Abstract

We develop the four proposed quantum-to-first-passage interfaces without treating an external catalogue entry as a verified theorem. For an explicitly controlled CNOT circuit, we prove a realization-wise identity between a normalized squared commutator and exponential first-passage infection on the undirected square lattice. For a Poisson random-Clifford circuit, we derive a finite-volume comparison with the same infection process, including its dependence on volume, time, and local dimension. An independent-pair quench realizes independent exponential local Rényi-2 entanglement costs; its declared membrane optimization is exactly first-passage percolation, and finite-dimensional truncations have explicit preparation and optimization errors. Local measurements produce an atom at zero and therefore invalidate an exact exponential reduction. An advertised-service, serial-teleportation network has additive first-passage route costs, whereas concurrent link generation already violates additivity on a two-link chain.

The macroscopic transfer theorems are proved from the standard first-passage shape theorem. Strict convexity and boundary differentiability remain conditional on independent verification of the corresponding claims in manuscript 212-02. These constructions do not establish a generic entanglement-growth membrane, a measurement-induced critical theory, or optimal routing in an unrestricted repeater network. They do not construct the separate characteristic quantum coefficient discussed in the supplied Cheung manuscripts. Thus the requested unconditional completion of the entire programme is not claimed.