Indefinite Spectral Certificate Calculus: A Framework for Signature Decomposition and Index-Jump Detection

Indefinite Spectral Certificate Calculus develops a standalone mathematical framework for indefinite symmetric and Hermitian quadratic forms. The method replaces universal positivity tests with finite inertia certificates, canonical removal of intrinsic negative and null sectors, and reduced quotient tests for detecting new negative directions.

The monograph introduces the core definitions, admissibility rules, signature certificates, quotient index-jump criterion, worked finite-dimensional examples, and representative applications to relativistic spectral theory, non-convex optimization, structural mechanics, metamaterial stability, and structured perturbation detection.