A direct remainder bound for the rotating force source
The trace identifies the source and its accumulated residual velocity as the next targets. Preserve the completed traces, including both time-limit failures. Do not shrink calibration, timing or observation-error assumptions.
Mathematical construction
The source has two pieces: A = R f and B = R (P - I) f. Here R is the
mount rotation, P the product of corrected gyro rotations up to the query,
and f the initial corrected specific force. All depend on the same calibration.
For each gyro rotation argument, independently bound its change d_i and its
first-order remainder r_i around the declared reference. Orthogonal rotation
products give a conservative prefix change at most min(2, sum(d_i)) and a
prefix remainder at most sum(r_i) + sum(d_i)**2 / 2. This follows from the
first- and second-derivative operator bounds for products of skew exponentials;
it must be checked on independent noncommuting-rotation controls.
For B, let s bound the norm of the fixed initial mounting matrix, d0
bound the nominal P - I, and f0 bound the nominal force. Let dR, dP,
df be change bounds, and rR, rP, rf be first-order remainders.
A conservative remainder about the ideal reference derivative is:
rR*d0*f0 + s*rP*f0 + s*d0*rf
+ dR*dP*f0 + dR*d0*df + s*dP*df + dR*dP*df
For A, the corresponding bound is rR*f0 + s*rf + dR*df.
These are norm bounds, not an assumption of independent calibration errors.
Include the measured numerical mounting-matrix defect; do not assume exact
orthogonality merely because a matrix is nearly a rotation.
Required implementation checks
- Establish exact rational change/remainder bounds for the bias/gain calibration map with strictly positive denominators. Include the query's clipped packet fractions, both boundary sides and the preserved reference derivative convention. Do not treat increments as instantaneous rates.
- Qualify the rotation-product and product-remainder inequalities independently, including zero motion, noncommuting axes, nonzero references, changing gains, clock-boundary crossings and deliberately conservative domains.
- Reconcile the ideal reference value and derivative with the actual supplied numerical center and all 27 rows. A certified point-derivative enclosure is needed; a floating Jacobian agreement tolerance is not an error bound.
- Intersect the new certified radius with existing valid radii only after source, reference and domain checks. Then require all native controls and a finite full-history usefulness screen before a complete comparison.
Each numerical study needs a new finite freeze. Component tests do not qualify recording-scale uncertainty or whole-program floating execution. Keep all six model/processing groups in eventual qualification and use training observations alone to constrain calibration. No aircraft or protected ESA work follows from this plan alone.