Finite-chain spectral drift and an exact normalized-current correction

We study the smallest singular frequency of a finite positive bidiagonal chain whose diagonal entries form an arithmetic progression. Its determinant is a rising factorial, but its smallest singular value is not proportional to that product at finite coupling. An explicit adjugate representation proves that their ratio decreases strictly with the spectral parameter for arbitrary fixed positive off-diagonal entries. We derive the leading large-coupling correction and distinguish the frequency from an exact static endpoint transfer scale. For the two-site chain, we construct a single homogeneous normalized-current interaction that restores exact proportionality on a finite regular band. The complete variation supplies both a self-consistent edge and a scalar frequency offset. The interaction has positive current curvature and admits a convex-conjugate representation; it therefore cannot arise solely by minimizing stable variables coupled affinely to that current. A positive squared-gradient action realizes the constructed frequencies as exact standing waves on its stated domain. Reproducible matrix, asymptotic, high-precision and full-variation checks accompany the proofs. The results establish properties of the specified finite-dimensional models, without asserting spatial localization, complete dynamical stability or a physical particle mass spectrum. This preprint includes explicit proofs and a standalone reproducibility package. It concerns a specified finite-dimensional spectral and inverse-variational model. It does not claim a derived physical particle spectrum or complete spatial stability. AI tools assisted mathematical development, code review and manuscript preparation; their use is disclosed in the paper.