A formal finite-size spectral-map hierarchy for free-fermion entanglement

Abstract

This record contains the preprint and the complete reproducibility package for “A formal finite-size spectral-map hierarchy for free-fermion entanglement”. The work studies the finite-size deformation of the spectral relation between the discrete prolate (Slepian) eigenvalue and the one-particle entanglement energy in a critical free-fermion chain. Using the exact DPSWF representation, Riccati/Dunham analysis and finite-part period calculus, an all-orders formal hierarchy is derived in which the finite-size coefficients satisfyFj=Pj(D)Φ∞F_j=P_j(D)\Phi_\infty, with PjP_j a rational function of the Euler operator D=t∂tD=t\partial_t. The work establishes universal denominator, degree and zero structures for these rational operators, including additional even-shift factors, and derives their Weber/Bernoulli leading behaviour. Independent checks include exact reconstruction of the half-filling Fisher–Hartwig phase polynomials Q20,Q40,Q60,Q80Q_{20},Q_{40},Q_{60},Q_{80}, exact Legendre reduction at the spectral centre, rational operator reconstruction through j=13j=13, and finite-matrix validation of the first two finite-size coefficients. The archive accompanying the manuscript contains the scripts, exact symbolic calculations, numerical data, regression tests and SHA-256 manifests required to reproduce the principal computational claims. The manuscript explicitly distinguishes the proved formal Dunham hierarchy from the still-open uniform physical connection/remainder theorem.