Theorem: Hawk04-A mathematical release

The following statements are proved:

1. complete unitary evolution preserves a finite code globally, with an explicit CPTP left inverse and all references retained;

2. exact recovery from one output is equivalent to a constant complete complementary channel, with the decoder (eq:exact-decoder);

3. exact no-hiding applies to full-channel bleaching, not arbitrary partial decoherence or incomplete environmental descriptions;

4. finite Choi and decoder certificates have the explicit dimension factors and complete-channel allowances of Theorem thm:certificate;

5. the regulated quartic SYK–scalar Hamiltonian has well-defined complete unitary dynamics and the finite-time regulator (eq:regulator);

6. one frozen four-Majorana, nonzero-coupling realization has the exact radiation-recovery obstruction (eq:exact-negative), despite exact global preservation;

7. a positive finite certificate can extend to a finite interval under Proposition prop:interval, but no such positive interacting certificate is reported here;

8. endpoint continuity requires the stated access-preserving identifications, and fixed finite proxies recur.

The gravitational implication is precisely the conditional theorem thm:BH-conditional, not an established physical evaporation solution.

Proof. The statements follow from the named theorems and propositions. Every microscopic object used by those proofs is specified in this source. The physical conclusion requires additional objects absent from the microscopic construction, as listed in Theorem thm:BH-conditional.