Bounded nonlinear scalar calibration controls

Question and fixed region

Replace the preceding unbounded local projection with feasible nonlinear calibration, without treating a finite optimization as a global certificate. Preserve all completed evidence and the 53-stretch main result.

For every epoch, use one shared correction vector: three acceleration gains ±0.15, three offsets ±0.25 m/s², and gravity adjustment ±0.05 m/s². The residual is the magnitude of (body minus offsets)/(1 plus gains), minus the magnitude of the candidate force with the gravity adjustment subtracted from its down component. These are assumed bounds, not measured device errors.

Use analytic derivatives for all seven coordinates. Optimize normalized bound coordinates with SciPy trust-region reflective least squares; x_scale=1, ftol=xtol=10⁻¹², internal gtol=2 times float64 machine epsilon, and maximum 200 residual evaluations. Count every residual and Jacobian evaluation, including one fresh residual/Jacobian at the endpoint. The hard combined allowance is 402 calls per start. Failure or budget exhaustion remains an unfinished outcome; no retry or tolerance relaxation is allowed.

Fresh acceptance requires a successful solver status and projected-gradient infinity norm at most 10⁻⁸. The projected gradient is x−clip(x−Jᵀr,−1,1) in normalized bound coordinates. This is a numerical stationarity check, not sensor accuracy, model adequacy or global optimality.

The internal scaled-gradient rule is deliberately near machine precision because it can otherwise terminate before the fresh projected-gradient check passes at exact boundaries. A positive internal tolerance still permits termination at zero residual in rank-deficient controls. The fresh acceptance limit, bounds and evaluation allowance remain independent of this internal numerical guard.

Qualification gate

First use four known-answer mathematical fixtures with 121 epochs: an excited interior solution, near-boundary solution, exact-boundary solution, and a rank-deficient constant-force solution. Their measured forces are constructed independently from declared proper rotations and planted calibrations.

Each uses three fixed starts: zero, the alternating vector (0.8,−0.8,0.8,0.8,−0.8,0.8,0.8), and its negative. All twelve must pass fresh stationarity, recover residual RMS at most 10⁻⁷ m/s², and agree in residual history within 10⁻⁷ m/s² RMS. Do not require unique calibration parameters when force magnitudes do not identify them. A failed gate blocks the trajectory matrix.

Matched matrix after qualification only

Read the 72 sealed constructed histories from the preceding local-calibration study, checking source receipt and artifact hashes before use. They are exact software fixtures, not original airborne measurements. Fit each from the same three fixed starts, retaining all candidates and failures.

For each identity report physical residuals, endpoint, stationarity, active bounds, all calls and the three-start residual agreement. Repeat agreement requires every start to pass, RMS scores within 10⁻⁸ m/s² and residual histories within 10⁻⁷ m/s² RMS. There is no required wrong-model separation. Failure of repeatability prevents promoting this diagnostic into recorded-data fitting.

Finite limits

Freeze sources, environment, fixture hashes, all task identities and allowances before launch. Charge each start before execution, preserve interruption charges, and never restart completed work. Publication reads saved results only. Passing software controls does not establish a real-recording noise model or authorize an observed-data campaign. Quantization, timing and uncertainty gates remain.