Camera-depth bias versus translation results¶

Protocol frozen at 9bee5c16 and run once from that commit: Linux, Python 3.12.3, NumPy 2.5.2, SciPy 1.18.0, one OpenBLAS thread, about 90 s of wall time. All 210 fits and 84 sensitivity evaluations completed, with no exceptions or retries. The 18 zero-bias stage A fits exactly reproduce the published one-sided results on a different machine. The β = 0 profile fits exactly reproduce the stage A truth starts.

Frozen verdict: not_supported. The joint bias variable keeps every pose within the gate, but it does not separate bias from camera-z translation to the declared 0.05 m. It also degrades exact-depth (zero-bias) accuracy.

A: pose-only fit under bias¶

Scene	Bias m	Pass (perturbed / truth start)	Rotation deg	Translation m	Δt_z / b
original	0	4/4 / 2/2	0.198–0.212	0.047–0.049	—
original	+0.25	0/4 / 0/2	0.466–0.563	0.262–0.267	0.96–0.97
original	−0.25	4/4 / 2/2	0.300–0.357	0.066–0.073	−0.16 to −0.15
near_foreground	0	4/4 / 2/2	0.109–0.132	0.032–0.035	—
near_foreground	+0.25	0/4 / 0/2	0.294–0.575	0.229–0.237	0.85–0.89
near_foreground	−0.25	4/4 / 2/2	0.244–0.246	0.048–0.049	−0.11
staggered	0	4/4 / 2/2	0.062–0.072	0.023–0.024	—
staggered	+0.25	0/4 / 0/2	0.079–0.250	0.223–0.227	0.89–0.91
staggered	−0.25	4/4 / 2/2	0.114–0.123	0.024–0.026	−0.08 to −0.07

Each row pools the visible and all-return populations. A positive bias (camera depth too long) fails every case, 0/18 including the truth starts. The translation moves along the camera axis by 85–97 % of the bias, and the error lands just above the 0.20 m gate. In 15 of these 18 failures the fit converges and retains support. The other 3 stop at max_nfev. A negative bias passes 18/18 for the same reason the frozen fit handles occlusion. The returns appear behind the surface, where the residual saturates, so the pose barely moves. This is masking, not robustness. Errors still grow relative to zero bias (original 0.20 deg / 4.7 cm → 0.30–0.36 deg / 6.6–7.3 cm), and nothing in the fit reports the bias.

The pre-declared pooled absorption label is false: the median Δt_z/b over the 36 biased fits is 0.39. The two signs behave in opposite ways, so the pooled median sits between them. The per-sign description above is post hoc and does not change any verdict.

B: identifiability¶

Local sensitivity at the reference (oracle diagnostic). The values agree across the three biases to 1e-14, as expected.

Scene	Unexplained fraction e	R² on pose	Alias Δt_z per m	Bias sd inflation	β–t_z correlation
original	0.899	0.19	0.18	1.11	−0.44
near_foreground	0.943	0.11	0.10	1.06	−0.33
staggered	0.929	0.14	0.12	1.08	−0.37

Visible and all-return populations agree to 0.001. The rows used are 11,255–11,520, with no probe-unstable rows. All 18 conditions pass the frozen e ≥ 0.04 rule and are labeled distinguishable, so no joint result is set aside as ambiguous.

The local measure contradicts stage A. It predicts a pose-only translation shift of 0.10–0.18 × b, but the actual nonlinear fit moved 0.85–0.97 × b under positive bias. A plausible explanation, not tested here, is that the local Jacobian gives silhouette rows full weight, and their strong pose sensitivity holds only for infinitesimal moves. A favorable local identifiability number therefore did not predict the finite-shift confounding, and it must not be used as a safety check on its own.

Bias profile (frozen pose-only fit from the truth start on depth − β). On the 0.25 m grid the lowest cost is at the true bias in 18/18 conditions, and that fit passes 18/18. The profile is strongly one-sided:

Bias corrected too little (β = b − 0.25). The cost is only 34–145 % higher than at the truth. The translation lands 0.223–0.262 m off, and 0/18 pass.
Bias corrected too much (β = b + 0.25). The cost is 266–400 % higher, yet the pose stays within the gate in 18/18.

So the objective does prefer the true bias. Yet a modest cost increase buys a gate-failing translation in one direction only. Nine profile fits stop at max_nfev. The grid is too coarse to locate a continuous minimum.

C: separate pose plus bias fit¶

Scene	Pose pass	Rotation deg	Translation m	β̂ − b m	Bias within 0.05 m	Pose-only zero-bias translation m
original	22/22	0.221–0.263	0.078–0.087	−0.069 to −0.059	0/22	0.047–0.049
near_foreground	22/22	0.154–0.200	0.061–0.068	−0.061 to −0.052	0/22	0.032–0.035
staggered	22/22	0.064–0.072	0.044–0.045	−0.044 to −0.042	22/22	0.023–0.024

Per scene, the 22 cases are 6 zero-bias fits and 16 biased fits, including 4 full-truth starts. Pose and bias outcomes are kept separate:

Pose: 66/66 pass. All fits converge and retain support, and the largest run takes 417 actual calls.
Bias: 22/66 within 0.05 m. Of the biased cases, 16/48 recover the bias; at zero bias, 6/18 report |β̂| < 0.05 m.
Trade-off: every estimate satisfies β̂ − b ≈ −Δt_z within ±2 mm. The joint fit slides 4–7 cm along a bias/camera-z trade direction. The slide is the same for every true bias, start and population. Adding b only reparameterizes the joint problem, so this is a property of the objective, not of the bias value.
Zero-bias damage: with exact depth, the joint fit's translation error is 4.4–8.5 cm, against 2.3–4.9 cm for the frozen pose-only fit. Rotation also worsens on original and near_foreground. Everything still passes the gate, but the extra variable costs exact-depth accuracy.
Reference-free check: the stage B sensitivity evaluated at each estimate says distinguishable every time (e 0.69–0.90), matching the oracle label in 66/66. Yet 44 of those estimates miss the bias tolerance. The diagnostic does not detect the systematic offset.

Decision¶

Rule 3 applies. There are 12 zero-bias regressions (|β̂| ≥ 0.05 m on original and near_foreground), and 32 of the 48 distinguishable biased cases miss the bias tolerance. Do not adopt the bias variable. Do not quote the 66/66 pose passes as solving depth bias, and do not widen the 0.05 m tolerance after seeing these estimates, which sit about 0.04–0.07 m away. The frozen one-sided pose-only fit remains the research baseline, outside production.

Answer to the question for these noiseless analytic scenes:

Pose-only fit: cannot tell a positive constant depth bias from a camera-z translation shift (0/18), and hides a negative bias behind its occlusion saturation.
Joint fit: the objective prefers the true bias, but the fit trades bias against translation almost 1:1, with a 4–7 cm offset.
Overall: bias and translation are not separated to the declared accuracy.

Positive camera-depth bias is an open blocker before any real-data rerun of this estimator.

Not run, and requiring a new frozen protocol:

Find what fixes the 4–7 cm β/t_z offset at zero bias. Candidates are the interpolation residual floor at silhouettes and the one-sided asymmetry.
Test whether geometry or views that break the trade direction separate them better.
Test detecting bias rather than estimating it. The sign-asymmetric front/behind residual populations are one candidate signal. It would need wrong-candidate controls, because earlier validators accepted wrong poses.

Real stereo bias is often range-dependent rather than constant, so a constant bias is the easiest case and a pass here would not have transferred directly.

Limits and evidence¶

Scenes: three noiseless sphere/plane layouts that were already used for development. They share one renderer, one extrinsic and the same views. They are not independent scenes, not real data and not SOTA evidence.
Oracle diagnostic: the local test is evaluated at the reference pose. It never fed an estimator or a selection.
Budget: actual residual calls are 12,534 for stage A, 22,705 for the profile, 15,229 for stage C and 1,260 for the sensitivity evaluations. That is 51,728 in total, against the 169,260 ceiling. The most calls in any fit was 628 (in the profile), so none reached the 800 cap. Fitting time summed to 21.3 s (A), 38.4 s (profile) and 26.2 s (C).

Full evidence and schema keep every fit: status, message, pose, camera-frame translation error, bias estimate, support per view, behind/front counts, actual calls, runtime and both sensitivity evaluations. Core code and defaults are unchanged.