Manuscript Oai265-A@2 · Oai265-A.tex · section · c0a34a45

The five obligations

Stage 1: audit of the five proof interfaces › The five obligations · lines 399–450

The supplied proof description has the following architecture. The audit column states what must actually be verified before a stage can be used.

Stage	Supplied description	Required interface and audit disposition
Stage	Supplied description	Required interface and audit disposition
1	A subvolume exponent from the global gap.	A box estimate, with the original cut and clearance from $Z_{A}$ retained; uniformity in induced-domain holes and shape; treatment of zero marginal eigenvalues; all optimization thresholds. Not established by a gap inequality alone in this manuscript.
2	Positive constraints with graph-local tails.	Positivity, annihilation of $Ω$ , a global comparison gap, summability, tail bounds in $d_{Λ}$ , and all interaction multiplicities. A nonlocal spectral projector is not a substitute.
3	Replica metrics and conditional-information transport.	Exact replica identities, support conventions, quantitative logarithmic norm bounds, justified limits, and dimension-independent constants where asserted. An identity for moments is not the transport estimate.
4	Collar amplification.	A precise buffer geometry, admissible collar widths, and a bound on mutual information or conditional mutual information with all processed regions and spectators retained. A collar is not a horizon.
5	Two ordered families and entropy cancellation.	An actual partition with $O (∣ \partial A ∣)$ exceptional sites and summable information charges. The cancellation implication is proved below; existence and quantitative pricing of the required partitions and bounds remain external obligations.

In particular, one must not replace the first stage by an area law for arbitrary regions, introduce a restricted-Hamiltonian gap in stage 2, or replace an unconditional information inequality by a normalized postselected branch.