The Harmonic Collapse Index and the Order of Vanishing of Elliptic Curve L-Functions

This work introduces a new analytic invariant, the harmonic collapse index, derived from base-filtered harmonic energies associated with modular forms of elliptic curves. The invariant detects structured spectral suppression at the critical point s = 1 and is shown numerically to align with the order of vanishing of the associated elliptic curve L-function across a broad range of examples. The construction is independent of Selmer group machinery and does not assume the Birch–Swinnerton–Dyer conjecture. Instead, it provides a spectral diagnostic invariant that is stable under base refinement and invariant under isogeny. Numerical evidence is presented together with structural lemmas establishing well-definedness and robustness.