Does numerical identifiability provide useful calibration?

An optimizer can recover an injected value exactly from noiseless data yet be unable to estimate that value usefully from uncertain measurements. The previous duration and rotation controls also found that a numerical rank changed when only the assumed GPS errors changed. This check asks how much calibration information survives, without using rank as a pass rule.

It uses the same 18 analytic fixtures: 2, 20 and 60 seconds, steady orientation or a one-second 0.1-degree pitch correction, under three fixed GPS-error assumptions. The baseline has an artificial millimetre-scale covariance. The other two add correlated horizontal/vertical errors of 1/3 metres and 10/30 metres, respectively, plus offsets and drift. These are sensitivity assumptions, not measured accuracy. Only the covariance changes between each set of three cases.

Eight quantities vary together: initial position, velocity and pitch; gyro bias and gain; acceleration bias and gain; and the GPS/IMU time offset. Their declared fixture bounds set the coordinate scales. Those bounds are not supported hardware limits. Every case uses four central derivative steps—0.0005, 0.001, 0.002 and 0.004 of each bound—with positive and negative perturbations. No step is selected after seeing its result. Independent analytic positions check the unchanged predictor.

How to read the diagnostic

The whitened Jacobian measures changes in the predicted GPS positions relative to the assumed measurement errors. Its singular values describe joint parameter directions. A singular value above one means a change of one normalized unit along that direction produces a local response larger than one whitened noise unit. This is a scale comparison, not a hypothesis test or an operational threshold.

For a smooth diagnostic, add an artificial unit reference penalty in the declared bound coordinates. The data-resolution matrix is R = JᵀJ (I + JᵀJ)⁻¹, calculated by SVD. Its diagonal describes how much the data reduce each coordinate's unit reference variance while allowing the other quantities to vary. The reported reference sensitivity is sqrt(diag((I + JᵀJ)⁻¹)), expressed as a fraction of the declared bound and in physical units. An entirely unobserved quantity retains sensitivity one. This avoids falsely assigning a precise estimate to a null direction through a pseudoinverse. The reference penalty is not a calibrated prior or a confidence interval, and these values are not data-only marginal standard deviations.

The report preserves all four derivative calculations. It compares their resolution matrices and flags operator changes above 0.02. That tolerance is a provisional software diagnostic, never a scientific gate. Three times the largest inter-step Jacobian difference supplies a numerical noise proxy; adding and subtracting it from each singular value shows whether the unit-information classification is sensitive to that proxy. This is not a rigorous derivative-error bound. Agreement across the tested steps cannot establish convergence at every smaller step or validate the physical instrument model.

Reproduction and scope

All 18 independent trajectory checks pass. All 72 derivative calculations are preserved, with no resolution-stability failures. The largest resolution-matrix change is 0.000510, below the declared 0.02 diagnostic tolerance. All 384 per-parameter comparisons between successive GPS-error assumptions obey the expected information decrease, allowing a numerical tolerance of 1e-9. Twenty focused tests pass: twelve new diagnostic/persistence tests and eight preceding analytic fixture tests. The run used 18 starts and 1,170 predictions; there were no optimizer calls. Each task's interruption allowance was two starts.

The correlated-error cases retain one joint direction above the unit-information scale at two seconds; two or three at twenty seconds; and three at sixty seconds. Those directions can mix several parameters and do not establish separate calibration of each one. The millimetre toy cases have four, seven and seven/eight, respectively. In the sixty-second toy pitch case, eight singular values exceed one, but the observed derivative-noise proxy leaves seven/eight possible. That boundary ambiguity is retained despite the small change in the smooth resolution diagnostic.

The table gives the largest reference sensitivity across the four steps, as a fraction of the declared bound. Values near one mean nearly all of the artificial reference uncertainty remains. These are neither measured errors nor data-only confidence intervals.

Without rotation, gyro gain is exactly unobserved and its remaining reference fraction stays one, up to floating-point roundoff. Longer duration helps gyro bias; the tiny pitch correction still supplies almost no gyro-gain calibration under the correlated assumptions. Reference regularization also limits nuisance variation, so its smaller sensitivities must not replace the earlier unbounded joint-precision results. The main limitation here is useful information under the assumed errors, rather than an unstable derivative calculation. No covariance or instrument bound was selected or calibrated by this result.

The parameter inventory retains every parameter and step. The completed evidence includes hashes of inputs and source snapshots, the numerical environment, per-task receipts, an interruption journal and a compressed full report. export_inventory.py verifies those records and packages them without rerunning predictions. A completed worker must not be resumed. Revised source or assumptions require a new separately named freeze.

No original was rescanned or deleted. All 232 stored originals remain. No archived IMU fit, Earth-model fit, flight campaign or eligibility change ran. The estimator and earlier evidence stay frozen. Instrument-specific processing, mounting, calibration and clock support are still required by the empirical fitting gate; software fixtures cannot supply them. See the instrument evidence matrix.