Abstract

We give a typed certification framework for a declared finite boundary channel and a specified deterministic decoder, without assuming that quantum corrections are perturbatively small. The framework distinguishes recovery of an unknown quantum code, reconstruction of a named bounded observable, and consistency with a separately declared geometric model.

For a complete finite isometry, we prove the constant-complement criterion, its residual-rank consequence, observable-specific certificates, and dimension-explicit approximate recovery bounds. If a confidence event supplies a normalized decoded Choi-overlap interval $F_L\le F_e\le F_U$, then an identified target channel has half-diamond error in

where $d$ is the logical dimension and $b$ is a separately justified half-diamond transfer allowance. Complete-complement and accessible-state witnesses give lower bounds applying to every decoder, whereas the Choi certificate concerns the specified decoder.

A separate operator-algebra module presents, with attribution and a local proof, the bounded modular-band witness statement of the supplied OpenAI manuscript dated 23 September 2026. We derive its one-way consequence for bounded asymptotic defect tests. This existential modular statement is neither a finite-sample channel certificate nor a bulk–boundary correspondence.

Two defects in the proposed synthesis are repaired. Nonconstant complements obstruct exact full-code recovery, but need not obstruct a named observable. Failure of a sufficient certificate is not an obstruction and does not imply failure of a geometric description. The resulting decision procedure has certified, obstructed, and inconclusive outcomes. The inherited characteristic-map obligations remain documentary open gates: no joint map, quantum coefficient, geometric emergence theorem, or gravitational endpoint is claimed.