diff --git a/notes/exact-foundations-correction-20260912.md b/notes/exact-foundations-correction-20260912.md
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+# Exact foundations: corrected topology, shape, and hidden criticality
+
+Date: 2026-09-12. Base read: `main@8b5bc840e4c499e3bfa31f61bfd6f2a713b2e1d5`.
+This note supersedes the affected arguments, not the historical measurements, in
+PRs #675, #690, #693, #666 and #688. No percolation production, threshold
+estimation, or large census is performed. New proofs below are mathematical
+consequences/constructions, not claims of literature novelty.
+
+## 1. All-size wrapping label map: no directional conjecture is needed
+
+Use exactly the scope of `notes/digital-alexander-duality-proof.md`: an honest
+periodic square-cell torus, with the NN black graph and complementary NN+NNN
+white graph. In a common homology basis let their rational ambient images be
+A,C in H_1(T^2;Q). The existing embedded-reduction and subsurface lemma gives
+C=A^perp for the intersection pairing omega((a,b),(x,y))=ay-bx.
+
+**Elementary correction.** If A=span(a,b) is a nonzero line, then
+
+    A^perp = {(x,y): ay-bx=0} = A.
+
+In particular span(1,0)^perp=span(1,0), NOT span(0,1). PR #690 confused
+symplectic orthogonality with Euclidean orthogonality; the same mistake entered
+its reviews and #693. No extra face-parity or 90-degree argument is required.
+
+**Theorem (allowed joint labels).** In the stated scope, the only possible pairs
+of the rank-refined labels none, x, y, spiral, cross are
+
+    (none,cross), (cross,none), (x,x), (y,y), (spiral,spiral).
+
+Proof. The rank-sum theorem permits (0,2),(2,0),(1,1) only. At rank one,
+C=A, so both sides have the same rational winding line. This forces the same
+single-axis label or the same both-coordinate spiral label. Distinct components
+of either embedded graph have zero intersection pairing. They cannot contribute
+two independent homology lines on a torus; if the total image has rank two, one
+component therefore carries it. The white embedded reduction preserves both
+connectivity and ambient classes by the facewise replacements in the existing
+proof. Thus rank two is a cross, not two disjoint transverse clusters. This
+also proves the both-two label is empty. QED.
+
+This is an **allowed-support** theorem, not a statement that every allowed cell
+has positive mass at every size. It equates rational lines and labels, not the
+integer subgroup, cluster counts, or a literal colour-flip operation. Degenerate
+periodic quotients outside the original proof remain outside this result.
+
+Consequences: the black-spiral and white-spiral events are the SAME configuration
+set at every size in scope; their cancellation is pointwise, not an involution
+between distinct configurations. Directional equality holds in the (1,1) case,
+not for all configurations (all-black already refutes that unqualified wording).
+For any probability measure on these configurations,
+
+    D = 1{r_b>0}-1{r_w>0} = r_b-1,
+    M = E[D] = P_2-P_0,
+    F := E[r_b]/2 = (1+M)/2.
+
+These need neither Bernoulli independence nor a census. Under the Bernoulli
+family F is strictly increasing, so its median parameter is the unique matching
+root: p*_N=Q_N(1/2). This says nothing about rates or a closed form for p_c.
+
+## 2. What the finite-operator objection actually establishes
+
+PR #675's probability/eigenvalue distinction is important but is not a general
+impossibility theorem. For example, on the space indexed by configurations,
+let W_p be diagonal with the Bernoulli weights and P_j the diagonal projector
+onto r_b=j. Then M=tr((P_2-P_0)W_p) is an exact single-operator representation.
+This is an exponentially large tautological representation, not an efficient
+row transfer, not pTL, and not a leading-eigenvalue identity.
+
+The useful unresolved question is therefore restricted: specify local weighted
+propagation, closure, the sector maps and the cylinder limit before comparing
+with Jacobsen's criterion. A finite probability difference need not equal a
+leading-eigenvalue difference. That does NOT imply that no operator can encode
+it. The black-only identity above also defeats a blanket claim that both graph
+connectivities must always be stored. #636 receives no automatic large-compute
+release from this correction; #681 should own the precise cylinder bridge.
+
+## 3. Shape symmetry is not reflection about the raw parameter 1/2
+
+PR #666 proposes C3: sup_{p in K}|M_N(p)+M_N(1-p)| -> 0 for every compact
+K in (0,1), and identifies it with normalized shape symmetry. Under the
+repository's location conclusion F_N(p)->0 below c and ->1 above c, this C3
+forces c=1/2: if c>1/2, M_N(1/2)->-1; if c<1/2 it tends to +1. Either way,
+for c!=1/2 the absolute sum at 1/2 tends to 2, not zero. Thus C3 cannot be
+used as the shape hypothesis for a differently centred transition.
+
+With symmetric anchors a,1-a and W_N=Q_N(1-a)-Q_N(a)>0, the correct diagnostic is
+
+    A_N(u) = [Q_N(u)+Q_N(1-u)-Q_N(a)-Q_N(1-a)]/W_N
+           = Z_N(u)+Z_N(1-u)-1.
+
+Its convergence to zero is a shape question independent of the limiting centre.
+It remains OPEN for the relevant percolation families. Self-duality constrains
+shape but does not select a unique symmetric profile. Two exact tiny sizes,
+a bond control and one site production are not a within-model asymptotic ladder.
+
+There is a second elementary correction to #622/#624 and their descendants.
+For an increasing affine map phi(p)=alpha*p+beta, ordinary pushforward gives
+F_phi=F composed with phi^{-1}, hence Q_phi=phi composed with Q. These actions
+DO intertwine. On a fixed [0,1] support one must track the transformed support;
+forcing both endpoints fixed leaves only the identity. A nonlinear kink is
+not an affine map and cannot distinguish two alleged Aff(1) actions.
+The valid location-without-shape constructions are unaffected by this correction.
+
+## 4. An irreducible exact no-go replacing the direct-sum overclaim
+
+The old `notes/bounded-task-rank-threshold-no-go-20260907.md` conflates Markov
+irreducibility and reducibility of a real linear representation. Already
+G=[[-1,1],[1,-1]] is an irreducible generator commuting with the nontrivial
+state swap R. Its even and odd eigenspaces are not disconnected Markov classes.
+Also, a direct sum of irreducible generators is not irreducible as a chain.
+
+**Theorem (product-chain repair).** For c in {1/2,1/3}, p in [0,1], L>=2,
+let H_L(p-c) be the reflecting nearest-neighbour walk with right rate
+exp(p-c), left rate exp(c-p). Set V=[[-2,2],[2,-2]] and
+
+    G_{L,c}=V tensor I_L + I_2 tensor H_L(p-c).
+
+Every finite chain is connected, irreducible, reversible, entrywise analytic,
+and local on a two-layer ladder (maximum degree three, uniformly bounded
+positive rates). Let e0=(1,0)^T, B=e0 tensor 1_L, and
+C=e0^T tensor (1_L^T/L), common to both families. Since H_L*1_L=0,
+
+    C exp(t G_{L,c}) B = e0^T exp(t V) e0 = (1+exp(-4t))/2
+
+for all L,p,t. The exact task order INCLUDING the stationary constant is two;
+the decaying contrast alone has order one. Yet the full-chain gap is
+
+    gamma_{L,c}(p)=2*cosh(p-c)-2*cos(pi/L),
+    gamma_{infinity,c}(p)=4*sinh((p-c)/2)^2.
+
+To verify the gap formula, similarity by the square root of the birth-death
+stationary weights makes each hidden off-diagonal equal sqrt(ab)=1; its
+nonzero eigenvalues are -(a+b)+2*sqrt(ab)*cos(k*pi/L), k=1,...,L-1.
+The product-chain rates are sums with 0 and 4. For the stated c,p,L,
+the smallest hidden rate is at most 2*cosh(2/3)<4, so it is also the full gap.
+Only p=c closes in the limit. QED.
+
+Thus exact task equality can coexist with different gap-closing locations
+EVEN FOR OVERALL IRREDUCIBLE CHAINS. This is an abstract spectral-threshold
+counterexample, not percolation, not a universality claim, and not robustness
+to arbitrary interactions that destroy the product structure. The construction
+is presented without a novelty claim. The accompanying mpmath check is a
+numerical diagnostic of the formula, not the proof.
+
+A related correction: a mode's response residue is (C v)(w^T B). It vanishes
+if either factor vanishes, not only when both do. In systems with repeated
+modes/couplings, use the minimal realization/Kalman decomposition rather than
+an unsupported if-and-only-if assertion about an arbitrary invariant subspace.
+
+## 5. The nonnegative-rank separation survives with logarithmic growth
+
+`notes/positive-vs-signed-slack-separation-20260906.md` and #688 state that a
+regular n-gon's slack matrix has nonnegative rank n. This is false. Its ordinary
+rank is three and its nonnegative rank is Theta(log n). The unbounded-versus-
+bounded separation remains valid; the claimed linear growth does not.
+
+PRIMARY_TEXT_READ, 2026-09-12: Fiorini, Rothvoss, Tiwary, *Extended formulations
+for polygons*, arXiv:1107.0371v2, sections 1-3 (HTML):
+https://arxiv.org/html/1107.0371v2
+Theorem 1 identifies extension complexity with slack nonnegative rank;
+Theorem 2 gives O(log n) for regular polygons; section 1 gives the matching
+Omega(log n) bound. No polygon-to-native-cut-network realization is proved here.
+
+## 6. Consequence for further work
+
+Do not commission larger wrapping censuses to prove the directional label map;
+it follows already. Do not test #666's raw-p C3 as though it were shape symmetry.
+Do not infer a universal one-operator no-go from #675, or a percolation threshold
+from the product-chain construction. Retain the valid exact data, location
+lemma, finite non-scalar failures and original-U identifiability problem.
+
+The next empirical target should be the corrected within-model shape diagnostic
+and full-vector residual on existing histograms, not another fitted exponent.
+P3 needs an explicit covariance-support policy before singular scores are used;
+its missing N580 covariance should first be recovered from committed `_deleted`
+arrays, not bought by another production.
+
+Validation: `python -m unittest discover -s tests -p 'test_research_control_exact.py'`.
+Five checks pass locally, including 54 product-chain response checks at 40 digits.
+No full repository test suite was run in this connector-only checkout.
diff --git a/results/research-control-20260912/irreducible-product.json b/results/research-control-20260912/irreducible-product.json
new file mode 100644
index 00000000..1d46f7c5
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+++ b/results/research-control-20260912/irreducible-product.json
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+{
+  "scope": "Abstract irreducible Markov control; not percolation",
+  "response_checks": 54,
+  "precision_decimal_digits": 40,
+  "max_response_error_diagnostic": "2.2958874e-41",
+  "exact_response": "(1+exp(-4*t))/2",
+  "finite_gap": "2*cosh(p-c)-2*cos(pi/L), L>=2, c=1/2 or 1/3",
+  "limiting_gap": "4*sinh((p-c)/2)^2",
+  "exact_task_order_including_constant_mode": 2,
+  "all_checked_graphs_connected": true
+}
diff --git a/scripts/theory/irreducible_hidden_threshold.py b/scripts/theory/irreducible_hidden_threshold.py
new file mode 100644
index 00000000..51c85d80
--- /dev/null
+++ b/scripts/theory/irreducible_hidden_threshold.py
@@ -0,0 +1,68 @@
+#!/usr/bin/env python3
+"""An irreducible product-chain no-go, not a percolation simulation.
+
+Row generators act on terminal observables. B=e0 tensor 1; C=e0^T tensor
+uniform initial law. Algebra proves the response; small mpmath checks are
+numerical diagnostics, never the evidence for an all-L theorem.
+"""
+from __future__ import annotations
+import json
+from mpmath import mp
+
+
+def generator(length, p, center):
+    if length < 2:
+        raise ValueError("length must be at least two")
+    a, b = mp.exp(p-center), mp.exp(center-p)
+    g = mp.zeros(2*length)
+    for hand in range(2):
+        for i in range(length):
+            row = hand*length+i
+            g[row, (1-hand)*length+i] = 2
+            if i+1 < length:
+                g[row, row+1] = a
+            if i:
+                g[row, row-1] = b
+            g[row, row] = -mp.fsum(g[row, j] for j in range(2*length) if j != row)
+    return g
+
+
+def check():
+    with mp.workdps(40):
+        error = mp.mpf(0)
+        count = 0
+        for length in (2, 3, 5):
+            B = mp.matrix([1]*length+[0]*length)
+            C = mp.matrix([[mp.mpf(1)/length]*length+[0]*length])
+            for p in (mp.mpf(0), mp.mpf('0.5'), mp.mpf(1)):
+                for center in (mp.mpf(1)/2, mp.mpf(1)/3):
+                    g = generator(length, p, center)
+                    assert mp.norm(g*mp.ones(2*length, 1)) < mp.mpf('1e-35')
+                    # All rails in both directions and every rung are positive.
+                    for hand in range(2):
+                        for i in range(length-1):
+                            j = hand*length+i
+                            assert g[j, j+1] > 0 and g[j+1, j] > 0
+                    for i in range(length):
+                        assert g[i, i+length] == g[i+length, i] == 2
+                    for t in (mp.mpf(0), mp.mpf('0.2'), mp.mpf(1)):
+                        got = (C*mp.expm(t*g)*B)[0]
+                        want = (1+mp.exp(-4*t))/2
+                        error = max(error, abs(got-want))
+                        count += 1
+        assert error < mp.mpf('1e-35')
+        return {
+            'scope': 'Abstract irreducible Markov control; not percolation',
+            'response_checks': count,
+            'precision_decimal_digits': 40,
+            'max_response_error_diagnostic': mp.nstr(error, 8),
+            'exact_response': '(1+exp(-4*t))/2',
+            'finite_gap': '2*cosh(p-c)-2*cos(pi/L), L>=2, c=1/2 or 1/3',
+            'limiting_gap': '4*sinh((p-c)/2)^2',
+            'exact_task_order_including_constant_mode': 2,
+            'all_checked_graphs_connected': True,
+        }
+
+
+if __name__ == '__main__':
+    print(json.dumps(check(), indent=2, allow_nan=False))
diff --git a/tests/test_research_control_exact.py b/tests/test_research_control_exact.py
new file mode 100644
index 00000000..2c2b86c9
--- /dev/null
+++ b/tests/test_research_control_exact.py
@@ -0,0 +1,54 @@
+"""Checks for concrete mathematical mistakes, not manuscript wording."""
+from fractions import Fraction as F
+from pathlib import Path
+import sys
+import unittest
+sys.path.insert(0, str(Path(__file__).resolve().parents[1]/'scripts'/'theory'))
+from irreducible_hidden_threshold import check
+
+
+class ExactControlTests(unittest.TestCase):
+    def test_symplectic_line_is_its_own_orthogonal(self):
+        # In a symplectic plane annihilator of (a,b) is ay-bx=0.
+        for a,b in ((1,0),(0,1),(1,1),(2,-3)):
+            for x in range(-5,6):
+                for y in range(-5,6):
+                    self.assertEqual(a*y-b*x == 0, F(a)*y == F(b)*x)
+        self.assertEqual(1*0-0*1, 0)  # (1,0) lies in its own annihilator.
+        self.assertNotEqual(1*1-0*0, 0)  # (0,1) does not.
+
+    def test_rank_identity_needs_no_direction_conjecture(self):
+        for r in range(3):
+            white = 2-r
+            d = int(r>0)-int(white>0)
+            self.assertEqual(d, r-1)
+            self.assertEqual(F(1+d,2), F(r,2))
+
+    def test_affine_pushforward_intertwines_inverse(self):
+        # F(p)=p^2, Q(u)=sqrt(u); exact rational points avoid roundoff.
+        alpha,beta = F(3,2),F(-1,7)
+        for q in (F(1,5),F(1,2),F(4,5)):
+            x = alpha*q+beta
+            self.assertEqual(((x-beta)/alpha)**2, q*q)
+
+    def test_shape_symmetry_does_not_force_center_half(self):
+        center,a,b = F(3,5), F(1,4), F(3,4)
+        for n in (10,20,100):
+            width,eps = F(1,n),F(1,n*n)
+            lower = center-width/2
+            def Q(u):
+                return (u+(1-eps)*lower/width)/((1-eps)/width+eps)
+            def Z(u):
+                return (Q(u)-Q(a))/(Q(b)-Q(a))
+            for i in range(1,10):
+                u=F(i,10)
+                self.assertEqual(Z(u)+Z(1-u),1)
+            # At 1/2 only the uniform background contributes to the CDF.
+            self.assertEqual(2*(2*eps*F(1,2)-1), -2+2*eps)
+
+    def test_irreducible_product_chain(self):
+        self.assertEqual(check()['response_checks'],54)
+
+
+if __name__ == '__main__':
+    unittest.main()