Half-integer corrections in finite Cayley–Laguerre weak compressions

We study a finite family of odd sine packets transported through a Cayley rational approximation and compressed against a fixed weak translation multiplier. The usual low-frequency calculation produces an even expansion in the mesh parameter ε = M⁻¹. We show that this is not the complete asymptotic expansion when the translation front c belongs to (0,1): a bulk saddle contributes an oscillatory rank-one term of order ε⁷ᐟ², followed by a term of order ε⁹ᐟ². An exact Laurent constant-term identity separates the residue sector, which carries the even coefficients, from the finite-section factors z±M, which carry the half-integer terms. In the exact local coordinate z = exp(εw), the coalescing poles become fixed and the complete local one-form is analytic in ε², which makes the parity separation rigorous. The saddle phase is identified exactly with the classical Plancherel–Rotach phase of L_M⁽⁻¹⁾(4cM). We give the first two half-integer coefficients in closed form and record a computer-assisted interval evaluation for an 80-mode example. The result concerns a finite-section transfer mechanism; it does not claim a new general Laguerre asymptotic or any result on the Riemann hypothesis. This preprint record contains the article PDF, the complete LaTeX source archive, and SHA-256 checksums. The companion reproducibility software and audit archive is available at https://doi.org/10.5281/zenodo.22198929. OpenAI ChatGPT and Codex assisted with preparation; Julien Lange is the sole author and assumes responsibility.