Analytic stopping diagnostic results¶
Protocol frozen at bf5d96e0. Probed initial and
final states of all 18 archived truth-start cases; no optimization was rerun and
no historical pose, convergence or recovery verdict was changed.
Six wrong negative-bias stops¶
| Scene | Population | Final correction m | Cost | Cost after -1 cm bias step | Exact bias gradient | Fixed-grid best bias m | Fixed-grid cost |
|---|---|---|---|---|---|---|---|
| original | exact_z_visible | -0.000104 | 96.80952 | 94.39428 | 233.327 | -0.25 | 0.000531 |
| original | exact_z_all | -0.000100 | 99.04593 | 96.63073 | 233.323 | -0.25 | 2.224961 |
| near_foreground | exact_z_visible | -0.003076 | 93.79572 | 91.23502 | 247.381 | -0.24 | 0.110754 |
| near_foreground | exact_z_all | -0.003082 | 97.05922 | 94.49786 | 247.444 | -0.24 | 3.353853 |
| staggered | exact_z_visible | -0.000561 | 96.17171 | 93.73983 | 234.933 | -0.25 | 0.009470 |
| staggered | exact_z_all | -0.000537 | 98.85625 | 96.42552 | 234.823 | -0.25 | 2.666471 |
A feasible descent remains at every failed stop¶
All six negative-bias truth-start fits terminated with SciPy status 3 (xtol),
with estimated correction still between approximately -.00010 and -.00308 m
instead of -.25 m. At those unchanged final poses, the analytic derivative of the
same objective with respect to correction is +233.3 to +247.4. The fixed -.01 m
probe decreases cost in all six cases; this is a feasible descent independent of
reference pose or target-bias selection. The full preregistered grid finds still
lower costs at -.25 m (original/staggered) or -.24 m (near-foreground).
The grid is a diagnostic, not a replacement estimate; no fit is retroactively called successful. Its minimum is not proof of global pose/bias optimality. Nevertheless, these results rule out "no better feasible correction exists" as an explanation for the six stops. Step-size termination did not certify stationarity.
Pose derivatives cross surface discontinuities¶
At a pose-coordinate perturbation of 1e-8 degree/metre, each failed final state has a raw-residual jump greater than 1 m for at least one point. All such probes retain exactly the same support population. In this continuous first-surface sampler that indicates a change in the sampled surface, not a missing-point count or a pixel interpolation boundary. The actual optical first-surface measurement can be discontinuous at silhouettes even with an exact raycaster.
The forward residual-Jacobian maximum singular value ranges from approximately 9.42e8 to 1.85e9 at that step. Forward versus central Jacobian relative differences are approximately one; the largest derivatives strongly depend on the finite- difference step. At 1e-6 they remain of order 1e7, and at 1e-4 of order 1e5, consistent with a finite jump divided by the step. This is not evidence of huge smooth physical sensitivity or a meaningful Gaussian pose covariance.
In contrast, the finite-difference bias-gradient component closely agrees with the analytic bias derivative at all probed steps, and bias perturbations have no >1 m residual jumps. Bias-only movement is smooth here while the jointly optimized pose coordinates encounter discontinuities. This identifies a concrete mismatch with smooth trust-region linearization near the archived stops. The probes do not record SciPy's internal trust-region state or exactly reproduce its machine-relative step policy, so they do not prove the entire causal chain inside that implementation.
Next controlled change¶
Test separating the smooth scalar bias update from pose updates, with the original objective and fixed samplers retained. Use a bounded bias-only update based on measured cost, count every call within a predeclared total budget, and never initialize with the injected correction or adopt the diagnostic grid as a truth lookup. Require actual objective decrease and retain all original correct/incorrect cases, including positive/zero biases and true starts. Compare returned pose and correction, not merely the solver's convergence label. Freeze this algorithm before scoring.
Do not simultaneously introduce boundary-aware image sampling or change tolerances: that would obscure whether the stopping failure was addressed. This diagnosis also reinforces that local Jacobian rank near a surface switch is not sufficient validation. No core/default change, production promotion, new recovery count or SOTA claim.
Full probes and schema retain all 18 cases, both states, all seven axes/three step sizes and all grid costs. Six focused tests, repository Ruff, strict documentation build, frozen hashes and complete-probe checks pass. The sign/cost formula has an independent finite- difference unit check; unsupported penalties do not contribute a bias derivative.