[math-04] Derive the charge-potential integral kernels

[marjohma]

Summary

• derive diagonal and off-diagonal charge-potential kernel formulas
• record normalization, Bessel, harmonic, and limiting checks
• commit deterministic kernel oracle fixtures and update formula provenance

Why

The Fourier charge kernel has a source-derived normalization and sign oracle.

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-194-susceptibility-oracle
Head: fix/kim-fp-196-charge-kernel-oracle

Closes #196

[marjohma]

Review (standards + spec vs #196) — keeping this in draft. Standout genuine win: the 1+b_+ vs 1−b_+ discrepancy is actually resolved from the upstream integral as the issue demanded — 04_charge_potential_kernels.wl:61-80 proves the (1−b_+−m) form symbolically from the Weber generating integral and cross-checks it by arbitrary-precision integration of the unreduced Bessel moment; it matches production core_rho_phi_sp. The FP I00/I20 ingredients are integrated independently of getIfunc, Krook/FP assumptions are kept separate, and all cited code paths are real. The 13 oracle rows cover species signs, diagonal/off-diagonal, finite k_s, the b_x=600 asymptotic branch, zero-FLR, Debye-only, and m_phi=0,±1,±2.

What keeps it in draft:

1. "Make [kim-fp-03] Verify the off-diagonal Fokker-Planck Fourier kernel #187 consume the generated oracle" is not done — nothing outside verification/ references rho_phi_kernels.dat; the pre-existing test_flr2_fourier_kernel.f90 never reads it, so the success criterion ctest -R test_flr2_fourier_kernel doesn't test this PR's output. The oracle is currently consumed only by the script that generated it. (If the intent is that [kim-fp-03] Verify the off-diagonal Fokker-Planck Fourier kernel #217 wires it up, the issue text should be amended or the PR should carry the reader.)
2. Kernel assembly is transcription: kernelParts reproduces core_rho_phi_sp's structure (prefactor i·vT²·ks/(λ²ω_c·ν), I00(A1F0+A2F2)+A2·I20·F0/2, phase, 1/4π) with no upstream derivation of that combination — the non-goal "do not define the Mathematica kernel by transcribing flr2_fourier_kernel.f90" is violated for the assembly, ingredients aside. Several Behavior items (perturbation-field representation, guiding-centre + Fourier integration, thermodynamic-force source, species sum, Debye term, charge sign entering the math) are asserted rather than derived.
3. "Every analytic reduction checked against the unreduced integral at nontrivial points": only the FLR Bessel factors, at one probe (a=0.7, c=1.1, m=1). The assembled kernel is never checked against an unreduced integral; the final oracle-row check compares the two total columns against the script's own formula — that's provenance, not verification. The other ~19 committed columns are unchecked.
4. 4 of 6 negative fixtures are tautological trip-wires (four-pi, fourier-measure, phase, hidden-ks2 are caught by re-stated-definition checks two lines after the assignment, not by independent physics; only bessel-index and bplus-sign are rejected by real integration). The early FinishChecks[] exit means mutations never reach the FP integrals or oracle comparison. FORMULA_INDEX's "six negative fixtures independently mutate…" overstates this.
5. verify-mathematica-charge-kernels-negative inherits the false-pass flaw from the inventory PR: with WOLFRAM_KERNEL unset, the inner make fails at the kernel guard for every fixture, all six print "rejected", and the target exits green without any Wolfram run.
6. Production-default mismatch undisclosed in the commit: the oracle assumes k_s² included, but the production default is kern_include_ks2=.false. — today's code default cannot match the .dat (deferred to [kim-fp-03] Verify the off-diagonal Fokker-Planck Fourier kernel #187 in the index, unmentioned in the commit message).

Minor: m_phi truncation "proof" is a self-referential demonstration on cases 9-13; homogeneous limit exists as a script-only identity at one point with no .dat row; all rows sit at moderate x1 (0.3–0.7). Commit type verify: still undocumented in AGENTS.md.