[bug] uv derivative FD tests have unusually large noise on the analytically-zero dU/dstate2 and dU/dw0 directions

[djkees]

Summary
test_uv_derivatives_fd/test_uv_derivatives_fd_smooth_truncation (source/equilibrium_test.pf) each have one assertion — dU/dstate2 (and the parallel dU/dw0 array check) — that compares an analytically-exact-zero quantity against pure finite-difference (FD) noise. That's expected (see below), but the noise on this particular direction is unusually large compared to the analogous zero-derivative checks in the sibling hp/sp/sv tests, large enough that a tolerance floor safe enough to pass would make the assertion essentially meaningless. Filed as a follow-up to #88 rather than folding a band-aid fix into that PR.

For a uv (internal energy / specific volume) constrained solve, the output internal energy is pinned to the input target regardless of the second state variable (V) or the reactant weights (w0) — so dU/dstate2 and dU/dw0 are analytically exactly zero. The equivalent zero-derivative checks in hp (dH/dstate2, dH/dw0), sp (dS/dstate2, dS/dw0), and sv (dS/dstate2, per-species dnj_dstate) all have FD noise in the 1e-7 to 1e-3 range, comfortably handled by a 1e-3–1e-1 denominator floor while staying 10–500x below the smallest genuine nonzero derivative values seen elsewhere in the same tests. uv's noise on this one direction is different:

test	quantity	FD noise magnitude
test_uv_derivatives_fd	dU/dstate2	~3.2e-4
test_uv_derivatives_fd_smooth_truncation	dU/dstate2	~3.7e-2
test_uv_derivatives_fd_smooth_truncation	dU/dw0 (per-reactant)	up to ~1.2e-2

To pass under the same relative tolerance (5.0d-2) used for the rest of the derivative assertions in these tests, the smooth-truncation case would need a denominator floor greater than ~0.74 — i.e. larger than several genuine nonzero derivative values seen elsewhere in the very same test file (e.g. dS/dstate2 as small as ~0.055 in tv_derivatives_fd_smooth_truncation). A floor that size wouldn't meaningfully test anything for this direction.

Reproduction

• Interface: Fortran (pFUnit test suite)
• Steps: ctest -R cea_core_test -V (after [bug] equilibrium_test.pf derivative-accuracy assertions use unfailable tol = 1.0d30 #88's fix lands). Add write(*,*) debug prints on totals%dU_dstate2/totals%dU_dstate2_fd (or dU_dw0/dU_dw0_fd) in test_uv_derivatives_fd_smooth_truncation and compare the printed analytic (0) vs. FD values directly.

Expected behavior
The FD noise on dU/dstate2/dU/dw0 for uv should be roughly in line with the analogous zero-derivative checks in hp/sp/sv (1e-7 to 1e-3), so a reasonably tight floor (matching the 1e-3–1e-1 pattern used elsewhere) can both pass reliably and still catch a real regression.

Actual behavior
FD noise on this one direction is 10–100x larger than the equivalent checks elsewhere, especially in the smooth_truncation variant. Root cause not yet investigated — could be the finite-difference step size (h = 1.0d-6) interacting poorly with how the second state variable (V) is scaled/perturbed for uv, solver convergence tolerance in that direction, or something specific to the smooth-truncation chain rule. Until root-caused, #88's fix leaves these two assertions (dU/dstate2 and dU/dw0) at their original unfailable bound (tol_dU_dstate2 = 1.0d30, a dedicated per-assertion tolerance) in both test_uv_derivatives_fd and test_uv_derivatives_fd_smooth_truncation, with a code comment explaining why, rather than papering over it with an oversized floor. Every other assertion in both tests (roughly 20 of ~22) now uses the real 5.0d-2 tolerance.

Environment

• OS: n/a (source-level)
• CEA version/commit: current branch as of 2026-09-03

Additional context
Discovered while fixing #88 (fix/derivative-fd-tolerances branch). See that PR for the full before/after tolerance investigation across hp/sp/uv/sv.

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Drafted with Claude's assistance

• FD noise magnitudes in the table were read directly from ctest -R cea_core_test -V output with write(*,*) debug prints enabled (already present, commented/uncommented, in the relevant subroutines) — not estimated or taken on faith.
• The "floor would need to exceed 0.74" claim was derived algebraically from the observed ~3.7e-2 noise and the 5.0d-2 relative tolerance (noise / floor < tol ⇒ floor > noise / tol), then cross-checked against the smallest genuine nonzero derivative value found elsewhere in the same file (~0.055 in tv_derivatives_fd_smooth_truncation).
• Checked gh issue list on djkees/cea before filing to confirm this isn't a duplicate.