Joint rotation-product qualification

Qualify necessary relationships between force-matrix columns before using them in a prediction comparison. Preserve every completed source and result.

The construction retains all original 51 coordinates and 27 physical parameters. Nine additional coordinates represent matrix-entry/gain products; eighteen represent squares and cross-products of the resulting rotation columns. These are arithmetic relaxations, not additional physical calibration freedom. Use four McCormick planes per product, a midpoint tangent for each square, and joint Gram bounds. There are 78 coordinates and 129 new inequalities.

Bound the exact binary-input initial mounting matrix's Gram defect by its Frobenius norm. Do not repair it to an exact identity. Add a separately declared dimensionless 1e-8 numerical-control allowance; it is not an IMU specification or a complete roundoff proof. Export coefficients and bounds outward over the complete extended box. Gram constraints and relaxed product planes remain necessary, not sufficient, for physical realizability or a proper rotation.

Local checks cover exact signed product corners/interiors, anisotropic gains, zero-width domains and caps, source mismatch, invalid gains, nonorthogonal initial matrices, joint-incompatible unit-length columns and reflected matrices. The adapter tests prohibit rerunning the old physical oracle and reject a changed source or extra probe. Twelve focused checks pass before this freeze.

Reuse exactly the 24 sealed column-orientation control identities: six generated candidate/processing configurations crossed with full, sector, fixed and near-gain- boundary domains. Reconstruct only the configuration-bound matrix component. Require its binding to match the sealed source, then apply the new inequalities to the existing 64 independently generated physical probes per identity. Do not regenerate probes, change their values or call the old oracle again.

Freeze 24 identities / 1,536 reused probes / 3,096 plane constructions / 198,144 exact inequality checks / 119,808 coordinate-bound checks. Reserve each identity's complete work before construction/checking. Four workers use one numerical thread each, with 180 cooperative seconds. There are zero LP proposals, native integrations or bounds, new physical-oracle calls, original-aircraft reads, observed fits or retries. Interrupted work remains charged; no expanded allowance.

Require all original and extended coordinates to stay in bounds and every exported inequality to retain every physical probe. Publish failures without replacements. A passed component may enter a separately frozen complete saved- prediction comparison; it does not establish narrower widths, calibrated observational coverage or permission for observed fitting.