Preserve nonlinear axial-gauge query laws beyond matched covariance

[FlorianPfaff]

Scientific purpose

Extend the observable-subspace factor and query-observability work (#332, #338) with a decision-relevant nonlinear mechanism: retaining the unobserved angular distribution, rather than only its tangent variance or even its exact Cartesian mean/covariance.

A uniform rotation and a threefold rotation law have identical first two Cartesian moments but give halfspace probabilities 1/2 and 1/3. Under the threefold law, collapsing both to probability 1/2 adds 1/36 expected Brier loss. Thus covariance repair alone cannot resolve this failure.

Implementation

• AxialGaugeOrbit: explicit reference-frame rotation orbit with strict exact-line geometry checks and rigid-frame equivariance.
• CircularQuadrature: preserves a caller-supplied conditional angular law, not an assumed uniform prior.
• Nonlinear correspondence likelihood update with complete fixed cross-point residual covariance. Exact-line observations preserve the conditional prior; informative off-axis geometry updates it.
• Shared-angle joint query pushforward, first/two-harmonic exact finite-law moments, full joint covariance, stable Gaussian-mixture density, and halfspace probabilities.
• No production provider-v2 or BayesianPhysTwin complete-belief/fallback behavior changes.

Source-frozen controlled experiment, completed locally

Scientific source revision: f9f63b9fbecb9a91b61ec532157c8405b368eeb5.

Protocol SHA-256: ce26865227378b548548e20d513843b978aab0003279a117a02e29f2eb4cf06d.

Five predefined regimes, two seeds, 16,384 independent continuous gauge/readout draws per seed and regime (163,840 total). Truth is not sampled from the prediction grid. The primary comparator has the same exact nonlinear mean and covariance as the orbit mixture.

Angular regime	Exact-moment Gaussian NLL	Orbit-mixture NLL	Paired difference, approximate 95% Monte Carlo interval
Narrow, 0.05 rad	7.825349	7.825184	-0.000166 [-0.000295, -0.000037]
Moderate, 0.6 rad	11.009556	9.864043	-1.145513 [-1.158874, -1.132153]
Broad, 1.2 rad	12.198044	10.523599	-1.674446 [-1.682318, -1.666573]
Uniform	12.499400	10.790085	-1.709315 [-1.717009, -1.701621]
Smooth threefold	12.501588	9.466914	-3.034674 [-3.045656, -3.023693]

NLL is in nats for Cartesian densities expressed in millimeters; lower is better. In the smooth threefold regime, the fixed halfspace Brier score changes from 0.250000 to 0.223182 (10.73% improvement). Decision gains are not universal: the uniform halfspace score is exactly tied, and the broad-regime Brier interval crosses zero. Point RMSE is identical for the two primary arms.

512-to-1024-node log-density refinement differs by at most 3.56e-15 nats on the fixed refinement observations. This is a numerical check, not a universal quadrature bound.

Finalized evidence bytes and interpretation are being retained separately in the private paper repository; this PR does not alter public scientific claim status.

Validation

• Local focused tests: 27 passed, Python 3.13.5 / NumPy 2.3.5.
• Both executed source-module Git blob identities match the pinned GitHub revision.
• Python compilation passed.
• Added hosted workflow runs Ruff, MyPy, the focused tests, and the full predefined controlled study on Python 3.12. Hosted checks are not yet claimed passed.

Claim and novelty boundary

This is conditional controlled mechanism evidence with known angular laws and readout noise, not an end-to-end gauge estimator, real-provider competence/calibration, BayesianPhysTwin/Causal4D benefit, or deployment safety. Geometry-only rank deficiency is not proof of full likelihood symmetry. The module does not assume independence between observable geometry and unobserved angle; a complete uncertain-geometry belief must preserve and mix the appropriate conditional laws.

Circular moments, non-Gaussian rotation models, symmetry-aware inference, and Bayesian conditioning are established. The candidate contribution is their specific connection to deficient learned-window gauges and query-level loss that remains after exact moment matching. Documentation cites primary prior work and distinguishes this method from the existing general quotient-lifting manuscript.

No PointWorld, Flat'n'Fold, MotionCrafter, Deform360, BayesianPhysTwin, Causal4D, or protected target outcome was opened. The separate source-qualification route in #333 remains unchanged.