[Superseded by #343 and #347] Audit finite gauge ambiguity beyond local query observability

docs/finite-gauge-query-ambiguity.md

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+# Finite axial-gauge ambiguity for physical queries
+
+## Purpose and status
+
+This experimental, NumPy-only analysis extends the **local**
+[query-observability diagnostic](query-conditioned-observability.md) with exact
+finite-orbit bounds for a restricted but useful geometry: one free rotation
+about a registered line per shared gauge group, with all other transformation
+parameters fixed. It also supplies sharp worst-case regret for actual affine
+action losses that share those same gauges.
+
+The existing local diagnostic is not mathematically incorrect. The new control
+shows why its direct-observability fraction must not be interpreted as a global
+invariance or decision certificate. A small first derivative does not establish
+that a query is constant on the finite observation-equivalence class.
+
+This is analytic development evidence, not real-provider competence,
+calibration, BayesianPhysTwin benefit, Causal4D benefit, or deployment safety.
+No provider, dataset, protected target, or physical execution is used. No
+existing result, production exporter, stable API, or admission rule is changed.
+
+## A falsifying control with a positive counterpart
+
+An exactly collinear overlap lies on the x-axis. Rotation about that axis leaves
+all its point coordinates unchanged. A probe initially at `(0, 0.1, 0)` has
+
+$$q(\phi)=0.1\cos\phi,\qquad q'(0)=0,\qquad q''(0)=-0.1.$$
+
+For the existing analytic rank-six factor, the scalar y-position Jacobian has
+zero projection onto the missing rotation-x direction. With its complete
+identity prior and observable precision 10, the local report gives direct
+support 1, variance reduction 10/11, and worst variance ratio 1/11. The existing
+illustrative `(0.8, 0.8, 0.5)` gate admits this scalar query. Nevertheless, the
+finite y-query range over a full rotation is `[-100, 100] mm`.
+
+These comparisons concern **different guarantees**. A full-circle ambiguity
+bound is deliberately stronger than a prior-weighted local uncertainty
+calculation. The control does not establish that the local posterior's actual
+credible interval undercovers or that its Bayesian decision is wrong.
+
+There are also useful positive controls. The axial coordinate of an off-axis
+probe remains fixed. Two points with the same transverse displacement have an
+exactly invariant y-difference when they share one angle. Treating their angles
+as separately variable destroys that cancellation; falsely merging unrelated
+angles invents it.
+
+## Domain assumptions
+
+The caller registers metric-space points, their persistent identities, a pivot
+and unit axis for each group, and which points share exactly the same angle.
+Each group angle ranges over a full circle; the joint domain is the Cartesian
+product of those circles. This is a set-valued domain, **not** an assertion of
+statistical independence. A smaller or coupled domain is conservatively
+contained by this product, but the bounds need not be sharp on that smaller
+domain.
+
+A rank-six information matrix alone does not prove the presence of an exact
+axial symmetry. The caller must establish that the finite transformations are
+admissible for the relevant observation model. An exactly collinear support
+has zero geometric motion under its axial orbit. A near-collinear support does
+not: `maximum_support_motion` returns twice its maximum distance to the axis.
+That is a geometric motion bound, not a bound on likelihood ratio, calibration,
+or posterior probability. Additional observations, metric anchors, covariance
+orientation, priors, or physical constraints can restrict or break the proposed
+symmetry and must not be ignored.
+
+Points are already expressed in the aligned metric frame. Uncertainty in scale,
+axis location, other gauge directions, identity, deformable dynamics, and
+nonlinear physical queries is outside this exact special case. The helper does
+not automatically turn a provider dependence-group label into a shared angle.
+
+## Exact query bounds
+
+Let `a_g` be a unit axis through `c_g`, and decompose each point in group `g` as
+
+$$d_i=p_i-c_g,\quad d_i^\parallel=a_ga_g^\top d_i,
+\quad d_i^\perp=d_i-d_i^\parallel.$$
+
+Rodrigues' formula gives
+
+$$p_i(\phi_g)=c_g+d_i^\parallel
+ +d_i^\perp\cos\phi_g+(a_g\times d_i^\perp)\sin\phi_g.$$
+
+For affine queries `q_k = b_k + sum_i w_ki^T p_i`, collect contributions
+**inside each shared group before taking an amplitude**:
+
+$$q_k(\phi)=C_k+\sum_g[A_{kg}\cos\phi_g+B_{kg}\sin\phi_g].$$
+
+Consequently the sharp componentwise interval is
+
+$$q_k\in\left[C_k-\sum_g\sqrt{A_{kg}^2+B_{kg}^2},
+                 C_k+\sum_g\sqrt{A_{kg}^2+B_{kg}^2}\right].$$
+
+Proof: each harmonic has extrema plus or minus its amplitude, attained at
+`atan2(B_kg, A_kg)` and that angle plus pi. Product-domain freedom lets the
+extrema be attained simultaneously for one query. Different query endpoints
+need not be jointly attainable: the interval collection is not an exact
+rectangular joint feasible set.
+
+Global invariance holds exactly when every `A_kg` and `B_kg` is zero. Stationarity
+at the reference only requires every `B_kg` to be zero. This isolates the
+first-order blind spot without a Monte Carlo tolerance or an arbitrary grid.
+
+## Sharp action regret preserves common uncertainty
+
+When the query family contains actual lower-is-better affine action losses
+`L_k`, define regret for a caller-selected action `k` by
+
+$$R_k=\sup_\phi\left[L_k(\phi)-\min_j L_j(\phi)\right].$$
+
+For finitely many actions this has the closed form
+
+$$R_k=\max_j\left\{C_k-C_j+
+ \sum_g\sqrt{(A_{kg}-A_{jg})^2+(B_{kg}-B_{jg})^2}\right\}.$$
+
+Proof: interchange the supremum and the finite maximum over `j`, and apply the
+harmonic bound to each loss difference. Including `j=k` ensures nonnegative
+regret. `regret_witness(k)` returns a competing action and an angle vector
+attaining this bound. Query contrasts share the same angles, so amplitudes are
+computed **after subtraction**, not from separate marginal interval endpoints.
+
+For `L_0=0.1-y` and `L_1=0.1+y`, the reference selects action 0 with zero loss.
+At half a turn, action 0 has loss 200 mm while action 1 has zero loss. Both
+worst-case regrets are 200 mm, so an illustrative 50 mm regret budget rejects
+that reference-selected action. This is a constructed loss, not robot-control
+or physical-state estimation performance.
+
+Conversely, `L_0=0.05+y` and `L_1=0.10+y` have overlapping marginal loss
+intervals but action 0 is uniformly preferable: its exact regret is zero.
+The common uncertain term cancels. The method therefore does not simply
+reject everything with a broad marginal interval.
+
+## Reproduce and use
+
+From an installed source checkout:
+
+```bash
+python -m pytest tests/test_axial_gauge_query.py
+python -m prob4d.axial_gauge_query_study \
+  --output outputs/axial-gauge-query-development-v1.json
+```
+
+The output is labelled deterministic development evidence. Repeating the command
+at the same path is allowed only for identical bytes; different existing output
+is never overwritten. The tests include an independent quaternion-rotation
+reference, attaining extrema, common-nuisance cancellation, frame and axis-sign
+invariance, malformed inputs, and parity with the actual existing local gate.
+The integration test must run in a complete checkout; it does not silently skip
+when the existing modules cannot be imported. Random algebraic test cases are
+development controls, not independent empirical trials.
+
+```python
+import numpy as np
+from prob4d.axial_gauge_query import AxialGaugeOrbit, affine_axial_queries
+
+orbit = AxialGaugeOrbit(np.zeros(3), np.array([1.0, 0.0, 0.0]))
+losses = affine_axial_queries(
+    np.array([[0.0, 0.1, 0.0]]),
+    np.array([[[0.0, -1.0, 0.0]], [[0.0, 1.0, 0.0]]]),
+    offsets=np.array([0.1, 0.1]),
+    point_group_ids=("window-gauge-0",),
+    orbits={"window-gauge-0": orbit},
+)
+regret = losses.worst_case_regrets()  # [0.2, 0.2], in these loss units
+within_budget = losses.within_regret_budget(0, maximum_regret=0.05)  # False
+```
+
+`within_regret_budget` executes no action and changes no belief. A downstream caller
+must retain every existing provider, calibration, identifiability, and support
+gate, then route rejection through its existing complete-belief fallback.
+Passing this additional check cannot rescue an upstream failure. Applying a
+position bound to a nonlinear simulator loss without a valid reduction does not
+produce an action-regret guarantee.
+
+The formulas are analytic, but the implementation evaluates them in float64,
+not certified interval arithmetic. A caller-frozen nonnegative numerical margin
+can make a threshold decision more conservative; it is not a proof of a
+floating-point enclosure. The implementation rejects nonfinite values rather
+than manufacturing a bounded result.
+
+## Relation to existing work and paper ownership
+
+Weak-direction-aware registration is established prior art: see Tuna et al.,
+[X-ICP](https://arxiv.org/abs/2211.16335). Group-invariant observable/unobservable
+decompositions are also established; see Shen and Leok,
+[Geometric Symmetry Reduction of the Unobservable Subspace for Kalman
+Filtering](https://arxiv.org/abs/1901.03474). Rodrigues' formula, harmonic support
+functions, and worst-case regret are not claimed as new generic mathematics.
+This note does not establish an exhaustive novelty claim.
+
+The bounded contribution candidate is the finite-gauge, shared-lineage query
+and decision audit for partially observable learned 4D windows, together with
+its stationary-derivative counterexample and cancellation-preserving positive
+controls. The existing [Gaussian linearization-closure
+diagnostic](gauge-linearization-closure.md) instead checks numerical moment
+closure under a declared joint Gaussian. Neither result subsumes the other's
+assumptions or gives empirical calibration.
+
+This belongs to the Prob4D geometric observation boundary. It does not duplicate
+the general query-quotient, Jeffrey-lift, or cross-intervention theorems owned by
+the existing theory companion in the paper repository. No fourth manuscript or
+new physical acquisition is proposed. Paper-facing development results and
+interpretation belong in `FlorianPfaff/BayesianPhysTwin-Paper`; this public
+repository owns the method, tests, and reproducer.
+
+Real-provider value remains a separate empirical question. The existing
+PointWorld/Flat'n'Fold source-qualification route in issue #333 remains governed
+by its own unresolved source requirements and target-access rules. This method
+neither authorizes its execution nor makes it a dependency for the already
+bounded BayesianPhysTwin and Causal4D manuscripts.