Abstract

We complete a source-qualified audit programme for the external OpenAI Family 265 area-law and polynomial-PEPS claims without using either claim as a theorem premise. The frozen class consists of finite induced square-lattice domains, bounded finite-range interactions, a unique ground state, and a positive full-system spectral gap. We prove the boundary conversions, an independently verified two-family entropy-cancellation lemma, and a quantitative terminal certificate that identifies exactly which geometric and information-theoretic constants would price a uniform area-law constant. The missing estimates are not supplied by a decoder resource ledger or a channel error budget.

We distinguish PEPS representability from tensor production, contraction, and physical preparation. Explicit models show that dropping uniqueness permits volume-law ground vectors even with a unit gap above the ground space, that a uniform full-system gap does not imply a uniform restricted-Hamiltonian gap, and that an entropy upper bound does not imply a strict area-law equality. We also correct two overstrong exclusions: complementary mutual information and Rényi entropies of order greater than one do have elementary bounds conditional on a von Neumann area law.

Locally proved channel counterexamples reproduce the separation between selected entropy and reference-preserving recovery: complete isometries can agree on every pure-input accessible spectrum while their optimal recovery errors are $0$ and $1-d^{-2}$. Uniformly gapped bond-dimension-two instantaneous ground states permit both turnover and no turnover; their preparation dynamics are separate data. A finite qubit-memory certificate is included only as a statistical comparison, with an explicit procedure-dependent forty-shot nonacceptance result. All stages receive a proved result, an obstruction, or an exact unresolved disposition. The single active successor is quantitative constant extraction within the frozen class. Family 265 remains unverified here, and no Page curve, gravitational dictionary, or inherited quantum-coefficient admission is constructed.