Theorem: Oai212-A assembly

For the models and interfaces specified in this paper:

1. The flag-controlled CNOT circuit has an exact realization-wise squared-commutator identity with undirected FPP, and an exact complete-channel identity against its declared lift.

2. The Poisson random-Clifford circuit has the support-history allowance $2\lambda Mt/(q^2+1)$ and an FPP macroscopic limit under the stated growing-dimension scaling. No fixed-dimension or full-payload-channel regularity transfer is proved.

3. The independent-pair quench has exact exponential local Rényi-2 costs. Its declared dual membrane optimization is FPP. Its finite-dimensional approximations have the stated global preparation and optimization allowances.

4. The monitored quench has an independent zero/exponential mixture cost law. For $p>0$ the exact positive exponential and Gamma interfaces fail.

5. The advertised-service serial teleportation model has additive FPP route cost and reference-preserving quantum transport. Concurrent generation does not obey this additive law even on a two-link chain.

6. Any asymptotically identical passage-time interface satisfying Theorem thm:transfer has the target macroscopic body. Strict convexity and $C^1$ regularity are transferred only conditionally on the corresponding external regularity assertion.

7. None of these results constructs the inherited characteristic quantum coefficient or discharges a gravitational admission gate.

Proof. Assertions (i)–(v) are respectively Theorem thm:exact-otoc and Corollary cor:cnotchannel, Theorem thm:clifford-coupling and Corollary cor:largeq, Theorems thm:quench–thm:membrane and Proposition prop:truncation, Theorem thm:measured and Proposition prop:zero-tv, and Theorem thm:serial with Proposition prop:parallel. Assertion (vi) is Theorem thm:transfer and Corollary cor:regularity. Section sec:analytic identifies the different domain and the unconstructed analytic objects, proving the release boundary in (vii).