Abstract

We develop a self-contained certificate framework for the external Family 272 claims while keeping the explicit dimension-ten geometric construction outside the verified release. For a precisely specified local-instrument model, we prove the common-factor form of Eve's complete-transcript output. A four-effect inequality then yields a uniform unhalved trace-norm distance of at least $1/5$ from every ideal secret bit for states represented by two completely positive, completely copositive local maps. The final numerical implication has an exact rational certificate, independent of copy number, communication length, and local processing cost.

We separately prove the Choi transfer identity, the product-free range criterion, and the conversion of a PPT pair into a trace-preserving PPT channel in dimension $2d+1$. We give a terminating, promise-dependent finite procedure producing an entanglement witness from an exact rational product-free support, and derive a complete-record statistical certificate for a specified witness measurement. These are conditional construction and verification interfaces, not an execution of the external dimension-ten arithmetic.

Several corrections are essential. Restricting an already excluded protocol class preserves the obstruction. Product-free range is sufficient for entanglement, not necessary for the zero-key argument. A Bell-overlap recovery test cannot detect PPT entanglement. The full channel Choi matrix need not have product-free range even when its compressed corner does. The inspected Comparator targets contain sorry; their statements are not machine-checked proofs, and neither target formalizes the secret-key theorem. The supplementary integer checks concern a separate combinatorial construction, not every argument in the paper.

The completed release is the local certificate and reduction package. It does not independently establish the external matrices' required geometric properties, minimal dimensions, or a gravitational interpretation. No Page curve, island, gravity-generated-entanglement test, or exclusion of classical gravity is constructed.