What is completed

The intended next theorem has a finite terminal form:

Theorems thm:transcript, thm:bit-gap, and thm:rational-gap prove this implication with its model and normalization exposed. The final rational certificate is (eq:rational-certificate).

This does not require a claim that the supplementary arithmetic appendix proves the entire paper. It also does not require simulation, a tomography interval, or a decoder-memory bound.

For the source's specified dimension-ten maps:

1. pin the exact source version and complete input data;

2. verify their CP and complete-copositivity certificates;

3. verify $Z=J({\Phi }_2{\Phi }_1)\ne 0$;

4. verify the complete product-free range argument, including every required arithmetic and geometric implication;

5. record which checks were proved analytically, recomputed exactly, or left unexecuted.

Only after these tasks may Corollary cor:state and Theorem thm:completion be instantiated as independently verified claims about those particular matrices.

This is one application-verification handoff, not a simultaneous opening of dimension minimization, new protocol classes, and failure of eventual entanglement breaking.