Global bounds for calibration and rotating-force increments
This study addresses two actual nonlinear components of the native prediction map: reciprocal bias/gain correction and finite rotation of measured velocity increments. It is a prerequisite for a native global error bound, not that bound in its entirety. No observed data or fitted Earth-model outcomes are used.
Exact mathematical bounds
For one corrected component, let q = (raw - bias*dt)/(1+gain). At a supplied
center write A = raw - bias0*dt and d = 1+gain0. If the complete box has
1+gain >= dmin > 0, with maximum center displacements B and G, the first-order
remainder is bounded by
abs(A)*G²/(d²*dmin) + dt*B*G/(d*dmin).
Compute corrected-value extrema at the bias/gain rectangle's corners and combine all three axes conservatively in Euclidean norm. These are bounds for the entire declared box, not maxima inferred from test probes. Preserve signed increments.
For the skew-matrix exponential, the derivative norm is at most one and the
second-order Taylor coefficient at most one half. For its velocity integral
J = integral_0^1 Exp(u*theta) du, the corresponding constants are one half and
one sixth; for the position integral
K = integral_0^1 (1-u)*Exp(u*theta) du, they are one sixth and one twenty-fourth.
The value norms are at most one, one and one half for rotation, J and K.
These follow from differentiating the exponential integral, with orthogonal
factors having norm one. They hold beyond a small-angle neighborhood.
For M(theta)*dv, propagate both actual parameter changes and the nonlinear
remainders of corrected theta and dv. Include their product term. A bound
that omits that term is not a bound for joint calibration uncertainty.
Propagate rotation, velocity and position change/remainder radii through a fixed-duration body-frame preintegration. Shared calibration across packets is allowed: triangle inequalities do not assume independent packet errors. Norm bounds may be conservative and are not statistical confidence intervals.
Qualification
Thirty fixed software identities:
- Twelve three-axis calibration boxes: angle/velocity increments, centers at lower/zero/upper bounds, and narrow/broad calibration. Evaluate all 64 corners of each six-dimensional box.
- Twelve rotating-force cases: rotation/J/K, small/large center rotation, and narrow/broad perturbations. Sixty-four fixed joint angle/velocity probes each.
- Six fixed-time histories: 1, 7 and 31 packets under narrow/broad calibration. Sixty-four shared twelve-parameter probes each.
Use a separate nine-dimensional block-matrix exponential and its Frechet derivative as the value/tangent reference for rotation/J/K. History references propagate those matrices and directional derivatives independently. Compare with the repository's existing within-packet/preintegration implementation; never generate a bound from a largest measured discrepancy.
Require each actual change and Taylor remainder to lie within the analytic
radius, allowing 1e-12*(1 + norm(reference) + norm(linear_term) + radius) for
the numerical comparison. This is a qualification tolerance, not a proof of a
whole-record floating-point rounding envelope. Preserve failures without retries.
Limits: 30 identities, 1,920 probes, at most 13,000 software packet/reference evaluations, 300 cooperative seconds, one worker/four threads, no original reads, optimizer starts, native-engine evaluations or retries. Charge before execution and preserve sources, inputs, environment, generated parameters and all results.
The task list reserves 12,726 reference/value evaluations before execution. Center derivatives are reused across probes. Each identity's complete allowance is charged up front; interruption does not refund its unused portion.
Remaining native terms
These bounds do not yet cover candidate-dependent transport, gravity, geographic coordinate conversion, lever arms, rotation-removal processing or uncertain receiver timing across packet boundaries. Nor do they establish a machine- roundoff bound for long integrations. The prediction interface must still return unsupported when any required global term is missing. No archive prediction fit or empirical coverage claim follows from passing these component controls.