Validated quadrature and residual records

A position certificate must contain:

• dimensions, bins, domain endpoints, and exact quadrature weights;

• the identified finite preparation and its error to the target;

• cellwise bounds for $G_d^{\prime \prime }$, quadrature totals, and arithmetic errors including wavefunction evaluation;

• numerical centers $A_k$, positivity shifts, total $\eta$, completed effects, and the outside effect;

• backend, precision, outward-rounding conventions, and scalar readout errors;

• target error, certified upper bound, and pass/fail outcome.

A residual certificate must contain:

• exact model data or certified enclosures for those data;

• time endpoints and concrete Hermitian polynomial coefficients;

• matching conditions or charged jump bounds;

• initial-error enclosure and whole-cell residual enclosures;

• each $B_k$, the final partial-cell contribution, and total $E$;

• survival-error budget, physicality status, and any charged repair;

• backend, precision, target accuracy, and pass/fail outcome.

Changing $d$, $L$, quadrature cell widths, time-cell widths, polynomial degree, and arithmetic precision addresses different terms. A failed or overly large certificate must remain visible. Ordinary agreement between two solvers does not replace these records.