Derivative and integration precision at fixed endpoints

The preceding six-endpoint diagnostic found identical repeated predictions and near-identical value/tangent outputs, while several full local corrections failed stringent stopping checks. This study separates arithmetic precision from changes in physical assumptions or fit selection.

Fixed scope

Reuse all six predetermined diagnostic identities and their original whole stretches. Each identity receives fourteen native integration evaluations, for 84 maximum overall, with zero optimizer starts and no retries. Four numerical threads are the total allocation. The old kernels, fits, bounds and result labels stay frozen.

An additional diagnostic kernel uses platform long double internally in the same automatic-derivative equations. The input/output ABI stays float64. The original float64 packet-clock accumulation and timestamp branch choices are retained; no sample is dropped, smoothed or interpolated. The actual mantissa width, original and generated source hashes, compiler flags and binary hash are recorded. A platform without wider arithmetic cannot run this comparison.

Evaluations per identity

1. One original and one extended-arithmetic prediction/derivative at the saved center.
2. Both implementations at the same frozen correction with fractions 1, 1/2, 1/4, 1/8 and 1/16: ten more evaluations.
3. A bounded affine correction calculated from the extended center's residual and derivatives, evaluated once with each implementation: two more evaluations.

The affine calculation allows at most 100 linear iterations and does not optimize the recorded data. No extended endpoint replaces an existing fitted result.

Measurements and controls

Record central coordinate and derivative differences, actual versus locally predicted motion/cost, and unchanged stationarity and stability-check quantities. Comparisons use identical inputs and feasible parameters under each arithmetic choice; only internal numerical representation changes. Separate wider-arithmetic corrections from the frozen-direction comparisons.

Known-answer software checks precede archival evaluations: exact straight-disc motion tests accumulation error and an analytic velocity derivative; finite differences test every active derivative for all three models and both processing cases. These controls are numerical checks, not new Earth-model simulations or statistical calibration campaigns.

The source, input, environment and original-storage hashes are frozen before any native call. Extraction separately verifies decoded-measurement provenance. Calls are charged durably before execution, and an exclusive lock prevents duplicate workers. Interrupted charged identities cannot retry. Completed work cannot resume.

The result can qualify a subsequent numerical proposal or reject a precision hypothesis. It does not establish a global optimum, weaken stopping tolerances, change the 30/0/23 findings, or calibrate a scientific decision.