ncid, "bsubvmnc");
+      NetcdfReadArray2D(ncid, "bsubvmnc").value();
   std::vector<std::vector<double>> reference_bsubsmns =
-      NetcdfReadArray2D(ncid, "bsubsmns");
+      NetcdfReadArray2D(ncid, "bsubsmns").value();
   std::vector<std::vector<double>> reference_bsupumnc =
-      NetcdfReadArray2D(ncid, "bsupumnc");
+      NetcdfReadArray2D(ncid, "bsupumnc").value();
   std::vector<std::vector<double>> reference_bsupvmnc =
-      NetcdfReadArray2D(ncid, "bsupvmnc");
+      NetcdfReadArray2D(ncid, "bsupvmnc").value();
   for (int jF = 0; jF < fc.ns; ++jF) {
     for (int mn_nyq = 0; mn_nyq < s.mnmax_nyq; ++mn_nyq) {
       EXPECT_TRUE(IsCloseRelAbs(reference_gmnc[jF][mn_nyq],
@@ -395,9 +424,9 @@ TEST_P(WOutFileContentsTest, CheckWOutFileContents) {
 
   if (s.lasym) {
     std::vector<double> reference_raxis_cs =
-        NetcdfReadArray1D(ncid, "raxis_cs");
+        NetcdfReadArray1D(ncid, "raxis_cs").value();
     std::vector<double> reference_zaxis_cc =
-        NetcdfReadArray1D(ncid, "zaxis_cc");
+        NetcdfReadArray1D(ncid, "zaxis_cc").value();
     for (int n = 0; n <= wout.ntor; ++n) {
       EXPECT_TRUE(
           IsCloseRelAbs(reference_raxis_cs[n], wout.raxis_cs[n], tolerance));
@@ -556,16 +585,17 @@ TEST(SplineProfileEquilibrium, CthLikeCubicSplinePressureMatchesFortranGolden) {
   };
 
   // scalar physics quantities
-  scalar("volume_p", NetcdfReadDouble(ncid, "volume_p"), wout.volume);
-  scalar("betatotal", NetcdfReadDouble(ncid, "betatotal"), wout.betatotal);
-  scalar("aspect", NetcdfReadDouble(ncid, "aspect"), wout.aspect);
-  scalar("b0", NetcdfReadDouble(ncid, "b0"), wout.b0);
-  scalar("wp", NetcdfReadDouble(ncid, "wp"), wout.wp);
-  scalar("wb", NetcdfReadDouble(ncid, "wb"), wout.wb);
-  scalar("rbtor", NetcdfReadDouble(ncid, "rbtor"), wout.rbtor);
-  scalar("ctor", NetcdfReadDouble(ncid, "ctor"), wout.ctor);
-  scalar("Aminor_p", NetcdfReadDouble(ncid, "Aminor_p"), wout.Aminor_p);
-  scalar("Rmajor_p", NetcdfReadDouble(ncid, "Rmajor_p"), wout.Rmajor_p);
+  scalar("volume_p", NetcdfReadDouble(ncid, "volume_p").value(), wout.volume);
+  scalar("betatotal", NetcdfReadDouble(ncid, "betatotal").value(),
+         wout.betatotal);
+  scalar("aspect", NetcdfReadDouble(ncid, "aspect").value(), wout.aspect);
+  scalar("b0", NetcdfReadDouble(ncid, "b0").value(), wout.b0);
+  scalar("wp", NetcdfReadDouble(ncid, "wp").value(), wout.wp);
+  scalar("wb", NetcdfReadDouble(ncid, "wb").value(), wout.wb);
+  scalar("rbtor", NetcdfReadDouble(ncid, "rbtor").value(), wout.rbtor);
+  scalar("ctor", NetcdfReadDouble(ncid, "ctor").value(), wout.ctor);
+  scalar("Aminor_p", NetcdfReadDouble(ncid, "Aminor_p").value(), wout.Aminor_p);
+  scalar("Rmajor_p", NetcdfReadDouble(ncid, "Rmajor_p").value(), wout.Rmajor_p);
 
   // 1D radial profiles, including the spline-driven pressure
   std::vector<double> wpresf(fc.ns), wpres(fc.ns), wmass(fc.ns), wiotaf(fc.ns),
@@ -578,38 +608,38 @@ TEST(SplineProfileEquilibrium, CthLikeCubicSplinePressureMatchesFortranGolden) {
     wiotas[jF] = wout.iotas[jF];
     wjcurv[jF] = wout.jcurv[jF];
   }
-  compare("presf", NetcdfReadArray1D(ncid, "presf"), wpresf);
-  compare("pres", NetcdfReadArray1D(ncid, "pres"), wpres);
-  compare("mass", NetcdfReadArray1D(ncid, "mass"), wmass);
-  compare("iotaf", NetcdfReadArray1D(ncid, "iotaf"), wiotaf);
-  compare("iotas", NetcdfReadArray1D(ncid, "iotas"), wiotas);
-  compare("jcurv", NetcdfReadArray1D(ncid, "jcurv"), wjcurv);
+  compare("presf", NetcdfReadArray1D(ncid, "presf").value(), wpresf);
+  compare("pres", NetcdfReadArray1D(ncid, "pres").value(), wpres);
+  compare("mass", NetcdfReadArray1D(ncid, "mass").value(), wmass);
+  compare("iotaf", NetcdfReadArray1D(ncid, "iotaf").value(), wiotaf);
+  compare("iotas", NetcdfReadArray1D(ncid, "iotas").value(), wiotas);
+  compare("jcurv", NetcdfReadArray1D(ncid, "jcurv").value(), wjcurv);
 
   // flux-surface geometry
   auto [r_ref, r_val] =
-      flatten(NetcdfReadArray2D(ncid, "rmnc"), fc.ns, s.mnmax,
+      flatten(NetcdfReadArray2D(ncid, "rmnc").value(), fc.ns, s.mnmax,
               [&](int mn, int jF) { return wout.rmnc(mn, jF); });
   compare("rmnc", r_ref, r_val);
   auto [z_ref, z_val] =
-      flatten(NetcdfReadArray2D(ncid, "zmns"), fc.ns, s.mnmax,
+      flatten(NetcdfReadArray2D(ncid, "zmns").value(), fc.ns, s.mnmax,
               [&](int mn, int jF) { return wout.zmns(mn, jF); });
   compare("zmns", z_ref, z_val);
   auto [l_ref, l_val] =
-      flatten(NetcdfReadArray2D(ncid, "lmns"), fc.ns, s.mnmax,
+      flatten(NetcdfReadArray2D(ncid, "lmns").value(), fc.ns, s.mnmax,
               [&](int mn, int jF) { return wout.lmns(mn, jF); });
   compare("lmns", l_ref, l_val);
 
   // magnetic field on the Nyquist mode set
   auto [b_ref, b_val] =
-      flatten(NetcdfReadArray2D(ncid, "bmnc"), fc.ns, s.mnmax_nyq,
+      flatten(NetcdfReadArray2D(ncid, "bmnc").value(), fc.ns, s.mnmax_nyq,
               [&](int mn, int jF) { return wout.bmnc(mn, jF); });
   compare("bmnc", b_ref, b_val);
   auto [bu_ref, bu_val] =
-      flatten(NetcdfReadArray2D(ncid, "bsubumnc"), fc.ns, s.mnmax_nyq,
+      flatten(NetcdfReadArray2D(ncid, "bsubumnc").value(), fc.ns, s.mnmax_nyq,
               [&](int mn, int jF) { return wout.bsubumnc(mn, jF); });
   compare("bsubumnc", bu_ref, bu_val);
   auto [bv_ref, bv_val] =
-      flatten(NetcdfReadArray2D(ncid, "bsubvmnc"), fc.ns, s.mnmax_nyq,
+      flatten(NetcdfReadArray2D(ncid, "bsubvmnc").value(), fc.ns, s.mnmax_nyq,
               [&](int mn, int jF) { return wout.bsubvmnc(mn, jF); });
   compare("bsubvmnc", bv_ref, bv_val);
 
@@ -704,17 +734,18 @@ TEST(SolovevFreeBoundary, MatchesEducationalVmecGolden) {
   };
 
   // scalar physics quantities
-  scalar("volume_p", NetcdfReadDouble(ncid, "volume_p"), wout.volume);
-  scalar("betatotal", NetcdfReadDouble(ncid, "betatotal"), wout.betatotal);
-  scalar("aspect", NetcdfReadDouble(ncid, "aspect"), wout.aspect);
-  scalar("b0", NetcdfReadDouble(ncid, "b0"), wout.b0);
-  scalar("wp", NetcdfReadDouble(ncid, "wp"), wout.wp);
-  scalar("wb", NetcdfReadDouble(ncid, "wb"), wout.wb);
-  scalar("rbtor", NetcdfReadDouble(ncid, "rbtor"), wout.rbtor);
-  scalar("ctor", NetcdfReadDouble(ncid, "ctor"), wout.ctor);
-  scalar("Aminor_p", NetcdfReadDouble(ncid, "Aminor_p"), wout.Aminor_p);
-  scalar("Rmajor_p", NetcdfReadDouble(ncid, "Rmajor_p"), wout.Rmajor_p);
-  scalar("volavgB", NetcdfReadDouble(ncid, "volavgB"), wout.volavgB);
+  scalar("volume_p", NetcdfReadDouble(ncid, "volume_p").value(), wout.volume);
+  scalar("betatotal", NetcdfReadDouble(ncid, "betatotal").value(),
+         wout.betatotal);
+  scalar("aspect", NetcdfReadDouble(ncid, "aspect").value(), wout.aspect);
+  scalar("b0", NetcdfReadDouble(ncid, "b0").value(), wout.b0);
+  scalar("wp", NetcdfReadDouble(ncid, "wp").value(), wout.wp);
+  scalar("wb", NetcdfReadDouble(ncid, "wb").value(), wout.wb);
+  scalar("rbtor", NetcdfReadDouble(ncid, "rbtor").value(), wout.rbtor);
+  scalar("ctor", NetcdfReadDouble(ncid, "ctor").value(), wout.ctor);
+  scalar("Aminor_p", NetcdfReadDouble(ncid, "Aminor_p").value(), wout.Aminor_p);
+  scalar("Rmajor_p", NetcdfReadDouble(ncid, "Rmajor_p").value(), wout.Rmajor_p);
+  scalar("volavgB", NetcdfReadDouble(ncid, "volavgB").value(), wout.volavgB);
 
   // 1D radial profiles (pressure and rotational transform)
   std::vector<double> wpresf(fc.ns), wpres(fc.ns), wiotaf(fc.ns), wiotas(fc.ns);
@@ -724,36 +755,36 @@ TEST(SolovevFreeBoundary, MatchesEducationalVmecGolden) {
     wiotaf[jF] = wout.iotaf[jF];
     wiotas[jF] = wout.iotas[jF];
   }
-  compare("presf", NetcdfReadArray1D(ncid, "presf"), wpresf);
-  compare("pres", NetcdfReadArray1D(ncid, "pres"), wpres);
-  compare("iotaf", NetcdfReadArray1D(ncid, "iotaf"), wiotaf);
-  compare("iotas", NetcdfReadArray1D(ncid, "iotas"), wiotas);
+  compare("presf", NetcdfReadArray1D(ncid, "presf").value(), wpresf);
+  compare("pres", NetcdfReadArray1D(ncid, "pres").value(), wpres);
+  compare("iotaf", NetcdfReadArray1D(ncid, "iotaf").value(), wiotaf);
+  compare("iotas", NetcdfReadArray1D(ncid, "iotas").value(), wiotas);
 
   // flux-surface geometry
   auto [r_ref, r_val] =
-      flatten(NetcdfReadArray2D(ncid, "rmnc"), fc.ns, s.mnmax,
+      flatten(NetcdfReadArray2D(ncid, "rmnc").value(), fc.ns, s.mnmax,
               [&](int mn, int jF) { return wout.rmnc(mn, jF); });
   compare("rmnc", r_ref, r_val);
   auto [z_ref, z_val] =
-      flatten(NetcdfReadArray2D(ncid, "zmns"), fc.ns, s.mnmax,
+      flatten(NetcdfReadArray2D(ncid, "zmns").value(), fc.ns, s.mnmax,
               [&](int mn, int jF) { return wout.zmns(mn, jF); });
   compare("zmns", z_ref, z_val);
   auto [l_ref, l_val] =
-      flatten(NetcdfReadArray2D(ncid, "lmns"), fc.ns, s.mnmax,
+      flatten(NetcdfReadArray2D(ncid, "lmns").value(), fc.ns, s.mnmax,
               [&](int mn, int jF) { return wout.lmns(mn, jF); });
   compare("lmns", l_ref, l_val);
 
   // magnetic field on the Nyquist mode set
   auto [b_ref, b_val] =
-      flatten(NetcdfReadArray2D(ncid, "bmnc"), fc.ns, s.mnmax_nyq,
+      flatten(NetcdfReadArray2D(ncid, "bmnc").value(), fc.ns, s.mnmax_nyq,
               [&](int mn, int jF) { return wout.bmnc(mn, jF); });
   compare("bmnc", b_ref, b_val);
   auto [bu_ref, bu_val] =
-      flatten(NetcdfReadArray2D(ncid, "bsubumnc"), fc.ns, s.mnmax_nyq,
+      flatten(NetcdfReadArray2D(ncid, "bsubumnc").value(), fc.ns, s.mnmax_nyq,
               [&](int mn, int jF) { return wout.bsubumnc(mn, jF); });
   compare("bsubumnc", bu_ref, bu_val);
   auto [bv_ref, bv_val] =
-      flatten(NetcdfReadArray2D(ncid, "bsubvmnc"), fc.ns, s.mnmax_nyq,
+      flatten(NetcdfReadArray2D(ncid, "bsubvmnc").value(), fc.ns, s.mnmax_nyq,
               [&](int mn, int jF) { return wout.bsubvmnc(mn, jF); });
   compare("bsubvmnc", bv_ref, bv_val);
 
@@ -764,8 +795,8 @@ TEST(SolovevFreeBoundary, MatchesEducationalVmecGolden) {
     wjcuru[jF] = wout.jcuru[jF];
     wjcurv[jF] = wout.jcurv[jF];
   }
-  compare("jcuru", NetcdfReadArray1D(ncid, "jcuru"), wjcuru);
-  compare("jcurv", NetcdfReadArray1D(ncid, "jcurv"), wjcurv);
+  compare("jcuru", NetcdfReadArray1D(ncid, "jcuru").value(), wjcuru);
+  compare("jcurv", NetcdfReadArray1D(ncid, "jcurv").value(), wjcurv);
 
   ASSERT_EQ(nc_close(ncid), NC_NOERR);
 
diff --git a/src/vmecpp/cpp/vmecpp_large_cpp_tests/free_boundary/mgrid_provider/mgrid_provider_test.cc b/src/vmecpp/cpp/vmecpp_large_cpp_tests/free_boundary/mgrid_provider/mgrid_provider_test.cc
index aa52e884b..5cc996a2c 100644
--- a/src/vmecpp/cpp/vmecpp_large_cpp_tests/free_boundary/mgrid_provider/mgrid_provider_test.cc
+++ b/src/vmecpp/cpp/vmecpp_large_cpp_tests/free_boundary/mgrid_provider/mgrid_provider_test.cc
@@ -104,21 +104,21 @@ TEST_P(LoadMGridTest, CheckLoadMGrid) {
   ASSERT_EQ(nc_open(vmec_indata->mgrid_file.c_str(), NC_NOWRITE, &ncid),
             NC_NOERR);
 
-  const int number_of_field_periods = NetcdfReadInt(ncid, "nfp");
+  const int number_of_field_periods = NetcdfReadInt(ncid, "nfp").value();
 
-  const int number_of_r_grid_points = NetcdfReadInt(ncid, "ir");
-  const double r_grid_minimum = NetcdfReadDouble(ncid, "rmin");
-  const double r_grid_maximum = NetcdfReadDouble(ncid, "rmax");
+  const int number_of_r_grid_points = NetcdfReadInt(ncid, "ir").value();
+  const double r_grid_minimum = NetcdfReadDouble(ncid, "rmin").value();
+  const double r_grid_maximum = NetcdfReadDouble(ncid, "rmax").value();
   const double r_grid_increment =
       (r_grid_maximum - r_grid_minimum) / (number_of_r_grid_points - 1.0);
 
-  const int number_of_z_grid_points = NetcdfReadInt(ncid, "jz");
-  const double z_grid_minimum = NetcdfReadDouble(ncid, "zmin");
-  const double z_grid_maximum = NetcdfReadDouble(ncid, "zmax");
+  const int number_of_z_grid_points = NetcdfReadInt(ncid, "jz").value();
+  const double z_grid_minimum = NetcdfReadDouble(ncid, "zmin").value();
+  const double z_grid_maximum = NetcdfReadDouble(ncid, "zmax").value();
   const double z_grid_increment =
       (z_grid_maximum - z_grid_minimum) / (number_of_z_grid_points - 1.0);
 
-  const int number_of_phi_grid_points = NetcdfReadInt(ncid, "kp");
+  const int number_of_phi_grid_points = NetcdfReadInt(ncid, "kp").value();
   const double phi_grid_increment =
       2.0 * M_PI / (number_of_phi_grid_points * number_of_field_periods);
 
diff --git a/src/vmecpp/cpp/vmecpp_large_cpp_tests/vmec/output_quantities/output_quantities_test.cc b/src/vmecpp/cpp/vmecpp_large_cpp_tests/vmec/output_quantities/output_quantities_test.cc
index 68b062c89..87cd81582 100644
--- a/src/vmecpp/cpp/vmecpp_large_cpp_tests/vmec/output_quantities/output_quantities_test.cc
+++ b/src/vmecpp/cpp/vmecpp_large_cpp_tests/vmec/output_quantities/output_quantities_test.cc
@@ -1358,9 +1358,9 @@ TEST_P(CurrentDensityTest, CheckCurrentDensityFourierCoefficients) {
 
   // Read 2D arrays: Fortran stores as (ns, mnmax_nyq), C++ as (mnmax_nyq, ns)
   const std::vector<std::vector<double>> ref_currumnc =
-      NetcdfReadArray2D(ncid, "currumnc");
+      NetcdfReadArray2D(ncid, "currumnc").value();
   const std::vector<std::vector<double>> ref_currvmnc =
-      NetcdfReadArray2D(ncid, "currvmnc");
+      NetcdfReadArray2D(ncid, "currvmnc").value();
   nc_close(ncid);
 
   const int ns = static_cast<int>(ref_currumnc.size());
diff --git a/tests/test_init.py b/tests/test_init.py
index 7a4779a1e..10604d6a5 100644
--- a/tests/test_init.py
+++ b/tests/test_init.py
@@ -51,8 +51,7 @@ def test_run(max_threads, input_file, verbose):
     [
         ("cma.json", RuntimeError),  # Invalid netcdf
         ("does_not_exist", RuntimeError),
-        # TODO(jurasic) Enable test after switching netcdf_io to absl::Status
-        # ("wout_cma.nc", RuntimeError),  # Valid netcdf, but invalid mgrid
+        ("wout_cma.nc", RuntimeError),  # Valid netcdf, but invalid mgrid
     ],
 )
 def test_raise_invalid_mgrid(mgrid_path: str, expected_exception):

From 2449ddc9c59d0a4cbc676fbe30b0dc1dfdc72089 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 14:55:50 +0200
Subject: [PATCH 54/79] pybind: Hessian-vector product inside VMEC++ + internal
 Newton-Krylov (#580)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

* build: bump CMake abseil pin to 20260107.1 for Clang >= 21

The CMake FetchContent abseil pin (2024-08) fails to compile under
Clang >= 21: absl::Nonnull SFINAE in absl/strings/ascii.cc and the
numbers.cc nullability annotations are rejected by the newer frontend.
Bump to the 20260107.1 LTS, which compiles cleanly under Clang 21.1.8
and GCC. Clang is the compiler required for the Enzyme autodiff build.

The Bazel build keeps its own (BCR) abseil pin and is unaffected.

* enzyme: opt-in Clang/Enzyme build option and AD smoke test

Add VMECPP_ENABLE_ENZYME (OFF by default), which requires a Clang
compiler and a ClangEnzyme plugin path and builds a self-contained
autodiff smoke test. The test differentiates a scalar objective written
over Eigen::Map'd caller buffers and checks reverse- and forward-mode
Enzyme gradients against the closed form and central finite differences.

enzyme.h documents the intrinsic ABI and the allocation constraint that
shapes the differentiable kernels: Enzyme cannot track Eigen's aligned
allocator, so differentiable paths use Eigen::Map over caller-owned
buffers and avoid heap expression temporaries.

With the option off the build is unchanged.

* pybind: expose the unpreconditioned internal-basis gradient

Add a precondition flag to VmecModel.evaluate (default true, unchanged
behaviour). With precondition=false the forward model returns at the
INVARIANT_RESIDUALS checkpoint, so get_forces() yields the raw,
unpreconditioned force: the gradient of VMEC's augmented functional (MHD
energy plus the spectral-condensation and lambda constraints) with
respect to the decomposed internal-basis state.

This is the consistent state/gradient pair an external optimizer needs
to minimise in VMEC's own basis. The native solver's preconditioned
search direction (precondition=true) is a different vector; the raw
gradient is the equilibrium residual and vanishes at convergence.

Tests: raw force is finite and differs in direction from the
preconditioned force, and drops by >1e6 from the initial guess to the
converged equilibrium.

* examples: drive VMEC++ from external optimizers in the internal basis

Treat the equilibrium as the root problem F(x) = 0, where F is the raw
internal-basis force (gradient of VMEC's augmented functional) exposed by
evaluate(precondition=False). Wire it to two solvers that reuse VMEC++'s
forward model: native-style preconditioned descent and Jacobian-free
Newton-Krylov (matrix-free Hessian information). Both reach the native
solver's equilibrium.

This is the external-differentiability path: VMEC++ as a differentiable
equilibrium component an outside optimizer can drive. Quasi-Newton
root-finders without a preconditioner diverge on this stiff system, which
motivates exposing VMEC's preconditioner as an operator next.

Tests assert both solvers reach force balance and recover the native
energy and state.

* pybind: expose VMEC preconditioner as an operator; preconditioned JFNK

Add VmecModel.apply_preconditioner(v): applies VMEC's preconditioner
M^-1 (m=1, radial, lambda steps) to a vector in the decomposed basis.
M^-1 is VMEC's hand-built approximate inverse Hessian; this exposes it
as a reusable linear operator for preconditioned Krylov / quasi-Newton
and for the Hessian solve in adjoint sensitivities. It requires a prior
evaluate(precondition=true), which assembles the radial preconditioner.

Validated exactly: apply_preconditioner(raw force) equals the native
preconditioned search direction; the operator is linear and, once
assembled, state-invariant.

Use it as the inner Krylov preconditioner in Newton-Krylov: on solovev
(ns=11) this cuts force evaluations from 2242 to 505 (4.4x) versus
unpreconditioned JFNK, converging to the same equilibrium.

* pybind: Hessian-vector product inside VMEC++; internal Newton-Krylov

Add VmecModel.hessian_vector_product(v): the curvature of VMEC's
augmented functional, computed inside VMEC++ as a central directional
derivative of the analytic force (its gradient). The force is exact; only
the directional step is finite-differenced. Add a force_eval_count for
fair cross-optimizer cost comparison (counts evaluations hidden in the
Hessian-vector products).

Drive a true Newton-Krylov from this HVP plus the preconditioner: it
reaches the equilibrium in ~7 outer iterations (second order) versus
~1300 descent steps. This is the inside-the-solver Hessian path; together
with the external optimizers it gives differentiability inside and out.

Benchmark (solovev, ns=11, force evals counted in VMEC++):
  preconditioned descent          2606 evals  1302 iters
  Newton-Krylov (JFNK)            2243 evals
  Newton-Krylov (preconditioned)   507 evals
  Newton (VMEC++ HVP + M^-1)      9194 evals     7 iters

The HVP-Newton's higher force-eval count (two evals per finite-difference
HVP) is what the exact Enzyme Hessian will remove.

* ideal_mhd_model: make computeMHDForces allocation-free

The force kernel allocated 17 dynamic Eigen vectors per radial surface (the
_o half-grid quantities and the avg/wavg surface averages). Move them to
preallocated per-thread ThreadLocalStorage scratch and assign in place, so
the radial loop allocates nothing.

Two benefits: it removes per-surface heap churn from the hot force loop, and
it makes the kernel differentiable by Enzyme, which cannot trace dynamic
Eigen temporaries (forward and reverse mode both abort on them). This is the
allocation-free prerequisite for an exact autodiff Hessian.

Pure refactor, identical arithmetic. Verified bit-for-bit: vmec_standalone
MHD energy unchanged on solovev (2.548352e+00) and cth_like_fixed_bdy
(5.057191e-02).

* examples: globalize HVP Newton with a backtracking line search

The full Newton step overshoots on stiff 3D equilibria (cth_like stalled
at the iteration cap with ||F|| ~ 5e-2). Add a backtracking line search on
||F|| so each step is damped to a decrease. With it the HVP-Newton
converges on cth_like in 9 outer iterations (||F|| = 1.8e-10) and still
converges solovev in 8.

* dft_toroidal: make ForcesToFourier allocation-free

The forces transform materialized two per-(surface,m,zeta) Eigen temporaries
(tempR_seg, tempZ_seg) inside the inner loop. Reuse per-thread scratch
instead, so the whole FFTX-off force path (geometryFromFourier,
computeJacobian/Metric/BContra/BCo, pressureAndEnergies, computeMHDForces,
forcesToFourier) is now allocation-free end to end.

Same arithmetic as the previous .eval(); verified bit-for-bit: solovev
2.548352e+00, cth_like_fixed_bdy 5.057191e-02.

* enzyme: exact autodiff of the VMEC Jacobian kernel (forward vs reverse)

Demonstrate exact automatic differentiation of a real VMEC nonlinear
kernel. JacobianKernel reproduces IdealMhdModel::computeJacobian (half-grid
r12/ru12/zu12/rs/zs and the Jacobian tau), written allocation-free over flat
buffers, which is the form Enzyme differentiates.

For L = 0.5||outputs||^2 the test computes dL/dgeom by reverse mode and the
directional derivative dL.v by forward mode, checks both against central
finite differences, and against each other:

  reverse dL.v vs FD : 1.9e-9
  forward dL.v vs FD : 1.9e-9
  forward vs reverse : 2.9e-15
  performance: reverse ~16 us/pass (full gradient), forward ~16 us/pass
               (one direction)

Reverse returns the whole gradient per pass and wins for a scalar gradient;
forward is the cheaper primitive for a single Jacobian/Hessian-vector
product. tau is nonlinear in the geometry, so this kernel's Jacobian is a
genuine building block of the exact MHD force Hessian; the remaining force
chain follows the same allocation-free pattern.

* ideal_mhd_model: share the Jacobian kernel between solver and autodiff

Move the half-grid Jacobian arithmetic into jacobian_kernel.h
(ComputeHalfGridJacobian), allocation-free over flat buffers. Production
computeJacobian now calls it (followed by the unchanged Jacobian-sign
check), and the Enzyme forward/reverse test differentiates the same
kernel: one implementation, no duplication.

Bit-exact: vmec_standalone MHD energy unchanged on solovev
(2.548352e+00) and cth_like_fixed_bdy (5.057191e-02). Autodiff test still
matches finite differences and agrees forward vs reverse to 3e-15.

* ideal_mhd_model: share the metric kernel (gsqrt, guu, guv, gvv)

Extract computeMetricElements into the shared, allocation-free kernel
ComputeMetricElements (metric_kernel.h), over flat buffers, and call it
from the solver. guv and the 3D part of gvv are computed only when
lthreed, matching the original. This is the second force-chain kernel made
Enzyme-differentiable (composed into the exact Hessian-vector product
later), following the Jacobian kernel pattern.

Bit-exact: vmec_standalone MHD energy unchanged on solovev (2.548352e+00,
2D) and cth_like_fixed_bdy (5.057191e-02, 3D path with guv/gvv).

* ideal_mhd_model: share the contravariant-field kernel (bsupu, bsupv)

Factor the bsupu/bsupv arithmetic out of computeBContra into the shared,
allocation-free kernel ComputeBsupContra (bcontra_kernel.h). The lambda
normalization (lamscale, + phi') and the chi'/iota profile and
toroidal-current-constraint logic stay in the solver verbatim, since they
mutate state and update profiles; only the differentiable field arithmetic
moves to the shared kernel.

Bit-exact across 1 and 4 threads (so the ghost-cell radial partitioning is
exercised) on solovev (2.548352e+00, 2D) and cth_like_fixed_bdy
(5.057191e-02, 3D).

* ideal_mhd_model: share the covariant-field kernel (bsubu, bsubv)

Extract the metric index-lowering (bsubu = guu B^u + guv B^v, bsubv = guv
B^u + gvv B^v; guv absent in 2D) from computeBCo into the shared,
allocation-free kernel ComputeBCo (bco_kernel.h).

Bit-exact across 1 and 4 threads on solovev (2.548352e+00) and
cth_like_fixed_bdy (5.057191e-02).

* ideal_mhd_model: share the magnetic-pressure kernel

Extract the field-dependent magnetic pressure |B|^2/2 = 0.5(B^u B_u + B^v
B_v) from pressureAndEnergies into the shared, allocation-free kernel
ComputeMagneticPressure (pressure_kernel.h). The kinetic-pressure profile
and the energy volume integrals stay in the solver.

Bit-exact across 1 and 4 threads on solovev (2.548352e+00) and
cth_like_fixed_bdy (5.057191e-02). Completes the point-local nonlinear
force-chain kernels (Jacobian, metric, B^contra, B_cov, pressure).

* ideal_mhd_model: share the MHD force-density kernel

Extract computeMHDForces' real-space force-density assembly (armn/azmn/
brmn/bzmn, and crmn/czmn in 3D, even+odd) into the shared, allocation-free
kernel ComputeMHDForceDensity (mhdforce_kernel.h). The Eigen arithmetic is
preserved verbatim over flat-buffer Eigen::Map views with caller-owned
handover/average scratch, so it is bit-for-bit identical.

This is the sixth and final point-local force-chain kernel; the six
(Jacobian, metric, B^contra, B_cov, pressure, force) now form the local map
geometry -> force density, ready to compose into the exact Hessian-vector
product. (This branch also merges the allocation-free force kernel, #12,
which removes the per-surface heap temporaries this extraction relies on.)

Bit-exact across 1 and 4 threads on solovev (2.548352e+00) and
cth_like_fixed_bdy (5.057191e-02).

* enzyme: exact Hessian of the composed local force map

Compose the six shared force-chain kernels (Jacobian, metric, B^contra,
B_cov, magnetic pressure, MHD force density) into the single local map
g: real-space geometry -> real-space force density, the nonlinear core of
VMEC's force. The full MHD force is T^T . g . T with the linear spectral
transforms; the exact force Hessian-vector product is therefore
T^T . J_g . T . v, and this provides J_g by autodiff.

The new test takes the Jacobian of g by forward and reverse Enzyme modes
over flat allocation-free buffers, checks both against central finite
differences and against each other, and times one forward Jacobian-vector
pass against the two force evaluations a finite-difference HVP costs.

* ideal_mhd_model: share the hybrid lambda-force kernel

Extract hybridLambdaForce's full-grid lambda force (blmn, and clmn in 3D)
into lambda_force_kernel.h (ComputeHybridLambdaForce), shared between the
solver and the Enzyme autodiff path. The method drops from 115 lines to a
single kernel call; the OpenMP barriers stay in the method.

The kernel is allocation-free over flat buffers and preserves the radial
sweep that carries the inside half-grid point in scratch and shifts it
outward each surface, plus the blend of the two bsubv interpolations.

This is the lambda-force piece of the augmented functional, the second
nonlinear force-density term after the MHD force chain.

* ideal_mhd_model: share the constraint-force kernels

Extract the two local (non-transform) pieces of the spectral-condensation
constraint force into constraint_force_kernel.h, shared between the solver
and the Enzyme autodiff path:

- ComputeEffectiveConstraintForce: gConEff = (rCon-rCon0) ru + (zCon-zCon0) zu
  (effectiveConstraintForce), skipping the axis surface.
- AddConstraintForces: add the bandpass-filtered gCon back into the MHD R/Z
  forces and write frcon/fzcon (the constraint part of assembleTotalForces).

The Fourier-space bandpass between them stays the shared free function
deAliasConstraintForce; the free-boundary rBSq contribution stays in
assembleTotalForces. Allocation-free over flat buffers.

This completes the local force-density terms of the augmented functional
(MHD + lambda + constraint), the nonlinear core of the exact Hessian.

* enzyme: extend the composed-force Hessian test with the lambda force

Add the hybrid lambda force (lambda_force_kernel.h) to the composed local
map g and differentiate the combined MHD-plus-lambda force density by
forward and reverse Enzyme modes. This proves J_g for the second nonlinear
force-density term, not just the MHD force chain.

The spectral-condensation constraint force also carries a linear Fourier
bandpass; it is validated end-to-end against the finite-difference HVP in
the pybind exact-HVP path rather than in this flat-buffer microtest.

* apply pre-commit formatting (ruff, docformatter, clang-format)

* apply pre-commit formatting (ruff, docformatter, clang-format)

* apply pre-commit formatting (ruff, docformatter, clang-format)

* apply pre-commit formatting (ruff, docformatter, clang-format)

* apply pre-commit formatting (ruff, docformatter, clang-format)

* apply pre-commit formatting (ruff, docformatter, clang-format)

* apply pre-commit formatting (ruff, docformatter, clang-format)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* bazel: declare force-chain kernel headers in ideal_mhd_model (sandbox fix)

* test: docformatter-format test_internal_gradient docstrings

Satisfies the docformatter pre-commit hook (was failing CI).

* test: docformatter-format external/internal optimizer test docstrings

Satisfies the docformatter pre-commit hook (was failing CI).

* ci: re-trigger (transient apt-403 on packages.microsoft.com)

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* ci: skip benchmark result upload on fork PRs (token is read-only)

The 'Compare benchmark result' step uses github-action-benchmark with
comment-on-alert and the GITHUB_TOKEN, which is read-only for pull requests from
forks -> 'Resource not accessible by integration'. Gate that step on the PR
coming from the same repo so fork PRs still run the benchmarks but skip the
write-back instead of failing.

* ci: build VMEC2000 from source so the compat test runs on numpy 2

The pinned vmec-0.0.6 cp310 wheel was f90wrapped against numpy 1.x. Under
the numpy 2.x that the test env now resolves, importing it dies in the
f90wrap array interface (f90wrap_vmec_input__array__rbc: 0-th dimension
must be fixed to 2 but got 4), so test_ensure_vmec2000_input_from_vmecpp_input
could never actually run on CI (and is currently red on main too, where the
wheel's runtime libs are not even installed).

Build VMEC2000 from upstream source with current f90wrap, which produces
numpy-2-compatible bindings. The recipe mirrors SIMSOPT's own CI
(hiddenSymmetries/VMEC2000, cmake/machines/ubuntu.json). An explicit
'import vmec' check in the install step surfaces any remaining problem here
rather than as a confusing test failure.

* test: skip vmecpp-only indata fields in the VMEC2000 compat subset

With VMEC2000 built from current upstream source, the compatibility test
runs for the first time and hits vmecpp indata fields that have no
counterpart in the legacy VMEC2000 INDATA namelist (e.g.
free_boundary_method), which raised AttributeError. The test explicitly
checks only the common subset, so guard the lookup with hasattr and skip
fields VMEC2000 does not have, instead of enumerating them one by one.

* build: pin abseil to the 20260107.1 commit hash

Pin the FetchContent abseil dependency to commit 255c84d (the exact
commit behind the 20260107.1 LTS tag) instead of the tag itself, so a
moved tag cannot change the dependency under us.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash, not the tag.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash for Clang >= 21.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash for Clang >= 21.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash for Clang >= 21.

* ci: sync VMEC2000-from-source build, benchmark fork guard, abseil commit pin

Bring this stack branch up to the corrected CI baseline (from #583/#564):
- tests.yaml: build VMEC2000 from the pinned source commit and cache the
  wheel; drop the unused FFTW/HDF5 dev packages.
- benchmarks.yaml: skip the result upload on fork PRs (read-only token).
- test_simsopt_compat.py: skip vmecpp-only INDATA fields.
- CMakeLists: pin abseil to the 20260107.1 commit hash for Clang >= 21.

* ci: cache and pin the VMEC2000-from-source build

Use the canonical recipe (cache the built wheel keyed on the pinned
source commit 728af8b, drop the unused FFTW/HDF5 dev packages) instead
of rebuilding VMEC2000 unpinned on every run.

* ideal_mhd_model: mark Jacobian kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: hoist ForcesToFourier scratch out of the inner loop

The allocation-free rewrite placed tempR_seg/tempZ_seg in a block-scope
thread_local inside the (jF, m, zeta) inner loop, which emits a
__tls_get_addr call and an init-guard branch every iteration. Declare
the two scratch vectors once at function scope instead: still
allocation-free in the hot loop and per-thread safe via the stack frame,
without the per-iteration TLS overhead. Same arithmetic; cma and w7x
wout are bit-for-bit unchanged.

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: hoist ForcesToFourier scratch out of the inner loop

The allocation-free rewrite placed tempR_seg/tempZ_seg in a block-scope
thread_local inside the (jF, m, zeta) inner loop, which emits a
__tls_get_addr call and an init-guard branch every iteration. Declare
the two scratch vectors once at function scope instead: still
allocation-free in the hot loop and per-thread safe via the stack frame,
without the per-iteration TLS overhead. Same arithmetic; cma and w7x
wout are bit-for-bit unchanged.

* ideal_mhd_model: hoist ForcesToFourier scratch out of the inner loop

The allocation-free rewrite placed tempR_seg/tempZ_seg in a block-scope
thread_local inside the (jF, m, zeta) inner loop, which emits a
__tls_get_addr call and an init-guard branch every iteration. Declare
the two scratch vectors once at function scope instead: still
allocation-free in the hot loop and per-thread safe via the stack frame,
without the per-iteration TLS overhead. Same arithmetic; cma and w7x
wout are bit-for-bit unchanged.

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: hoist ForcesToFourier scratch out of the inner loop

The allocation-free rewrite placed tempR_seg/tempZ_seg in a block-scope
thread_local inside the (jF, m, zeta) inner loop, which emits a
__tls_get_addr call and an init-guard branch every iteration. Declare
the two scratch vectors once at function scope instead: still
allocation-free in the hot loop and per-thread safe via the stack frame,
without the per-iteration TLS overhead. Same arithmetic; cma and w7x
wout are bit-for-bit unchanged.

* ideal_mhd_model: hoist ForcesToFourier scratch out of the inner loop

The allocation-free rewrite placed tempR_seg/tempZ_seg in a block-scope
thread_local inside the (jF, m, zeta) inner loop, which emits a
__tls_get_addr call and an init-guard branch every iteration. Declare
the two scratch vectors once at function scope instead: still
allocation-free in the hot loop and per-thread safe via the stack frame,
without the per-iteration TLS overhead. Same arithmetic; cma and w7x
wout are bit-for-bit unchanged.

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* ideal_mhd_model: mark Jacobian  metric kernel buffers __restrict

Raw double* kernel params over the same flat layout prevent the compiler
from vectorizing the pointwise loop (assumed aliasing), so on w7x these
kernels ran ~2x slower than the Eigen-expression code they replaced.
The buffers never overlap; mark them __restrict to restore SIMD. Enzyme
derivatives are unchanged (jacobian_kernel_autodiff + QS GN benchmark).

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* enzyme: run the AD smoke test through bazel instead of ctest

Move the Enzyme autodiff smoke test into the bazel test framework, which
owns every other C++ test in this repository, and drop the separate CMake
ctest path that nothing in CI exercised.

- vmecpp/common/enzyme/BUILD.bazel: an `enzyme` header library plus an
  `enzyme_smoke_test` cc_test. The test is tagged `manual` so the default
  GCC `bazel test //...` skips it (the Enzyme intrinsics only resolve under
  Clang with the plugin attached) and never tries to compile it with GCC.
- .bazelrc: a `--config=enzyme` that sets -O2 so the Enzyme optimization
  pass fires. Select Clang with CC/CXX and pass the plugin path the way
  -DVMECPP_ENZYME_PLUGIN did under CMake:
    CC=clang CXX=clang++ bazel test --config=enzyme \
      --copt=-fplugin=/path/to/ClangEnzyme-NN.so \
      //vmecpp/common/enzyme:enzyme_smoke_test
- CMakeLists.txt: remove the VMECPP_ENABLE_ENZYME option and the ctest
  registration it only existed to drive.

* ci: build ClangEnzyme and run the enzyme smoke test in CI

Add a GitHub Actions job that gives the Enzyme autodiff smoke test actual CI
coverage. It mirrors the EnzymeAD upstream recipe: install Clang/LLVM 21 from
apt.llvm.org, build a pinned ClangEnzyme-21 plugin (v0.0.264, the version this
stack is developed against) against the installed LLVM and Clang, then run the
bazel target under --config=enzyme with the plugin attached. The plugin build
is cached on the pinned ref so only the first run pays for it.

This is what the enzyme test needed beyond the bazel move: the default GCC
test_bazel job skips the manual-tagged target, so without a Clang/Enzyme job
nothing exercised it.

* output_quantities: compare jcuru/jcurv at the standard tolerance

The Jacobian-kernel refactor is structure-only, so drop the opt-in
current_density_tolerance loosening and compare current densities at the
same relabs tolerance as every other wout quantity.

* test: address review nits in test_internal_gradient

Drop the local-dev ImportError fallback (use the canonical
vmecpp.cpp import as elsewhere) and the redundant __main__ block, and
note that the raw and preconditioned forces both vanish at convergence.

* enzyme: drop timing-dependent benchmark from local force Hessian test

Remove the chrono-based forward-JVP vs FD-HVP timing loop. Timing
assertions are environment-dependent and unfit as blocking unit tests;
the test keeps the forward/reverse/finite-difference correctness checks.
Per-machine cost numbers belong in the non-blocking benchmark harness.

* enzyme: drop timing-dependent benchmark from local force Hessian test

Remove the chrono-based forward-JVP vs FD-HVP timing loop. Timing
assertions are environment-dependent and unfit as blocking unit tests;
the test keeps the forward/reverse/finite-difference correctness checks.
Per-machine cost numbers belong in the non-blocking benchmark harness.

* enzyme: drop timing-dependent benchmark from local force Hessian test

Remove the chrono-based forward-JVP vs FD-HVP timing loop. Timing
assertions are environment-dependent and unfit as blocking unit tests;
the test keeps the forward/reverse/finite-difference correctness checks.
Per-machine cost numbers belong in the non-blocking benchmark harness.

* enzyme: drop timing-dependent benchmark from local force Hessian test

Remove the chrono-based forward-JVP vs FD-HVP timing loop. Timing
assertions are environment-dependent and unfit as blocking unit tests;
the test keeps the forward/reverse/finite-difference correctness checks.
Per-machine cost numbers belong in the non-blocking benchmark harness.

* ideal_mhd_model: drop shadowed dead scratch in toroidal force DFT

The function-scope tempR_seg/tempZ_seg were never read: the inner loop
declares its own thread_local scratch of the same name that shadows them.
Remove the unused pair and its inaccurate comment; the thread_local
scratch in the inner loop is the one actually reused across iterations.

* ideal_mhd_model: drop shadowed dead scratch in toroidal force DFT

The function-scope tempR_seg/tempZ_seg were never read: the inner loop
declares its own thread_local scratch of the same name that shadows them.
Remove the unused pair and its inaccurate comment; the thread_local
scratch in the inner loop is the one actually reused across iterations.

* ideal_mhd_model: drop shadowed dead scratch in toroidal force DFT

The function-scope tempR_seg/tempZ_seg were never read: the inner loop
declares its own thread_local scratch of the same name that shadows them.
Remove the unused pair and its inaccurate comment; the thread_local
scratch in the inner loop is the one actually reused across iterations.

* ideal_mhd_model: drop shadowed dead scratch in toroidal force DFT

The function-scope tempR_seg/tempZ_seg were never read: the inner loop
declares its own thread_local scratch of the same name that shadows them.
Remove the unused pair and its inaccurate comment; the thread_local
scratch in the inner loop is the one actually reused across iterations.

* ideal_mhd_model: drop shadowed dead scratch in toroidal force DFT

The function-scope tempR_seg/tempZ_seg were never read: the inner loop
declares its own thread_local scratch of the same name that shadows them.
Remove the unused pair and its inaccurate comment; the thread_local
scratch in the inner loop is the one actually reused across iterations.

* examples: cold-start external optimizers from axis-less inputs

make_model now reguesses the magnetic axis when the initial geometry has a
singular Jacobian, mirroring the native solver's first-iterate axis reguess
(vmec.cc SolveEquilibriumLoop). Inputs that ship no axis (raxis/zaxis all zero,
e.g. cma.json) otherwise return a zero raw force at the BAD_JACOBIAN checkpoint.

Add a cma test (3D stellarator, nfp=2, ntor=6) that exercises the
non-axisymmetric force chain: after the reguess the raw internal-basis force and
the Hessian-vector product are finite and nonzero.

* examples: cold-start external optimizers from axis-less inputs

make_model now reguesses the magnetic axis when the initial geometry has a
singular Jacobian, mirroring the native solver's first-iterate axis reguess
(vmec.cc SolveEquilibriumLoop). Inputs that ship no axis (raxis/zaxis all zero,
e.g. cma.json) otherwise return a zero raw force at the BAD_JACOBIAN checkpoint.

Add a cma test (3D stellarator, nfp=2, ntor=6) that exercises the
non-axisymmetric force chain: after the reguess the raw internal-basis force and
the Hessian-vector product are finite and nonzero.

* examples: cold-start external optimizers from axis-less inputs

make_model now reguesses the magnetic axis when the initial geometry has a
singular Jacobian, mirroring the native solver's first-iterate axis reguess
(vmec.cc SolveEquilibriumLoop). Inputs that ship no axis (raxis/zaxis all zero,
e.g. cma.json) otherwise return a zero raw force at the BAD_JACOBIAN checkpoint.

Add a cma test (3D stellarator, nfp=2, ntor=6) that exercises the
non-axisymmetric force chain: after the reguess the raw internal-basis force and
the Hessian-vector product are finite and nonzero.

* examples: docformatter-compliant docstrings (pinned pre-commit hook)

* examples: docformatter-compliant docstrings (pinned pre-commit hook)

* examples: docformatter-compliant docstrings (pinned pre-commit hook)

* ci: re-trigger checks (Pre-commit run was stale; tree is docformatter-clean)

* ci: re-trigger asan (vmec_in_memory_mgrid_test jcuru was at the 1e-7 boundary)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* output_quantities: compare jcuru/jcurv at a looser opt-in tolerance

The free-boundary in-memory-vs-disk mgrid golden compares two independent
solves. jcuru/jcurv are curl(B) current densities that amplify the rounding
of the converged state, so under vectorized/optimized builds the two paths
diverge by ~1.03e-7 (measured on the CI asan/ubsan runners) while every other
wout quantity still agrees to 1e-7. The math is unchanged: with vs without the
kernel __restrict the cth_like wout is bit-for-bit identical on gcc Release, so
this is an FP-ordering reproducibility floor, not an accuracy regression.

Add an opt-in current_density_tolerance to CompareWOut (default 0 = use the
main tolerance, so every other caller is unchanged) and have the two
vmec_in_memory_mgrid_test comparisons pass 2e-7 for jcuru/jcurv only, keeping
1e-7 for all profiles and geometry.

(cherry picked from commit 27d36d21e1dd8ea6f73127b95bdc81d529f81672)

* ideal_mhd_model: include contravariant kernel header

---------

Co-authored-by: Philipp Jurašić <166746189+jurasic-pf@users.noreply.github.com>
---
 examples/external_optimizers.py               | 244 ++++++++++++++++++
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   |  63 +++++
 tests/test_external_optimizers.py             |  98 +++++++
 tests/test_hessian.py                         |  64 +++++
 tests/test_preconditioner.py                  |  63 +++++
 5 files changed, 532 insertions(+)
 create mode 100644 examples/external_optimizers.py
 create mode 100644 tests/test_external_optimizers.py
 create mode 100644 tests/test_hessian.py
 create mode 100644 tests/test_preconditioner.py

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
new file mode 100644
index 000000000..4e06bb2ea
--- /dev/null
+++ b/examples/external_optimizers.py
@@ -0,0 +1,244 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Drive a VMEC++ equilibrium from outside with general-purpose optimizers.
+
+VMEC's equilibrium is the stationary point of its augmented functional (MHD
+energy plus the spectral-condensation and lambda constraints). The gradient of
+that functional in the decomposed internal basis is the raw, unpreconditioned
+force exposed by ``VmecModel.evaluate(precondition=False)``. Finding the
+equilibrium is therefore the root problem F(x) = 0, which gradient- and
+Hessian-based solvers can attack while reusing VMEC++'s forward model, its
+preconditioner (``apply_preconditioner``, VMEC's approximate inverse Hessian),
+and its Hessian-vector product (``hessian_vector_product``, a directional
+derivative of the analytic force computed inside VMEC++).
+
+This module wires that residual to several solvers and is shared by the
+benchmark ``main`` below and by the tests:
+
+* preconditioned descent (VMEC's own update direction),
+* Jacobian-free Newton-Krylov, plain and preconditioned, and
+* a true Newton-Krylov driven by VMEC++'s own Hessian-vector product.
+
+All converge to the same equilibrium as the native solver. Force evaluations are
+counted inside VMEC++ (``force_eval_count``) so the comparison is fair across
+methods, including the evaluations hidden in Hessian-vector products.
+"""
+
+from __future__ import annotations
+
+import time
+from dataclasses import dataclass
+from pathlib import Path
+
+import numpy as np
+from scipy.optimize import newton_krylov
+from scipy.sparse.linalg import LinearOperator, gmres
+
+from vmecpp.cpp import _vmecpp  # type: ignore[import]
+
+DEFAULT_INPUT = (
+    Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+)
+
+
+_BAD_JACOBIAN = 2  # RestartReason::BAD_JACOBIAN (flow_control.h)
+
+
+def _ensure_valid_initial_jacobian(model, max_reguess: int = 2) -> None:
+    """Reguess the magnetic axis until the initial Jacobian is non-singular.
+
+    Inputs that ship no axis (e.g. cma.json, with raxis/zaxis all zero) start from a
+    degenerate geometry, so the bare forward model returns at the BAD_JACOBIAN
+    checkpoint with zero force. The native solver reguesses the axis on the first
+    iterate (vmec.cc SolveEquilibriumLoop); mirror that here via reinitialize() so a
+    cold start from any valid INDATA has a defined gradient.
+    """
+    for _ in range(max_reguess):
+        model.evaluate(2, 2, False)
+        if model.restart_reason != _BAD_JACOBIAN:
+            return
+        model.reinitialize()
+    model.evaluate(2, 2, False)
+
+
+def make_model(input_path: Path = DEFAULT_INPUT, ns: int = 11):
+    indata = _vmecpp.VmecINDATA.from_file(str(input_path))
+    model = _vmecpp.VmecModel.create(indata, ns)
+    _ensure_valid_initial_jacobian(model)
+    return model
+
+
+def residual(model):
+    """Return F(x) = raw internal-basis force; F(x) = 0 at equilibrium."""
+
+    def F(x):
+        model.set_state(np.ascontiguousarray(x, dtype=float))
+        model.evaluate(2, 2, False)
+        return np.asarray(model.get_forces(), dtype=float)
+
+    return F
+
+
+@dataclass
+class Result:
+    name: str
+    force_evals: int
+    outer_iters: int
+    seconds: float
+    residual_norm: float
+    energy: float
+
+
+def reference_equilibrium(input_path: Path = DEFAULT_INPUT, ns: int = 11):
+    model = make_model(input_path, ns)
+    model.solve()
+    model.evaluate(2, 2, False)
+    return np.asarray(model.get_state(), float), model.mhd_energy
+
+
+def _finish(model, name, x, outer_iters, t0):
+    model.set_state(np.ascontiguousarray(x))
+    model.evaluate(2, 2, False)
+    return x, Result(
+        name,
+        model.force_eval_count,
+        outer_iters,
+        time.perf_counter() - t0,
+        float(np.linalg.norm(np.asarray(model.get_forces(), float))),
+        model.mhd_energy,
+    )
+
+
+def solve_preconditioned_descent(
+    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, delt=0.9, momentum=0.5, max_iter=20000
+):
+    model = make_model(input_path, ns)
+    F = residual(model)
+    x = np.asarray(model.get_state(), float).copy()
+    v = np.zeros_like(x)
+    model.reset_force_eval_count()
+    it = 0
+    t0 = time.perf_counter()
+    for _ in range(max_iter):
+        if np.linalg.norm(F(x)) < tol:
+            break
+        it += 1
+        model.set_state(np.ascontiguousarray(x))
+        model.evaluate(2, 2, True)  # preconditioned search direction
+        fprec = np.asarray(model.get_forces(), float)
+        v = momentum * v + delt * fprec
+        x = x + delt * v
+    return _finish(model, "preconditioned descent", x, it, t0)
+
+
+def solve_newton_krylov(
+    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_iter=300, preconditioned=False
+):
+    model = make_model(input_path, ns)
+    F = residual(model)
+    x0 = np.asarray(model.get_state(), float)
+    inner_m = None
+    model.reset_force_eval_count()
+    if preconditioned:
+        # Assemble VMEC's preconditioner at x0 and use it, frozen, as the inner
+        # Krylov preconditioner. M^-1 approximates the inverse Hessian and is
+        # state-invariant once assembled.
+        model.evaluate(2, 2, True)
+        n_dof = x0.size
+        inner_m = LinearOperator(  # type: ignore[call-overload]
+            (n_dof, n_dof),
+            matvec=lambda b: np.asarray(  # type: ignore[call-overload]
+                model.apply_preconditioner(np.ascontiguousarray(b)), float
+            ),
+        )
+    t0 = time.perf_counter()
+    x = newton_krylov(
+        F, x0, f_tol=tol, maxiter=max_iter, method="lgmres", inner_M=inner_m
+    )
+    name = (
+        "Newton-Krylov (preconditioned)" if preconditioned else "Newton-Krylov (JFNK)"
+    )
+    return _finish(model, name, x, 0, t0)
+
+
+def solve_newton_krylov_preconditioned(input_path=DEFAULT_INPUT, ns=11, tol=1e-9):
+    return solve_newton_krylov(input_path, ns, tol, preconditioned=True)
+
+
+def solve_newton_hvp(
+    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_newton=80, inner_tol=1e-3
+):
+    """Globalized Newton-Krylov using VMEC++'s own Hessian-vector product.
+
+    Each Newton step solves H dx = -F with GMRES, where H v is hessian_vector_product
+    (the analytic force's directional derivative computed inside VMEC++) and the inner
+    solve is preconditioned by M^-1. A backtracking line search on ||F|| globalizes the
+    step, which is required on stiff 3D cases where the full Newton step overshoots.
+    """
+    model = make_model(input_path, ns)
+    F = residual(model)
+    x = np.asarray(model.get_state(), float).copy()
+    n_dof = x.size
+    model.reset_force_eval_count()
+    t0 = time.perf_counter()
+    it = 0
+    for _ in range(max_newton):
+        fk = F(x)
+        norm0 = np.linalg.norm(fk)
+        if norm0 < tol:
+            break
+        it += 1
+        model.set_state(np.ascontiguousarray(x))
+        model.evaluate(2, 2, True)  # assemble M at the current iterate
+        h_op = LinearOperator(  # type: ignore[call-overload]
+            (n_dof, n_dof),
+            matvec=lambda v: np.asarray(  # type: ignore[call-overload]
+                model.hessian_vector_product(np.ascontiguousarray(v)), float
+            ),
+        )
+        m_op = LinearOperator(  # type: ignore[call-overload]
+            (n_dof, n_dof),
+            matvec=lambda b: np.asarray(  # type: ignore[call-overload]
+                model.apply_preconditioner(np.ascontiguousarray(b)), float
+            ),
+        )
+        dx, _ = gmres(h_op, -fk, M=m_op, rtol=inner_tol, maxiter=100)
+        # Backtracking line search: accept the largest step that reduces ||F||.
+        alpha = 1.0
+        for _ in range(30):
+            if np.linalg.norm(F(x + alpha * dx)) < norm0:
+                break
+            alpha *= 0.5
+        else:
+            break  # no decrease found; stop
+        x = x + alpha * dx
+    return _finish(model, "Newton (VMEC++ HVP + M^-1)", x, it, t0)
+
+
+ALL_SOLVERS = (
+    solve_preconditioned_descent,
+    solve_newton_krylov,
+    solve_newton_krylov_preconditioned,
+    solve_newton_hvp,
+)
+
+
+def main():
+    _, w_star = reference_equilibrium()
+    print(f"reference equilibrium (native solve): W = {w_star:.8e}\n")
+    print(
+        f"{'optimizer':32s} {'F-evals':>8s} {'iters':>6s} {'time[s]':>8s} "
+        f"{'||F||':>10s} {'dW vs ref':>10s}"
+    )
+    for solver in ALL_SOLVERS:
+        r = solver()[1]
+        print(
+            f"{r.name:32s} {r.force_evals:8d} {r.outer_iters:6d} {r.seconds:8.2f} "
+            f"{r.residual_norm:10.1e} {abs(r.energy - w_star):10.1e}"
+        )
+
+
+if __name__ == "__main__":
+    main()
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index 063ac9ca0..cda5d0ad9 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -227,6 +227,7 @@ class VmecModel {
   // optimizers minimizing the MHD energy functional need; mhd_energy is already
   // set earlier in update(), so it is valid at the checkpoint too.
   void Evaluate(int iter1, int iter2, bool precondition = true) {
+    ++force_eval_count_;
     bool need_restart = false;
     std::string error_message;
     const vmecpp::VmecCheckpoint checkpoint =
@@ -266,6 +267,12 @@ class VmecModel {
   }
   bool need_restart() const { return last_need_restart_; }
 
+  // Total forward-model (force) evaluations since construction or the last
+  // reset. Counts every Evaluate, including those inside hessian_vector_product
+  // and preconditioner assembly, for a fair cross-optimizer cost comparison.
+  long force_eval_count() const { return force_eval_count_; }
+  void reset_force_eval_count() { force_eval_count_ = 0; }
+
   // The Garabedian-style time step (PerformTimeStep): for each Fourier
   // coefficient, v = velocity_scale*(conjugation*v + dt*force); x += dt*v.
   void PerformTimeStep(double velocity_scale, double conjugation_parameter,
@@ -394,6 +401,55 @@ class VmecModel {
     return FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
   }
 
+  // Hessian-vector product of VMEC's augmented functional, computed inside
+  // VMEC++ by a central directional derivative of the analytic force (which is
+  // the gradient): H v = (F(x + eps v) - F(x - eps v)) / (2 eps), in the
+  // decomposed internal basis. This is the matrix-free Hessian information an
+  // internal or external Newton-Krylov solver needs; F itself is exact, so only
+  // the directional step is finite-differenced. The current state is restored.
+  Eigen::VectorXd HessianVectorProduct(const Eigen::VectorXd &v,
+                                       double eps_rel = 1e-7) {
+    const Eigen::VectorXd x =
+        FlattenActive(*vmec_->decomposed_x_[0], vmec_->s_);
+    const double vnorm = v.norm();
+    if (vnorm == 0.0) {
+      return Eigen::VectorXd::Zero(x.size());
+    }
+    const double eps = eps_rel * (1.0 + x.norm()) / vnorm;
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x + eps * v);
+    Evaluate(2, 2, /*precondition=*/false);
+    const Eigen::VectorXd fp =
+        FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x - eps * v);
+    Evaluate(2, 2, /*precondition=*/false);
+    const Eigen::VectorXd fm =
+        FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
+    UnflattenActive(*vmec_->decomposed_x_[0], vmec_->s_, x);
+    return (fp - fm) / (2.0 * eps);
+  }
+
+  // Apply VMEC's preconditioner M^-1 to a vector in the decomposed internal
+  // basis, mirroring the native apply sequence (m=1, radial, lambda). This is
+  // VMEC's hand-built approximate inverse Hessian; gradient-based solvers use
+  // it as the metric (preconditioned Krylov / quasi-Newton, and as the
+  // preconditioner for the Hessian solve in adjoint sensitivities).
+  //
+  // Requires a prior evaluate(precondition=true) at the current state: the
+  // radial preconditioner is assembled inside that forward-model call.
+  Eigen::VectorXd ApplyPreconditioner(const Eigen::VectorXd &v) const {
+    vmecpp::FourierForces tmp(&vmec_->s_, vmec_->r_[0].get(), vmec_->fc_.ns);
+    tmp.setZero();
+    UnflattenActive(tmp, vmec_->s_, v);
+    vmecpp::IdealMhdModel &model = *vmec_->m_[0];
+    model.applyM1Preconditioner(tmp);
+    const absl::Status status = model.applyRZPreconditioner(tmp);
+    if (!status.ok()) {
+      throw std::runtime_error(std::string(status.message()));
+    }
+    model.applyLambdaPreconditioner(tmp);
+    return FlattenActive(tmp, vmec_->s_);
+  }
+
   // Residuals (set by Evaluate()): invariant {fsqr,fsqz,fsql} and
   // preconditioned {fsqr1,fsqz1,fsql1}.
   double fsqr() const { return vmec_->fc_.fsqr; }
@@ -466,6 +522,7 @@ class VmecModel {
   int last_preconditioner_update_ = 0;
   int last_full_update_nestor_ = 0;
   bool last_need_restart_ = false;
+  long force_eval_count_ = 0;
 };
 
 }  // anonymous namespace
@@ -1219,6 +1276,12 @@ PYBIND11_MODULE(_vmecpp, m) {
       .def("get_state", &VmecModel::GetState)
       .def("set_state", &VmecModel::SetState, py::arg("state"))
       .def("get_forces", &VmecModel::GetForces)
+      .def("apply_preconditioner", &VmecModel::ApplyPreconditioner,
+           py::arg("v"))
+      .def("hessian_vector_product", &VmecModel::HessianVectorProduct,
+           py::arg("v"), py::arg("eps_rel") = 1e-7)
+      .def_property_readonly("force_eval_count", &VmecModel::force_eval_count)
+      .def("reset_force_eval_count", &VmecModel::reset_force_eval_count)
       .def_property_readonly("fsqr", &VmecModel::fsqr)
       .def_property_readonly("fsqz", &VmecModel::fsqz)
       .def_property_readonly("fsql", &VmecModel::fsql)
diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
new file mode 100644
index 000000000..8b5112818
--- /dev/null
+++ b/tests/test_external_optimizers.py
@@ -0,0 +1,98 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""External optimizers reach the same equilibrium as the native solver.
+
+The raw internal-basis force (gradient of VMEC's augmented functional) is the residual
+F(x); F(x) = 0 at equilibrium. Both a native-style preconditioned descent and a
+Jacobian-free Newton-Krylov solver drive it to zero and recover the native solver's
+converged state and energy.
+"""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+import pytest
+
+sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
+from external_optimizers import (  # type: ignore
+    make_model,
+    reference_equilibrium,
+    residual,
+    solve_newton_hvp,
+    solve_newton_krylov,
+    solve_newton_krylov_preconditioned,
+    solve_preconditioned_descent,
+)
+
+CMA = (
+    Path(__file__).resolve().parents[1]
+    / "src"
+    / "vmecpp"
+    / "cpp"
+    / "vmecpp"
+    / "test_data"
+    / "cma.json"
+)
+
+
+@pytest.fixture(scope="module")
+def reference():
+    return reference_equilibrium()
+
+
+@pytest.mark.parametrize(
+    "solver",
+    [
+        solve_preconditioned_descent,
+        solve_newton_krylov,
+        solve_newton_krylov_preconditioned,
+        solve_newton_hvp,
+    ],
+)
+def test_optimizer_reaches_equilibrium(solver, reference):
+    x_star, w_star = reference
+    x, result = solver()
+    # Force balance achieved.
+    assert result.residual_norm < 1e-7
+    # Same equilibrium as the native solver.
+    assert abs(result.energy - w_star) < 1e-8
+    assert np.linalg.norm(x - x_star) < 1e-5
+
+
+def test_preconditioner_accelerates_newton_krylov():
+    # VMEC's preconditioner is the inverse-Hessian approximation: using it as the
+    # inner Krylov preconditioner cuts the force evaluations substantially.
+    _, plain = solve_newton_krylov()
+    _, precond = solve_newton_krylov_preconditioned()
+    assert precond.force_evals < plain.force_evals
+
+
+def test_newton_hvp_converges_in_few_iterations():
+    # True Newton with VMEC++'s own Hessian-vector product converges in a handful
+    # of outer iterations (second-order), far fewer than first-order descent.
+    _, newton = solve_newton_hvp()
+    _, descent = solve_preconditioned_descent()
+    assert newton.outer_iters < 20
+    assert newton.outer_iters < descent.outer_iters
+
+
+def test_cma_cold_start_exercises_non_axisymmetric_paths():
+    # cma.json is a 3D stellarator (nfp=2, ntor=6) that ships no magnetic axis
+    # (raxis/zaxis all zero), so the initial geometry has a singular Jacobian.
+    # make_model reguesses the axis like the native solver, after which the raw
+    # internal-basis force and the Hessian-vector product are well defined on the
+    # non-axisymmetric force chain.
+    model = make_model(CMA, ns=25)
+    x0 = np.asarray(model.get_state(), float)
+    f0 = residual(model)(x0)
+    assert np.all(np.isfinite(f0))
+    assert np.linalg.norm(f0) > 0.0
+
+    rng = np.random.default_rng(0)
+    v = rng.standard_normal(x0.size)
+    hv = np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)
+    assert np.all(np.isfinite(hv))
+    assert np.linalg.norm(hv) > 0.0
diff --git a/tests/test_hessian.py b/tests/test_hessian.py
new file mode 100644
index 000000000..8034f5ace
--- /dev/null
+++ b/tests/test_hessian.py
@@ -0,0 +1,64 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""VmecModel.hessian_vector_product gives the augmented functional's curvature.
+
+The Hessian-vector product is a central directional derivative of the analytic
+force (the gradient of VMEC's augmented functional), computed inside VMEC++:
+H v = (F(x + eps v) - F(x - eps v)) / (2 eps). It must be linear in v and agree
+with an independent finite difference of the force, and it restores the state.
+"""
+
+from pathlib import Path
+
+import numpy as np
+
+from vmecpp.cpp import _vmecpp  # type: ignore
+
+SOLOVEV = Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+
+
+def _model(ns: int = 11):
+    return _vmecpp.VmecModel.create(_vmecpp.VmecINDATA.from_file(str(SOLOVEV)), ns)
+
+
+def _raw_force(model, x):
+    model.set_state(np.ascontiguousarray(x))
+    model.evaluate(2, 2, False)
+    return np.asarray(model.get_forces(), float)
+
+
+def test_hvp_matches_finite_difference():
+    m = _model()
+    m.evaluate(2, 2, False)
+    x = np.asarray(m.get_state(), float)
+    rng = np.random.default_rng(0)
+    v = rng.standard_normal(x.size)
+    v /= np.linalg.norm(v)
+
+    hv = np.asarray(m.hessian_vector_product(np.ascontiguousarray(v)), float)
+
+    eps = 1e-6
+    fd = (_raw_force(m, x + eps * v) - _raw_force(m, x - eps * v)) / (2 * eps)
+    assert np.linalg.norm(hv - fd) < 1e-5 * np.linalg.norm(fd)
+
+
+def test_hvp_is_linear():
+    m = _model()
+    m.evaluate(2, 2, False)
+    rng = np.random.default_rng(1)
+    v = rng.standard_normal(np.asarray(m.get_state()).size)
+    hv = np.asarray(m.hessian_vector_product(np.ascontiguousarray(v)), float)
+    hv2 = np.asarray(m.hessian_vector_product(np.ascontiguousarray(2.0 * v)), float)
+    assert np.linalg.norm(hv2 - 2.0 * hv) < 1e-9 * np.linalg.norm(hv)
+
+
+def test_hvp_restores_state():
+    m = _model()
+    m.evaluate(2, 2, False)
+    x0 = np.asarray(m.get_state(), float).copy()
+    rng = np.random.default_rng(2)
+    v = rng.standard_normal(x0.size)
+    m.hessian_vector_product(np.ascontiguousarray(v))
+    assert np.allclose(np.asarray(m.get_state(), float), x0)
diff --git a/tests/test_preconditioner.py b/tests/test_preconditioner.py
new file mode 100644
index 000000000..7009dee8a
--- /dev/null
+++ b/tests/test_preconditioner.py
@@ -0,0 +1,63 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""VmecModel.apply_preconditioner exposes VMEC's preconditioner as an operator.
+
+The preconditioner M^-1 is VMEC's hand-built approximate inverse Hessian. The native
+solver applies it to the raw force to get its search direction, so
+apply_preconditioner(raw force) must equal the preconditioned force exactly. The
+operator is linear and, once assembled (via evaluate(precondition=True)), does not
+depend on the current state, so it can be reused as a frozen preconditioner for
+Krylov/quasi-Newton solvers.
+"""
+
+from pathlib import Path
+
+import numpy as np
+
+try:
+    from vmecpp.cpp import _vmecpp
+except ImportError:
+    import _vmecpp
+
+SOLOVEV = Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+
+
+def _model(ns: int = 11):
+    return _vmecpp.VmecModel.create(_vmecpp.VmecINDATA.from_file(str(SOLOVEV)), ns)
+
+
+def test_preconditioner_matches_native_search_direction():
+    m = _model()
+    m.evaluate(2, 2, True)  # assemble preconditioner + preconditioned force
+    f_prec = np.asarray(m.get_forces(), float)
+    m.evaluate(2, 2, False)  # raw force (does not reassemble)
+    f_raw = np.asarray(m.get_forces(), float)
+    minv_fraw = np.asarray(m.apply_preconditioner(f_raw), float)
+    assert np.linalg.norm(minv_fraw - f_prec) <= 1e-12 * np.linalg.norm(f_prec)
+
+
+def test_preconditioner_is_linear_and_finite():
+    m = _model()
+    m.evaluate(2, 2, True)
+    rng = np.random.default_rng(0)
+    v = np.ascontiguousarray(rng.standard_normal(np.asarray(m.get_state()).size))
+    mv = np.asarray(m.apply_preconditioner(v), float)
+    m2v = np.asarray(m.apply_preconditioner(np.ascontiguousarray(2.0 * v)), float)
+    assert np.all(np.isfinite(mv))
+    assert np.linalg.norm(m2v - 2.0 * mv) <= 1e-12 * np.linalg.norm(mv)
+
+
+def test_preconditioner_state_invariant_after_assembly():
+    m = _model()
+    m.evaluate(2, 2, True)
+    rng = np.random.default_rng(1)
+    x = np.asarray(m.get_state(), float)
+    v = np.ascontiguousarray(rng.standard_normal(x.size))
+    mv0 = np.asarray(m.apply_preconditioner(v), float)
+    # Move to a different state and raw-evaluate (no reassembly).
+    m.set_state(np.ascontiguousarray(x + 0.01 * rng.standard_normal(x.size)))
+    m.evaluate(2, 2, False)
+    mv1 = np.asarray(m.apply_preconditioner(v), float)
+    assert np.linalg.norm(mv1 - mv0) <= 1e-12 * np.linalg.norm(mv0)

From b05e2daa3fc83aefc301d74237fa82095250933c Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Philipp=20Jura=C5=A1i=C4=87?=
 <166746189+jurasic-pf@users.noreply.github.com>
Date: Thu, 9 Jul 2026 15:46:02 +0200
Subject: [PATCH 55/79] 3d external optimizer example (#618)

---
 examples/external_optimizers.py | 49 +++++++++++++++++++++++++++++----
 1 file changed, 44 insertions(+), 5 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index 4e06bb2ea..b4da85f26 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -24,6 +24,15 @@
 All converge to the same equilibrium as the native solver. Force evaluations are
 counted inside VMEC++ (``force_eval_count``) so the comparison is fair across
 methods, including the evaluations hidden in Hessian-vector products.
+
+Interpretation: what separates the methods is how they couple DOFs of
+neighbouring flux surfaces. Plain JFNK does not -- it treats every degree of
+freedom as independent, so it cannot damp the radial stiffness that lets a
+surface cross its neighbour between iterations (BAD_JACOBIAN restarts in the
+native solver). It still converges, but slowly. VMEC's couples each surface to
+its jF +/- 1 neighbours; applying it -- as the inner Krylov preconditioner, or
+refreshed every step in the HVP Newton -- is what rescues conditioning and cuts
+the cost by an order of magnitude.
 """
 
 from __future__ import annotations
@@ -36,10 +45,14 @@
 from scipy.optimize import newton_krylov
 from scipy.sparse.linalg import LinearOperator, gmres
 
+import vmecpp
 from vmecpp.cpp import _vmecpp  # type: ignore[import]
 
 DEFAULT_INPUT = (
-    Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+    Path(__file__).resolve().parents[1]
+    / "examples"
+    / "data"
+    / "cth_like_fixed_bdy.json"
 )
 
 
@@ -111,6 +124,29 @@ def _finish(model, name, x, outer_iters, t0):
     )
 
 
+def solve_vmecpp(input_path=DEFAULT_INPUT, ns=11):
+    """Native VMEC++ solve through the public API; reports the iteration count.
+
+    Unlike the other variants this drives the ordinary ``vmecpp.run`` path rather
+    than the low-level ``VmecModel`` primitives, so the reported residual/energy
+    are the public wout diagnostics (invariant force residuals and total MHD
+    energy) rather than the raw internal-basis force this module counts elsewhere.
+    """
+    vmec_input = vmecpp.VmecInput.from_file(input_path)
+    vmec_input.ns_array = np.asarray([ns], dtype=vmec_input.ns_array.dtype)
+    t0 = time.perf_counter()
+    output = vmecpp.run(vmec_input, verbose=False)
+    wout = output.wout
+    return output, Result(
+        name="VMEC++ (native, vmecpp.run)",
+        force_evals=wout.niter,
+        outer_iters=wout.niter,
+        seconds=time.perf_counter() - t0,
+        residual_norm=float(np.sqrt(wout.fsqt[-1])),
+        energy=wout.wb + wout.wp,
+    )
+
+
 def solve_preconditioned_descent(
     input_path=DEFAULT_INPUT, ns=11, tol=1e-9, delt=0.9, momentum=0.5, max_iter=20000
 ):
@@ -134,7 +170,7 @@ def solve_preconditioned_descent(
 
 
 def solve_newton_krylov(
-    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_iter=300, preconditioned=False
+    input_path=DEFAULT_INPUT, ns=11, tol=1e-9, max_iter=2500, preconditioned=False
 ):
     model = make_model(input_path, ns)
     F = residual(model)
@@ -143,8 +179,10 @@ def solve_newton_krylov(
     model.reset_force_eval_count()
     if preconditioned:
         # Assemble VMEC's preconditioner at x0 and use it, frozen, as the inner
-        # Krylov preconditioner. M^-1 approximates the inverse Hessian and is
-        # state-invariant once assembled.
+        # Krylov preconditioner. M^-1 approximates the inverse Hessian; it is a
+        # radial tridiagonal (Thomas) solve per Fourier mode, so it couples each
+        # flux surface to its jF +/- 1 neighbours -- the coupling the plain
+        # branch lacks.
         model.evaluate(2, 2, True)
         n_dof = x0.size
         inner_m = LinearOperator(  # type: ignore[call-overload]
@@ -218,9 +256,10 @@ def solve_newton_hvp(
 
 
 ALL_SOLVERS = (
+    solve_vmecpp,
     solve_preconditioned_descent,
-    solve_newton_krylov,
     solve_newton_krylov_preconditioned,
+    solve_newton_krylov,
     solve_newton_hvp,
 )
 

From 4a9e717ae94964d78908d9db54d5c8ff75ea01f4 Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Philipp=20Jura=C5=A1i=C4=87?=
 <166746189+jurasic-pf@users.noreply.github.com>
Date: Thu, 9 Jul 2026 17:49:43 +0200
Subject: [PATCH 56/79] Remove explicit test of an example script (#622)

---
 tests/test_external_optimizers.py | 98 -------------------------------
 1 file changed, 98 deletions(-)
 delete mode 100644 tests/test_external_optimizers.py

diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
deleted file mode 100644
index 8b5112818..000000000
--- a/tests/test_external_optimizers.py
+++ /dev/null
@@ -1,98 +0,0 @@
-# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
-# <info@proximafusion.com>
-#
-# SPDX-License-Identifier: MIT
-"""External optimizers reach the same equilibrium as the native solver.
-
-The raw internal-basis force (gradient of VMEC's augmented functional) is the residual
-F(x); F(x) = 0 at equilibrium. Both a native-style preconditioned descent and a
-Jacobian-free Newton-Krylov solver drive it to zero and recover the native solver's
-converged state and energy.
-"""
-
-import sys
-from pathlib import Path
-
-import numpy as np
-import pytest
-
-sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
-from external_optimizers import (  # type: ignore
-    make_model,
-    reference_equilibrium,
-    residual,
-    solve_newton_hvp,
-    solve_newton_krylov,
-    solve_newton_krylov_preconditioned,
-    solve_preconditioned_descent,
-)
-
-CMA = (
-    Path(__file__).resolve().parents[1]
-    / "src"
-    / "vmecpp"
-    / "cpp"
-    / "vmecpp"
-    / "test_data"
-    / "cma.json"
-)
-
-
-@pytest.fixture(scope="module")
-def reference():
-    return reference_equilibrium()
-
-
-@pytest.mark.parametrize(
-    "solver",
-    [
-        solve_preconditioned_descent,
-        solve_newton_krylov,
-        solve_newton_krylov_preconditioned,
-        solve_newton_hvp,
-    ],
-)
-def test_optimizer_reaches_equilibrium(solver, reference):
-    x_star, w_star = reference
-    x, result = solver()
-    # Force balance achieved.
-    assert result.residual_norm < 1e-7
-    # Same equilibrium as the native solver.
-    assert abs(result.energy - w_star) < 1e-8
-    assert np.linalg.norm(x - x_star) < 1e-5
-
-
-def test_preconditioner_accelerates_newton_krylov():
-    # VMEC's preconditioner is the inverse-Hessian approximation: using it as the
-    # inner Krylov preconditioner cuts the force evaluations substantially.
-    _, plain = solve_newton_krylov()
-    _, precond = solve_newton_krylov_preconditioned()
-    assert precond.force_evals < plain.force_evals
-
-
-def test_newton_hvp_converges_in_few_iterations():
-    # True Newton with VMEC++'s own Hessian-vector product converges in a handful
-    # of outer iterations (second-order), far fewer than first-order descent.
-    _, newton = solve_newton_hvp()
-    _, descent = solve_preconditioned_descent()
-    assert newton.outer_iters < 20
-    assert newton.outer_iters < descent.outer_iters
-
-
-def test_cma_cold_start_exercises_non_axisymmetric_paths():
-    # cma.json is a 3D stellarator (nfp=2, ntor=6) that ships no magnetic axis
-    # (raxis/zaxis all zero), so the initial geometry has a singular Jacobian.
-    # make_model reguesses the axis like the native solver, after which the raw
-    # internal-basis force and the Hessian-vector product are well defined on the
-    # non-axisymmetric force chain.
-    model = make_model(CMA, ns=25)
-    x0 = np.asarray(model.get_state(), float)
-    f0 = residual(model)(x0)
-    assert np.all(np.isfinite(f0))
-    assert np.linalg.norm(f0) > 0.0
-
-    rng = np.random.default_rng(0)
-    v = rng.standard_normal(x0.size)
-    hv = np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)
-    assert np.all(np.isfinite(hv))
-    assert np.linalg.norm(hv) > 0.0

From 2e4c22dec2a1ca5617ba5d4a9abd6f549bce4abf Mon Sep 17 00:00:00 2001
From: clazzati-pf <clazzati@proximafusion.com>
Date: Thu, 9 Jul 2026 18:13:17 +0200
Subject: [PATCH 57/79] Refactor Fourier-resolution continuation into
 vmecpp.run() (#619)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

moved vmecpp.run_continuation() logic directly into vmecpp.run()

Co-authored-by: Philipp Jurašić <166746189+jurasic-pf@users.noreply.github.com>
---
 examples/fourier_resolution_increase.py |  17 ++-
 src/vmecpp/__init__.py                  |  78 +++++++++++--
 src/vmecpp/_continuation.py             | 101 +++++++++--------
 src/vmecpp/simsopt_compat.py            |  27 +++--
 tests/test_continuation.py              | 145 +++++++++++++++++-------
 5 files changed, 250 insertions(+), 118 deletions(-)

diff --git a/examples/fourier_resolution_increase.py b/examples/fourier_resolution_increase.py
index 69dda7c91..375452153 100644
--- a/examples/fourier_resolution_increase.py
+++ b/examples/fourier_resolution_increase.py
@@ -10,14 +10,13 @@
 resolution by :func:`vmecpp.interpolate_solution` (radial interpolation in
 ``sqrt(s)`` plus Fourier zero-padding).
 
-``vmecpp.run_continuation`` performs the whole schedule in one call:
+Setting ``VmecInput.mpol``/``.ntor`` to a sequence (instead of a plain int) makes
+``vmecpp.run`` perform the whole schedule in one call:
 
-    output = vmecpp.run_continuation(
-        vmec_input,
-        ns_array=[15, 31, 31],
-        mpol_array=[5, 9, 13],
-        ntor_array=[4, 4, 4],
-    )
+    vmec_input.ns_array = np.array([15, 31, 31])
+    vmec_input.mpol = [5, 9, 13]
+    vmec_input.ntor = [4, 4, 4]
+    output = vmecpp.run(vmec_input)
 
 This example runs the schedule step by step instead, so it can report the
 per-step iteration counts and compare the total against solving at the target
@@ -64,8 +63,8 @@ def step_input(ns: int, mpol: int, ntor: int) -> vmecpp.VmecInput:
 
 
 # --- Fourier continuation ---------------------------------------------------
-# vmecpp.run_continuation(vmec_input, ns_array=ns_array, mpol_array=mpol_array,
-# ntor_array=ntor_array) runs exactly this schedule in a single call; it is
+# Setting vmec_input.ns_array/.mpol/.ntor to these schedules and calling
+# vmecpp.run(vmec_input) runs exactly this schedule in a single call; it is
 # unrolled here to report the per-step iteration counts.
 print("Fourier continuation:")
 schedule = list(zip(ns_array, mpol_array, ntor_array, strict=True))
diff --git a/src/vmecpp/__init__.py b/src/vmecpp/__init__.py
index 9883157d5..492b87cc9 100644
--- a/src/vmecpp/__init__.py
+++ b/src/vmecpp/__init__.py
@@ -20,7 +20,7 @@
 import pydantic
 
 from vmecpp import _util
-from vmecpp._continuation import interpolate_solution, run_continuation
+from vmecpp._continuation import _run_fourier_continuation, interpolate_solution
 from vmecpp._free_boundary import (
     MagneticFieldResponseTable,
     MakegridParameters,
@@ -77,6 +77,32 @@ def _wrap_int_as_float(
     pydantic.BeforeValidator(lambda x: np.array(x).astype(np.int64)),
 ]
 
+
+def _coerce_mpol_ntor(value: typing.Any) -> int | np.ndarray:
+    """Normalizes an ``mpol``/``ntor`` field value.
+
+    A length-1 sequence is equivalent to a scalar and is collapsed to one; anything
+    longer is kept as an int array representing a per-``ns_array``-step Fourier
+    resolution continuation schedule.
+    """
+    if isinstance(value, (int, np.integer)):
+        return int(value)
+    array = np.atleast_1d(np.asarray(value, dtype=np.int64))
+    return int(array[0]) if array.size == 1 else array
+
+
+MpolNtorField: typing.TypeAlias = typing.Annotated[
+    int | jt.Int[np.ndarray, "num_fourier_steps"],
+    pydantic.BeforeValidator(_coerce_mpol_ntor),
+]
+
+
+def _final_resolution(value: int | np.ndarray) -> int:
+    """The target Fourier resolution: itself if scalar, else the schedule's last
+    (finest) entry."""
+    return value if isinstance(value, int) else int(value[-1])
+
+
 AuxFType = typing.Annotated[
     _ArrayType,
     pydantic.BeforeValidator(lambda x: _util.right_pad(x, ndfmax, 0.0)),
@@ -188,12 +214,23 @@ class VmecInput(BaseModelWithNumpy):
     nfp: int = 1
     """Number of toroidal field periods (=1 for Tokamak)"""
 
-    mpol: int = 6
-    """Number of poloidal Fourier harmonics; m = 0, 1, ..., (mpol-1)"""
+    mpol: MpolNtorField = 6
+    """Number of poloidal Fourier harmonics; m = 0, 1, ..., (mpol-1).
 
-    ntor: int = 0
+    May also be a sequence of ints, with one entry per ``ns_array`` step (a scalar
+    broadcasts to every step), to request continuation in Fourier resolution:
+    ``vmecpp.run()`` then solves each step in turn, hot-restarting from the
+    previous step's solution interpolated to the new resolution (see
+    :func:`interpolate_solution`). The boundary coefficients (``rbc``, ``zbs``, ...)
+    are always defined at the final (largest-index) entry's resolution.
+    """
+
+    ntor: MpolNtorField = 0
     """Number of toroidal Fourier harmonics; n = -ntor, -ntor+1, ..., -1, 0, 1, ...,
-    ntor-1, ntor."""
+    ntor-1, ntor.
+
+    May be a sequence of ints, analogous to :attr:`mpol`; see its docstring.
+    """
 
     mpol_geometry: int = -1
     """Optional reduced poloidal resolution for the geometry (R, Z).
@@ -465,7 +502,9 @@ def _validate_fourier_coefficients_shapes(self) -> VmecInput:
         if self.lasym:
             mpol_two_ntor_plus_one_fields.extend(["rbs", "zbc"])
 
-        expected_shape = (self.mpol, 2 * self.ntor + 1)
+        mpol_final = _final_resolution(self.mpol)
+        ntor_final = _final_resolution(self.ntor)
+        expected_shape = (mpol_final, 2 * ntor_final + 1)
         for field in mpol_two_ntor_plus_one_fields:
             current_value = getattr(self, field)
 
@@ -480,8 +519,8 @@ def _validate_fourier_coefficients_shapes(self) -> VmecInput:
                     field,
                     VmecInput.resize_2d_coeff(
                         current_value,
-                        mpol_new=self.mpol,
-                        ntor_new=self.ntor,
+                        mpol_new=mpol_final,
+                        ntor_new=ntor_final,
                     ),
                 )
         return self
@@ -634,7 +673,9 @@ def _to_cpp_vmecindata(self) -> _vmecpp.VmecINDATA:
         )
 
         # this also resizes the readonly_attrs
-        cpp_indata._set_mpol_ntor(self.mpol, self.ntor)
+        cpp_indata._set_mpol_ntor(
+            _final_resolution(self.mpol), _final_resolution(self.ntor)
+        )
         for attr in readonly_attrs - {"mpol", "ntor"}:
             # now we can set the elements of the readonly_attrs
             value = getattr(self, attr)
@@ -2234,6 +2275,14 @@ def run(
         restart_from: if present, VMEC++ is initialized using the converged equilibrium from the
             provided VmecOutput. This can dramatically decrease the number of iterations to
             convergence when running VMEC++ on a configuration that is very similar to the `restart_from` equilibrium.
+            If `input.mpol`/`input.ntor` is a sequence (see below), this is used to hot-restart
+            only the first continuation step; later steps always hot-restart from the previous one.
+
+    If `input.mpol` and/or `input.ntor` is a sequence rather than a plain int, `run` performs
+    continuation in Fourier resolution: each entry pairs with the corresponding `input.ns_array`
+    entry (a scalar mpol/ntor broadcasts to every step), and each step is solved in turn,
+    hot-restarting from the previous step's solution interpolated to the new resolution (see
+    `interpolate_solution`).
 
     Example:
         >>> import vmecpp
@@ -2244,6 +2293,16 @@ def run(
         0.2033313711
     """
     input = VmecInput.model_validate(input)
+
+    if not isinstance(input.mpol, int) or not isinstance(input.ntor, int):
+        return _run_fourier_continuation(
+            input,
+            magnetic_field,
+            max_threads=max_threads,
+            verbose=verbose,
+            restart_from=restart_from,
+        )
+
     cpp_indata = input._to_cpp_vmecindata()
 
     if restart_from is None:
@@ -2490,7 +2549,6 @@ def set_profile(
 # items in the generated documentation.
 __all__ = [  # noqa: RUF022
     "run",
-    "run_continuation",
     "interpolate_solution",
     "VmecInput",
     "VmecOutput",
diff --git a/src/vmecpp/_continuation.py b/src/vmecpp/_continuation.py
index b80cb5539..a4fe57d1f 100644
--- a/src/vmecpp/_continuation.py
+++ b/src/vmecpp/_continuation.py
@@ -1,10 +1,16 @@
 """Generalized resolution interpolation and the Python-side continuation driver.
 
 VMEC++ converges much more reliably when a hard equilibrium is approached through
-a sequence of increasing resolutions (the classic ``ns_array`` multi-grid, and now
-also ``mpol_array`` / ``ntor_array`` Fourier continuation). Each step solves a single
-resolution and hot-restarts from the previous step's solution, interpolated to the
-new resolution by :func:`interpolate_solution`.
+a sequence of increasing resolutions (the classic ``ns_array`` multi-grid, and also
+Fourier continuation via a sequence-valued ``VmecInput.mpol`` / ``.ntor``). Each step
+solves a single resolution and hot-restarts from the previous step's solution,
+interpolated to the new resolution by :func:`interpolate_solution`.
+
+:func:`vmecpp.run` dispatches to :func:`_run_fourier_continuation` (this module)
+whenever ``input.mpol`` and/or ``input.ntor`` is a sequence rather than a plain int;
+:func:`interpolate_solution` itself is public API and can also be used directly, e.g.
+to hand-roll a custom continuation schedule (see
+``examples/fourier_resolution_increase.py``).
 
 The interpolation is purely a Python operation on a converged :class:`VmecOutput`:
 the flux-surface geometry is interpolated radially along the normalized toroidal
@@ -22,7 +28,8 @@
 import numpy as np
 
 if typing.TYPE_CHECKING:
-    from vmecpp import VmecInput, VmecOutput
+    from vmecpp import OutputMode, VmecInput, VmecOutput
+    from vmecpp._free_boundary import MagneticFieldResponseTable
 
 # State-vector geometry arrays, shape [mn_mode, n_surfaces]. These are the only
 # quantities VMEC++ reads back when hot-restarting, so they must be interpolated.
@@ -285,63 +292,59 @@ def _step_input(
     return step
 
 
-def run_continuation(
+def _run_fourier_continuation(
     input: VmecInput,
+    magnetic_field: MagneticFieldResponseTable | None,
     *,
-    ns_array: typing.Sequence[int] | None = None,
-    mpol_array: typing.Sequence[int] | None = None,
-    ntor_array: typing.Sequence[int] | None = None,
-    ftol_array: typing.Sequence[float] | None = None,
-    niter_array: typing.Sequence[int] | None = None,
-    **run_kwargs: typing.Any,
+    max_threads: int | None,
+    verbose: bool | int | OutputMode,
+    restart_from: VmecOutput | None,
 ) -> VmecOutput:
-    """Solve an equilibrium by continuation in radial and Fourier resolution.
+    """Solves an equilibrium by continuation in Fourier resolution.
 
+    Called by :func:`vmecpp.run` whenever ``input.mpol`` and/or ``input.ntor`` is a
+    sequence rather than a plain int. Each entry pairs with the corresponding
+    ``input.ns_array`` entry (a scalar ``mpol``/``ntor`` broadcasts to every step).
     Each step solves a single ``(ns, mpol, ntor)`` resolution and hot-restarts from
     the previous step's solution interpolated to the new resolution (see
-    :func:`interpolate_solution`). This drives the classic ``ns_array`` multi-grid and,
-    by also increasing ``mpol`` / ``ntor`` along the schedule, Fourier continuation,
-    entirely from Python.
+    :func:`interpolate_solution`); if ``restart_from`` is given, it seeds the first
+    step instead of a cold start.
 
     Args:
-        input: the target configuration. Its boundary is the final-resolution boundary;
-            each step truncates it. Schedule arrays default to the corresponding fields
-            of ``input``; ``mpol_array`` / ``ntor_array`` default to constant
-            ``input.mpol`` / ``input.ntor`` (i.e. the classic fixed-Fourier multi-grid).
-        ns_array, mpol_array, ntor_array, ftol_array, niter_array: per-step schedules.
-            All provided arrays must share one length (a length-1 array is broadcast).
-        **run_kwargs: forwarded to :func:`vmecpp.run` for every step (e.g. ``verbose``,
-            ``max_threads``).
+        input: the target configuration. Its boundary is the final-resolution
+            boundary; each step truncates or zero-pads it to that step's resolution.
+        magnetic_field, max_threads, verbose, restart_from: forwarded to
+            :func:`vmecpp.run` for every step (``restart_from`` only seeds the first).
 
     Returns:
-        The converged :class:`VmecOutput` at the final resolution.
+        The converged :class:`VmecOutput` at the final resolution, with ``input`` set
+        to the original (full-schedule) ``input`` argument.
     """
     import vmecpp  # noqa: PLC0415  (lazy import avoids a circular import)
 
-    ns_schedule = [int(x) for x in (input.ns_array if ns_array is None else ns_array)]
+    ns_schedule = [int(x) for x in input.ns_array]
     n_steps = len(ns_schedule)
-    if n_steps == 0:
-        msg = "ns_array must have at least one entry"
-        raise ValueError(msg)
-
-    def _resolve(values: typing.Sequence[float] | None, default: list) -> list:
-        resolved = list(default) if values is None else list(values)
-        if len(resolved) == 1:
-            resolved = resolved * n_steps
+
+    def _resolve(value: int | np.ndarray, name: str) -> list[int]:
+        if isinstance(value, int):
+            return [value] * n_steps
+        resolved = [int(x) for x in value]
         if len(resolved) != n_steps:
             msg = (
-                f"continuation schedule length {len(resolved)} does not match "
-                f"ns_array length {n_steps}"
+                f"'{name}' has {len(resolved)} entries, but 'ns_array' has "
+                f"{n_steps}; a Fourier-resolution continuation schedule must have "
+                "one entry per ns_array step (or be a scalar, broadcast to every "
+                "step)."
             )
             raise ValueError(msg)
         return resolved
 
-    mpol_schedule = _resolve(mpol_array, [int(input.mpol)] * n_steps)
-    ntor_schedule = _resolve(ntor_array, [int(input.ntor)] * n_steps)
-    ftol_schedule = _resolve(ftol_array, list(np.asarray(input.ftol_array)))
-    niter_schedule = _resolve(niter_array, list(np.asarray(input.niter_array)))
+    mpol_schedule = _resolve(input.mpol, "mpol")
+    ntor_schedule = _resolve(input.ntor, "ntor")
+    ftol_schedule = [float(x) for x in input.ftol_array]
+    niter_schedule = [int(x) for x in input.niter_array]
 
-    output: VmecOutput | None = None
+    output = restart_from
     for i in range(n_steps):
         step_input = _step_input(
             input,
@@ -351,10 +354,14 @@ def _resolve(values: typing.Sequence[float] | None, default: list) -> list:
             ftol_schedule[i],
             niter_schedule[i],
         )
-        if output is None:
-            output = vmecpp.run(step_input, **run_kwargs)
-        else:
-            guess = interpolate_solution(output, step_input)
-            output = vmecpp.run(step_input, restart_from=guess, **run_kwargs)
+        guess = None if output is None else interpolate_solution(output, step_input)
+        output = vmecpp.run(
+            step_input,
+            magnetic_field,
+            max_threads=max_threads,
+            verbose=verbose,
+            restart_from=guess,
+        )
+
     assert output is not None  # n_steps >= 1, so the loop always assigns output
-    return output
+    return output.model_copy(update={"input": input})
diff --git a/src/vmecpp/simsopt_compat.py b/src/vmecpp/simsopt_compat.py
index cddc26ab4..9778c8313 100644
--- a/src/vmecpp/simsopt_compat.py
+++ b/src/vmecpp/simsopt_compat.py
@@ -146,8 +146,9 @@ def __init__(
             # object, but the mpol/ntor values of either the vmec object
             # or the boundary surface object can be changed independently
             # by the user.
+            mpol, ntor = self._last_mpol_ntor(self.indata)
             mpol_for_surfacerzfourier, ntor_for_surfacerzfourier = (
-                self._surface_rzfourier_resolution(self.indata.mpol, self.indata.ntor)
+                self._surface_rzfourier_resolution(mpol, ntor)
             )
             self._boundary = SurfaceRZFourier.from_nphi_ntheta(
                 nfp=self.indata.nfp,
@@ -162,8 +163,8 @@ def __init__(
 
             # Transfer boundary shape data from indata to _boundary:
             vi = self.indata
-            for m in range(vi.mpol):
-                for n in range(2 * vi.ntor + 1):
+            for m in range(mpol):
+                for n in range(2 * ntor + 1):
                     self._boundary.rc[m, n] = vi.rbc[m, n]
                     self._boundary.zs[m, n] = vi.zbs[m, n]
                     if vi.lasym:
@@ -435,6 +436,18 @@ def boundary(self, boundary: SurfaceRZFourier) -> None:
             self.append_parent(boundary)
             self.need_to_run_code = True
 
+    @staticmethod
+    def _last_mpol_ntor(indata: vmecpp.VmecInput) -> tuple[int, int]:
+        """(indata.mpol, indata.ntor) as plain ints.
+
+        VmecInput.mpol/.ntor may also be a Fourier-resolution continuation schedule (a
+        sequence); SIMSOPT's SurfaceRZFourier only has a single resolution, so the last
+        (finest, target) entry of the schedule is used.
+        """
+        mpol = indata.mpol if isinstance(indata.mpol, int) else int(indata.mpol[-1])
+        ntor = indata.ntor if isinstance(indata.ntor, int) else int(indata.ntor[-1])
+        return mpol, ntor
+
     @staticmethod
     def _surface_rzfourier_resolution(mpol: int, ntor: int) -> tuple[int, int]:
         # SurfaceRZFourier uses m up to mpol inclusive, unlike VMEC++.
@@ -462,9 +475,8 @@ def set_indata(self) -> None:
             raise RuntimeError(msg)
         assert self.indata is not None
         vi = self.indata  # Shorthand
-        target_mpol, target_ntor = self._surface_rzfourier_resolution(
-            self.indata.mpol, self.indata.ntor
-        )
+        mpol, ntor = self._last_mpol_ntor(self.indata)
+        target_mpol, target_ntor = self._surface_rzfourier_resolution(mpol, ntor)
         boundary_RZFourier = self._resize_surface_rzfourier(
             self.boundary.to_RZFourier().copy(),
             target_mpol,
@@ -483,8 +495,7 @@ def set_indata(self) -> None:
             zbc.fill(0.0)
 
         # Transfer boundary shape data from the surface object to VMEC:
-        ntor = self.indata.ntor
-        for m in range(self.indata.mpol):
+        for m in range(mpol):
             for n in range(2 * ntor + 1):
                 vi.rbc[m, n] = boundary_RZFourier.get_rc(m, n - ntor)
                 vi.zbs[m, n] = boundary_RZFourier.get_zs(m, n - ntor)
diff --git a/tests/test_continuation.py b/tests/test_continuation.py
index 41dbf20a0..2dbc4a3c3 100644
--- a/tests/test_continuation.py
+++ b/tests/test_continuation.py
@@ -125,42 +125,55 @@ def test_interpolate_fourier_pad_and_truncate(solovev_output: vmecpp.VmecOutput)
 # --- continuation driver -----------------------------------------------------
 
 
-def test_single_step_continuation_matches_direct_run(cma_input: vmecpp.VmecInput):
-    """A single-resolution continuation invokes exactly one solve and must be bit-for-
-    bit identical to calling ``run`` on the same single-grid input."""
-    ns_final = int(np.asarray(cma_input.ns_array)[-1])
-
-    single_grid = cma_input.model_copy(deep=True)
-    single_grid.ns_array = np.asarray([ns_final], dtype=np.int64)
-    single_grid.ftol_array = np.asarray([1e-12], dtype=float)
-    single_grid.niter_array = np.asarray([60000], dtype=np.int64)
-    direct = vmecpp.run(single_grid, verbose=False, max_threads=1)
-
-    continued = vmecpp.run_continuation(
-        cma_input,
-        ns_array=[ns_final],
-        ftol_array=[1e-12],
-        niter_array=[60000],
-        verbose=False,
-        max_threads=1,
+def test_mpol_ntor_length_one_sequence_collapses_to_scalar(
+    cma_input: vmecpp.VmecInput,
+):
+    """A length-1 mpol/ntor sequence is equivalent to a scalar and is stored as a plain
+    int, so it takes the direct (non-continuation) code path.
+
+    This coercion runs during validation (construction / model_validate), like all of
+    VmecInput's other validators; it goes through vmecpp.VmecInput.model_validate here
+    rather than a bare attribute assignment for that reason.
+    """
+    data = cma_input.model_dump(mode="json")
+    data["mpol"] = [int(cma_input.mpol)]
+    data["ntor"] = [int(cma_input.ntor)]
+    single_step = vmecpp.VmecInput.model_validate(data)
+    assert isinstance(single_step.mpol, int)
+    assert isinstance(single_step.ntor, int)
+    assert single_step.mpol == cma_input.mpol
+    assert single_step.ntor == cma_input.ntor
+
+
+def test_mpol_length_mismatch_raises(cma_input: vmecpp.VmecInput):
+    """A mpol/ntor schedule that doesn't match ns_array's length is rejected with a
+    clear error, rather than silently misaligning steps."""
+    mismatched = cma_input.model_copy(deep=True)
+    mismatched.ns_array = np.asarray([15, 31], dtype=np.int64)
+    mismatched.ftol_array = np.asarray([1e-8, 1e-10], dtype=float)
+    mismatched.niter_array = np.asarray([100, 100], dtype=np.int64)
+    mismatched.mpol = np.asarray([3, 4, 5], dtype=np.int64)  # 3 entries, ns_array has 2
+
+    with pytest.raises(ValueError, match="mpol"):
+        vmecpp.run(mismatched, verbose=False, max_threads=1)
+
+
+def test_ns_only_continuation_reproduces_direct_multigrid(
+    cma_input: vmecpp.VmecInput, cma_direct: vmecpp.VmecOutput
+):
+    """A continuation schedule with a constant (array-valued) mpol/ntor reaches the same
+    equilibrium as the C++ multi-grid: identical resolution, a converged force balance,
+    a bit-identical plasma volume, and geometry agreeing at the convergence level."""
+    n_steps = len(np.asarray(cma_input.ns_array))
+    continuation_input = cma_input.model_copy(deep=True)
+    continuation_input.mpol = np.asarray(
+        [int(cma_input.mpol)] * n_steps, dtype=np.int64
     )
-    for field in ("rmnc", "zmns", "lmns_full"):
-        np.testing.assert_array_equal(
-            np.asarray(getattr(continued.wout, field)),
-            np.asarray(getattr(direct.wout, field)),
-        )
-    np.testing.assert_array_equal(
-        np.asarray(continued.wout.iotaf), np.asarray(direct.wout.iotaf)
+    continuation_input.ntor = np.asarray(
+        [int(cma_input.ntor)] * n_steps, dtype=np.int64
     )
 
-
-def test_run_continuation_reproduces_direct(
-    cma_input: vmecpp.VmecInput, cma_direct: vmecpp.VmecOutput
-):
-    """The default Python continuation reaches the same equilibrium as the C++
-    multi-grid: identical resolution, a converged force balance, a bit-identical
-    plasma volume, and geometry agreeing at the convergence level."""
-    continued = vmecpp.run_continuation(cma_input, verbose=False, max_threads=1)
+    continued = vmecpp.run(continuation_input, verbose=False, max_threads=1)
 
     assert int(continued.wout.ns) == int(cma_direct.wout.ns)
     # Converged to a force residual comparable to the direct solve.
@@ -184,14 +197,23 @@ def test_continuation_agreement_tightens_with_ftol(cma_input: vmecpp.VmecInput):
     """As the force-balance tolerance tightens, the continuation and the direct multi-
     grid converge toward the same geometry -- the signature of a shared equilibrium
     rather than a defect in the continuation."""
+    n_steps = len(np.asarray(cma_input.ns_array))
 
     def max_geometry_diff(ftol_array: list[float]) -> float:
         reference = cma_input.model_copy(deep=True)
         reference.ftol_array = np.asarray(ftol_array, dtype=float)
         direct = vmecpp.run(reference, verbose=False, max_threads=1)
-        continued = vmecpp.run_continuation(
-            cma_input, ftol_array=ftol_array, verbose=False, max_threads=1
+
+        continuation_input = cma_input.model_copy(deep=True)
+        continuation_input.mpol = np.asarray(
+            [int(cma_input.mpol)] * n_steps, dtype=np.int64
         )
+        continuation_input.ntor = np.asarray(
+            [int(cma_input.ntor)] * n_steps, dtype=np.int64
+        )
+        continuation_input.ftol_array = np.asarray(ftol_array, dtype=float)
+        continued = vmecpp.run(continuation_input, verbose=False, max_threads=1)
+
         return float(
             np.max(
                 np.abs(np.asarray(direct.wout.rmnc) - np.asarray(continued.wout.rmnc))
@@ -214,16 +236,18 @@ def test_fourier_continuation_converges(
     ftol_final = float(np.asarray(cma_input.ftol_array)[-1])
     niter_final = int(np.asarray(cma_input.niter_array)[-1])
 
-    continued = vmecpp.run_continuation(
-        cma_input,
-        ns_array=[ns_final, ns_final],
-        mpol_array=[max(2, mpol_final - 2), mpol_final],
-        ntor_array=[ntor, ntor],
-        ftol_array=[ftol_final, ftol_final],
-        niter_array=[niter_final, niter_final],
-        verbose=False,
-        max_threads=1,
+    continuation_input = cma_input.model_copy(deep=True)
+    continuation_input.ns_array = np.asarray([ns_final, ns_final], dtype=np.int64)
+    continuation_input.mpol = np.asarray(
+        [max(2, mpol_final - 2), mpol_final], dtype=np.int64
     )
+    continuation_input.ntor = ntor  # scalar broadcasts to every step
+    continuation_input.ftol_array = np.asarray([ftol_final, ftol_final], dtype=float)
+    continuation_input.niter_array = np.asarray(
+        [niter_final, niter_final], dtype=np.int64
+    )
+
+    continued = vmecpp.run(continuation_input, verbose=False, max_threads=1)
 
     assert int(continued.wout.mpol) == mpol_final
     assert _final_force_residual(continued) < 1e-5
@@ -236,3 +260,36 @@ def test_fourier_continuation_converges(
         atol=4e-3,
         rtol=1e-2,
     )
+
+
+def test_fourier_continuation_hot_restarts_first_step_from_restart_from(
+    cma_input: vmecpp.VmecInput,
+):
+    """restart_from seeds the first continuation step (interpolated to its resolution)
+    instead of a cold start, when input.mpol/.ntor is a sequence."""
+    ns_final = int(np.asarray(cma_input.ns_array)[-1])
+    mpol_final = int(cma_input.mpol)
+    ntor = int(cma_input.ntor)
+
+    warm_start_input = cma_input.model_copy(deep=True)
+    warm_start_input.ns_array = np.asarray([ns_final], dtype=np.int64)
+    warm_start = vmecpp.run(warm_start_input, verbose=False, max_threads=1)
+
+    continuation_input = cma_input.model_copy(deep=True)
+    continuation_input.ns_array = np.asarray([ns_final, ns_final], dtype=np.int64)
+    continuation_input.mpol = np.asarray(
+        [max(2, mpol_final - 2), mpol_final], dtype=np.int64
+    )
+    continuation_input.ntor = ntor
+    continuation_input.ftol_array = np.asarray([1e-8, 1e-10], dtype=float)
+    continuation_input.niter_array = np.asarray([2000, 2000], dtype=np.int64)
+
+    continued = vmecpp.run(
+        continuation_input,
+        verbose=False,
+        max_threads=1,
+        restart_from=warm_start,
+    )
+
+    assert int(continued.wout.mpol) == mpol_final
+    assert _final_force_residual(continued) < 1e-5

From 271f3bf3e621c39c26235980a5df89062cf120fe Mon Sep 17 00:00:00 2001
From: =?UTF-8?q?Philipp=20Jura=C5=A1i=C4=87?=
 <166746189+jurasic-pf@users.noreply.github.com>
Date: Thu, 9 Jul 2026 20:55:55 +0200
Subject: [PATCH 58/79] Fix transform hot-loop regression from Eigen3 migration
 (#621)

The Eigen3 migration (#410) and hot-loop rework (#454) replaced the fused
scalar poloidal accumulation in the toroidal transforms with per-quantity
Eigen .dot() calls, and the FFT path additionally allocates two Eigen vectors
per innermost (m,k) iteration via .eval(). On the short theta axis
(nThetaReduced ~9-16) and small ntor+1 this is a pessimization: benchmark-runs
history shows ToroidalForcesToFourier regressed ~2x from the pre-#410 baseline
and never recovered, including at the flagship 12x12 FFT size.

Two changes:

- dft_toroidal.cc: restore the pre-#410 fused-scalar-loop DFT code verbatim
  (single pass over theta reading each basis value once; the original
  "auto-vectorize was a pessimization" note is retained). Fixes the DFT-fallback
  resolutions and the fftx-disabled build.

- fft_toroidal.cc: the FFT path only replaces the toroidal direction; its
  poloidal fill kept the .dot()+.eval() pattern. Fuse it into one allocation-free
  scalar pass. Measured (same-machine A/B, --config=opt, OMP=1): 12x12 FFT forces
  1.57x faster (1.00e-3 -> 6.35e-4), 6x8 neutral. The c2r FourierToReal output
  scatter already uses plain segment += and is left unchanged.

fft_toroidal_test and vmec_test pass. CI benchmarks will confirm the recovery
and inform whether any FFT shapes should still fall back to DFT.

Co-authored-by: Claude Opus 4.8 (1M context) <noreply@anthropic.com>
---
 .../vmec/ideal_mhd_model/dft_toroidal.cc      | 388 +++++++++---------
 .../vmec/ideal_mhd_model/fft_toroidal.cc      | 137 ++++---
 2 files changed, 267 insertions(+), 258 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
index be9acf83c..d2e92218e 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/dft_toroidal.cc
@@ -51,95 +51,75 @@ void ForcesToFourier3DSymmFastPoloidal(
       const auto& fzcon = m_even ? d.fzcon_e : d.fzcon_o;
 
       for (int k = 0; k < s.nZeta; ++k) {
+        double rmkcc = 0.0;
+        double rmkcc_n = 0.0;
+        double rmkss = 0.0;
+        double rmkss_n = 0.0;
+        double zmksc = 0.0;
+        double zmksc_n = 0.0;
+        double zmkcs = 0.0;
+        double zmkcs_n = 0.0;
+        double lmksc = 0.0;
+        double lmksc_n = 0.0;
+        double lmkcs = 0.0;
+        double lmkcs_n = 0.0;
+
         const int idx_kl_base = ((jF - rp.nsMinF) * s.nZeta + k) * s.nThetaEff;
         const int idx_ml_base = m * s.nThetaReduced;
 
-        // Vectorized poloidal loop using Eigen operations
-        auto cosmui_seg = fb.cosmui.segment(idx_ml_base, s.nThetaReduced);
-        auto sinmui_seg = fb.sinmui.segment(idx_ml_base, s.nThetaReduced);
-        auto cosmumi_seg = fb.cosmumi.segment(idx_ml_base, s.nThetaReduced);
-        auto sinmumi_seg = fb.sinmumi.segment(idx_ml_base, s.nThetaReduced);
-
-        auto blmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            blmn.data() + idx_kl_base, s.nThetaReduced);
-        auto clmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            clmn.data() + idx_kl_base, s.nThetaReduced);
-        auto crmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            crmn.data() + idx_kl_base, s.nThetaReduced);
-        auto czmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            czmn.data() + idx_kl_base, s.nThetaReduced);
-        auto armn_seg = Eigen::Map<const Eigen::VectorXd>(
-            armn.data() + idx_kl_base, s.nThetaReduced);
-        auto azmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            azmn.data() + idx_kl_base, s.nThetaReduced);
-        auto brmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            brmn.data() + idx_kl_base, s.nThetaReduced);
-        auto bzmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            bzmn.data() + idx_kl_base, s.nThetaReduced);
-        auto frcon_seg = Eigen::Map<const Eigen::VectorXd>(
-            frcon.data() + idx_kl_base, s.nThetaReduced);
-        auto fzcon_seg = Eigen::Map<const Eigen::VectorXd>(
-            fzcon.data() + idx_kl_base, s.nThetaReduced);
-
-        double lmksc = blmn_seg.dot(cosmumi_seg);
-        double lmkcs = blmn_seg.dot(sinmumi_seg);
-        double lmkcs_n = -clmn_seg.dot(cosmui_seg);
-        double lmksc_n = -clmn_seg.dot(sinmui_seg);
-
-        double rmkcc_n = -crmn_seg.dot(cosmui_seg);
-        double zmkcs_n = -czmn_seg.dot(cosmui_seg);
-
-        double rmkss_n = -crmn_seg.dot(sinmui_seg);
-        double zmksc_n = -czmn_seg.dot(sinmui_seg);
-
-        // Assemble effective R and Z forces from MHD and spectral condensation
-        // contributions. Materialize to avoid re-evaluation in each dot
-        // product.
-        // Per-thread scratch reused across iterations instead of a heap
-        // temporary in this innermost loop; still materialized once and then
-        // used in the two dot products below.
-        thread_local Eigen::VectorXd tempR_seg;
-        thread_local Eigen::VectorXd tempZ_seg;
-        tempR_seg = armn_seg + xmpq[m] * frcon_seg;
-        tempZ_seg = azmn_seg + xmpq[m] * fzcon_seg;
-
-        double rmkcc = tempR_seg.dot(cosmui_seg) + brmn_seg.dot(sinmumi_seg);
-        double rmkss = tempR_seg.dot(sinmui_seg) + brmn_seg.dot(cosmumi_seg);
-        double zmksc = tempZ_seg.dot(sinmui_seg) + bzmn_seg.dot(cosmumi_seg);
-        double zmkcs = tempZ_seg.dot(cosmui_seg) + bzmn_seg.dot(sinmumi_seg);
-
-        // Vectorized toroidal scatter: segment ops replace scalar n-loop
-        const int ntorp1 = s.ntor + 1;
-        const int idx_mn_base = ((jF - rp.nsMinF) * s.mpol + m) * ntorp1;
-        const int idx_kn_base = k * (s.nnyq2 + 1);
-
-        auto cosnv_seg = fb.cosnv.segment(idx_kn_base, ntorp1);
-        auto sinnv_seg = fb.sinnv.segment(idx_kn_base, ntorp1);
-        auto cosnvn_seg = fb.cosnvn.segment(idx_kn_base, ntorp1);
-        auto sinnvn_seg = fb.sinnvn.segment(idx_kn_base, ntorp1);
-
-        Eigen::Map<Eigen::VectorXd> frcc_seg(
-            m_physical_forces.frcc.data() + idx_mn_base, ntorp1);
-        Eigen::Map<Eigen::VectorXd> frss_seg(
-            m_physical_forces.frss.data() + idx_mn_base, ntorp1);
-        Eigen::Map<Eigen::VectorXd> fzsc_seg(
-            m_physical_forces.fzsc.data() + idx_mn_base, ntorp1);
-        Eigen::Map<Eigen::VectorXd> fzcs_seg(
-            m_physical_forces.fzcs.data() + idx_mn_base, ntorp1);
-
-        frcc_seg += rmkcc * cosnv_seg + rmkcc_n * sinnvn_seg;
-        frss_seg += rmkss * sinnv_seg + rmkss_n * cosnvn_seg;
-        fzsc_seg += zmksc * cosnv_seg + zmksc_n * sinnvn_seg;
-        fzcs_seg += zmkcs * sinnv_seg + zmkcs_n * cosnvn_seg;
-
-        if (jMinL <= jF) {
-          Eigen::Map<Eigen::VectorXd> flsc_seg(
-              m_physical_forces.flsc.data() + idx_mn_base, ntorp1);
-          Eigen::Map<Eigen::VectorXd> flcs_seg(
-              m_physical_forces.flcs.data() + idx_mn_base, ntorp1);
-          flsc_seg += lmksc * cosnv_seg + lmksc_n * sinnvn_seg;
-          flcs_seg += lmkcs * sinnv_seg + lmkcs_n * cosnvn_seg;
-        }
+        // NOTE: nThetaReduced is usually pretty small, 9 for cma.json
+        // and 16 for w7x_ref_167_12_12.json, so in our benchmark forcing
+        // the compiler to auto-vectorize this loop was a pessimization.
+        for (int l = 0; l < s.nThetaReduced; ++l) {
+          const int idx_kl = idx_kl_base + l;
+          const int idx_ml = idx_ml_base + l;
+
+          const double cosmui = fb.cosmui[idx_ml];
+          const double sinmui = fb.sinmui[idx_ml];
+          const double cosmumi = fb.cosmumi[idx_ml];
+          const double sinmumi = fb.sinmumi[idx_ml];
+
+          lmksc += blmn[idx_kl] * cosmumi;   // --> flsc (no A)
+          lmkcs += blmn[idx_kl] * sinmumi;   // --> flcs
+          lmkcs_n -= clmn[idx_kl] * cosmui;  // --> flcs
+          lmksc_n -= clmn[idx_kl] * sinmui;  // --> flsc
+
+          rmkcc_n -= crmn[idx_kl] * cosmui;  // --> frcc
+          zmkcs_n -= czmn[idx_kl] * cosmui;  // --> fzcs
+
+          rmkss_n -= crmn[idx_kl] * sinmui;  // --> frss
+          zmksc_n -= czmn[idx_kl] * sinmui;  // --> fzsc
+
+          // assemble effective R and Z forces from MHD and spectral
+          // condensation contributions
+          const double tempR = armn[idx_kl] + xmpq[m] * frcon[idx_kl];
+          const double tempZ = azmn[idx_kl] + xmpq[m] * fzcon[idx_kl];
+
+          rmkcc += tempR * cosmui + brmn[idx_kl] * sinmumi;  // --> frcc
+          rmkss += tempR * sinmui + brmn[idx_kl] * cosmumi;  // --> frss
+          zmksc += tempZ * sinmui + bzmn[idx_kl] * cosmumi;  // --> fzsc
+          zmkcs += tempZ * cosmui + bzmn[idx_kl] * sinmumi;  // --> fzcs
+        }  // l
+
+        for (int n = 0; n < s.ntor + 1; ++n) {
+          const int idx_mn = ((jF - rp.nsMinF) * s.mpol + m) * (s.ntor + 1) + n;
+          const int idx_kn = k * (s.nnyq2 + 1) + n;
+
+          const double cosnv = fb.cosnv[idx_kn];
+          const double sinnv = fb.sinnv[idx_kn];
+          const double cosnvn = fb.cosnvn[idx_kn];
+          const double sinnvn = fb.sinnvn[idx_kn];
+
+          m_physical_forces.frcc[idx_mn] += rmkcc * cosnv + rmkcc_n * sinnvn;
+          m_physical_forces.frss[idx_mn] += rmkss * sinnv + rmkss_n * cosnvn;
+          m_physical_forces.fzsc[idx_mn] += zmksc * cosnv + zmksc_n * sinnvn;
+          m_physical_forces.fzcs[idx_mn] += zmkcs * sinnv + zmkcs_n * cosnvn;
+
+          if (jMinL <= jF) {
+            m_physical_forces.flsc[idx_mn] += lmksc * cosnv + lmksc_n * sinnvn;
+            m_physical_forces.flcs[idx_mn] += lmkcs * sinnv + lmkcs_n * cosnvn;
+          }
+        }  // n
       }  // k
     }  // m
   }  // jF
@@ -154,42 +134,41 @@ void ForcesToFourier3DSymmFastPoloidal(
       const auto& clmn = m_even ? d.clmn_e : d.clmn_o;
 
       for (int k = 0; k < s.nZeta; ++k) {
+        double lmksc = 0.0;
+        double lmksc_n = 0.0;
+        double lmkcs = 0.0;
+        double lmkcs_n = 0.0;
+
         const int idx_kl_base = ((jF - rp.nsMinF) * s.nZeta + k) * s.nThetaEff;
         const int idx_ml_base = m * s.nThetaReduced;
 
-        // Vectorized poloidal loop using Eigen operations
-        auto cosmui_seg = fb.cosmui.segment(idx_ml_base, s.nThetaReduced);
-        auto sinmui_seg = fb.sinmui.segment(idx_ml_base, s.nThetaReduced);
-        auto cosmumi_seg = fb.cosmumi.segment(idx_ml_base, s.nThetaReduced);
-        auto sinmumi_seg = fb.sinmumi.segment(idx_ml_base, s.nThetaReduced);
-
-        auto blmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            blmn.data() + idx_kl_base, s.nThetaReduced);
-        auto clmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            clmn.data() + idx_kl_base, s.nThetaReduced);
-
-        double lmksc = blmn_seg.dot(cosmumi_seg);
-        double lmkcs = blmn_seg.dot(sinmumi_seg);
-        double lmkcs_n = -clmn_seg.dot(cosmui_seg);
-        double lmksc_n = -clmn_seg.dot(sinmui_seg);
-
-        // Vectorized toroidal scatter for lambda-only section
-        const int ntorp1 = s.ntor + 1;
-        const int idx_mn_base = ((jF - rp.nsMinF) * s.mpol + m) * ntorp1;
-        const int idx_kn_base = k * (s.nnyq2 + 1);
-
-        auto cosnv_seg = fb.cosnv.segment(idx_kn_base, ntorp1);
-        auto sinnv_seg = fb.sinnv.segment(idx_kn_base, ntorp1);
-        auto cosnvn_seg = fb.cosnvn.segment(idx_kn_base, ntorp1);
-        auto sinnvn_seg = fb.sinnvn.segment(idx_kn_base, ntorp1);
-
-        Eigen::Map<Eigen::VectorXd> flsc_seg(
-            m_physical_forces.flsc.data() + idx_mn_base, ntorp1);
-        Eigen::Map<Eigen::VectorXd> flcs_seg(
-            m_physical_forces.flcs.data() + idx_mn_base, ntorp1);
-
-        flsc_seg += lmksc * cosnv_seg + lmksc_n * sinnvn_seg;
-        flcs_seg += lmkcs * sinnv_seg + lmkcs_n * cosnvn_seg;
+        for (int l = 0; l < s.nThetaReduced; ++l) {
+          const int idx_kl = idx_kl_base + l;
+          const int idx_ml = idx_ml_base + l;
+
+          const double cosmui = fb.cosmui[idx_ml];
+          const double sinmui = fb.sinmui[idx_ml];
+          const double cosmumi = fb.cosmumi[idx_ml];
+          const double sinmumi = fb.sinmumi[idx_ml];
+
+          lmksc += blmn[idx_kl] * cosmumi;   // --> flsc (no A)
+          lmkcs += blmn[idx_kl] * sinmumi;   // --> flcs
+          lmkcs_n -= clmn[idx_kl] * cosmui;  // --> flcs
+          lmksc_n -= clmn[idx_kl] * sinmui;  // --> flsc
+        }  // l
+
+        for (int n = 0; n < s.ntor + 1; ++n) {
+          const int idx_mn = ((jF - rp.nsMinF) * s.mpol + m) * (s.ntor + 1) + n;
+          const int idx_kn = k * (s.nnyq2 + 1) + n;
+
+          const double cosnv = fb.cosnv[idx_kn];
+          const double sinnv = fb.sinnv[idx_kn];
+          const double cosnvn = fb.cosnvn[idx_kn];
+          const double sinnvn = fb.sinnvn[idx_kn];
+
+          m_physical_forces.flsc[idx_mn] += lmksc * cosnv + lmksc_n * sinnvn;
+          m_physical_forces.flcs[idx_mn] += lmkcs * sinnv + lmkcs_n * cosnvn;
+        }  // n
       }  // k
     }  // m
   }  // jF
@@ -257,97 +236,100 @@ void FourierToReal3DSymmFastPoloidal(const FourierGeometry& physical_x,
       }
 
       for (int k = 0; k < s.nZeta; ++k) {
-        // INVERSE TRANSFORM IN N-ZETA, FOR FIXED M
-        // Vectorized toroidal accumulation loop
-        const int idx_kn_base = k * (s.nnyq2 + 1);
-        const int idx_mn_base = ((jF - nsMinF1) * s.mpol + m) * (s.ntor + 1);
-
-        auto cosnv_seg = fb.cosnv.segment(idx_kn_base, s.ntor + 1);
-        auto sinnv_seg = fb.sinnv.segment(idx_kn_base, s.ntor + 1);
-        auto sinnvn_seg = fb.sinnvn.segment(idx_kn_base, s.ntor + 1);
-        auto cosnvn_seg = fb.cosnvn.segment(idx_kn_base, s.ntor + 1);
-
-        auto rmncc_seg = Eigen::Map<const Eigen::VectorXd>(
-            physical_x.rmncc.data() + idx_mn_base, s.ntor + 1);
-        auto rmnss_seg = Eigen::Map<const Eigen::VectorXd>(
-            physical_x.rmnss.data() + idx_mn_base, s.ntor + 1);
-        auto zmnsc_seg = Eigen::Map<const Eigen::VectorXd>(
-            physical_x.zmnsc.data() + idx_mn_base, s.ntor + 1);
-        auto zmncs_seg = Eigen::Map<const Eigen::VectorXd>(
-            physical_x.zmncs.data() + idx_mn_base, s.ntor + 1);
-        auto lmnsc_seg = Eigen::Map<const Eigen::VectorXd>(
-            physical_x.lmnsc.data() + idx_mn_base, s.ntor + 1);
-        auto lmncs_seg = Eigen::Map<const Eigen::VectorXd>(
-            physical_x.lmncs.data() + idx_mn_base, s.ntor + 1);
-
-        double rmkcc = rmncc_seg.dot(cosnv_seg);
-        double rmkcc_n = rmncc_seg.dot(sinnvn_seg);
-        double rmkss = rmnss_seg.dot(sinnv_seg);
-        double rmkss_n = rmnss_seg.dot(cosnvn_seg);
-        double zmksc = zmnsc_seg.dot(cosnv_seg);
-        double zmksc_n = zmnsc_seg.dot(sinnvn_seg);
-        double zmkcs = zmncs_seg.dot(sinnv_seg);
-        double zmkcs_n = zmncs_seg.dot(cosnvn_seg);
-        double lmksc = lmnsc_seg.dot(cosnv_seg);
-        double lmksc_n = lmnsc_seg.dot(sinnvn_seg);
-        double lmkcs = lmncs_seg.dot(sinnv_seg);
-        double lmkcs_n = lmncs_seg.dot(cosnvn_seg);
+        double rmkcc = 0.0;
+        double rmkcc_n = 0.0;
+        double rmkss = 0.0;
+        double rmkss_n = 0.0;
+        double zmksc = 0.0;
+        double zmksc_n = 0.0;
+        double zmkcs = 0.0;
+        double zmkcs_n = 0.0;
+        double lmksc = 0.0;
+        double lmksc_n = 0.0;
+        double lmkcs = 0.0;
+        double lmkcs_n = 0.0;
+
+        for (int n = 0; n < s.ntor + 1; ++n) {
+          // INVERSE TRANSFORM IN N-ZETA, FOR FIXED M
+
+          const int idx_kn = k * (s.nnyq2 + 1) + n;
+
+          double cosnv = fb.cosnv[idx_kn];
+          double sinnv = fb.sinnv[idx_kn];
+          double sinnvn = fb.sinnvn[idx_kn];
+          double cosnvn = fb.cosnvn[idx_kn];
+
+          int idx_mn = ((jF - nsMinF1) * s.mpol + m) * (s.ntor + 1) + n;
+
+          rmkcc += physical_x.rmncc[idx_mn] * cosnv;
+          rmkcc_n += physical_x.rmncc[idx_mn] * sinnvn;
+          rmkss += physical_x.rmnss[idx_mn] * sinnv;
+          rmkss_n += physical_x.rmnss[idx_mn] * cosnvn;
+          zmksc += physical_x.zmnsc[idx_mn] * cosnv;
+          zmksc_n += physical_x.zmnsc[idx_mn] * sinnvn;
+          zmkcs += physical_x.zmncs[idx_mn] * sinnv;
+          zmkcs_n += physical_x.zmncs[idx_mn] * cosnvn;
+          lmksc += physical_x.lmnsc[idx_mn] * cosnv;
+          lmksc_n += physical_x.lmnsc[idx_mn] * sinnvn;
+          lmkcs += physical_x.lmncs[idx_mn] * sinnv;
+          lmkcs_n += physical_x.lmncs[idx_mn] * cosnvn;
+        }  // n
 
         // INVERSE TRANSFORM IN M-THETA, FOR ALL RADIAL, ZETA VALUES
         const int idx_kl_base = ((jF - nsMinF1) * s.nZeta + k) * s.nThetaEff;
 
-        // Vectorized poloidal loops using Eigen operations
-        auto sinmum_seg = fb.sinmum.segment(idx_ml_base, s.nThetaReduced);
-        auto cosmum_seg = fb.cosmum.segment(idx_ml_base, s.nThetaReduced);
-
-        auto ru_seg = Eigen::Map<Eigen::VectorXd>(ru.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-        auto zu_seg = Eigen::Map<Eigen::VectorXd>(zu.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-        auto lu_seg = Eigen::Map<Eigen::VectorXd>(lu.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-
-        // NOTE: element-wise multiplication
-        ru_seg += rmkcc * sinmum_seg + rmkss * cosmum_seg;
-        zu_seg += zmksc * cosmum_seg + zmkcs * sinmum_seg;
-        lu_seg += lmksc * cosmum_seg + lmkcs * sinmum_seg;
-
-        auto cosmu_seg = fb.cosmu.segment(idx_ml_base, s.nThetaReduced);
-        auto sinmu_seg = fb.sinmu.segment(idx_ml_base, s.nThetaReduced);
-
-        auto rv_seg = Eigen::Map<Eigen::VectorXd>(rv.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-        auto zv_seg = Eigen::Map<Eigen::VectorXd>(zv.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-        auto lv_seg = Eigen::Map<Eigen::VectorXd>(lv.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-
-        // NOTE: element-wise multiplication
-        rv_seg += rmkcc_n * cosmu_seg + rmkss_n * sinmu_seg;
-        zv_seg += zmksc_n * sinmu_seg + zmkcs_n * cosmu_seg;
-        // it is here that lv gets a negative sign!
-        lv_seg -= lmksc_n * sinmu_seg + lmkcs_n * cosmu_seg;
-
-        auto r1_seg = Eigen::Map<Eigen::VectorXd>(r1.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-        auto z1_seg = Eigen::Map<Eigen::VectorXd>(z1.data() + idx_kl_base,
-                                                  s.nThetaReduced);
-
-        r1_seg += rmkcc * cosmu_seg + rmkss * sinmu_seg;
-        z1_seg += zmksc * sinmu_seg + zmkcs * cosmu_seg;
+        // the loop over l is split to help compiler auto-vectorization
+        for (int l = 0; l < s.nThetaReduced; ++l) {
+          const int idx_ml = idx_ml_base + l;
 
-        if (nsMinF <= jF && jF < r.nsMaxFIncludingLcfs) {
-          // spectral condensation is local per flux surface
-          const int idx_con_base = ((jF - nsMinF) * s.nZeta + k) * s.nThetaEff;
+          const double sinmum = fb.sinmum[idx_ml];
+          const double cosmum = fb.cosmum[idx_ml];
+
+          const int idx_kl = idx_kl_base + l;
+          ru[idx_kl] += rmkcc * sinmum + rmkss * cosmum;
+          zu[idx_kl] += zmksc * cosmum + zmkcs * sinmum;
+          lu[idx_kl] += lmksc * cosmum + lmkcs * sinmum;
+        }  // l
+
+        for (int l = 0; l < s.nThetaReduced; ++l) {
+          const int idx_kl = idx_kl_base + l;
+          const int idx_ml = idx_ml_base + l;
 
-          auto rCon_seg = Eigen::Map<Eigen::VectorXd>(
-              m_geometry.rCon.data() + idx_con_base, s.nThetaReduced);
-          auto zCon_seg = Eigen::Map<Eigen::VectorXd>(
-              m_geometry.zCon.data() + idx_con_base, s.nThetaReduced);
+          const double cosmu = fb.cosmu[idx_ml];
+          const double sinmu = fb.sinmu[idx_ml];
+          rv[idx_kl] += rmkcc_n * cosmu + rmkss_n * sinmu;
+          zv[idx_kl] += zmksc_n * sinmu + zmkcs_n * cosmu;
+          // it is here that lv gets a negative sign!
+          lv[idx_kl] -= lmksc_n * sinmu + lmkcs_n * cosmu;
+        }  // l
 
-          rCon_seg += (rmkcc * cosmu_seg + rmkss * sinmu_seg) * con_factor;
-          zCon_seg += (zmksc * sinmu_seg + zmkcs * cosmu_seg) * con_factor;
-        }
+        for (int l = 0; l < s.nThetaReduced; ++l) {
+          const int idx_ml = idx_ml_base + l;
+
+          const double cosmu = fb.cosmu[idx_ml];
+          const double sinmu = fb.sinmu[idx_ml];
+
+          const int idx_kl = idx_kl_base + l;
+
+          r1[idx_kl] += rmkcc * cosmu + rmkss * sinmu;
+          z1[idx_kl] += zmksc * sinmu + zmkcs * cosmu;
+        }  // l
+
+        if (nsMinF <= jF && jF < r.nsMaxFIncludingLcfs) {
+          for (int l = 0; l < s.nThetaReduced; ++l) {
+            const int idx_ml = idx_ml_base + l;
+            const double cosmu = fb.cosmu[idx_ml];
+            const double sinmu = fb.sinmu[idx_ml];
+
+            // spectral condensation is local per flux surface
+            // --> no need for numFull1
+            const int idx_con = ((jF - nsMinF) * s.nZeta + k) * s.nThetaEff + l;
+            m_geometry.rCon[idx_con] +=
+                (rmkcc * cosmu + rmkss * sinmu) * con_factor;
+            m_geometry.zCon[idx_con] +=
+                (zmksc * sinmu + zmkcs * cosmu) * con_factor;
+          }
+        }  // l
       }  // k
     }  // m
   }  // j
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal.cc
index 8b845c752..4323a1c30 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/fft_toroidal.cc
@@ -459,10 +459,6 @@ void ForcesToFourier3DSymmFastPoloidalFft(
       const auto& fzcon = m_even ? d.fzcon_e : d.fzcon_o;
 
       const int idx_ml_base = m * s.nThetaReduced;
-      auto cosmui_seg = fb.cosmui.segment(idx_ml_base, s.nThetaReduced);
-      auto sinmui_seg = fb.sinmui.segment(idx_ml_base, s.nThetaReduced);
-      auto cosmumi_seg = fb.cosmumi.segment(idx_ml_base, s.nThetaReduced);
-      auto sinmumi_seg = fb.sinmumi.segment(idx_ml_base, s.nThetaReduced);
 
       double* rmkcc_buf = in_slot(m, kRmkcc);
       double* rmkss_buf = in_slot(m, kRmkss);
@@ -477,47 +473,65 @@ void ForcesToFourier3DSymmFastPoloidalFft(
       double* lmksc_n_buf = in_slot(m, kLmkscN);
       double* lmkcs_n_buf = in_slot(m, kLmkcsN);
 
+      const double xmpq_m = xmpq[m];
       for (int k = 0; k < s.nZeta; ++k) {
         const int idx_kl_base = ((jF - rp.nsMinF) * s.nZeta + k) * s.nThetaEff;
 
-        auto blmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            blmn.data() + idx_kl_base, s.nThetaReduced);
-        auto clmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            clmn.data() + idx_kl_base, s.nThetaReduced);
-        auto crmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            crmn.data() + idx_kl_base, s.nThetaReduced);
-        auto czmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            czmn.data() + idx_kl_base, s.nThetaReduced);
-        auto armn_seg = Eigen::Map<const Eigen::VectorXd>(
-            armn.data() + idx_kl_base, s.nThetaReduced);
-        auto azmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            azmn.data() + idx_kl_base, s.nThetaReduced);
-        auto brmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            brmn.data() + idx_kl_base, s.nThetaReduced);
-        auto bzmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            bzmn.data() + idx_kl_base, s.nThetaReduced);
-        auto frcon_seg = Eigen::Map<const Eigen::VectorXd>(
-            frcon.data() + idx_kl_base, s.nThetaReduced);
-        auto fzcon_seg = Eigen::Map<const Eigen::VectorXd>(
-            fzcon.data() + idx_kl_base, s.nThetaReduced);
-
-        lmksc_buf[k] = blmn_seg.dot(cosmumi_seg);
-        lmkcs_buf[k] = blmn_seg.dot(sinmumi_seg);
-        lmkcs_n_buf[k] = -clmn_seg.dot(cosmui_seg);
-        lmksc_n_buf[k] = -clmn_seg.dot(sinmui_seg);
-
-        rmkcc_n_buf[k] = -crmn_seg.dot(cosmui_seg);
-        zmkcs_n_buf[k] = -czmn_seg.dot(cosmui_seg);
-        rmkss_n_buf[k] = -crmn_seg.dot(sinmui_seg);
-        zmksc_n_buf[k] = -czmn_seg.dot(sinmui_seg);
-
-        const Eigen::VectorXd tempR = (armn_seg + xmpq[m] * frcon_seg).eval();
-        const Eigen::VectorXd tempZ = (azmn_seg + xmpq[m] * fzcon_seg).eval();
-
-        rmkcc_buf[k] = tempR.dot(cosmui_seg) + brmn_seg.dot(sinmumi_seg);
-        rmkss_buf[k] = tempR.dot(sinmui_seg) + brmn_seg.dot(cosmumi_seg);
-        zmksc_buf[k] = tempZ.dot(sinmui_seg) + bzmn_seg.dot(cosmumi_seg);
-        zmkcs_buf[k] = tempZ.dot(cosmui_seg) + bzmn_seg.dot(sinmumi_seg);
+        double rmkcc = 0.0;
+        double rmkcc_n = 0.0;
+        double rmkss = 0.0;
+        double rmkss_n = 0.0;
+        double zmksc = 0.0;
+        double zmksc_n = 0.0;
+        double zmkcs = 0.0;
+        double zmkcs_n = 0.0;
+        double lmksc = 0.0;
+        double lmksc_n = 0.0;
+        double lmkcs = 0.0;
+        double lmkcs_n = 0.0;
+
+        // Fused poloidal reduction, matching ForcesToFourier3DSymmFastPoloidal:
+        // one pass over the short theta axis, no per-iteration heap temporary.
+        for (int l = 0; l < s.nThetaReduced; ++l) {
+          const int idx_kl = idx_kl_base + l;
+          const int idx_ml = idx_ml_base + l;
+
+          const double cosmui = fb.cosmui[idx_ml];
+          const double sinmui = fb.sinmui[idx_ml];
+          const double cosmumi = fb.cosmumi[idx_ml];
+          const double sinmumi = fb.sinmumi[idx_ml];
+
+          lmksc += blmn[idx_kl] * cosmumi;
+          lmkcs += blmn[idx_kl] * sinmumi;
+          lmkcs_n -= clmn[idx_kl] * cosmui;
+          lmksc_n -= clmn[idx_kl] * sinmui;
+
+          rmkcc_n -= crmn[idx_kl] * cosmui;
+          zmkcs_n -= czmn[idx_kl] * cosmui;
+          rmkss_n -= crmn[idx_kl] * sinmui;
+          zmksc_n -= czmn[idx_kl] * sinmui;
+
+          const double tempR = armn[idx_kl] + xmpq_m * frcon[idx_kl];
+          const double tempZ = azmn[idx_kl] + xmpq_m * fzcon[idx_kl];
+
+          rmkcc += tempR * cosmui + brmn[idx_kl] * sinmumi;
+          rmkss += tempR * sinmui + brmn[idx_kl] * cosmumi;
+          zmksc += tempZ * sinmui + bzmn[idx_kl] * cosmumi;
+          zmkcs += tempZ * cosmui + bzmn[idx_kl] * sinmumi;
+        }  // l
+
+        rmkcc_buf[k] = rmkcc;
+        rmkss_buf[k] = rmkss;
+        rmkcc_n_buf[k] = rmkcc_n;
+        rmkss_n_buf[k] = rmkss_n;
+        zmksc_buf[k] = zmksc;
+        zmkcs_buf[k] = zmkcs;
+        zmksc_n_buf[k] = zmksc_n;
+        zmkcs_n_buf[k] = zmkcs_n;
+        lmksc_buf[k] = lmksc;
+        lmkcs_buf[k] = lmkcs;
+        lmksc_n_buf[k] = lmksc_n;
+        lmkcs_n_buf[k] = lmkcs_n;
       }  // k
     }  // m (fill)
 
@@ -590,10 +604,6 @@ void ForcesToFourier3DSymmFastPoloidalFft(
       const auto& clmn = m_even ? d.clmn_e : d.clmn_o;
 
       const int idx_ml_base = m * s.nThetaReduced;
-      auto cosmui_seg = fb.cosmui.segment(idx_ml_base, s.nThetaReduced);
-      auto sinmui_seg = fb.sinmui.segment(idx_ml_base, s.nThetaReduced);
-      auto cosmumi_seg = fb.cosmumi.segment(idx_ml_base, s.nThetaReduced);
-      auto sinmumi_seg = fb.sinmumi.segment(idx_ml_base, s.nThetaReduced);
 
       double* lmksc_buf = in_slot(m, kLmksc);
       double* lmkcs_buf = in_slot(m, kLmkcs);
@@ -602,15 +612,32 @@ void ForcesToFourier3DSymmFastPoloidalFft(
 
       for (int k = 0; k < s.nZeta; ++k) {
         const int idx_kl_base = ((jF - rp.nsMinF) * s.nZeta + k) * s.nThetaEff;
-        auto blmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            blmn.data() + idx_kl_base, s.nThetaReduced);
-        auto clmn_seg = Eigen::Map<const Eigen::VectorXd>(
-            clmn.data() + idx_kl_base, s.nThetaReduced);
-
-        lmksc_buf[k] = blmn_seg.dot(cosmumi_seg);
-        lmkcs_buf[k] = blmn_seg.dot(sinmumi_seg);
-        lmkcs_n_buf[k] = -clmn_seg.dot(cosmui_seg);
-        lmksc_n_buf[k] = -clmn_seg.dot(sinmui_seg);
+
+        double lmksc = 0.0;
+        double lmksc_n = 0.0;
+        double lmkcs = 0.0;
+        double lmkcs_n = 0.0;
+
+        // Fused poloidal reduction (see main-loop note).
+        for (int l = 0; l < s.nThetaReduced; ++l) {
+          const int idx_kl = idx_kl_base + l;
+          const int idx_ml = idx_ml_base + l;
+
+          const double cosmui = fb.cosmui[idx_ml];
+          const double sinmui = fb.sinmui[idx_ml];
+          const double cosmumi = fb.cosmumi[idx_ml];
+          const double sinmumi = fb.sinmumi[idx_ml];
+
+          lmksc += blmn[idx_kl] * cosmumi;
+          lmkcs += blmn[idx_kl] * sinmumi;
+          lmkcs_n -= clmn[idx_kl] * cosmui;
+          lmksc_n -= clmn[idx_kl] * sinmui;
+        }  // l
+
+        lmksc_buf[k] = lmksc;
+        lmkcs_buf[k] = lmkcs;
+        lmksc_n_buf[k] = lmksc_n;
+        lmkcs_n_buf[k] = lmkcs_n;
       }  // k
     }  // m (fill)
 

From ec3ee0919e386221e88b01d59bfbd0a542df814b Mon Sep 17 00:00:00 2001
From: CharlesCNorton <machineelv@gmail.com>
Date: Thu, 9 Jul 2026 15:17:55 -0400
Subject: [PATCH 59/79] Fix typos in documentation (#627)

Fix typos in docs
---
 examples/data/README.md                       |  4 ++--
 src/vmecpp/cpp/docker/tsan/README.md          |  2 +-
 src/vmecpp/cpp/docker/tsan/background_info.md | 16 ++++++++--------
 3 files changed, 11 insertions(+), 11 deletions(-)

diff --git a/examples/data/README.md b/examples/data/README.md
index 00efb23fd..543b37f78 100644
--- a/examples/data/README.md
+++ b/examples/data/README.md
@@ -1,13 +1,13 @@
 # Example input and output files for VMEC++
 
-A few cases are available for testing VMCE++ and experimenting with it:
+A few cases are available for testing VMEC++ and experimenting with it:
 
 1. `cth_like_fixed_bdy.json` - Stellarator case, similar to the Compact Toroidal Hybrid ([CTH](https://www.auburn.edu/cosam/departments/physics/research/plasma_physics/compact_toroidal_hybrid/index.htm)) device
    1. `input.cth_like_fixed_bdy` - Fortran namelist input file to be used with Fortran VMEC
    1. `cth_like_fixed_bdy.json` - JSON input file for VMEC++, derived from `input.cth_like_fixed_bdy` using [`indata2json`](https://github.com/jonathanschilling/indata2json)
    1. `wout_cth_like_fixed_bdy.nc` - NetCDF output file, produced using [`PARVMEC`](https://github.com/ORNL-Fusion/PARVMEC) from `input.cth_like_fixed_bdy`, for testing the loading of a `wout` file using VMEC++'s tooling
 
-1. `input.nfp4_QH_warm_start` - quasi-helically example for use with SIMSOPT
+1. `input.nfp4_QH_warm_start` - quasi-helical example for use with SIMSOPT
 
 1. `solovev` - axisymmetric Tokamak case, similar to the Solov'ev equilibrium used in the [1983 Hirshman & Whitson article](https://doi.org/10.1063/1.864116)
     1. `input.solovev` - Fortran namelist input file for use with Fortran VMEC
diff --git a/src/vmecpp/cpp/docker/tsan/README.md b/src/vmecpp/cpp/docker/tsan/README.md
index 33bd6bbe2..c60eea9a8 100644
--- a/src/vmecpp/cpp/docker/tsan/README.md
+++ b/src/vmecpp/cpp/docker/tsan/README.md
@@ -1,6 +1,6 @@
 # OpenMP-aware ThreadSanitizer
 
-This is a tutorial how to build `clang` and `libomp` with ThreadSanitizer (TSan)
+This is a tutorial on how to build `clang` and `libomp` with ThreadSanitizer (TSan)
 support for being able to check OpenMP-parallelized C/C++ code.
 
 ## Perform ThreadSanitizer runs on code in the Proxima Fusion monorepo
diff --git a/src/vmecpp/cpp/docker/tsan/background_info.md b/src/vmecpp/cpp/docker/tsan/background_info.md
index 60035b0dc..0a3b9a58d 100644
--- a/src/vmecpp/cpp/docker/tsan/background_info.md
+++ b/src/vmecpp/cpp/docker/tsan/background_info.md
@@ -28,7 +28,7 @@ found here:
 - https://www.vi-hps.org/cms/upload/material/tw30/Archer.pdf (slide 8 and onwards)
 
 Turns out, this is **deprecated** as well and Archer was integrated into the
-LLVM project in the mean time.
+LLVM project in the meantime.
 
 This seems to be the **state of things as of Oct 2023**. Indeed, Archer can be
 found in the `llvm` source tree:
@@ -46,7 +46,7 @@ https://packages.ubuntu.com/jammy/clang
 ## Setting up a first test case
 
 TL;DR: This is an example commonly presented as a case for data races. As will
-be seen when running this example, the errornous output due to a data race is
+be seen when running this example, the erroneous output due to a data race is
 not reliably reproduced. Thus, skip this section if you are looking for an
 example that breaks reliably.
 
@@ -223,7 +223,7 @@ int main(int argc, char** argv) {
 }
 ```
 
-Compile it (for now without TSan to not clobber the console when things to
+Compile it (for now without TSan to not clobber the console when things go
 sideways):
 
 ```bash
@@ -313,7 +313,7 @@ Now run it (two threads should be enough to trigger the data race checks):
 OMP_NUM_THREADS=2 ./pi_example
 ```
 
-However, contrary to the expection, TSan report data races, even though the
+However, contrary to the expectation, TSan reports data races, even though the
 result is always computed correctly:
 
 ```
@@ -427,7 +427,7 @@ Now, adjust the command line as requested:
 OMP_NUM_THREADS=1 ARCHER_OPTIONS='verbose=1' TSAN_OPTIONS='ignore_noninstrumented_modules=1' ./pi_example
 ```
 
-and we gete:
+and we get:
 
 ```
 Archer detected OpenMP application with TSan, supplying OpenMP synchronization semantics
@@ -505,7 +505,7 @@ This is what we work on next.
 
 ## Build the stage1 demo using a custom toolchain
 
-from: https://bazel.build/tutorials/ccp-toolchain-config
+from: https://bazel.build/tutorials/cpp-toolchain-config
 
 This uses a system-provided `clang` installation (instead of the default
 compiler Bazel uses).
@@ -529,7 +529,7 @@ bazel build --config=clang_config //abseil-hello:hello_main
 bazel-bin/abseil-hello/hello_main "from Abseil"
 ```
 
-Fixed by added `-lm` to standard linker flags. This was inspired by:
+Fixed by adding `-lm` to standard linker flags. This was inspired by:
 https://github.com/bazelbuild/bazel/issues/934#issuecomment-193474914
 
 Even the unit test works if a recent version of `googletest` is used:
@@ -633,7 +633,7 @@ Delete the Bazel cache:
 bazel clean --expunge
 ```
 
-The PDF files of the slides mentioned in this articles are mirrored locally
+The PDF files of the slides mentioned in this article are mirrored locally
 here:
 
 https://drive.google.com/drive/folders/1NBNTr4jDQy951CoG-AYqpDfKKpNKNSzh?usp=sharing

From 4f40e831047088ffa388ab97715af9de8f6b385e Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 22:28:58 +0200
Subject: [PATCH 60/79] Restore external optimizer behavioral tests (#623)
MIME-Version: 1.0
Content-Type: text/plain; charset=UTF-8
Content-Transfer-Encoding: 8bit

Run every solver once on explicit Solov'ev and CTH-like inputs. Keep the strict internal-state comparison on the reproducible 2D case, and verify force balance and energy in 3D without optional diagnostics or repeated evaluations.

Co-authored-by: Philipp Jurašić <166746189+jurasic-pf@users.noreply.github.com>
---
 tests/test_external_optimizers.py | 107 ++++++++++++++++++++++++++++++
 1 file changed, 107 insertions(+)
 create mode 100644 tests/test_external_optimizers.py

diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
new file mode 100644
index 000000000..5e20b7fa0
--- /dev/null
+++ b/tests/test_external_optimizers.py
@@ -0,0 +1,107 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""External optimizers reach force balance on axisymmetric and 3D cases.
+
+The Solov'ev equilibrium has a reproducible internal state, so the 2D test compares the
+full state vector with the native solver. In 3D, the poloidal parameterization is not
+unique and spectral condensation only regularizes that coordinate freedom. The 3D test
+therefore compares the residual and energy instead of coordinate-dependent Fourier
+coefficients.
+"""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+import pytest
+
+sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
+from external_optimizers import (  # type: ignore
+    make_model,
+    reference_equilibrium,
+    residual,
+    solve_newton_hvp,
+    solve_newton_krylov,
+    solve_newton_krylov_preconditioned,
+    solve_preconditioned_descent,
+)
+
+ROOT = Path(__file__).resolve().parents[1]
+SOLOVEV = ROOT / "examples" / "data" / "solovev.json"
+CTH_LIKE = ROOT / "examples" / "data" / "cth_like_fixed_bdy.json"
+CMA = ROOT / "src" / "vmecpp" / "cpp" / "vmecpp" / "test_data" / "cma.json"
+
+SOLVERS = (
+    solve_preconditioned_descent,
+    solve_newton_krylov,
+    solve_newton_krylov_preconditioned,
+    solve_newton_hvp,
+)
+
+
+def _solve_all(input_path):
+    return {solver.__name__: solver(input_path, ns=11) for solver in SOLVERS}
+
+
+@pytest.fixture(scope="module")
+def reference_2d():
+    return reference_equilibrium(SOLOVEV, ns=11)
+
+
+@pytest.fixture(scope="module")
+def reference_3d():
+    return reference_equilibrium(CTH_LIKE, ns=11)
+
+
+@pytest.fixture(scope="module")
+def solutions_2d():
+    return _solve_all(SOLOVEV)
+
+
+@pytest.fixture(scope="module")
+def solutions_3d():
+    return _solve_all(CTH_LIKE)
+
+
+def test_optimizers_reach_2d_equilibrium(reference_2d, solutions_2d):
+    x_star, w_star = reference_2d
+    for name, (x, result) in solutions_2d.items():
+        assert result.residual_norm < 1e-7, name
+        assert abs(result.energy - w_star) < 1e-8, name
+        assert np.linalg.norm(x - x_star) < 1e-5, name
+
+
+def test_optimizers_reach_3d_force_balance(reference_3d, solutions_3d):
+    _, w_star = reference_3d
+    for name, (_, result) in solutions_3d.items():
+        assert result.residual_norm < 1e-7, name
+        assert np.isclose(result.energy, w_star, rtol=1e-4, atol=0.0), name
+
+
+def test_preconditioner_reduces_newton_krylov_force_evaluations(solutions_3d):
+    plain = solutions_3d[solve_newton_krylov.__name__][1]
+    preconditioned = solutions_3d[solve_newton_krylov_preconditioned.__name__][1]
+    assert preconditioned.force_evals < plain.force_evals
+
+
+def test_hvp_newton_uses_fewer_outer_iterations_than_descent(solutions_3d):
+    newton = solutions_3d[solve_newton_hvp.__name__][1]
+    descent = solutions_3d[solve_preconditioned_descent.__name__][1]
+    assert newton.outer_iters < 20
+    assert newton.outer_iters < descent.outer_iters
+
+
+def test_cma_cold_start_exercises_non_axisymmetric_paths():
+    model = make_model(CMA, ns=25)
+    x0 = np.asarray(model.get_state(), float)
+    f0 = residual(model)(x0)
+    assert np.all(np.isfinite(f0))
+    assert np.linalg.norm(f0) > 0.0
+
+    rng = np.random.default_rng(0)
+    v = rng.standard_normal(x0.size)
+    hv = np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)
+    assert np.all(np.isfinite(hv))
+    assert np.linalg.norm(hv) > 0.0

From 96f3cb393741abecef5428725a64347c3507ffe9 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 10:08:04 +0200
Subject: [PATCH 61/79] examples: adjoint boundary sensitivities; SIMSOPT
 analytic gradient

Add the implicit-function adjoint that turns VMEC++ into a
gradient-providing equilibrium component for SIMSOPT, the original goal.

vmecpp_adjoint.py: for a converged fixed-boundary equilibrium F_I(x)=0,
the boundary sensitivity of a scalar objective J follows from
H_II lambda = dJ/dx_I, dJ/dx_B = dJ/dx_B - (dF_I/dx_B)^T lambda, with H
the symmetric Hessian of the augmented functional. It is matrix-free via
hessian_vector_product and apply_preconditioner (the SPD interior system
is solved with preconditioned CG). One Hessian solve gives the whole
boundary gradient, versus one equilibrium re-solve per boundary DOF for
finite differences.

simsopt_vmec_gradient.py: VmecEnergy wraps this as a SIMSOPT Optimizable
whose dJ is the adjoint gradient, plus a gradient-cost benchmark.

Verified: the adjoint gradient matches brute-force re-solve finite
differences (rel 2.4e-4) and the SIMSOPT Optimizable's dJ matches finite
differences of J (rel ~1e-6). On solovev (ns=11, 18 boundary DOFs) the
adjoint boundary gradient costs 762 force evaluations versus 9112 for
finite differences (12x), and the gap grows with the boundary DOF count.
---
 examples/simsopt_vmec_gradient.py | 154 +++++++++++++++++++++++++++
 examples/vmecpp_adjoint.py        | 169 ++++++++++++++++++++++++++++++
 tests/test_adjoint.py             |  51 +++++++++
 tests/test_simsopt_gradient.py    |  57 ++++++++++
 4 files changed, 431 insertions(+)
 create mode 100644 examples/simsopt_vmec_gradient.py
 create mode 100644 examples/vmecpp_adjoint.py
 create mode 100644 tests/test_adjoint.py
 create mode 100644 tests/test_simsopt_gradient.py

diff --git a/examples/simsopt_vmec_gradient.py b/examples/simsopt_vmec_gradient.py
new file mode 100644
index 000000000..2240d2a66
--- /dev/null
+++ b/examples/simsopt_vmec_gradient.py
@@ -0,0 +1,154 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Use VMEC++ as a gradient-providing equilibrium component for SIMSOPT.
+
+A SIMSOPT optimization over a plasma boundary normally differentiates VMEC by
+finite differences: one equilibrium re-solve per boundary degree of freedom and
+per outer iteration. VMEC++ instead provides the boundary gradient analytically
+through the implicit-function adjoint (``vmecpp_adjoint.boundary_gradient``): one
+extra Hessian solve, independent of the number of boundary DOFs.
+
+``VmecEnergy`` wraps that as a SIMSOPT ``Optimizable`` whose objective is the MHD
+energy of the converged equilibrium and whose ``dJ`` is the adjoint gradient.
+``optimize_to_target`` runs a gradient-based optimization of the boundary toward
+a target energy, with the analytic gradient or with finite differences, and
+reports the cost (forward-model evaluations counted inside VMEC++, outer
+iterations, wall time) so the two can be compared.
+"""
+
+from __future__ import annotations
+
+import time
+from dataclasses import dataclass
+from pathlib import Path
+
+import numpy as np
+from vmecpp_adjoint import (
+    DEFAULT_INPUT,
+    boundary_gradient,
+    finite_difference_boundary_gradient,
+    make_model,
+    mhd_energy,
+    partition,
+    solve_interior,
+)
+
+
+class VmecBoundaryProblem:
+    """Equilibrium energy and its boundary gradient, cached per boundary state."""
+
+    def __init__(self, input_path: Path = DEFAULT_INPUT, ns: int = 11):
+        self.model = make_model(input_path, ns)
+        self.model.solve()
+        self.ns = ns
+        self.interior, self.boundary = partition(self.model, ns)
+        self._x_full = np.asarray(self.model.get_state(), float).copy()
+        self._cached_p = self._x_full[self.boundary].copy()
+
+    @property
+    def x0(self):
+        return self._x_full[self.boundary].copy()
+
+    def _resolve(self, p):
+        p = np.asarray(p, float)
+        if not np.array_equal(p, self._cached_p):
+            self._x_full = solve_interior(
+                self.model, self._x_full, self.interior, self.boundary, p
+            )
+            self._cached_p = p.copy()
+
+    def value(self, p):
+        self._resolve(p)
+        self.model.set_state(np.ascontiguousarray(self._x_full))
+        self.model.evaluate(2, 2, False)
+        return self.model.mhd_energy
+
+    def gradient(self, p):
+        self._resolve(p)
+        return boundary_gradient(
+            self.model, self._x_full, self.interior, self.boundary, mhd_energy
+        )
+
+
+def make_simsopt_optimizable(problem: VmecBoundaryProblem):
+    """Wrap the problem as a SIMSOPT Optimizable exposing an analytic gradient."""
+    # Imported lazily so the rest of the module (and the gradient benchmark) work
+    # without SIMSOPT installed.
+    from simsopt._core import Optimizable  # noqa: PLC0415
+    from simsopt._core.derivative import Derivative, derivative_dec  # noqa: PLC0415
+
+    class VmecEnergy(Optimizable):
+        def __init__(self):
+            x0 = problem.x0
+            super().__init__(x0=x0, names=[f"boundary{i}" for i in range(x0.size)])
+
+        def J(self):
+            return problem.value(self.local_full_x)
+
+        @derivative_dec
+        def dJ(self):
+            return Derivative({self: problem.gradient(self.local_full_x)})
+
+    return VmecEnergy()
+
+
+@dataclass
+class GradResult:
+    method: str
+    force_evals: int
+    seconds: float
+    gradient: np.ndarray
+
+
+def gradient_cost(input_path: Path = DEFAULT_INPUT, ns: int = 11, analytic=True):
+    """Cost of one full boundary gradient at the converged equilibrium.
+
+    This is what an external optimizer pays per iteration. The analytic adjoint
+    needs one Hessian solve regardless of the number of boundary DOFs; finite
+    differences re-converge the equilibrium twice per boundary DOF.
+    """
+    problem = VmecBoundaryProblem(input_path, ns)
+    x_star = problem._x_full.copy()
+    interior, boundary = problem.interior, problem.boundary
+    problem.model.reset_force_eval_count()
+    t0 = time.perf_counter()
+    if analytic:
+        g_dict = None
+        g = boundary_gradient(problem.model, x_star, interior, boundary, mhd_energy)
+    else:
+        g_dict = finite_difference_boundary_gradient(
+            problem.model,
+            x_star,
+            interior,
+            boundary,
+            mhd_energy,
+            range(boundary.size),
+        )
+        g = np.array([g_dict[j] for j in range(boundary.size)])
+    return GradResult(
+        "analytic adjoint" if analytic else "finite differences",
+        problem.model.force_eval_count,
+        time.perf_counter() - t0,
+        g,
+    )
+
+
+def main():
+    analytic = gradient_cost(analytic=True)
+    fd = gradient_cost(analytic=False)
+    rel = np.linalg.norm(analytic.gradient - fd.gradient) / np.linalg.norm(fd.gradient)
+    n_boundary = analytic.gradient.size
+    print(f"boundary gradient cost ({n_boundary} boundary DOFs, solovev ns=11)\n")
+    print(f"{'method':20s} {'F-evals':>9s} {'time[s]':>8s}")
+    for r in (fd, analytic):
+        print(f"{r.method:20s} {r.force_evals:9d} {r.seconds:8.2f}")
+    print(
+        f"\nspeedup (force evals): {fd.force_evals / max(analytic.force_evals, 1):.1f}x"
+        f"   gradient agreement: {rel:.1e}"
+    )
+
+
+if __name__ == "__main__":
+    main()
diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
new file mode 100644
index 000000000..e4b27f3bf
--- /dev/null
+++ b/examples/vmecpp_adjoint.py
@@ -0,0 +1,169 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Adjoint sensitivity of a converged VMEC++ equilibrium to its boundary.
+
+A fixed-boundary equilibrium satisfies the interior force balance F_I(x) = 0,
+where x is the decomposed internal-basis state and F is the gradient of VMEC's
+augmented functional. The outermost flux surface (the boundary) is the last
+radial block of the state and is held fixed during the solve. For a scalar
+objective J(x), the sensitivity to the boundary degrees of freedom follows from
+the implicit function theorem:
+
+    dJ/dx_B = dJ/dx_B|_x - (dF_I/dx_B)^T lambda,   H_II lambda = dJ/dx_I,
+
+with H = dF/dx the (symmetric) Hessian of the augmented functional. Every
+operator is matrix-free and already exposed by VmecModel: the Hessian-vector
+product (``hessian_vector_product``) and the preconditioner
+(``apply_preconditioner``), used to solve the adjoint system. Only one Hessian
+solve is needed for the full boundary gradient, versus one equilibrium re-solve
+per boundary degree of freedom for finite differences.
+"""
+
+from __future__ import annotations
+
+from pathlib import Path
+
+import numpy as np
+from scipy.sparse.linalg import LinearOperator, cg
+
+try:
+    from vmecpp.cpp import _vmecpp
+except ImportError:
+    import _vmecpp
+
+DEFAULT_INPUT = (
+    Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+)
+
+
+def make_model(input_path: Path = DEFAULT_INPUT, ns: int = 11):
+    return _vmecpp.VmecModel.create(_vmecpp.VmecINDATA.from_file(str(input_path)), ns)
+
+
+def partition(model, ns: int):
+    """Indices of the interior (free) and boundary (LCFS) state components."""
+    k = model.mpol * (model.ntor + 1)
+    n = np.asarray(model.get_state()).size
+    per_span = ns * k
+    n_span = n // per_span
+    boundary = []
+    for s in range(n_span):
+        boundary.extend(range(s * per_span + (ns - 1) * k, s * per_span + ns * k))
+    boundary = np.array(sorted(boundary))
+    interior = np.setdiff1d(np.arange(n), boundary)
+    return interior, boundary
+
+
+def _raw_force(model, x):
+    model.set_state(np.ascontiguousarray(x))
+    model.evaluate(2, 2, False)
+    return np.asarray(model.get_forces(), float)
+
+
+def _interior_operators(model, x, interior):
+    # The caller must set the base state to x and assemble the preconditioner
+    # (evaluate(2, 2, True)) before using these. hessian_vector_product uses the
+    # current state as its base point and restores it, so no per-matvec state
+    # update is needed: that keeps each Hessian matvec at two force evaluations.
+    n = x.size
+    ni = interior.size
+
+    def hii(vi):
+        v = np.zeros(n)
+        v[interior] = vi
+        return np.asarray(model.hessian_vector_product(np.ascontiguousarray(v)), float)[
+            interior
+        ]
+
+    def mii(bi):
+        v = np.zeros(n)
+        v[interior] = bi
+        return np.asarray(model.apply_preconditioner(np.ascontiguousarray(v)), float)[
+            interior
+        ]
+
+    return (
+        LinearOperator((ni, ni), matvec=hii),
+        LinearOperator((ni, ni), matvec=mii),
+    )
+
+
+def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=40):
+    """Converge the interior to force balance with the boundary held fixed."""
+    x = np.asarray(x0, float).copy()
+    x[boundary] = x_boundary
+    for _ in range(max_newton):
+        f = _raw_force(model, x)
+        if np.linalg.norm(f[interior]) < tol:
+            break
+        model.set_state(np.ascontiguousarray(x))
+        model.evaluate(2, 2, True)  # assemble preconditioner + set base state
+        h_op, m_op = _interior_operators(model, x, interior)
+        dxi, _ = cg(h_op, -f[interior], M=m_op, rtol=1e-4, maxiter=200)
+        x[interior] += dxi
+    return x
+
+
+def objective_state_gradient(model, x, objective, h=1e-6):
+    """Partial derivative dJ/dx at fixed state, by central finite differences."""
+    n = x.size
+    g = np.zeros(n)
+    for i in range(n):
+        xp = x.copy()
+        xp[i] += h
+        model.set_state(np.ascontiguousarray(xp))
+        model.evaluate(2, 2, False)
+        jp = objective(model)
+        xm = x.copy()
+        xm[i] -= h
+        model.set_state(np.ascontiguousarray(xm))
+        model.evaluate(2, 2, False)
+        jm = objective(model)
+        g[i] = (jp - jm) / (2 * h)
+    return g
+
+
+def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6):
+    """Adjoint gradient dJ/dx_B at the converged equilibrium x_star."""
+    n = x_star.size
+    dj = objective_state_gradient(model, x_star, objective, h)
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, True)  # assemble preconditioner + set base state to x_star
+    h_op, m_op = _interior_operators(model, x_star, interior)
+    lam, _ = cg(h_op, dj[interior], M=m_op, rtol=1e-7, maxiter=400)
+    embedded = np.zeros(n)
+    embedded[interior] = lam
+    model.set_state(np.ascontiguousarray(x_star))
+    model.evaluate(2, 2, False)
+    coupling = np.asarray(
+        model.hessian_vector_product(np.ascontiguousarray(embedded)), float
+    )[boundary]
+    return dj[boundary] - coupling
+
+
+def finite_difference_boundary_gradient(
+    model, x_star, interior, boundary, objective, dofs, h=1e-5
+):
+    """Reference gradient: re-solve the interior for each perturbed boundary DOF."""
+    g = {}
+    for j in dofs:
+        xbp = x_star[boundary].copy()
+        xbp[j] += h
+        xp = solve_interior(model, x_star, interior, boundary, xbp)
+        model.set_state(np.ascontiguousarray(xp))
+        model.evaluate(2, 2, False)
+        jp = objective(model)
+        xbm = x_star[boundary].copy()
+        xbm[j] -= h
+        xm = solve_interior(model, x_star, interior, boundary, xbm)
+        model.set_state(np.ascontiguousarray(xm))
+        model.evaluate(2, 2, False)
+        jm = objective(model)
+        g[j] = (jp - jm) / (2 * h)
+    return g
+
+
+def mhd_energy(model):
+    return model.mhd_energy
diff --git a/tests/test_adjoint.py b/tests/test_adjoint.py
new file mode 100644
index 000000000..0d740d6be
--- /dev/null
+++ b/tests/test_adjoint.py
@@ -0,0 +1,51 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""The adjoint boundary gradient matches brute-force finite differences.
+
+dJ/d(boundary) from one Hessian solve (implicit-function adjoint) agrees with the
+reference gradient obtained by re-converging the interior equilibrium for each
+perturbed boundary degree of freedom. J here is the MHD energy of the converged
+equilibrium.
+"""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+
+sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
+from vmecpp_adjoint import (
+    boundary_gradient,
+    finite_difference_boundary_gradient,
+    make_model,
+    mhd_energy,
+    partition,
+)
+
+
+def test_adjoint_matches_finite_difference():
+    ns = 11
+    model = make_model(ns=ns)
+    model.solve()
+    x_star = np.asarray(model.get_state(), float).copy()
+    interior, boundary = partition(model, ns)
+
+    g_adjoint = boundary_gradient(model, x_star, interior, boundary, mhd_energy)
+
+    # Reference on a representative subset (each requires interior re-solves).
+    dofs = [0, 2, 9]
+    g_fd = finite_difference_boundary_gradient(
+        model, x_star, interior, boundary, mhd_energy, dofs
+    )
+
+    scale = max(np.linalg.norm(g_adjoint), 1e-30)
+    for j in dofs:
+        assert abs(g_adjoint[j] - g_fd[j]) < 1e-3 * scale
+
+
+if __name__ == "__main__":
+    import pytest
+
+    raise SystemExit(pytest.main([__file__, "-v"]))
diff --git a/tests/test_simsopt_gradient.py b/tests/test_simsopt_gradient.py
new file mode 100644
index 000000000..4080079fe
--- /dev/null
+++ b/tests/test_simsopt_gradient.py
@@ -0,0 +1,57 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""VMEC++ exposes an analytic boundary gradient to SIMSOPT.
+
+The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches
+finite differences of its objective, and computing it is much cheaper than the
+conventional finite-difference boundary gradient (which re-solves the
+equilibrium per boundary degree of freedom).
+"""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+import pytest
+
+sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
+
+pytest.importorskip("simsopt")
+
+from simsopt_vmec_gradient import (
+    VmecBoundaryProblem,
+    gradient_cost,
+    make_simsopt_optimizable,
+)
+
+
+def test_simsopt_optimizable_gradient_matches_fd():
+    problem = VmecBoundaryProblem(ns=11)
+    opt = make_simsopt_optimizable(problem)
+    g = np.asarray(opt.dJ(), float)
+
+    p0 = np.asarray(opt.local_full_x, float)
+    h = 1e-5
+    scale = max(np.linalg.norm(g), 1e-30)
+    for j in (0, 2, 9):
+        pp = p0.copy()
+        pp[j] += h
+        opt.local_full_x = pp
+        jp = opt.J()
+        pm = p0.copy()
+        pm[j] -= h
+        opt.local_full_x = pm
+        jm = opt.J()
+        opt.local_full_x = p0
+        assert abs(g[j] - (jp - jm) / (2 * h)) < 1e-3 * scale
+
+
+def test_adjoint_gradient_cheaper_than_finite_difference():
+    analytic = gradient_cost(analytic=True)
+    fd = gradient_cost(analytic=False)
+    # Same gradient, far fewer force evaluations (advantage grows with #DOFs).
+    rel = np.linalg.norm(analytic.gradient - fd.gradient) / np.linalg.norm(fd.gradient)
+    assert rel < 1e-2
+    assert analytic.force_evals < fd.force_evals

From d6d109127f5919196c0a822fd6c7578a1e93c715 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Sun, 14 Jun 2026 10:52:21 +0200
Subject: [PATCH 62/79] examples: fix adjoint on 3D (GMRES for indefinite
 Hessian) + globalize

Two correctness fixes for stiff 3D equilibria (cth_like):

- VMEC's augmented-Lagrangian Hessian is symmetric *indefinite* (the lambda
  constraint makes it a saddle, not a minimum), so CG silently gives the
  wrong adjoint there. Use GMRES, which handles indefinite systems, for the
  H_II solve and the interior Newton solve. With a loose, restarted tolerance
  the adjoint solve stays cheap.
- Add a backtracking line search to solve_interior so the interior re-solve
  (used by the SIMSOPT wrapper and the finite-difference reference) converges
  on 3D instead of overshooting.

Verified with a directional-derivative check against a re-converged
finite-difference reference: solovev 1.5e-4, cth_like 2.2e-2 relative; both
previously agreed only in 2D. Boundary-gradient cost on solovev: 626 force
evaluations (analytic adjoint) versus 10460 (finite differences).
---
 examples/vmecpp_adjoint.py | 29 ++++++++++++++++++++++-------
 1 file changed, 22 insertions(+), 7 deletions(-)

diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index e4b27f3bf..a85ea2006 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -26,7 +26,7 @@
 from pathlib import Path
 
 import numpy as np
-from scipy.sparse.linalg import LinearOperator, cg
+from scipy.sparse.linalg import LinearOperator, gmres
 
 try:
     from vmecpp.cpp import _vmecpp
@@ -90,19 +90,34 @@ def mii(bi):
     )
 
 
-def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=40):
-    """Converge the interior to force balance with the boundary held fixed."""
+def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=80):
+    """Converge the interior to force balance with the boundary held fixed.
+
+    Preconditioned Newton-Krylov on the interior residual with a backtracking
+    line search; the line search is required for stiff 3D equilibria, where the
+    full Newton step overshoots.
+    """
     x = np.asarray(x0, float).copy()
     x[boundary] = x_boundary
     for _ in range(max_newton):
         f = _raw_force(model, x)
-        if np.linalg.norm(f[interior]) < tol:
+        norm0 = np.linalg.norm(f[interior])
+        if norm0 < tol:
             break
         model.set_state(np.ascontiguousarray(x))
         model.evaluate(2, 2, True)  # assemble preconditioner + set base state
         h_op, m_op = _interior_operators(model, x, interior)
-        dxi, _ = cg(h_op, -f[interior], M=m_op, rtol=1e-4, maxiter=200)
-        x[interior] += dxi
+        dxi, _ = gmres(h_op, -f[interior], M=m_op, rtol=1e-4, maxiter=300)
+        alpha = 1.0
+        for _ in range(30):
+            xt = x.copy()
+            xt[interior] += alpha * dxi
+            if np.linalg.norm(_raw_force(model, xt)[interior]) < norm0:
+                break
+            alpha *= 0.5
+        else:
+            break  # no decrease found; stop
+        x[interior] += alpha * dxi
     return x
 
 
@@ -132,7 +147,7 @@ def boundary_gradient(model, x_star, interior, boundary, objective, h=1e-6):
     model.set_state(np.ascontiguousarray(x_star))
     model.evaluate(2, 2, True)  # assemble preconditioner + set base state to x_star
     h_op, m_op = _interior_operators(model, x_star, interior)
-    lam, _ = cg(h_op, dj[interior], M=m_op, rtol=1e-7, maxiter=400)
+    lam, _ = gmres(h_op, dj[interior], M=m_op, rtol=1e-6, restart=100, maxiter=30)
     embedded = np.zeros(n)
     embedded[interior] = lam
     model.set_state(np.ascontiguousarray(x_star))

From 2f258ed37acfbf829a6b99c36fcaeae0a67fdd11 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 22:30:32 +0200
Subject: [PATCH 63/79] Fix adjoint checks on current main

---
 examples/simsopt_vmec_gradient.py | 15 ++++++++++-----
 examples/vmecpp_adjoint.py        | 15 ++++++---------
 tests/test_adjoint.py             |  7 +++----
 tests/test_simsopt_gradient.py    | 10 +++++-----
 4 files changed, 24 insertions(+), 23 deletions(-)

diff --git a/examples/simsopt_vmec_gradient.py b/examples/simsopt_vmec_gradient.py
index 2240d2a66..54f188789 100644
--- a/examples/simsopt_vmec_gradient.py
+++ b/examples/simsopt_vmec_gradient.py
@@ -77,7 +77,11 @@ def make_simsopt_optimizable(problem: VmecBoundaryProblem):
     # Imported lazily so the rest of the module (and the gradient benchmark) work
     # without SIMSOPT installed.
     from simsopt._core import Optimizable  # noqa: PLC0415
-    from simsopt._core.derivative import Derivative, derivative_dec  # noqa: PLC0415
+    from simsopt._core.derivative import (  # noqa: PLC0415
+        Derivative,
+        OptimizableDefaultDict,
+        derivative_dec,
+    )
 
     class VmecEnergy(Optimizable):
         def __init__(self):
@@ -89,7 +93,8 @@ def J(self):
 
         @derivative_dec
         def dJ(self):
-            return Derivative({self: problem.gradient(self.local_full_x)})
+            data = OptimizableDefaultDict({self: problem.gradient(self.local_full_x)})
+            return Derivative(data)
 
     return VmecEnergy()
 
@@ -105,9 +110,9 @@ class GradResult:
 def gradient_cost(input_path: Path = DEFAULT_INPUT, ns: int = 11, analytic=True):
     """Cost of one full boundary gradient at the converged equilibrium.
 
-    This is what an external optimizer pays per iteration. The analytic adjoint
-    needs one Hessian solve regardless of the number of boundary DOFs; finite
-    differences re-converge the equilibrium twice per boundary DOF.
+    This is what an external optimizer pays per iteration. The analytic adjoint needs
+    one Hessian solve regardless of the number of boundary DOFs; finite differences re-
+    converge the equilibrium twice per boundary DOF.
     """
     problem = VmecBoundaryProblem(input_path, ns)
     x_star = problem._x_full.copy()
diff --git a/examples/vmecpp_adjoint.py b/examples/vmecpp_adjoint.py
index a85ea2006..f10b63d12 100644
--- a/examples/vmecpp_adjoint.py
+++ b/examples/vmecpp_adjoint.py
@@ -28,10 +28,7 @@
 import numpy as np
 from scipy.sparse.linalg import LinearOperator, gmres
 
-try:
-    from vmecpp.cpp import _vmecpp
-except ImportError:
-    import _vmecpp
+from vmecpp.cpp import _vmecpp  # type: ignore
 
 DEFAULT_INPUT = (
     Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
@@ -85,17 +82,17 @@ def mii(bi):
         ]
 
     return (
-        LinearOperator((ni, ni), matvec=hii),
-        LinearOperator((ni, ni), matvec=mii),
+        LinearOperator((ni, ni), matvec=hii),  # type: ignore[call-overload]
+        LinearOperator((ni, ni), matvec=mii),  # type: ignore[call-overload]
     )
 
 
 def solve_interior(model, x0, interior, boundary, x_boundary, tol=1e-10, max_newton=80):
     """Converge the interior to force balance with the boundary held fixed.
 
-    Preconditioned Newton-Krylov on the interior residual with a backtracking
-    line search; the line search is required for stiff 3D equilibria, where the
-    full Newton step overshoots.
+    Preconditioned Newton-Krylov on the interior residual with a backtracking line
+    search; the line search is required for stiff 3D equilibria, where the full Newton
+    step overshoots.
     """
     x = np.asarray(x0, float).copy()
     x[boundary] = x_boundary
diff --git a/tests/test_adjoint.py b/tests/test_adjoint.py
index 0d740d6be..54a45c1a1 100644
--- a/tests/test_adjoint.py
+++ b/tests/test_adjoint.py
@@ -5,9 +5,8 @@
 """The adjoint boundary gradient matches brute-force finite differences.
 
 dJ/d(boundary) from one Hessian solve (implicit-function adjoint) agrees with the
-reference gradient obtained by re-converging the interior equilibrium for each
-perturbed boundary degree of freedom. J here is the MHD energy of the converged
-equilibrium.
+reference gradient obtained by re-converging the interior equilibrium for each perturbed
+boundary degree of freedom. J here is the MHD energy of the converged equilibrium.
 """
 
 import sys
@@ -16,7 +15,7 @@
 import numpy as np
 
 sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
-from vmecpp_adjoint import (
+from vmecpp_adjoint import (  # type: ignore
     boundary_gradient,
     finite_difference_boundary_gradient,
     make_model,
diff --git a/tests/test_simsopt_gradient.py b/tests/test_simsopt_gradient.py
index 4080079fe..90f4f97c5 100644
--- a/tests/test_simsopt_gradient.py
+++ b/tests/test_simsopt_gradient.py
@@ -4,10 +4,10 @@
 # SPDX-License-Identifier: MIT
 """VMEC++ exposes an analytic boundary gradient to SIMSOPT.
 
-The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches
-finite differences of its objective, and computing it is much cheaper than the
-conventional finite-difference boundary gradient (which re-solves the
-equilibrium per boundary degree of freedom).
+The VmecEnergy Optimizable's analytic dJ (the implicit-function adjoint) matches finite
+differences of its objective, and computing it is much cheaper than the conventional
+finite-difference boundary gradient (which re-solves the equilibrium per boundary degree
+of freedom).
 """
 
 import sys
@@ -20,7 +20,7 @@
 
 pytest.importorskip("simsopt")
 
-from simsopt_vmec_gradient import (
+from simsopt_vmec_gradient import (  # type: ignore
     VmecBoundaryProblem,
     gradient_cost,
     make_simsopt_optimizable,

From a9f7cf026501af702433d6db0d0ef4d8b3f662c5 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 20:22:02 +0200
Subject: [PATCH 64/79] Make raw VMEC forces history independent

Separate the legacy previous-residual m=1 projection used by native iteration from the exact constrained projection used by external force evaluations. Cover both directions across the residual threshold.
---
 src/vmecpp/_iteration.py                      |  2 +-
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   |  6 ++-
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    |  5 ++-
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 13 ++++--
 tests/test_internal_gradient.py               | 42 +++++++++++++++++++
 5 files changed, 61 insertions(+), 7 deletions(-)

diff --git a/src/vmecpp/_iteration.py b/src/vmecpp/_iteration.py
index b019a79fa..7e6e0f8dc 100644
--- a/src/vmecpp/_iteration.py
+++ b/src/vmecpp/_iteration.py
@@ -259,7 +259,7 @@ def solve_equilibrium(
             iter2 = force_iteration - bad_resets
 
             # Evolve: forward model, then convergence / damping / time step.
-            model.evaluate(iter1, iter2)
+            model.evaluate(iter1, iter2, precondition=True, legacy_m1_constraint=True)
             fsqr, fsqz, fsql = model.fsqr, model.fsqz, model.fsql
             rr = model.restart_reason
             finite = math.isfinite(fsqr) and math.isfinite(fsqz) and math.isfinite(fsql)
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 1ec40ce51..824631c0a 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -421,7 +421,8 @@ absl::StatusOr<bool> IdealMhdModel::update(
     bool& m_need_restart, int& m_last_preconditioner_update,
     int& m_last_full_update_nestor, FlowControl& m_fc, const int iter1,
     const int iter2, const VmecCheckpoint& checkpoint,
-    const int iterations_before_checkpointing, bool verbose) {
+    const int iterations_before_checkpointing, bool verbose,
+    M1ConstraintMode m1_constraint_mode) {
   // An axis re-guess after a bad Jacobian can repopulate high geometry modes
   // directly, bypassing the force mask; clear them on the state each iteration.
   if (s_.mpolGeometry < s_.mpol || s_.ntorGeometry < s_.ntor) {
@@ -813,7 +814,8 @@ absl::StatusOr<bool> IdealMhdModel::update(
   m_decomposed_f.m1Constraint(1.0 / std::numbers::sqrt2);
 
   // v8.50: ADD iter2<2 so reset=<WOUT_FILE> works
-  if (m_fc.fsqz < 1.0e-6 || iter2 < 2) {
+  if (m1_constraint_mode == M1ConstraintMode::kEnforce || m_fc.fsqz < 1.0e-6 ||
+      iter2 < 2) {
     // ensure that the m=1 constraint is satisfied exactly
     // --> the corresponding m=1 coeffs of R,Z are constrained to be zero
     //     and thus must not be "forced" (by the time evol using gc) away from
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index b753e53f5..423203151 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -34,6 +34,8 @@
 
 namespace vmecpp {
 
+enum class M1ConstraintMode { kLegacy, kEnforce };
+
 // Implemented as a free function for easier testing and benchmarking.
 void deAliasConstraintForce(const RadialPartitioning& rp,
                             const FourierBasisFastPoloidal& fb, const Sizes& s_,
@@ -70,7 +72,8 @@ class IdealMhdModel {
       bool& m_need_restart, int& m_last_preconditioner_update,
       int& m_last_full_update_nestor, FlowControl& m_fc, const int iter1,
       const int iter2, const VmecCheckpoint& checkpoint = VmecCheckpoint::NONE,
-      const int iterations_before_checkpointing = INT_MAX, bool verbose = true);
+      const int iterations_before_checkpointing = INT_MAX, bool verbose = true,
+      M1ConstraintMode m1_constraint_mode = M1ConstraintMode::kLegacy);
 
   // Coordinates which inverse-DFT routine to call for computing
   // the flux surface geometry and lambda on it from the provided Fourier
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index cda5d0ad9..56fd65359 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -226,7 +226,11 @@ class VmecModel {
   // lambda-constraint components. That raw gradient is what gradient-based
   // optimizers minimizing the MHD energy functional need; mhd_energy is already
   // set earlier in update(), so it is valid at the checkpoint too.
-  void Evaluate(int iter1, int iter2, bool precondition = true) {
+  // The native iteration delays the exact m=1 projection based on the previous
+  // Z residual. External evaluations enforce it immediately so F(x) does not
+  // depend on the previously evaluated state.
+  void Evaluate(int iter1, int iter2, bool precondition = true,
+                bool legacy_m1_constraint = false) {
     ++force_eval_count_;
     bool need_restart = false;
     std::string error_message;
@@ -255,7 +259,9 @@ class VmecModel {
           *vmec_->decomposed_f_[0], *vmec_->physical_f_[0], need_restart,
           last_preconditioner_update_, last_full_update_nestor_, vmec_->fc_,
           iter1, iter2, checkpoint, checkpoint_after,
-          /*verbose=*/false);
+          /*verbose=*/false,
+          legacy_m1_constraint ? vmecpp::M1ConstraintMode::kLegacy
+                               : vmecpp::M1ConstraintMode::kEnforce);
       if (!s.ok()) {
         error_message = std::string(s.status().message());
       }
@@ -1260,7 +1266,8 @@ PYBIND11_MODULE(_vmecpp, m) {
       .def_static("create", &VmecModel::Create, py::arg("indata"),
                   py::arg("ns"), py::arg("initial_state") = std::nullopt)
       .def("evaluate", &VmecModel::Evaluate, py::arg("iter1"), py::arg("iter2"),
-           py::arg("precondition") = true)
+           py::arg("precondition") = true,
+           py::arg("legacy_m1_constraint") = false)
       .def_property_readonly("need_restart", &VmecModel::need_restart)
       .def("perform_time_step", &VmecModel::PerformTimeStep,
            py::arg("velocity_scale"), py::arg("conjugation_parameter"),
diff --git a/tests/test_internal_gradient.py b/tests/test_internal_gradient.py
index 17d36f65f..16d26825e 100644
--- a/tests/test_internal_gradient.py
+++ b/tests/test_internal_gradient.py
@@ -17,10 +17,17 @@
 from pathlib import Path
 
 import numpy as np
+import pytest
 
 from vmecpp.cpp import _vmecpp  # type: ignore
 
 SOLOVEV = Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+CTH_LIKE = (
+    Path(__file__).resolve().parents[1]
+    / "examples"
+    / "data"
+    / "cth_like_fixed_bdy.json"
+)
 
 
 def _model(ns: int = 11):
@@ -63,3 +70,38 @@ def test_cold_start_is_excluded():
     m = _model()
     m.evaluate(2, 2, False)
     assert np.all(np.isfinite(np.asarray(m.get_forces(), float)))
+
+
+@pytest.fixture(scope="module")
+def force_histories():
+    indata = _vmecpp.VmecINDATA.from_file(str(CTH_LIKE))
+    reference = _vmecpp.VmecModel.create(indata, 11)
+    reference.solve()
+    low_residual_state = np.asarray(reference.get_state(), float).copy()
+
+    model = _vmecpp.VmecModel.create(indata, 11)
+    high_residual_state = np.asarray(model.get_state(), float).copy()
+    model.evaluate(2, 2, False)
+    high_after_high = np.asarray(model.get_forces(), float).copy()
+
+    model.set_state(np.ascontiguousarray(low_residual_state))
+    model.evaluate(2, 2, False)
+    low_after_high = np.asarray(model.get_forces(), float).copy()
+    model.evaluate(2, 2, False)
+    low_after_low = np.asarray(model.get_forces(), float).copy()
+
+    model.set_state(np.ascontiguousarray(high_residual_state))
+    model.evaluate(2, 2, False)
+    high_after_low = np.asarray(model.get_forces(), float).copy()
+
+    return high_after_high, high_after_low, low_after_high, low_after_low
+
+
+def test_raw_force_is_independent_of_high_residual_history(force_histories):
+    _, _, low_after_high, low_after_low = force_histories
+    np.testing.assert_array_equal(low_after_high, low_after_low)
+
+
+def test_raw_force_is_independent_of_low_residual_history(force_histories):
+    high_after_high, high_after_low, _, _ = force_histories
+    np.testing.assert_array_equal(high_after_high, high_after_low)

From 8c8a46ef7b0047f5918c0a6904f1c62864ba44f3 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 21:01:50 +0200
Subject: [PATCH 65/79] Use compact storage for constraint policy

---
 src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h | 3 ++-
 1 file changed, 2 insertions(+), 1 deletion(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index 423203151..c1c9d3560 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -7,6 +7,7 @@
 
 #include <Eigen/Dense>
 #include <climits>
+#include <cstdint>
 #include <span>
 
 #ifdef _OPENMP
@@ -34,7 +35,7 @@
 
 namespace vmecpp {
 
-enum class M1ConstraintMode { kLegacy, kEnforce };
+enum class M1ConstraintMode : std::uint8_t { kLegacy, kEnforce };
 
 // Implemented as a free function for easier testing and benchmarking.
 void deAliasConstraintForce(const RadialPartitioning& rp,

From 2def2d5aa2cc5d39db9f8996b25b3cd7cc48f696 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 20:15:55 +0200
Subject: [PATCH 66/79] Report comparable external optimizer metrics

Use the low-level VMEC model for the native benchmark row so every solver reports the same force norm and MHD functional. Count SciPy Newton-Krylov outer iterations through its callback and correct the preconditioner description.
---
 examples/external_optimizers.py            | 94 +++++++++++-----------
 tests/test_external_optimizer_benchmark.py | 37 +++++++++
 2 files changed, 85 insertions(+), 46 deletions(-)
 create mode 100644 tests/test_external_optimizer_benchmark.py

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index b4da85f26..e8dd622c2 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -19,20 +19,16 @@
 
 * preconditioned descent (VMEC's own update direction),
 * Jacobian-free Newton-Krylov, plain and preconditioned, and
-* a true Newton-Krylov driven by VMEC++'s own Hessian-vector product.
-
-All converge to the same equilibrium as the native solver. Force evaluations are
-counted inside VMEC++ (``force_eval_count``) so the comparison is fair across
-methods, including the evaluations hidden in Hessian-vector products.
-
-Interpretation: what separates the methods is how they couple DOFs of
-neighbouring flux surfaces. Plain JFNK does not -- it treats every degree of
-freedom as independent, so it cannot damp the radial stiffness that lets a
-surface cross its neighbour between iterations (BAD_JACOBIAN restarts in the
-native solver). It still converges, but slowly. VMEC's couples each surface to
-its jF +/- 1 neighbours; applying it -- as the inner Krylov preconditioner, or
-refreshed every step in the HVP Newton -- is what rescues conditioning and cuts
-the cost by an order of magnitude.
+* Newton-Krylov driven by VMEC++'s finite-difference Hessian-vector product.
+
+All methods report the same raw internal-basis force norm and MHD energy. Force
+evaluations are counted inside VMEC++ (``force_eval_count``), including the
+evaluations hidden in Hessian-vector products.
+
+Both Newton-Krylov variants obtain radial coupling from the full residual's
+Jacobian-vector product. VMEC's preconditioner approximates the inverse Hessian
+with a radial tridiagonal solve per Fourier mode. Applying it to the inner Krylov
+solve reduces the radial stiffness seen by the linear solver.
 """
 
 from __future__ import annotations
@@ -45,7 +41,6 @@
 from scipy.optimize import newton_krylov
 from scipy.sparse.linalg import LinearOperator, gmres
 
-import vmecpp
 from vmecpp.cpp import _vmecpp  # type: ignore[import]
 
 DEFAULT_INPUT = (
@@ -111,12 +106,14 @@ def reference_equilibrium(input_path: Path = DEFAULT_INPUT, ns: int = 11):
     return np.asarray(model.get_state(), float), model.mhd_energy
 
 
-def _finish(model, name, x, outer_iters, t0):
+def _finish(model, name, x, outer_iters, t0, force_evals=None):
     model.set_state(np.ascontiguousarray(x))
     model.evaluate(2, 2, False)
+    if force_evals is None:
+        force_evals = model.force_eval_count
     return x, Result(
         name,
-        model.force_eval_count,
+        force_evals,
         outer_iters,
         time.perf_counter() - t0,
         float(np.linalg.norm(np.asarray(model.get_forces(), float))),
@@ -125,25 +122,19 @@ def _finish(model, name, x, outer_iters, t0):
 
 
 def solve_vmecpp(input_path=DEFAULT_INPUT, ns=11):
-    """Native VMEC++ solve through the public API; reports the iteration count.
-
-    Unlike the other variants this drives the ordinary ``vmecpp.run`` path rather
-    than the low-level ``VmecModel`` primitives, so the reported residual/energy
-    are the public wout diagnostics (invariant force residuals and total MHD
-    energy) rather than the raw internal-basis force this module counts elsewhere.
-    """
-    vmec_input = vmecpp.VmecInput.from_file(input_path)
-    vmec_input.ns_array = np.asarray([ns], dtype=vmec_input.ns_array.dtype)
+    """Native VMEC++ solve reported with the benchmark's force and energy metrics."""
+    model = make_model(input_path, ns)
     t0 = time.perf_counter()
-    output = vmecpp.run(vmec_input, verbose=False)
-    wout = output.wout
-    return output, Result(
-        name="VMEC++ (native, vmecpp.run)",
-        force_evals=wout.niter,
-        outer_iters=wout.niter,
-        seconds=time.perf_counter() - t0,
-        residual_norm=float(np.sqrt(wout.fsqt[-1])),
-        energy=wout.wb + wout.wp,
+    model.solve()
+    x = np.asarray(model.get_state(), float).copy()
+    iterations = len(model.force_residual_r)
+    return _finish(
+        model,
+        "VMEC++ (native)",
+        x,
+        iterations,
+        t0,
+        force_evals=iterations,
     )
 
 
@@ -179,10 +170,8 @@ def solve_newton_krylov(
     model.reset_force_eval_count()
     if preconditioned:
         # Assemble VMEC's preconditioner at x0 and use it, frozen, as the inner
-        # Krylov preconditioner. M^-1 approximates the inverse Hessian; it is a
-        # radial tridiagonal (Thomas) solve per Fourier mode, so it couples each
-        # flux surface to its jF +/- 1 neighbours -- the coupling the plain
-        # branch lacks.
+        # Krylov preconditioner. M^-1 approximates the inverse Hessian with a
+        # radial tridiagonal solve per Fourier mode.
         model.evaluate(2, 2, True)
         n_dof = x0.size
         inner_m = LinearOperator(  # type: ignore[call-overload]
@@ -192,13 +181,25 @@ def solve_newton_krylov(
             ),
         )
     t0 = time.perf_counter()
+    outer_iters = 0
+
+    def count_outer_iteration(_x, _f):
+        nonlocal outer_iters
+        outer_iters += 1
+
     x = newton_krylov(
-        F, x0, f_tol=tol, maxiter=max_iter, method="lgmres", inner_M=inner_m
+        F,
+        x0,
+        f_tol=tol,
+        maxiter=max_iter,
+        method="lgmres",
+        inner_M=inner_m,
+        callback=count_outer_iteration,
     )
     name = (
         "Newton-Krylov (preconditioned)" if preconditioned else "Newton-Krylov (JFNK)"
     )
-    return _finish(model, name, x, 0, t0)
+    return _finish(model, name, x, outer_iters, t0)
 
 
 def solve_newton_krylov_preconditioned(input_path=DEFAULT_INPUT, ns=11, tol=1e-9):
@@ -255,8 +256,7 @@ def solve_newton_hvp(
     return _finish(model, "Newton (VMEC++ HVP + M^-1)", x, it, t0)
 
 
-ALL_SOLVERS = (
-    solve_vmecpp,
+EXTERNAL_SOLVERS = (
     solve_preconditioned_descent,
     solve_newton_krylov_preconditioned,
     solve_newton_krylov,
@@ -265,14 +265,16 @@ def solve_newton_hvp(
 
 
 def main():
-    _, w_star = reference_equilibrium()
+    _, native = solve_vmecpp()
+    w_star = native.energy
     print(f"reference equilibrium (native solve): W = {w_star:.8e}\n")
     print(
         f"{'optimizer':32s} {'F-evals':>8s} {'iters':>6s} {'time[s]':>8s} "
         f"{'||F||':>10s} {'dW vs ref':>10s}"
     )
-    for solver in ALL_SOLVERS:
-        r = solver()[1]
+    results = [native]
+    results.extend(solver()[1] for solver in EXTERNAL_SOLVERS)
+    for r in results:
         print(
             f"{r.name:32s} {r.force_evals:8d} {r.outer_iters:6d} {r.seconds:8.2f} "
             f"{r.residual_norm:10.1e} {abs(r.energy - w_star):10.1e}"
diff --git a/tests/test_external_optimizer_benchmark.py b/tests/test_external_optimizer_benchmark.py
new file mode 100644
index 000000000..7d97d2a05
--- /dev/null
+++ b/tests/test_external_optimizer_benchmark.py
@@ -0,0 +1,37 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""The external-optimizer benchmark reports comparable solver metrics."""
+
+import sys
+from pathlib import Path
+
+import numpy as np
+
+ROOT = Path(__file__).resolve().parents[1]
+sys.path.insert(0, str(ROOT / "examples"))
+from external_optimizers import (  # type: ignore  # noqa: E402
+    reference_equilibrium,
+    solve_newton_krylov,
+    solve_vmecpp,
+)
+
+SOLOVEV = ROOT / "examples" / "data" / "solovev.json"
+CTH_LIKE = ROOT / "examples" / "data" / "cth_like_fixed_bdy.json"
+
+
+def test_native_metrics_match_external_metric_definitions():
+    x_star, w_star = reference_equilibrium(CTH_LIKE, ns=11)
+    x, result = solve_vmecpp(CTH_LIKE, ns=11)
+
+    assert result.energy == w_star
+    np.testing.assert_array_equal(x, x_star)
+    assert result.residual_norm < 1e-8
+    assert result.force_evals == result.outer_iters
+    assert result.outer_iters > 0
+
+
+def test_newton_krylov_reports_outer_iterations():
+    _, result = solve_newton_krylov(SOLOVEV, ns=11)
+    assert result.outer_iters > 0

From 0fd57f0657aacd0b7d2eeb56147454c59323315b Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 22:53:58 +0200
Subject: [PATCH 67/79] Count native force evaluations directly

---
 examples/external_optimizers.py                           | 4 +++-
 .../cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc    | 2 ++
 .../cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h     | 4 ++++
 src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc        | 8 ++++----
 tests/test_external_optimizer_benchmark.py                | 2 +-
 5 files changed, 14 insertions(+), 6 deletions(-)

diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index e8dd622c2..1779f2ce4 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -124,17 +124,19 @@ def _finish(model, name, x, outer_iters, t0, force_evals=None):
 def solve_vmecpp(input_path=DEFAULT_INPUT, ns=11):
     """Native VMEC++ solve reported with the benchmark's force and energy metrics."""
     model = make_model(input_path, ns)
+    model.reset_force_eval_count()
     t0 = time.perf_counter()
     model.solve()
     x = np.asarray(model.get_state(), float).copy()
     iterations = len(model.force_residual_r)
+    force_evals = model.force_eval_count
     return _finish(
         model,
         "VMEC++ (native)",
         x,
         iterations,
         t0,
-        force_evals=iterations,
+        force_evals=force_evals,
     )
 
 
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 1ec40ce51..1b2cd5d2e 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -422,6 +422,8 @@ absl::StatusOr<bool> IdealMhdModel::update(
     int& m_last_full_update_nestor, FlowControl& m_fc, const int iter1,
     const int iter2, const VmecCheckpoint& checkpoint,
     const int iterations_before_checkpointing, bool verbose) {
+  ++force_evaluation_count_;
+
   // An axis re-guess after a bad Jacobian can repopulate high geometry modes
   // directly, bypassing the force mask; clear them on the state each iteration.
   if (s_.mpolGeometry < s_.mpol || s_.ntorGeometry < s_.ntor) {
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index b753e53f5..86939f593 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -72,6 +72,9 @@ class IdealMhdModel {
       const int iter2, const VmecCheckpoint& checkpoint = VmecCheckpoint::NONE,
       const int iterations_before_checkpointing = INT_MAX, bool verbose = true);
 
+  std::int64_t forceEvaluationCount() const { return force_evaluation_count_; }
+  void resetForceEvaluationCount() { force_evaluation_count_ = 0; }
+
   // Coordinates which inverse-DFT routine to call for computing
   // the flux surface geometry and lambda on it from the provided Fourier
   // coefficients. Also computes the net dR/dTheta and dZ/dTheta, without the
@@ -408,6 +411,7 @@ class IdealMhdModel {
   const RadialPartitioning& r_;
   FreeBoundaryBase* m_fb_;
   VacuumPressureState& m_vacuum_pressure_state_;
+  std::int64_t force_evaluation_count_ = 0;
 
 #ifdef VMECPP_USE_FFTX
   // Pre-computed FFTX kernels for the toroidal (zeta) Fourier transforms.
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index cda5d0ad9..e603b3976 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -227,7 +227,6 @@ class VmecModel {
   // optimizers minimizing the MHD energy functional need; mhd_energy is already
   // set earlier in update(), so it is valid at the checkpoint too.
   void Evaluate(int iter1, int iter2, bool precondition = true) {
-    ++force_eval_count_;
     bool need_restart = false;
     std::string error_message;
     const vmecpp::VmecCheckpoint checkpoint =
@@ -270,8 +269,10 @@ class VmecModel {
   // Total forward-model (force) evaluations since construction or the last
   // reset. Counts every Evaluate, including those inside hessian_vector_product
   // and preconditioner assembly, for a fair cross-optimizer cost comparison.
-  long force_eval_count() const { return force_eval_count_; }
-  void reset_force_eval_count() { force_eval_count_ = 0; }
+  std::int64_t force_eval_count() const {
+    return vmec_->m_[0]->forceEvaluationCount();
+  }
+  void reset_force_eval_count() { vmec_->m_[0]->resetForceEvaluationCount(); }
 
   // The Garabedian-style time step (PerformTimeStep): for each Fourier
   // coefficient, v = velocity_scale*(conjugation*v + dt*force); x += dt*v.
@@ -522,7 +523,6 @@ class VmecModel {
   int last_preconditioner_update_ = 0;
   int last_full_update_nestor_ = 0;
   bool last_need_restart_ = false;
-  long force_eval_count_ = 0;
 };
 
 }  // anonymous namespace
diff --git a/tests/test_external_optimizer_benchmark.py b/tests/test_external_optimizer_benchmark.py
index 7d97d2a05..23e7042bc 100644
--- a/tests/test_external_optimizer_benchmark.py
+++ b/tests/test_external_optimizer_benchmark.py
@@ -28,7 +28,7 @@ def test_native_metrics_match_external_metric_definitions():
     assert result.energy == w_star
     np.testing.assert_array_equal(x, x_star)
     assert result.residual_norm < 1e-8
-    assert result.force_evals == result.outer_iters
+    assert result.force_evals == result.outer_iters + 1
     assert result.outer_iters > 0
 
 

From 6355da174df06c008f8bbdae41f9731ed9bbc8e4 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 23:02:36 +0200
Subject: [PATCH 68/79] Satisfy const wrapper contract

---
 src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc | 4 +++-
 1 file changed, 3 insertions(+), 1 deletion(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index e603b3976..6fe8742cc 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -272,7 +272,9 @@ class VmecModel {
   std::int64_t force_eval_count() const {
     return vmec_->m_[0]->forceEvaluationCount();
   }
-  void reset_force_eval_count() { vmec_->m_[0]->resetForceEvaluationCount(); }
+  void reset_force_eval_count() const {
+    vmec_->m_[0]->resetForceEvaluationCount();
+  }
 
   // The Garabedian-style time step (PerformTimeStep): for each Fourier
   // coefficient, v = velocity_scale*(conjugation*v + dt*force); x += dt*v.

From 4b2a9adaf63fba06da0c138dcc5d34e368964ded Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 23:02:52 +0200
Subject: [PATCH 69/79] Keep exact force derivatives policy-consistent

---
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   | 26 ++++++++++--------
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    |  6 +++--
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   |  8 +++---
 tests/test_hessian.py                         | 27 ++++++++++++++++++-
 4 files changed, 49 insertions(+), 18 deletions(-)

diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index f7ce0b93e..f5fa66f36 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -41,6 +41,13 @@ using vmecpp::vmec_algorithm_constants::kOddParity;
 
 namespace {
 
+constexpr double kM1ConstraintResidualThreshold = 1.0e-6;
+
+bool ShouldProjectM1Force(vmecpp::M1ConstraintMode mode, double fsqz) {
+  return mode == vmecpp::M1ConstraintMode::kEnforce ||
+         fsqz < kM1ConstraintResidualThreshold;
+}
+
 // Set m_h.rCC_LCFS etc. to the corresponding values in the FourierGeometry
 // of the last surface, also transposing m and n dimensions to make the
 // data layout what Nestor expects.
@@ -756,8 +763,7 @@ absl::StatusOr<bool> IdealMhdModel::update(
   m_decomposed_f.m1Constraint(1.0 / std::numbers::sqrt2);
 
   // v8.50: ADD iter2<2 so reset=<WOUT_FILE> works
-  if (m1_constraint_mode == M1ConstraintMode::kEnforce || m_fc.fsqz < 1.0e-6 ||
-      iter2 < 2) {
+  if (ShouldProjectM1Force(m1_constraint_mode, m_fc.fsqz) || iter2 < 2) {
     // ensure that the m=1 constraint is satisfied exactly
     // --> the corresponding m=1 coeffs of R,Z are constrained to be zero
     //     and thus must not be "forced" (by the time evol using gc) away from
@@ -2162,11 +2168,10 @@ LocalForceComposition IdealMhdModel::makeLocalForceComposition(
   return comp;
 }
 
-void IdealMhdModel::applyExactForceJacobian(const double* geomP,
-                                            const double* dgeom,
-                                            int geom_stride,
-                                            FourierForces& m_physical_f,
-                                            FourierForces& m_decomposed_hv) {
+void IdealMhdModel::applyExactForceJacobian(
+    const double* geomP, const double* dgeom, int geom_stride,
+    FourierForces& m_physical_f, FourierForces& m_decomposed_hv,
+    M1ConstraintMode m1_constraint_mode) {
   LocalForceComposition comp = makeLocalForceComposition(geom_stride);
   const int nForce = comp.force_stride;
 
@@ -2218,8 +2223,7 @@ void IdealMhdModel::applyExactForceJacobian(const double* geomP,
   forcesToFourier(m_physical_f);
   m_physical_f.decomposeInto(m_decomposed_hv, m_p_.scalxc);
   m_decomposed_hv.m1Constraint(1.0 / std::numbers::sqrt2);
-  // Match update(): zeroZForceForM1 is gated on the converged-state condition.
-  if (m_fc_.fsqz < 1.0e-6) {
+  if (ShouldProjectM1Force(m1_constraint_mode, m_fc_.fsqz)) {
     m_decomposed_hv.zeroZForceForM1();
   }
 }
@@ -2572,12 +2576,12 @@ void IdealMhdModel::dft_FourierToRealTranspose_3d_symm(
 void IdealMhdModel::applyExactForceJacobianTranspose(
     const double* geomP, int geom_stride, FourierForces& m_decomposed_in,
     FourierForces& m_physical_f, FourierGeometry& m_physical_scratch,
-    FourierGeometry& m_decomposed_out) {
+    FourierGeometry& m_decomposed_out, M1ConstraintMode m1_constraint_mode) {
   const int gS = geom_stride;
   const int nForce = (r_.nsMaxFIncludingLcfs - r_.nsMinF) * s_.nZnT;
 
   // C^T: transpose of [scatter -> forcesToFourier -> decompose -> m1 -> zeroZ].
-  if (m_fc_.fsqz < 1.0e-6) {
+  if (ShouldProjectM1Force(m1_constraint_mode, m_fc_.fsqz)) {
     m_decomposed_in.zeroZForceForM1();
   }
   m_decomposed_in.m1Constraint(1.0 / std::numbers::sqrt2);
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index c55d4aa89..649000d45 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -172,7 +172,8 @@ class IdealMhdModel {
   // the exact internal Newton-Krylov Hessian-vector product.
   void applyExactForceJacobian(const double* geomP, const double* dgeom,
                                int geom_stride, FourierForces& m_physical_f,
-                               FourierForces& m_decomposed_hv);
+                               FourierForces& m_decomposed_hv,
+                               M1ConstraintMode m1_constraint_mode);
 
   // Linear pre-chain decomposed -> real-space geometry (decomposeInto,
   // m1Constraint, extrapolate, geometryFromFourier) packed into the 20-block
@@ -222,7 +223,8 @@ class IdealMhdModel {
                                         FourierForces& m_decomposed_in,
                                         FourierForces& m_physical_f,
                                         FourierGeometry& m_physical_scratch,
-                                        FourierGeometry& m_decomposed_out);
+                                        FourierGeometry& m_decomposed_out,
+                                        M1ConstraintMode m1_constraint_mode);
 
   // Transposes of the spectral transforms, for the transposed exact Hessian.
   // dft_ForcesToFourierTranspose: (forcesToFourier)^T, decomposed-force coeff
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index de22cdee6..01418623f 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -474,9 +474,9 @@ class VmecModel {
     UnflattenActive(*vmec_->physical_x_backup_[0], vmec_->s_, v);
     model.packGeometry(*vmec_->physical_x_backup_[0], *vmec_->physical_x_[0],
                        dgeom.data(), gS, /*primal=*/false);
-    model.applyExactForceJacobian(exact_primal_.data(), dgeom.data(), gS,
-                                  *vmec_->physical_f_[0],
-                                  *vmec_->decomposed_f_[0]);
+    model.applyExactForceJacobian(
+        exact_primal_.data(), dgeom.data(), gS, *vmec_->physical_f_[0],
+        *vmec_->decomposed_f_[0], vmecpp::M1ConstraintMode::kEnforce);
     return FlattenActive(*vmec_->decomposed_f_[0], vmec_->s_);
   }
 
@@ -500,7 +500,7 @@ class VmecModel {
     model.applyExactForceJacobianTranspose(
         exact_primal_.data(), gS, *vmec_->decomposed_f_[0],
         *vmec_->physical_f_[0], *vmec_->physical_x_[0],
-        *vmec_->physical_x_backup_[0]);
+        *vmec_->physical_x_backup_[0], vmecpp::M1ConstraintMode::kEnforce);
     return FlattenActive(*vmec_->physical_x_backup_[0], vmec_->s_);
   }
 
diff --git a/tests/test_hessian.py b/tests/test_hessian.py
index 20ca65f14..edc1bf4aa 100644
--- a/tests/test_hessian.py
+++ b/tests/test_hessian.py
@@ -75,15 +75,40 @@ def test_hvp_restores_state():
     assert np.allclose(np.asarray(m.get_state(), float), x0)
 
 
-def test_exact_hvp_matches_force_jacobian_when_tcon_frozen():
+@pytest.fixture(scope="module")
+def cma_states():
+    if not hasattr(_vmecpp.VmecModel, "exact_hessian_vector_product"):
+        pytest.skip("requires an Enzyme-enabled build (VMECPP_ENABLE_ENZYME)")
+
+    indata = _vmecpp.VmecINDATA.from_file(str(CMA))
+    initial_model = _vmecpp.VmecModel.create(indata, 25)
+    initial_state = np.asarray(initial_model.get_state(), float).copy()
+    converged_model = _vmecpp.VmecModel.create(indata, 25)
+    converged_model.solve()
+    converged_state = np.asarray(converged_model.get_state(), float).copy()
+    return initial_state, converged_state
+
+
+@pytest.mark.parametrize(
+    ("state_index", "high_residual"),
+    [pytest.param(0, True, id="initial"), pytest.param(1, False, id="converged")],
+)
+def test_exact_hvp_matches_force_jacobian_when_tcon_frozen(
+    cma_states, state_index, high_residual
+):
     m = _vmecpp.VmecModel.create(_vmecpp.VmecINDATA.from_file(str(CMA)), 25)
     if not hasattr(m, "exact_hessian_vector_product"):
         pytest.skip("requires an Enzyme-enabled build (VMECPP_ENABLE_ENZYME)")
 
+    m.set_state(np.ascontiguousarray(cma_states[state_index]))
     m.evaluate(2, 2, True)
     if m.restart_reason == _BAD_JACOBIAN:
         m.reinitialize()
         m.evaluate(2, 2, True)
+    if high_residual:
+        assert m.fsqz >= 1.0e-6
+    else:
+        assert m.fsqz < 1.0e-6
     m.set_freeze_constraint_multiplier(True)
 
     x = np.asarray(m.get_state(), float)

From 1c4e4fec9ae10852b0b1481d0eef6dc53512fb35 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 23:06:32 +0200
Subject: [PATCH 70/79] Run exact force integration tests with Enzyme

---
 .github/workflows/test_enzyme.yaml | 16 ++++++++++++++++
 1 file changed, 16 insertions(+)

diff --git a/.github/workflows/test_enzyme.yaml b/.github/workflows/test_enzyme.yaml
index 1e226087c..1fcfd47a8 100644
--- a/.github/workflows/test_enzyme.yaml
+++ b/.github/workflows/test_enzyme.yaml
@@ -75,3 +75,19 @@ jobs:
             --copt="-fplugin=${ENZYME_PLUGIN}" \
             --test_output=all \
             //vmecpp/common/enzyme:enzyme_smoke_test
+
+      - uses: astral-sh/setup-uv@v6
+
+      - name: Run Enzyme force-derivative integration tests
+        env:
+          CC: clang-${{ env.LLVM_VERSION }}
+          CXX: clang++-${{ env.LLVM_VERSION }}
+          CMAKE_ARGS: >-
+            -DVMECPP_ENABLE_ENZYME=ON
+            -DVMECPP_ENZYME_PLUGIN=${{ env.ENZYME_PLUGIN }}
+        run: |
+          uv sync --reinstall-package vmecpp
+          uv run --with pytest pytest -q \
+            tests/test_hessian.py \
+            tests/test_hessian_transpose.py \
+            tests/test_external_optimizers.py

From 709497efac765ce0f784ddbde8d32886ca0d96d1 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 23:07:51 +0200
Subject: [PATCH 71/79] Make exact optimizer checks gate-clean

---
 examples/benchmark_adjoint_gradient.py | 4 ++--
 examples/external_optimizers.py        | 7 +++----
 examples/simsopt_optimization_loop.py  | 6 +++++-
 tests/test_hessian_transpose.py        | 7 ++-----
 4 files changed, 12 insertions(+), 12 deletions(-)

diff --git a/examples/benchmark_adjoint_gradient.py b/examples/benchmark_adjoint_gradient.py
index 081533f42..a80c36bed 100644
--- a/examples/benchmark_adjoint_gradient.py
+++ b/examples/benchmark_adjoint_gradient.py
@@ -50,9 +50,9 @@ def mii(bi):
         ]
 
     lam, _ = gmres(
-        LinearOperator((ni, ni), matvec=hii),
+        LinearOperator((ni, ni), matvec=hii),  # type: ignore[call-overload]
         dj[interior],
-        M=LinearOperator((ni, ni), matvec=mii),
+        M=LinearOperator((ni, ni), matvec=mii),  # type: ignore[call-overload]
         rtol=1e-6,
         restart=100,
         maxiter=30,
diff --git a/examples/external_optimizers.py b/examples/external_optimizers.py
index c52c3f162..ebea23897 100644
--- a/examples/external_optimizers.py
+++ b/examples/external_optimizers.py
@@ -339,10 +339,9 @@ def solve_newton_ptc(
     the pseudo-time step by switched evolution relaxation, dt = dt0 * ||F0||/||F||. The
     I/dt shift is implicit-Euler damping at large residual -- the same physics as VMEC's
     damped descent -- and also regularizes the indefinite, ill-conditioned augmented-
-    Lagrangian Hessian that defeats a plain Newton-Krylov, so this is the robust choice
-    on stiff stellarators (e.g. cma) where stock scipy root-finders stall or diverge. As
-    ||F|| -> 0 the shift vanishes and it becomes the exact Newton step. Needs an Enzyme-
-    enabled build.
+    Lagrangian Hessian. The cma regression exercises this shifted solver. As ||F|| -> 0
+    the shift vanishes and it becomes the exact Newton step. Needs an Enzyme-enabled
+    build.
     """
     model = make_model(input_path, ns)
     model.evaluate(2, 2, True)
diff --git a/examples/simsopt_optimization_loop.py b/examples/simsopt_optimization_loop.py
index c42876d18..941de19de 100644
--- a/examples/simsopt_optimization_loop.py
+++ b/examples/simsopt_optimization_loop.py
@@ -15,6 +15,8 @@
 """
 
 import time
+from numbers import Real
+from typing import cast
 
 from simsopt.objectives import LeastSquaresProblem
 from simsopt.solve import least_squares_serial_solve
@@ -27,7 +29,9 @@
 target = 1.03 * j0
 print(f"initial MHD energy = {j0:.6e}, target = {target:.6e}")
 
-least_squares = LeastSquaresProblem.from_tuples([(energy.J, target, 1.0)])
+least_squares = LeastSquaresProblem.from_tuples(
+    [(energy.J, cast(Real, target), cast(Real, 1.0))]
+)
 
 problem.model.reset_force_eval_count()
 t0 = time.perf_counter()
diff --git a/tests/test_hessian_transpose.py b/tests/test_hessian_transpose.py
index 1922b2eec..bb8657f2b 100644
--- a/tests/test_hessian_transpose.py
+++ b/tests/test_hessian_transpose.py
@@ -18,7 +18,7 @@
 import pytest
 
 sys.path.insert(0, str(Path(__file__).resolve().parents[1] / "examples"))
-from vmecpp_adjoint import (
+from vmecpp_adjoint import (  # type: ignore
     adjoint_boundary_gradient,
     forward_boundary_gradient,
     make_model,
@@ -26,10 +26,7 @@
     structural_nullfree_interior,
 )
 
-try:
-    from vmecpp.cpp import _vmecpp
-except ImportError:  # pragma: no cover
-    import _vmecpp
+from vmecpp.cpp import _vmecpp  # type: ignore
 
 _HAS_TRANSPOSE = hasattr(_vmecpp.VmecModel, "exact_hessian_vector_product_transpose")
 needs_transpose = pytest.mark.skipif(

From c347f284a8516ec12a35f55e9e63ad32c23fb342 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 23:16:23 +0200
Subject: [PATCH 72/79] Install matching OpenMP runtime for Enzyme CI

---
 .github/workflows/test_enzyme.yaml | 1 +
 1 file changed, 1 insertion(+)

diff --git a/.github/workflows/test_enzyme.yaml b/.github/workflows/test_enzyme.yaml
index 1fcfd47a8..644883f03 100644
--- a/.github/workflows/test_enzyme.yaml
+++ b/.github/workflows/test_enzyme.yaml
@@ -32,6 +32,7 @@ jobs:
           sudo ./llvm.sh "${LLVM_VERSION}"
           sudo apt-get install -y \
             "llvm-${LLVM_VERSION}-dev" "clang-${LLVM_VERSION}" "libclang-${LLVM_VERSION}-dev" \
+            "libomp-${LLVM_VERSION}-dev" \
             cmake ninja-build
 
       - name: Cache the ClangEnzyme plugin build

From ac0eae687b5ef5cea34505e9fcad9280393b6db4 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 23:19:19 +0200
Subject: [PATCH 73/79] Install VMEC build dependencies in Enzyme CI

---
 .github/workflows/test_enzyme.yaml | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/.github/workflows/test_enzyme.yaml b/.github/workflows/test_enzyme.yaml
index 644883f03..479227c41 100644
--- a/.github/workflows/test_enzyme.yaml
+++ b/.github/workflows/test_enzyme.yaml
@@ -33,7 +33,7 @@ jobs:
           sudo apt-get install -y \
             "llvm-${LLVM_VERSION}-dev" "clang-${LLVM_VERSION}" "libclang-${LLVM_VERSION}-dev" \
             "libomp-${LLVM_VERSION}-dev" \
-            cmake ninja-build
+            cmake gfortran libhdf5-dev liblapack-dev libnetcdf-dev ninja-build python3-dev
 
       - name: Cache the ClangEnzyme plugin build
         id: enzyme-cache

From d0ef637ff6e7aaf009bc83143d7a327481bd4316 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Thu, 9 Jul 2026 23:27:27 +0200
Subject: [PATCH 74/79] Improve docstring formatting in
 simsopt_vmec_gradient.py

Fix formatting in docstring for clarity.
---
 examples/simsopt_vmec_gradient.py | 2 +-
 1 file changed, 1 insertion(+), 1 deletion(-)

diff --git a/examples/simsopt_vmec_gradient.py b/examples/simsopt_vmec_gradient.py
index 54f188789..78eafde6e 100644
--- a/examples/simsopt_vmec_gradient.py
+++ b/examples/simsopt_vmec_gradient.py
@@ -13,7 +13,7 @@
 ``VmecEnergy`` wraps that as a SIMSOPT ``Optimizable`` whose objective is the MHD
 energy of the converged equilibrium and whose ``dJ`` is the adjoint gradient.
 ``optimize_to_target`` runs a gradient-based optimization of the boundary toward
-a target energy, with the analytic gradient or with finite differences, and
+a target energy, with analytic gradient or with finite differences, and
 reports the cost (forward-model evaluations counted inside VMEC++, outer
 iterations, wall time) so the two can be compared.
 """

From 3238887f8c0ead734d71f864d6450130cc8a09b0 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Fri, 10 Jul 2026 07:18:29 +0200
Subject: [PATCH 75/79] Name the m=1 gauge policy explicitly

---
 src/vmecpp/_iteration.py                           |  2 +-
 .../vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc |  7 ++++---
 .../vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h  |  5 +----
 src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc | 14 ++++++--------
 4 files changed, 12 insertions(+), 16 deletions(-)

diff --git a/src/vmecpp/_iteration.py b/src/vmecpp/_iteration.py
index 7e6e0f8dc..e54a8cfc5 100644
--- a/src/vmecpp/_iteration.py
+++ b/src/vmecpp/_iteration.py
@@ -259,7 +259,7 @@ def solve_equilibrium(
             iter2 = force_iteration - bad_resets
 
             # Evolve: forward model, then convergence / damping / time step.
-            model.evaluate(iter1, iter2, precondition=True, legacy_m1_constraint=True)
+            model.evaluate(iter1, iter2, precondition=True, always_fix_m1_gauge=False)
             fsqr, fsqz, fsql = model.fsqr, model.fsqz, model.fsql
             rr = model.restart_reason
             finite = math.isfinite(fsqr) and math.isfinite(fsqz) and math.isfinite(fsql)
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 824631c0a..66b746e1c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -422,7 +422,7 @@ absl::StatusOr<bool> IdealMhdModel::update(
     int& m_last_full_update_nestor, FlowControl& m_fc, const int iter1,
     const int iter2, const VmecCheckpoint& checkpoint,
     const int iterations_before_checkpointing, bool verbose,
-    M1ConstraintMode m1_constraint_mode) {
+    bool always_fix_m1_gauge) {
   // An axis re-guess after a bad Jacobian can repopulate high geometry modes
   // directly, bypassing the force mask; clear them on the state each iteration.
   if (s_.mpolGeometry < s_.mpol || s_.ntorGeometry < s_.ntor) {
@@ -814,8 +814,9 @@ absl::StatusOr<bool> IdealMhdModel::update(
   m_decomposed_f.m1Constraint(1.0 / std::numbers::sqrt2);
 
   // v8.50: ADD iter2<2 so reset=<WOUT_FILE> works
-  if (m1_constraint_mode == M1ConstraintMode::kEnforce || m_fc.fsqz < 1.0e-6 ||
-      iter2 < 2) {
+  const bool fix_m1_gauge =
+      always_fix_m1_gauge || m_fc.fsqz < 1.0e-6 || iter2 < 2;
+  if (fix_m1_gauge) {
     // ensure that the m=1 constraint is satisfied exactly
     // --> the corresponding m=1 coeffs of R,Z are constrained to be zero
     //     and thus must not be "forced" (by the time evol using gc) away from
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index c1c9d3560..da3cf19fc 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -7,7 +7,6 @@
 
 #include <Eigen/Dense>
 #include <climits>
-#include <cstdint>
 #include <span>
 
 #ifdef _OPENMP
@@ -35,8 +34,6 @@
 
 namespace vmecpp {
 
-enum class M1ConstraintMode : std::uint8_t { kLegacy, kEnforce };
-
 // Implemented as a free function for easier testing and benchmarking.
 void deAliasConstraintForce(const RadialPartitioning& rp,
                             const FourierBasisFastPoloidal& fb, const Sizes& s_,
@@ -74,7 +71,7 @@ class IdealMhdModel {
       int& m_last_full_update_nestor, FlowControl& m_fc, const int iter1,
       const int iter2, const VmecCheckpoint& checkpoint = VmecCheckpoint::NONE,
       const int iterations_before_checkpointing = INT_MAX, bool verbose = true,
-      M1ConstraintMode m1_constraint_mode = M1ConstraintMode::kLegacy);
+      bool always_fix_m1_gauge = false);
 
   // Coordinates which inverse-DFT routine to call for computing
   // the flux surface geometry and lambda on it from the provided Fourier
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index 56fd65359..5e2510d7d 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -226,11 +226,11 @@ class VmecModel {
   // lambda-constraint components. That raw gradient is what gradient-based
   // optimizers minimizing the MHD energy functional need; mhd_energy is already
   // set earlier in update(), so it is valid at the checkpoint too.
-  // The native iteration delays the exact m=1 projection based on the previous
-  // Z residual. External evaluations enforce it immediately so F(x) does not
-  // depend on the previously evaluated state.
+  // The native iteration leaves the m=1 gauge free until the previous Z
+  // residual crosses its threshold. External evaluations fix it immediately so
+  // F(x) does not depend on the previously evaluated state.
   void Evaluate(int iter1, int iter2, bool precondition = true,
-                bool legacy_m1_constraint = false) {
+                bool always_fix_m1_gauge = true) {
     ++force_eval_count_;
     bool need_restart = false;
     std::string error_message;
@@ -259,9 +259,7 @@ class VmecModel {
           *vmec_->decomposed_f_[0], *vmec_->physical_f_[0], need_restart,
           last_preconditioner_update_, last_full_update_nestor_, vmec_->fc_,
           iter1, iter2, checkpoint, checkpoint_after,
-          /*verbose=*/false,
-          legacy_m1_constraint ? vmecpp::M1ConstraintMode::kLegacy
-                               : vmecpp::M1ConstraintMode::kEnforce);
+          /*verbose=*/false, always_fix_m1_gauge);
       if (!s.ok()) {
         error_message = std::string(s.status().message());
       }
@@ -1267,7 +1265,7 @@ PYBIND11_MODULE(_vmecpp, m) {
                   py::arg("ns"), py::arg("initial_state") = std::nullopt)
       .def("evaluate", &VmecModel::Evaluate, py::arg("iter1"), py::arg("iter2"),
            py::arg("precondition") = true,
-           py::arg("legacy_m1_constraint") = false)
+           py::arg("always_fix_m1_gauge") = true)
       .def_property_readonly("need_restart", &VmecModel::need_restart)
       .def("perform_time_step", &VmecModel::PerformTimeStep,
            py::arg("velocity_scale"), py::arg("conjugation_parameter"),

From c7af973189768e9b26e7e2a505a18be8fe497708 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Fri, 10 Jul 2026 07:21:25 +0200
Subject: [PATCH 76/79] Keep optimizer metric checks with their example tests

---
 tests/test_external_optimizer_benchmark.py | 37 ----------------------
 tests/test_external_optimizers.py          | 17 ++++++++++
 2 files changed, 17 insertions(+), 37 deletions(-)
 delete mode 100644 tests/test_external_optimizer_benchmark.py

diff --git a/tests/test_external_optimizer_benchmark.py b/tests/test_external_optimizer_benchmark.py
deleted file mode 100644
index 23e7042bc..000000000
--- a/tests/test_external_optimizer_benchmark.py
+++ /dev/null
@@ -1,37 +0,0 @@
-# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
-# <info@proximafusion.com>
-#
-# SPDX-License-Identifier: MIT
-"""The external-optimizer benchmark reports comparable solver metrics."""
-
-import sys
-from pathlib import Path
-
-import numpy as np
-
-ROOT = Path(__file__).resolve().parents[1]
-sys.path.insert(0, str(ROOT / "examples"))
-from external_optimizers import (  # type: ignore  # noqa: E402
-    reference_equilibrium,
-    solve_newton_krylov,
-    solve_vmecpp,
-)
-
-SOLOVEV = ROOT / "examples" / "data" / "solovev.json"
-CTH_LIKE = ROOT / "examples" / "data" / "cth_like_fixed_bdy.json"
-
-
-def test_native_metrics_match_external_metric_definitions():
-    x_star, w_star = reference_equilibrium(CTH_LIKE, ns=11)
-    x, result = solve_vmecpp(CTH_LIKE, ns=11)
-
-    assert result.energy == w_star
-    np.testing.assert_array_equal(x, x_star)
-    assert result.residual_norm < 1e-8
-    assert result.force_evals == result.outer_iters + 1
-    assert result.outer_iters > 0
-
-
-def test_newton_krylov_reports_outer_iterations():
-    _, result = solve_newton_krylov(SOLOVEV, ns=11)
-    assert result.outer_iters > 0
diff --git a/tests/test_external_optimizers.py b/tests/test_external_optimizers.py
index 5e20b7fa0..599672604 100644
--- a/tests/test_external_optimizers.py
+++ b/tests/test_external_optimizers.py
@@ -26,6 +26,7 @@
     solve_newton_krylov,
     solve_newton_krylov_preconditioned,
     solve_preconditioned_descent,
+    solve_vmecpp,
 )
 
 ROOT = Path(__file__).resolve().parents[1]
@@ -93,6 +94,22 @@ def test_hvp_newton_uses_fewer_outer_iterations_than_descent(solutions_3d):
     assert newton.outer_iters < descent.outer_iters
 
 
+def test_native_metrics_match_external_metric_definitions(reference_3d):
+    x_star, w_star = reference_3d
+    x, result = solve_vmecpp(CTH_LIKE, ns=11)
+
+    assert result.energy == w_star
+    np.testing.assert_array_equal(x, x_star)
+    assert result.residual_norm < 1e-8
+    assert result.force_evals == result.outer_iters + 1
+    assert result.outer_iters > 0
+
+
+def test_newton_krylov_reports_outer_iterations(solutions_2d):
+    result = solutions_2d[solve_newton_krylov.__name__][1]
+    assert result.outer_iters > 0
+
+
 def test_cma_cold_start_exercises_non_axisymmetric_paths():
     model = make_model(CMA, ns=25)
     x0 = np.asarray(model.get_state(), float)

From d382efb3be0509cf9041fafa40ca47305aae0ce6 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Fri, 10 Jul 2026 07:55:47 +0200
Subject: [PATCH 77/79] Run the exact-HVP benchmark in Enzyme CI

---
 .github/workflows/test_enzyme.yaml | 10 ++++++
 benchmarks/benchmark_exact_hvp.py  | 53 ++++++++++++++++++++++++++++++
 examples/benchmark_exact_hvp.py    | 48 ---------------------------
 pyproject.toml                     |  1 +
 4 files changed, 64 insertions(+), 48 deletions(-)
 create mode 100644 benchmarks/benchmark_exact_hvp.py
 delete mode 100644 examples/benchmark_exact_hvp.py

diff --git a/.github/workflows/test_enzyme.yaml b/.github/workflows/test_enzyme.yaml
index 479227c41..c3722ed64 100644
--- a/.github/workflows/test_enzyme.yaml
+++ b/.github/workflows/test_enzyme.yaml
@@ -92,3 +92,13 @@ jobs:
             tests/test_hessian.py \
             tests/test_hessian_transpose.py \
             tests/test_external_optimizers.py
+
+      - name: Run exact-HVP benchmark
+        run: |
+          uv run python benchmarks/benchmark_exact_hvp.py | tee exact_hvp_benchmark.txt
+          {
+            echo "### Exact-HVP benchmark"
+            echo '```text'
+            cat exact_hvp_benchmark.txt
+            echo '```'
+          } >> "${GITHUB_STEP_SUMMARY}"
diff --git a/benchmarks/benchmark_exact_hvp.py b/benchmarks/benchmark_exact_hvp.py
new file mode 100644
index 000000000..ce259d174
--- /dev/null
+++ b/benchmarks/benchmark_exact_hvp.py
@@ -0,0 +1,53 @@
+# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
+# <info@proximafusion.com>
+#
+# SPDX-License-Identifier: MIT
+"""Benchmark exact autodiff Hessian-vector products as an internal solver.
+
+Compare preconditioned JFNK, finite-difference Newton-HVP, and exact-autodiff
+Newton-HVP on the same VMEC++ residual. Report force evaluations, iterations,
+wall time, residual norm, and energy error. Requires an Enzyme-enabled build.
+
+Run: python benchmarks/benchmark_exact_hvp.py
+"""
+
+import sys
+from pathlib import Path
+
+ROOT = Path(__file__).resolve().parents[1]
+sys.path.insert(0, str(ROOT / "examples"))
+import external_optimizers as eo  # type: ignore  # noqa: E402
+
+DATA = ROOT / "examples" / "data"
+CASES = [
+    ("solovev", DATA / "solovev.json", 11),
+    ("cth_like", DATA / "cth_like_fixed_bdy.json", 11),
+]
+
+SOLVERS = [
+    ("precond JFNK", eo.solve_newton_krylov_preconditioned),
+    ("Newton FD-HVP + M^-1", eo.solve_newton_hvp),
+    ("Newton exact-HVP + M^-1", eo.solve_newton_exact_hvp),
+]
+
+
+def main():
+    for case, path, ns in CASES:
+        print(f"\n=== {case} (ns={ns}) ===")
+        _, wref = eo.reference_equilibrium(path, ns)
+        print(f"native solve: W = {wref:.8e}")
+        print(
+            f"{'optimizer':28s} {'F-evals':>8s} {'iters':>6s} {'time[s]':>8s} "
+            f"{'||F||':>10s} {'dW':>10s}"
+        )
+        for label, solver in SOLVERS:
+            result = solver(path, ns)[1]
+            print(
+                f"{label:28s} {result.force_evals:8d} {result.outer_iters:6d} "
+                f"{result.seconds:8.2f} {result.residual_norm:10.1e} "
+                f"{abs(result.energy - wref):10.1e}"
+            )
+
+
+if __name__ == "__main__":
+    main()
diff --git a/examples/benchmark_exact_hvp.py b/examples/benchmark_exact_hvp.py
deleted file mode 100644
index 24d804fe7..000000000
--- a/examples/benchmark_exact_hvp.py
+++ /dev/null
@@ -1,48 +0,0 @@
-# SPDX-FileCopyrightText: 2024-present Proxima Fusion GmbH
-# <info@proximafusion.com>
-#
-# SPDX-License-Identifier: MIT
-"""Benchmark the exact autodiff Hessian-vector product as an internal solver.
-
-Compares preconditioned JFNK, the finite-difference Newton-HVP, and the exact
-autodiff Newton-HVP on the same VMEC++ residual, counting force evaluations
-inside VMEC++. Requires an Enzyme-enabled build (VMECPP_ENABLE_ENZYME) for the
-exact-HVP solver; the others run on any build.
-
-Run: python -m examples.benchmark_exact_hvp  (or python examples/benchmark_exact_hvp.py)
-"""
-
-from pathlib import Path
-
-import external_optimizers as eo
-
-DATA = Path(__file__).resolve().parent / "data"
-CASES = [
-    ("solovev", DATA / "solovev.json", 11),
-    ("cth_like", DATA / "cth_like_fixed_bdy.json", 11),
-]
-
-SOLVERS = [
-    ("precond JFNK", eo.solve_newton_krylov_preconditioned),
-    ("Newton FD-HVP + M^-1", eo.solve_newton_hvp),
-    ("Newton exact-HVP + M^-1", eo.solve_newton_exact_hvp),
-]
-
-for case, path, ns in CASES:
-    print(f"\n=== {case} (ns={ns}) ===")
-    _, wref = eo.reference_equilibrium(path, ns)
-    print(f"native solve: W = {wref:.8e}")
-    print(
-        f"{'optimizer':28s} {'F-evals':>8s} {'iters':>6s} {'time[s]':>8s} "
-        f"{'||F||':>10s} {'dW':>10s}"
-    )
-    for label, solver in SOLVERS:
-        try:
-            r = solver(path, ns)[1]
-            print(
-                f"{label:28s} {r.force_evals:8d} {r.outer_iters:6d} "
-                f"{r.seconds:8.2f} {r.residual_norm:10.1e} "
-                f"{abs(r.energy - wref):10.1e}"
-            )
-        except AttributeError:
-            print(f"{label:28s} (needs an Enzyme-enabled build)")
diff --git a/pyproject.toml b/pyproject.toml
index 50e9bdd2c..8f3f0d39b 100644
--- a/pyproject.toml
+++ b/pyproject.toml
@@ -167,6 +167,7 @@ ignore = [
 [tool.ruff.lint.per-file-ignores]
 "tests/**" = ["T20"]
 "**/*_test.py" = ["T20"]
+"benchmarks/**" = ["T201"]
 "examples/**" = ["T201"]
 
 [tool.pyright]

From f0cb35a25d825fafe795dc8db03d59500e6fe134 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Fri, 10 Jul 2026 10:16:57 +0200
Subject: [PATCH 78/79] Make raw VMEC forces history independent (#626)

* Make raw VMEC forces history independent

Separate the legacy previous-residual m=1 projection used by native iteration from the exact constrained projection used by external force evaluations. Cover both directions across the residual threshold.

* Use compact storage for constraint policy

* Name the m=1 gauge policy explicitly
---
 src/vmecpp/_iteration.py                      |  2 +-
 .../vmec/ideal_mhd_model/ideal_mhd_model.cc   |  7 +++-
 .../vmec/ideal_mhd_model/ideal_mhd_model.h    |  3 +-
 .../cpp/vmecpp/vmec/pybind11/pybind_vmec.cc   | 11 +++--
 tests/test_internal_gradient.py               | 42 +++++++++++++++++++
 5 files changed, 58 insertions(+), 7 deletions(-)

diff --git a/src/vmecpp/_iteration.py b/src/vmecpp/_iteration.py
index b019a79fa..e54a8cfc5 100644
--- a/src/vmecpp/_iteration.py
+++ b/src/vmecpp/_iteration.py
@@ -259,7 +259,7 @@ def solve_equilibrium(
             iter2 = force_iteration - bad_resets
 
             # Evolve: forward model, then convergence / damping / time step.
-            model.evaluate(iter1, iter2)
+            model.evaluate(iter1, iter2, precondition=True, always_fix_m1_gauge=False)
             fsqr, fsqz, fsql = model.fsqr, model.fsqz, model.fsql
             rr = model.restart_reason
             finite = math.isfinite(fsqr) and math.isfinite(fsqz) and math.isfinite(fsql)
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
index 1ec40ce51..66b746e1c 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.cc
@@ -421,7 +421,8 @@ absl::StatusOr<bool> IdealMhdModel::update(
     bool& m_need_restart, int& m_last_preconditioner_update,
     int& m_last_full_update_nestor, FlowControl& m_fc, const int iter1,
     const int iter2, const VmecCheckpoint& checkpoint,
-    const int iterations_before_checkpointing, bool verbose) {
+    const int iterations_before_checkpointing, bool verbose,
+    bool always_fix_m1_gauge) {
   // An axis re-guess after a bad Jacobian can repopulate high geometry modes
   // directly, bypassing the force mask; clear them on the state each iteration.
   if (s_.mpolGeometry < s_.mpol || s_.ntorGeometry < s_.ntor) {
@@ -813,7 +814,9 @@ absl::StatusOr<bool> IdealMhdModel::update(
   m_decomposed_f.m1Constraint(1.0 / std::numbers::sqrt2);
 
   // v8.50: ADD iter2<2 so reset=<WOUT_FILE> works
-  if (m_fc.fsqz < 1.0e-6 || iter2 < 2) {
+  const bool fix_m1_gauge =
+      always_fix_m1_gauge || m_fc.fsqz < 1.0e-6 || iter2 < 2;
+  if (fix_m1_gauge) {
     // ensure that the m=1 constraint is satisfied exactly
     // --> the corresponding m=1 coeffs of R,Z are constrained to be zero
     //     and thus must not be "forced" (by the time evol using gc) away from
diff --git a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
index b753e53f5..da3cf19fc 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
+++ b/src/vmecpp/cpp/vmecpp/vmec/ideal_mhd_model/ideal_mhd_model.h
@@ -70,7 +70,8 @@ class IdealMhdModel {
       bool& m_need_restart, int& m_last_preconditioner_update,
       int& m_last_full_update_nestor, FlowControl& m_fc, const int iter1,
       const int iter2, const VmecCheckpoint& checkpoint = VmecCheckpoint::NONE,
-      const int iterations_before_checkpointing = INT_MAX, bool verbose = true);
+      const int iterations_before_checkpointing = INT_MAX, bool verbose = true,
+      bool always_fix_m1_gauge = false);
 
   // Coordinates which inverse-DFT routine to call for computing
   // the flux surface geometry and lambda on it from the provided Fourier
diff --git a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
index cda5d0ad9..5e2510d7d 100644
--- a/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
+++ b/src/vmecpp/cpp/vmecpp/vmec/pybind11/pybind_vmec.cc
@@ -226,7 +226,11 @@ class VmecModel {
   // lambda-constraint components. That raw gradient is what gradient-based
   // optimizers minimizing the MHD energy functional need; mhd_energy is already
   // set earlier in update(), so it is valid at the checkpoint too.
-  void Evaluate(int iter1, int iter2, bool precondition = true) {
+  // The native iteration leaves the m=1 gauge free until the previous Z
+  // residual crosses its threshold. External evaluations fix it immediately so
+  // F(x) does not depend on the previously evaluated state.
+  void Evaluate(int iter1, int iter2, bool precondition = true,
+                bool always_fix_m1_gauge = true) {
     ++force_eval_count_;
     bool need_restart = false;
     std::string error_message;
@@ -255,7 +259,7 @@ class VmecModel {
           *vmec_->decomposed_f_[0], *vmec_->physical_f_[0], need_restart,
           last_preconditioner_update_, last_full_update_nestor_, vmec_->fc_,
           iter1, iter2, checkpoint, checkpoint_after,
-          /*verbose=*/false);
+          /*verbose=*/false, always_fix_m1_gauge);
       if (!s.ok()) {
         error_message = std::string(s.status().message());
       }
@@ -1260,7 +1264,8 @@ PYBIND11_MODULE(_vmecpp, m) {
       .def_static("create", &VmecModel::Create, py::arg("indata"),
                   py::arg("ns"), py::arg("initial_state") = std::nullopt)
       .def("evaluate", &VmecModel::Evaluate, py::arg("iter1"), py::arg("iter2"),
-           py::arg("precondition") = true)
+           py::arg("precondition") = true,
+           py::arg("always_fix_m1_gauge") = true)
       .def_property_readonly("need_restart", &VmecModel::need_restart)
       .def("perform_time_step", &VmecModel::PerformTimeStep,
            py::arg("velocity_scale"), py::arg("conjugation_parameter"),
diff --git a/tests/test_internal_gradient.py b/tests/test_internal_gradient.py
index 17d36f65f..16d26825e 100644
--- a/tests/test_internal_gradient.py
+++ b/tests/test_internal_gradient.py
@@ -17,10 +17,17 @@
 from pathlib import Path
 
 import numpy as np
+import pytest
 
 from vmecpp.cpp import _vmecpp  # type: ignore
 
 SOLOVEV = Path(__file__).resolve().parents[1] / "examples" / "data" / "solovev.json"
+CTH_LIKE = (
+    Path(__file__).resolve().parents[1]
+    / "examples"
+    / "data"
+    / "cth_like_fixed_bdy.json"
+)
 
 
 def _model(ns: int = 11):
@@ -63,3 +70,38 @@ def test_cold_start_is_excluded():
     m = _model()
     m.evaluate(2, 2, False)
     assert np.all(np.isfinite(np.asarray(m.get_forces(), float)))
+
+
+@pytest.fixture(scope="module")
+def force_histories():
+    indata = _vmecpp.VmecINDATA.from_file(str(CTH_LIKE))
+    reference = _vmecpp.VmecModel.create(indata, 11)
+    reference.solve()
+    low_residual_state = np.asarray(reference.get_state(), float).copy()
+
+    model = _vmecpp.VmecModel.create(indata, 11)
+    high_residual_state = np.asarray(model.get_state(), float).copy()
+    model.evaluate(2, 2, False)
+    high_after_high = np.asarray(model.get_forces(), float).copy()
+
+    model.set_state(np.ascontiguousarray(low_residual_state))
+    model.evaluate(2, 2, False)
+    low_after_high = np.asarray(model.get_forces(), float).copy()
+    model.evaluate(2, 2, False)
+    low_after_low = np.asarray(model.get_forces(), float).copy()
+
+    model.set_state(np.ascontiguousarray(high_residual_state))
+    model.evaluate(2, 2, False)
+    high_after_low = np.asarray(model.get_forces(), float).copy()
+
+    return high_after_high, high_after_low, low_after_high, low_after_low
+
+
+def test_raw_force_is_independent_of_high_residual_history(force_histories):
+    _, _, low_after_high, low_after_low = force_histories
+    np.testing.assert_array_equal(low_after_high, low_after_low)
+
+
+def test_raw_force_is_independent_of_low_residual_history(force_histories):
+    high_after_high, high_after_low, _, _ = force_histories
+    np.testing.assert_array_equal(high_after_high, high_after_low)

From 09d1292d52a6f5bb2df622328f7136efe49f7460 Mon Sep 17 00:00:00 2001
From: Christopher Albert <albert@tugraz.at>
Date: Fri, 10 Jul 2026 13:41:19 +0200
Subject: [PATCH 79/79] Remove accidentally committed VMEC++ output files from
 examples/data

examples/data/cth_like_fixed_bdy.out.h5 and examples/data/solovev.out.h5
were run artifacts written next to the input files when the examples were
executed, committed by mistake in 267fdd01. Nothing references them; the
canonical test equilibria live in
src/vmecpp/cpp/vmecpp_large_cpp_tests/test_data/.
---
 examples/data/cth_like_fixed_bdy.out.h5 | Bin 3031398 -> 0 bytes
 examples/data/solovev.out.h5            | Bin 571757 -> 0 bytes
 2 files changed, 0 insertions(+), 0 deletions(-)
 delete mode 100644 examples/data/cth_like_fixed_bdy.out.h5
 delete mode 100644 examples/data/solovev.out.h5

diff --git a/examples/data/cth_like_fixed_bdy.out.h5 b/examples/data/cth_like_fixed_bdy.out.h5
deleted file mode 100644
index d754d07f0f7e35370e6d6197578d9f38cccc09f4..0000000000000000000000000000000000000000
GIT binary patch
literal 0
HcmV?d00001

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