Second–Order Spectral Rigidity and an Effective Convexity Criterion

We develop a spectral–dynamic framework in which all first–order constraints arising from the explicit formula for the Riemann xi function—symmetry, density, and trace identities—are automatically satisfied. Once these constraints are saturated, the stability of admissible spectral configurations reduces to a single second–order object: the Hessian of a renormalized logarithmic energy restricted to a natural mass–zero subspace.  

This reduction isolates weak convexity of an effective potential as the unique remaining structural degree of freedom. A computable finite–scale criterion is obtained, and numerical evaluations on structured, perturbed, and random configurations exhibit sharply distinct behavior. The variational structure closes at second order, with no higher–order obstructions.