Theorem: Oai211-A operational assembly

For the independently defined marked conditioned tree satisfying (eq:offspring), the following claims are proved:

1. fixed offspring-bounded scalar labels satisfy the uniform path law (eq:pathmain);

2. fixed finite-dimensional channel marks define a CPTP spine average and a path-average operation with the finite-size and asymptotic half-diamond bounds of Theorem thm:channel;

3. the designed weak depolarizing serial family has the exact compositions, uniform approximation, and random excursion profile of Theorem thm:serial;

4. a declared selected-path qubit experiment satisfying (eq:calibration) has the complete confidence guarantee (eq:finitecertificate);

5. under the weighted block and endpoint contracts, these statements transfer to the exact $N=2n+1$ enriched map model, with degree-sampling transport (eq:degreechannel).

These statements certify only their named operations. They do not determine a bulk–boundary map, a complete complement for an unspecified physical code, an entanglement area law, a Page curve, or a gravitational action.

Acceptance, exclusion of the specified channel at a threshold, and an inconclusive outcome are separate decisions. The supplied FK application remains conditional as stated in Corollary cor:FK.

Proof. Items 1–5 are, respectively, Theorem thm:path, Theorem thm:channel, Theorem thm:serial, Theorem thm:experiment, and Theorem thm:blockrepresentation together with Proposition prop:corners. The nonidentification claims follow from Definition def:completion and Proposition prop:goodbad, and from the distinct types of objects required for those physical interpretations.

If the upper bound in (eq:finitecertificate) is at most $\epsilon$, the specified implemented channel is certified. If its lower bound is greater than $\epsilon$, that channel is excluded at the threshold; another decoder has not thereby been excluded. If neither inequality separates the threshold, the certificate is inconclusive.