Abstract

We complete the proposed finite-certificate programme for one specified generalized amplitude-damping channel, without assuming the external Family 276 repeated-use or arbitrary-partner additivity claims. We first prove the one-use Holevo formula directly from qubit output determinants, entropy concavity, and a scalar convexity argument. For the genuinely thermal, nonunital parameter pair

we specify the unique maximizing population as an isolated zero of an explicit strictly decreasing derivative and prove

The proof uses finite rational sums with rigorous tails, not floating-point optimization. Both the exact maximizing phase pair and a fully algebraic near-optimal pair are written out.

The remaining stages audit the block-entropy, thermal-extension, channel-substitution, ensemble, and partner-channel interfaces; price specified implementation and statistical comparisons; and separate one-use information, asymptotic classical transmission, quantum recovery, and secrecy. The inspected Comparator target contains a proof placeholder, so its scope description is not treated as an executed formal proof. The finite certificate does not verify the complete external capacity theorem, prove arbitrary-partner additivity, or establish optimality of separate measurements. No Page curve, gravitational test, quantum-capacity formula, secret-key formula, or spacetime emergence is inferred.