The Harmonic Collapse Index and a One-Direction Analytic Inequality at s = 1

This paper introduces a computable analytic invariant, the harmonic collapse index—extracted from base-filtered Fourier coefficient energies of modular forms attached to elliptic curves over \mathbb{Q}. The construction is numerically stable under truncation and benign changes of the base family, and is empirically consistent with known analytic ranks. A cautious one-direction inequality relating the collapse index to the analytic order of vanishing at s = 1 is recorded. No claims are made regarding equality with arithmetic rank, Selmer groups, or the full Birch–Swinnerton–Dyer conjecture. The paper is intended as a minimal analytic record of the invariant and its observed stability properties.