[math-05] Derive magnetic kernels, basis transforms, and Poisson systems

[marjohma]

Summary

• derive magnetic-drive/current kernels and Fourier/hat basis transforms
• derive weak and periodic Poisson operators, boundary terms, gauge rules, and field reconstruction
• commit deterministic field-operator matrix fixtures and formula mappings

Why

The field solve and reconstruction conventions have an independent symbolic derivation.

Validation

Validated cumulatively at the stack head:

• cmake --build build -j2
• ctest --test-dir build -E 'test_rhs_balance|test_periodic_convergence|test_periodic_vs_global' --output-on-failure — 35/35 passed

Known red gates are intentionally excluded from that command: the pre-existing test_rhs_balance baseline failure, the periodization-deformation acceptance gate, and the unresolved periodic-vs-global 5% agreement gate.

Stack

Base: fix/kim-fp-187-offdiagonal-kernel
Head: fix/kim-fp-197-poisson-oracle

Closes #197

[marjohma]

Review (standards + spec vs #197) — keeping this in draft. Positives first: all new FORMULA_INDEX citations resolve on the branch (no phantom paths), the Makefile targets mirror sibling style, and a minority of checks are genuine derivations (cylindrical harmonic Laplacian via D[], the Green's-identity probe, the FP generator symmetry). But this is the weakest of the math PRs against its own issue:

Blocking

1. "Make periodic/hat matrix tests read the committed oracle" is not done. Nothing in Fortran references field_operator_matrices.dat; test_periodic_{assembly,solve,vs_global} and KIM/tests/CMakeLists.txt are untouched. The oracle's only consumer is the generating script's own read-back — pure self-comparison, drift detection at best. The second success criterion (ctest -R 'test_periodic_(assembly|solve|vs_global)') runs pre-existing tests that never see the oracle.
2. Most of the Behavior list is absent: no canonical-magnetic-perturbation or radial-field derivation; K^{ρB} (FP_kernel_plasma_prefacs.f90:187-249) never derived or checked; no Fourier-to-hat transform or its measure (there is a 4-point DFT and flat-1D hat rows, but no transform between bases and no cylindrical measure); no manufactured Φ_MA or localized ψ (the kind-1 ψ is a smooth per-mode constant and source = aa*total is circular); "decide the radial versus helical Laplacian" is not decided — radialSymbol returns −k_r² by construction and both symbols are merely labeled.
3. K^{jB} verification proves the wrong thing: the check |jBCorrect − jBPrinted| > 1e-6 establishes that I11/I13 differs from I02/I22, not that I11/I13 is correct; jBCorrect transcribes the production formula (FP_G1_j_B at kernel.f90:335) — the same transcription failure mode flagged on [math-04] Derive the charge-potential integral kernels #216.
4. Check theater: hat mass/Laplace matrix rows are hardcoded literals checked against hardcoded arithmetic (never Integrated over hat functions); i10=i01; i12=i21 followed by checks that compare the value to itself (the .dat has no (1,0)/(1,2) rows); screenedZero, radialSymbol, aligned-potential residual, kernelAction == kernelProbe.phiProbe, and the source-sign check are all definitional when unmutated. The hardcoded row count ==14 repeats the [math-01] Inventory formulas and lock KAMEL conventions #193 counting pattern.

Inherited/minor

• verify-mathematica-field-operators-negative copies the false-pass-when-WOLFRAM_KERNEL-unset flaw a fourth time (all six fixtures print "rejected" without Wolfram ever running). Worth fixing once, centrally, for all four targets.
• All six negative fixtures exist but FinishChecks[] exits early, so mutations never reach the current-kernel or oracle sections.
• FORMULA_INDEX marks KIM-07/08/09 "verified by [math-05] Derive magnetic kernels, basis transforms, and Poisson systems #197" though the .wl never touches Fortran and no Fortran test reads the oracle — verified-derivation ≠ verified-production-code; poisson.f90's actual Dirichlet lifting (solve_poisson.f90:197-208) and calc_mass_matrix are never checked.
• .dat header omits the precision/generation provenance both sibling oracles state; 05_...wl:200-202 exceed the 100-col limit.