[math-02] Derive kinetic response and Fokker-Planck susceptibilities

[marjohma]

Summary

• derive the FP susceptibility identities and limiting cases independently
• commit deterministic arbitrary-precision oracle values
• add a Fortran oracle consumer covering values, symmetries, and mutation sensitivity

Why

KIM susceptibility functions are checked against an independent symbolic reference rather than implementation self-comparison.

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-193-conventions
Head: fix/kim-fp-194-susceptibility-oracle

Closes #194

[marjohma]

Review (standards + spec vs #194) — keeping this in draft. The core deliverable is genuinely good: the WL script defines I^kl from an independent generating integral (OU propagator, 02_kinetic_and_susceptibilities.wl:14-22) rather than the Fortran recurrences — the anti-circularity non-goal is satisfied — and test_fp_susceptibility_oracle.f90 exercises production getIfunc/W2_arr against the committed oracle at a real, failable 1e-12 scaled tolerance (all 8 (k,l) pairs, m_phi=−1/0/+1, 56 rows enforced, provenance header present, runs in CI). That part can stand as-is.

What keeps it in draft is that several spec-required proofs are missing or are check-theater, and the index overstates them:

1. No derivation of the linearization / action-angle response denominator / dimensionless variables — the integral is asserted via comment, while FORMULA_INDEX.md:592-596 claims the script "traces x1/x2 through (4.1), (4.26)…(A.1)–(A.29)". It doesn't. Please make the claim match the artifact (either derive, or state "asserted from thesis eq. X").
2. Collision-moment checks are tautologies: ouEigenvalues is a hardcoded association checked against its own literals two lines below (.wl:118-127) — nothing integrates the operator, so "particle/momentum/energy moments required by each collision model" is not verified, and the FP-vs-Krook separation check is vacuous.
3. "Prove recurrence relations and derivatives": two numeric recurrence checks at a single probe point (.wl:131-146), applied to the energy-corrected functions; no symbolic proof and no derivative checks at all.
4. Low/high-collisionality and zero/large-x1/x2 limits are fixture points only — never compared against analytic limits (the collisionless closed form in getIfunc.f90:24-35 would be a free cross-check).
5. Negative fixtures: spec requires each mutation to fail "at least one symbolic and one oracle check". All four mutations Exit via FinishChecks[] before the oracle-comparison block is reached — the oracle side never participates; the Fortran side has no negative fixtures.
6. KiLCA unverified (calc_Imn_array, calc_I_array_drift_serg) although the goal is "every susceptibility function used by KIM and KiLCA" — the disclosure in FORMULA_INDEX.md:573-575 is honest, but the issue isn't satisfied by disclosure.

Minor:

• The oracle test recomputes the x1/x2 mapping inline (test:52) rather than exercising the production mapping (calculate_thermodynamic_forces_and_susc) — the mapping is verified by duplication; production mapping coverage currently rests on the Add parallel flow to x2 in KIM-FP #37/Add parallel flow to x2 in KIM-FP #213 flow test. (Note: an earlier draft of this review claimed the kpar·Vpar term was absent from production x2 — that is incorrect; it is present via the conditional Doppler correction in species_mod.f90:410-417.)
• Cell-center path also consumes I10/I12 (species_mod.f90:433-434), covered only via untested symmetrization.
• Line width >100 in checklib.wl:4, 02_kinetic_and_susceptibilities.wl:77, and three Makefile recipe lines (AGENTS.md: 100 max).
• Commit type verify: still outside AGENTS.md's documented set — consider documenting it, it's now stack convention.