Theorem: Hawk03-A mathematical release

On the finite domains specified in this manuscript:

1. the independent-factor entropy benchmark and conditional fresh-pair growth bound are proved;

2. the sharp two-purifier projection bound is proved;

3. the exact complete-complement recovery criterion and half-diamond information–disturbance consequences are established;

4. the fixed unitary benchmark has a complete emission interface, a state-independent encoded partner algebra, exact message recovery, and exact prepared-state pair-proxy instrument statistics;

5. two commuting copies cannot exactly intertwine the same noncommutative logical factor, with the quantitative bound (eq:commutatorbound);

6. exterior purifier distillation and a proxy-destroying unitary are calculated explicitly;

7. comparison, instrument, and complete-output errors propagate according to Theorem thm:physical.

The finite mathematical ending is Constructed. The independent-copy requirement is Obstructed under Theorem thm:commuting's hypotheses. The physical firewall and infalling-diary targets are Unresolved: Requirements req:horizon and req:diary are not discharged.

Proof. Items 1 and 2 are Theorem thm:amps, Corollary cor:pairgrowth, and Theorem thm:twopurifiers. Item 3 is Theorems thm:exact and thm:approx. Item 4 is Theorems thm:modelchannels, thm:partner, and thm:compatibility. Item 5 is Theorem thm:commuting. Item 6 is Propositions prop:distill and prop:unitarydeparture. Item 7 is the comparison lemmas and Theorem thm:physical.

Every constructed object is finite dimensional and has the explicit domain stated above. No metric, continuum field-state construction, physical crossing trajectory, or controlled gravitational backreaction is supplied. Accordingly the physical requirements remain unresolved.