Finite-blocklength limits of the pure-loss bosonic channel

We determine the finite-blocklength asymptotics of classical communication over the pure-loss bosonic channel under a maximum-photon-number occupation constraint. Wilde, Renes and Guha previously derived a second-order achievable rate governed by thermal photon-number fluctuations and left its optimality open. We show that fixed-total-photon-number encoding removes input-number fluctuations while preserving the first-order capacity, leaving binomial photon-survival fluctuations as the operational second-order contribution. For fixed transmissivity 0 < η < 1, signal photon number NS > 0, and error probability 0 < ε < 1, we determine the optimal fixed-error expansion through third order. The operational dispersion is Vsh = η(1 − η)NS [log2(1 + 1/(ηNS))]^2, and the optimal code size contains the third-order correction −(1/2)log2 n up to a bounded O(1) remainder. The achievability construction uses fixed-photon-number shell codes, while the converse applies to arbitrary multimode quantum encodings and arbitrary joint decoding measurements within the stated occupation constraint. This Zenodo record contains the preprint, LaTeX source, vector figures, source data, and reproducibility code used for the numerical checks and figure generation.