Mean-Square Rigidity of Edge-Phase Sums and Spectral-Edge Instability in Li-Type Data

Version 4.0 update: This version introduces numerical validation on 500 zeta zeros (k = 2000 phases), spectral edge analysis A(δ), and a bridge toward infinite configurations via resonance counting and pair correlation heuristics. We investigate the behaviour of Li-type coefficients under symmetric off-critical deformations of the nontrivial zeros of the Riemann zeta function. We identify a spectral-edge instability mechanism: once the associated factors leave the unit circle, exponentially amplified contributions emerge, governed by an oscillatory edge-phase sum. We prove that exact cancellation of this oscillatory term is structurally rigid and generically impossible, reducing the Riemann Hypothesis to a phase-rigidity problem. Numerical experiments reveal a sharp transition between local stability and global instability under increasing deformation. This work provides a structural and dynamical perspective on the Riemann Hypothesis, isolating a concrete instability mechanism and a precise analytic target for future investigation.