test(sae): #2720 chart-gauge orbit baseline re-measurement at current main

[HomunculusLabs]

What

Re-measures the |gᵀvᵢ| directional derivatives of the penalized objective along the chart-gauge orbit directions at the ard_saddle_state fixture (#2336 geometry), which is the same fixture the original 8.24×/10.45× measurement was taken on.

Why

The issue body and three comment threads warned that the acceptance criteria numbers (8.24× and 10.45× over tolerance) were measured at a stall state produced by a solver that no longer exists — a386c1e8b deleted exact_hessian_physical_complement (zero remaining references crate-wide), which made the Newton step minimum-norm on a complement holding 87–93% of the gradient. 18 of #2674's 19 fixtures flipped green on that change.

⚠ THE ACCEPTANCE CRITERIA BELOW MUST BE RE-MEASURED BEFORE ANY FIX IS GRADED AGAINST THEM
— from the issue body

This PR takes that measurement.

Measurement

Direction	Pre-a386c1e8b	Post-a386c1e8b (this PR)
v₀	4.592e-3 (8.24× tol)	2.403e-4 (0.43× tol — below!)
v₁	5.823e-3 (10.45× tol)	1.067e-3 (1.89× tol)
tolerance	5.575e-4	5.643e-4

What changed

• v₀ dropped below tolerance entirely — the orbit is now flat in that direction at the current solver's stall state
• v₁ dropped from 10.45× to 1.89× but remains above tolerance
• The modelling question (orbit is a likelihood symmetry but not a posterior symmetry) is still live — that statement is about the objective, not about any solver
• But the witness is much weaker (1.89× vs 10.45×), and grading a fix against the old 8.24×/10.45× targets is no longer meaningful

What is NOT invalidated

• The argument. The orbit is an exact first-order symmetry of the likelihood (1.29e-16 … 1.11e-15 over 1333 constructions) and not of the posterior. That is a statement about the objective and no solver change can touch it.
• The reconstruction measurement. dense_step_gauge_vector_from_field in fit_drivers.rs was not touched by a386c1e8b.
• The framed round-trip result. tests_gauge_frame_roundtrip_2720.rs reads the same untouched file.
• The accept-side defect. quotient_residual_norm_sq was not touched. The second disjunct of quasi_laplace_kkt_stationary remains an unguarded weakening in a direction the solver is structurally biased to satisfy.

The -4.054 warning does not apply

The #2674 lane warned that anything computed against log|A| on the old quotiented block is short by -4.054. These are |gᵀv| projections of the raw gradient onto a fixed basis, not log-determinants. exact_joint_chart_gauge_basis is unchanged and retained explicitly as diagnostic-only.

Method

The test creates the ard_saddle_state from #2336, runs penalized_quasi_laplace_criterion_with_cache to reach the inner solve state, assembles the Arrow-Schur system, extracts the joint KKT gradient [g_t (per row); g_β], gets the chart-gauge basis from joint_chart_gauge_basis_for_arrow_layout, and projects |gᵀvᵢ| against SAE_MANIFOLD_INNER_GRAD_REL_TOL * iterate_scale.

The test is marked #[ignore] — it is a manual diagnostic reproducer, not a regression gate. Run with:

cargo test -p gam-sae measure_chart_gauge_orbit -- --ignored --nocapture

Prerequisite failures (inner solve refusal, assembly failure, empty gauge basis) panic rather than silently passing, so fixture drift is caught immediately.

Review

Reviewed through the RepoPrompt pipeline (engineer agent + oracle review):

• Engineer (gpt-5.6-sol): Verified method soundness, fixture correctness (matches converged_state_with_residual_a_saddle_2336 in construction_tests.rs), gradient layout (matches ArrowSchurSystem ordering). Identified P2: all failure paths silently returned Ok, making the test always-green. Identified that the 40-iteration solve should not run in normal CI.
• Oracle: Confirmed assessment. Recommended #[ignore] for the manual reproducer, panicking on prerequisite failures, and asserting finiteness of gradient/tolerance/projections. Applied.

Not established

• Measured on the gamma_fd_tiny_fixture (K=2, d=1, n=10, p=3, m=3 Periodic atoms) with ARD saddle perturbation — same as SAE dense route: terminal exact-curvature saddle-escape (B-converged modes can be A-saddles) #2336. Whether other chart kinds (Euclidean, Poincaré, Duchon) or framed atoms show different behavior is unmeasured.
• The inner solve converges to a stationary point on this fixture (criterion value finite, ‖g‖ = 1.19e-3). The pre-a386c1e8b measurements were taken at a stall state where the solver could not reduce the orbit component; whether a fixture exists where the current solver also stalls with materially higher orbit derivatives is an open question.