Two-Place Stratified Visibility and Catalan's Constant: A Tor–Smith Determinant Construction

This preprint develops a determinant construction for Catalan’s constant within a stratified-visibility framework based on Tor modules and Smith normal forms. At finite primes, denominator defects are analyzed through successive Tor-length filtrations and local Smith-depth profiles. At the archimedean place, finite-dimensional Hardy-space quotients and compressed multipliers provide the analytic framework for determinant estimates. The quadratic Catalan cover yields integral Chebyshev polynomial pairs and polynomial contact identities. A rational parameter family gives rise to determinants of size 98N. The manuscript derives a stratified odd-prime coefficient of 73373/38416 and a normalized finite-place bound below 2.33088. A two-layer Smith extraction further refines local bounds in a fixed 98-row instance. The proposed irrationality argument combines prime-scale nonvanishing, finite-place denominator estimates, and an archimedean energy bound below -2.33118, leading to incompatible inequalities under the assumption that Catalan’s constant is rational. The preprint includes mathematical derivations, explicit rational data, and computational certificates. It is made available for independent mathematical examination and has not undergone journal peer review.