Why the rectangular recovery trial stalled

The completed trial passed 8 of 24 archival fits: all six completed controls, two of fifteen unfinished primary fits, and none of three perturbed repeats. No test stretch became fully resolved. The preceding continuation method passed 15/24; the rectangular method did not improve recovery. Original findings remain 12 globe preferences, zero completed flat preferences and 41 unresolved stretches.

This audit verifies sealed summaries, journals and every saved local residual/Jacobian. It reads no original aircraft observations, performs no native integrations or new archival fits, and changes no scientific labels. Small affine-system solves below are numerical diagnostics, not additional flight results.

Two failure mechanisms

10 early gradient failures: the optimizer stopped on parameter-step size while the independent gradient check still failed.
3 early stability failures: the gradient passed, but the saved local correction changed predicted observations beyond the stability limit.
3 exhausted fits: perturbed starts spent their allowance without satisfying stationarity.

Of the 13 early failures, 11 have at least one outward-pointing gradient on a coordinate less than 1e-12 inside its bound. Many distances are one floating-point step. Such a coordinate can restrict the optimizer's initial feasible step even when other coordinates still need to move. Passed controls can also have near-bound coordinates; proximity alone does not explain or invalidate a fit.

Independently reproduced boundary stall

A two-coordinate control has residuals (x−2, 2y−0.4) inside the unchanged box −1 ≤ x,y ≤ 1. Its exact bounded answer is (1, 0.2), with squared residual cost 1. Starting one floating-point step below x=1 with y=0 makes the rectangular solver stop after two evaluations at cost 1.16, before correcting y. Starting exactly at x=1 recovers cost 1. This isolates a numerical boundary effect without aircraft physics, unknown instruments or an Earth-model preference.

Does the same explanation hold on the saved flight tangents?

For each saved point u₀, the audit holds the recorded residual r₀ and Jacobian J fixed and solves r(u) = r₀ + J(u−u₀). The two starts differ only by moving coordinates within 1e-12 of ±1 to their exact bounds. The physical parameter box, gradient threshold, optimizer settings and 200-evaluation limit remain unchanged. A bounded linear least-squares calculation supplies a separate local reference.

Across all 24 saved systems, the original start passes the gradient check in 12/24 cases; the exact-bound start passes in 16/24. These are local linear completion counts, not recovered archival fits or passes of the full nonlinear stability gate.

Stretch / processing	Model / start	Archival evaluations	Gradient / limit	Local gradient: original → exact-bound
03454752 / profiled_removal	flat_still / primary	3	5.15	False → True
03454752 / profiled_removal	flat_still / repeat	598	482	False → False
03454752 / profiled_removal	sphere_rotating / primary	97	3.39	False → False
03454752 / profiled_removal	sphere_still / primary	3	72.1	False → False
03454752 / unsubtracted	sphere_rotating / primary	3	46.8	True → True
03454752 / unsubtracted	sphere_still / primary	4	29.8	False → False
233183f6 / profiled_removal	sphere_rotating / primary	4	0.000222	True → True
233183f6 / unsubtracted	sphere_rotating / primary	3	0.00128	True → True
233183f6 / unsubtracted	sphere_rotating / repeat	598	3.78e+03	False → False
b365bdf3 / profiled_removal	flat_still / primary	4	44.9	False → True
b365bdf3 / profiled_removal	sphere_rotating / primary	4	0.059	True → True
b365bdf3 / profiled_removal	sphere_still / primary	3	4.42	False → True
b365bdf3 / unsubtracted	flat_still / primary	4	107	False → False
b365bdf3 / unsubtracted	sphere_rotating / primary	3	56.1	False → True
b365bdf3 / unsubtracted	sphere_still / primary	4	3.37	False → False
b365bdf3 / unsubtracted	sphere_still / repeat	598	4.79e+03	False → False

True means the local affine solve passed the same gradient threshold. CSV and JSON retain all 24 cases, objective values, bound movements and reference costs, including local failures.

Weak directions and interpretation

The failed saved Jacobians have a median condition ratio of 9.51e+07. Strong and weak parameter combinations therefore act on markedly different scales. The most frequent coordinates among the three largest components of the weakest directions are: gravity_delta (16/16), gyro_gain_x (14/16), gyro_gain_y (9/16), gyro_gain_z (9/16). Those directions describe correlated local combinations, not individually measured instrument errors. Column-normalized spectra and column correlations are also saved.

An affine model is exact only for these saved linear equations. It cannot establish that a sizeable local correction improves the actual nonlinear trajectory. The audit does not test derivative accuracy, receiver-time discontinuities, changing calibration or global optimality. It cannot promote any of the sixteen failed fits.

Next qualification

Test boundary handling on known nonlinear optima and correlated, nearly singular controls while retaining the existing feasible bounds and acceptance tolerances. Compare objective and predicted-observation recovery from predetermined starts. Only a separately frozen and budgeted observed-data trial can establish whether a numerical repair recovers the unfinished stretches. The exhausted perturbed starts also require attention; exact-bound initialization is not assumed to solve them.

Reproduce

From the repository root:

env PYTHONPATH=analysis OPENBLAS_NUM_THREADS=2 OMP_NUM_THREADS=2 MKL_NUM_THREADS=2 ~/venv/bin/python docs/ilvis0-rectangular-diagnosis-20261012/build_report.py

The report receipt hashes both completed input studies, all local tangent artifacts, audit software and generated outputs, and records the numerical environment. Local optimizer counts describe that environment; they are not guaranteed across platforms. The audit preserves the original trial's source snapshots.