Joint IMU/GPS checks with correlated position errors

The existing joint IMU/GPS estimator now has a separate deterministic check of how correlated position errors affect calibration and timing. Five analytic fixtures are evaluated under all thirteen previously declared GPS-error assumptions. The fixtures have known motion and sensor errors; they are not simulations of complete flights or fits to the archived IMU data. The two structural offset/drift checks are separate.

All 65 cases completed with their expected outcomes: 52 individually identifiable cases recover the known value, while all thirteen orientation/bias cases retain an unresolved direction. Both structural checks pass. The completed run used 65 control starts, 1,309 fit residual evaluations, 247 additional precision predictions and three structural predictions. No failed cases were replaced or selected away.

Recovery and useful precision are different

An IMU measures angular increments and specific force. The estimator integrates those measurements and compares its predicted antenna positions with GPS coordinates. These controls vary acceleration bias, effective initial orientation, accelerometer gain and GPS/IMU timing individually. A fifth fixture leaves orientation and two acceleration bias components free together, where the observations cannot resolve all three.

Both recoverability and precision are reported. Recovering a known value from noiseless positions does not establish that the value can be measured accurately with noisy data. The local precision calculation propagates the complete assumed GPS covariance through the nuisance Jacobian. It preserves correlations across times and axes. Marginal uncertainties are not assigned when the Jacobian has an unresolved direction.

The tests use two-second translation fixtures with 200 native IMU intervals and seven actual GPS epochs. The baseline one-millimetre position uncertainty and cross-axis terms are software fixtures, not receiver or instrument specifications. The twelve added cases use the previous 1/3, 3/10 and 10/30 metre horizontal/vertical scales, each with independent noise or 10/60/300-second correlation, plus common offsets and endpoint drift. Those scales are assumptions, not independently supported error bounds. The label formal_only here refers to the toy baseline, not the archived position solutions.

Only the covariance changes between the thirteen cases; native IMU increments, GPS coordinates, timestamps, parameter bounds and truth stay fixed. This tests correlated weighting without interpolating measurements or smoothing away brief motion. All fits use the existing frozen estimator, which is unchanged. The chosen analytic coordinate metric is declared explicitly and is used only for these controls, never to infer Earth's shape or assign a different observation covariance to each scientific candidate.

For illustration, the following local standard deviations compare the millimetre toy baseline with the smallest 60-second correlated case, including its offset and drift. Each parameter varies alone, with the other quantities fixed.

The noiseless fitter recovers these parameters in every one of the thirteen covariance cases. Nevertheless, each illustrated correlated uncertainty exceeds that parameter's declared control bound. These are unbounded local linearizations, not bounded intervals; extrapolation outside the bounds indicates weak information, not a calibrated error bar. Two-second recovery cannot establish useful archival calibration precision. Longer records, appropriate motion excitation and independent instrument constraints require separate checks.

Position and velocity have structural ambiguities

A one-metre GPS offset has the same observation response as a one-metre initial-position shift in the translation fixture. A linear GPS drift has the same response as an initial-velocity shift. Both checks compare the independently specified response with the IMU/GPS predictor to within 1e-8 metres. Weighting correlated errors correctly does not create information that separates these quantities. Independent constraints or explicit nuisance treatment are necessary.

These are specific analytic controls. They do not prove universal ambiguity over every aircraft maneuver, establish IMU mounting independently, or validate onboard corrections. The effective orientation and mounting remain inseparable as independent parameters.

Reproducibility and limits

Implementation: control module, worker, and focused tests. Thirty-three distinct focused tests pass: eight new control/persistence tests, ten preceding GPS sensitivity tests and fifteen estimator tests. They verify full-covariance whitening, an independently calculated bias Jacobian/uncertainty, unchanged observations, unresolved marginal precision, offset/drift ambiguities and interrupted/corrupted journals.

Each fit allows at most 160 residual evaluations, including the estimator's numerical derivatives. Local precision uses at most seven additional predictor calls per case. All counts are preserved. A separate append-only, fsynced control journal charges interrupted starts before work; each of the 65 cases permits at most two starts. This does not consume the empirical six-file/24-start Earth-fit allowance.

The run uses data/ilvis0-correlated-controls-20261007/, with its own frozen source/input/ environment manifest, source snapshots, atomic per-case results and hash checks. The compact handoff includes the full compressed summary, start journal and parameter inventory. export_inventory.py verifies the evidence without repeating fits. Do not resume the completed worker or revise its sources under this manifest. Every earlier completed study stays frozen.

These results do not establish archive calibration limits, physical integration/latency semantics, or whether Group4 retains Earth rate. The short translation fixtures do not characterize gyro performance or general aircraft rotations. All scientific decisions remain abstentions. No archived observation, original file or scientific gate was changed; all 232 originals remain and no empirical Earth-model fit ran.