Bound the complete calibration relationship

The completed map diagnostic found that all 36 tested linear extremes fail when their 24 error terms are recomputed from one physical calibration. The next V01 method must preserve that relationship, rather than add more matrix-only cuts.

Implementation sequence

1. Implement interval values and all 27 first derivatives for scalar arithmetic and a skew-matrix exponential. Bound the exponential polynomial's value and derivative tails separately. Test signs, reciprocal domains and independent trigonometric references.
2. Use those components to enclose the complete 24-component static calibration map and its derivatives on a declared physical box. Preserve source bindings, normalization scales, zero-error conventions and explicit numerical allowances. The center is a fixed box center, never an opened known-truth calibration.
3. Convert the derivative enclosures into necessary affine relationships between the existing physical and common-error coordinates. Keep all coordinates; quantify interval and export error. Subdivision may tighten a bound but cannot drop untested physical regions.
4. Qualify this construction against independent generated controls before a separately frozen complete six-group/orientation prediction comparison.

The initial primitive tests are ordinary local checks, not a launched numerical study. A finite matrix of source-map evaluations, probes, bounds, LP calls and threads must be recorded before the new study launches. This plan creates no native, observed-data or retry allowance. Completed studies remain unchanged.

Mean-value relationships require valid derivative enclosures throughout the declared box. A locally fitted Jacobian, sampled extrema or a feasible nonlinear point is insufficient. Real-arithmetic bounds, floating-point treatment, prediction usefulness and calibrated observational coverage remain distinct.

First source-map qualification allowance

Use the six preserved generated candidate/processing cases from the completed joint-product prediction study. Cross them with four predeclared domains: the full original normalized box, orientation sector 4, a fixed point at 0.1, and a box with the first accelerometer gain fixed at its upper boundary. Each source reference retains every parameter and derives its own unchanged radii. Do not open the generated truth calibration to construct a bound or center.

There are 24 identities, 24 source setups and 48 interval-map evaluations: one complete-box evaluation and one at its fixed midpoint per identity. Full, sector and gain-boundary cases each use 64 probes (both ends, midpoint, then 61 independently seeded points). Fixed domains need only one probe. Total: 1,158 oracle evaluations and 1,158 separate analytic Jacobians.

The analytic reference uses explicit calculus and SciPy Frechet derivatives, not the interval differentiation implementation. For each probe, verify all 24 values, all 24×27 derivatives and all 48 exported coupling inequalities. This is 48 value endpoint inequalities, 1,296 derivative endpoint inequalities and 48 coupled inequalities: 1,611,936 scalar inequalities total. Perform coupled checks with exact rational arithmetic on the stored binary inputs.

The declared source-value numerical allowance is 3e-10 per normalized error coordinate. Analytic derivative comparisons allow 3e-8 in normalized derivative units. That comparison tolerance does not enlarge the exported coupling rows or supply their gradient bounds. Interval differentiation and analytic tails provide the mathematical enclosures; finite agreement checks do not prove all floating-point behavior. Zero-radius source invariants must be tested, never replaced by artificial denominators. These are numerical assumptions, not sensor-accuracy claims.

Freeze sources, inputs, environment, seeds and the finite per-identity counters. Reserve every setup, bound, oracle/Jacobian pair and inequality batch before execution. Four workers use one numerical thread each, with a 300-second cooperative wall ceiling. No retries, LP calls, new native predictions, original-aircraft reads or observed fits. Preserve every failed probe. All 24 identities must pass before a new prediction-support comparison can follow. Observational coverage and useful full-record widths remain separate gates.