Theorem: Existence of finite operational wormhole certificates

For every $0<\epsilon <1$, there is a declared finite Enter model with a diary of dimension $16$, an independent entangled preparation, one Haar unitary, a threshold $k_{\ast }$, and a positive-residual cutoff $K$, such that a set of sampled unitaries of Haar measure at least $1/64$ has the following properties:

1. for every retained cut $k_{\ast }\le k\le K$,

   $$({V_k}^U,R_k,B_k,{\mathcal{D}}_k,\epsilon )$$

   is an operational wormhole certificate;

2. ${\mathcal{D}}_k$ uses one fixed decoder core and the nested discard policy, never physical $B_k$ or the test reference;

3. for one fixed pure input independent of the sampled unitary, late-radiation entropy rises and falls strictly within the three-cut retained window;

4. every append-only destructive Enter record has nondecreasing Shannon entropy;

5. the cutoff retains one physical vault qubit;

6. no second commuting, code-preserving representation of the same noncommutative diary algebra is produced;

7. terminal residual readout and forgetting leave the already returned accessible channel unchanged.

This is not a gravitational island, a quantum extremal surface, a replica wormhole, an Einstein–Rosen bridge, a Morris–Thorne throat, a gravitational evaporation Page curve, or spacetime emergence. The separately inherited analytic admission gates are not opened or discharged by this theorem.

Proof. Use the parameter choice in Corollary cor:tolerance. The complete prepared isometry is (eq:V), the accessible interface is $R_k=P_{k}E$, and the residual is $B_k$. The exact Haar calculation, single-cut probability estimate, and nested monotonicity give the complementary certificate.

Theorem thm:decoder supplies one fixed receiver core and full-diamond error at most $\epsilon$ throughout the window. Theorem thm:joint gives strict same-device entropy turnover and the positive cutoff. Proposition prop:record gives record monotonicity. Propositions prop:noduplicate and prop:drop give the last two assertions.

None of the declared data contains a spacetime metric, geometric throat, gravitational stress tensor, or causal crossing through such a throat. The excluded identifications are therefore not conclusions of the construction.