Bounding receiver-trajectory approximation error

All 13 deterministic derivative-bound checks pass. The existing derivative adapter remains unchanged. The new helper adds an explicit truncation/error budget around its position, velocity and acceleration estimates; it does not infer that budget from small fitted residuals.

Receiver positions are noisy and a local polynomial is only an approximation. Propagating the coordinate covariance accounts for the declared random errors, but not the trajectory structure omitted by that polynomial. This distinction matters because acceleration involves differentiating positions twice.

A declared derivative envelope

For a degree-\(p\) polynomial centered at \(t_0\), suppose each coordinate's \((p+1)\)-th derivative is bounded in magnitude by \(M_j\) throughout its support. Taylor's theorem bounds the omitted coordinate contribution at epoch \(t_i\) by

If \(H\) is the saved linear operator estimating the nine position/rate/ acceleration components, the corresponding error box is bounded by \(|H|r\). An explicit coordinate measurement-error box \(e\) contributes \(|H|e\). The helper also reports the operator's polynomial-reproduction defect, using declared bounds on lower derivatives; small numerical defects are not silently set to zero.

All cross-coordinate weights are retained. Latitude/longitude remain in radians and height in metres, with corresponding derivative units. The helper does not mix these units into a single norm. Longitude must be consistently unwrapped within the supported coordinate chart.

Verification and limits

Independent polynomial and trigonometric trajectories check containment of known derivative errors. Other controls cover irregular times, sharp coordinate error boxes, cross-axis weights, an intentionally perturbed operator, invalid envelopes and extrapolation. These are inexpensive software checks, not new airborne fits.

The derivative envelope is an assumption to be justified or stress-tested, not a measured bound established by this helper. A coordinate-error box is not automatically a covariance standard deviation or a calibrated confidence region. These deterministic bounds do not shrink with an assumed number of independent samples. Computations retain the reproduction defect but do not constitute a formal floating-point interval proof.

The next step combines these derivative errors with nonlinear force response, time support, increment quantization, within-packet averaging and calibration. A wide or overlapping result is retained as uninformative; the error assumptions cannot be tightened merely to create model separation.

Implementation · Checks · Derivative/covariance adapter · Roadmap.