Qualify joint rotation relationships before another prediction study

Four groups pass the fixed prediction target; both specified-field globe groups remain above it. Preserve this completed freeze and its full results.

The existing mathematical region admits impossible calibration matrices. Separate column support constraints do not express all the relationships between columns. Next qualify a necessary relaxation of the joint rotation identities, using extra arithmetic coordinates rather than extra physical nuisance parameters.

Mathematical component

Keep all original 51 coordinates. The saved affine expression defines each entry of the force transform M. For physical gains g, B=M diag(1+g) is the original mounting matrix times a proper rotation. Its column inner products are therefore constrained. The stored initial mounting matrix is only numerically orthogonal; bound its exact binary-input Gram defect rather than silently replacing it with a perfect identity.

Represent the nine products M_ij g_j with auxiliary coordinates. The entries of B are then affine in the original coordinates and these products. Introduce eighteen additional coordinates for the nine squares and nine cross-products needed by B-transpose B. Use exact interval arithmetic and the four McCormick inequalities for each product, with outward coefficient export and whole-box allowances for rounding. This gives 78 coordinates, including every original coordinate. Keep any explicit numerical allowance separate from sensor accuracy.

These constraints are necessary, not sufficient: Gram identities alone do not prove a proper rotation, and their linear relaxation need not recover physical feasibility. Do not advertise this as an exact nonlinear solver.

Qualification

Test exact product corners, signed intervals, zero-width domains, gains, shifted matrix expansions, nonorthogonal initial mounting and all original columns. Verify each exported inequality with independent exact arithmetic at known physical points. Reject incompatible source bindings and invalid domains. Include examples with individually plausible columns but impossible joint geometry, and a reflected matrix demonstrating the limitation of Gram checks.

Then freeze a finite component assessment on the existing generated control families, with input/source/environment hashes and explicit probe/inequality limits. It performs no archive reads, native integration or observed fits. Do not consume the completed prediction study's unused time or start allowance.

Only a qualified component may enter a separately frozen complete prediction comparison. That comparison must extend support objectives with zeros for the new arithmetic coordinates, preserve all original coefficients/remainders, independently audit every signed endpoint and retain the unchanged all-axis, all-group twofold target. Reuse sealed native results; stronger necessary constraints alone do not authorize another native calculation or observed fit.