Scalar calibration with an algebraic feasibility check

Purpose and preserved evidence

The preceding nonlinear study completed all 228 identities but qualified only 32/72 comparisons. Many unfinished endpoints had almost-zero residuals in flat calibration directions. Preserve those failures; this is a separate candidate, not a restart or an expansion of its allowance.

Before optimizing, test a simple feasible construction. Let m be the measured body-force magnitude history and q the candidate's required-force magnitude. Compute one constant isotropic gain g=(qᵀm)/(qᵀq)−1, with zero offsets and zero gravity adjustment. Reject this construction if g exceeds any axis gain bound. Do not clip it. It is a sufficient construction, not an exhaustive search.

Recompute residuals and the complete seven-column Jacobian at that point. Accept an algebraic near-zero certificate only when physical RMS is at most 10⁻¹² m/s², maximum absolute residual at most 10⁻¹¹ m/s² and fresh projected stationarity at most 10⁻⁸. Publish the feasible objective and its nonnegative lower bound of zero, not an assertion of a unique calibration or exact global optimum. Repeated starts reaching the same algebraic construction are identified as certificates, not independent optimizer convergence.

Otherwise run the original bounded nonlinear candidate with the original requested start and 200-evaluation limit. Preserve its analytic derivatives, seven bounds, numerical tolerances and fresh endpoint gate unchanged. There is no retry or automatic replacement of an unsuccessful result.

Fixed qualification and matrix

Use the same four independent planted fixtures and three starts as the prior study: twelve known-answer identities, followed only after their qualification by the same 72 sealed constructed histories with three starts each. No new trajectory is chosen to improve the result.

Qualification still requires all twelve endpoints, RMS ≤10⁻⁷ m/s² and residual history agreement ≤10⁻⁷ m/s². Matrix acceptance still requires all three endpoints, RMS-score spread ≤10⁻⁸ m/s² and residual-history spread ≤10⁻⁷ m/s². No wrong-model separation is required. Report certificate use separately from actual optimization.

Finite frozen allowance

• 228 charged identities, with at most 228 mathematical optimizer starts.
• At most 404 residual/Jacobian calls per identity: two feasibility calls plus the original optimizer/fresh-check ceiling of 402. Accepted certificates use two initial and two fresh calls, without running an optimizer.
• At most 92,112 residual/Jacobian calls overall, plus two explicit magnitude history constructions per identity. No original-file reads, observed starts or native integrations. No retries.
• One worker, four total numerical threads, 300-second wall-time ceiling.
• Seal source, environment, input and policy hashes before launch; charge before execution. Qualification failure blocks the matrix. Completed work cannot resume.

The analytic branch cannot declare a different physical model correct. It can only exhibit one allowed calibration matching a scalar necessary constraint. It supplies neither a complete gyro-free motion history nor statistical coverage. Recorded-data use still needs C03's timing, averaging, calibration and uncertainty qualification. The main 53-stretch results and all earlier failures remain intact.