Indefinite Spectral Certificate Calculus: A Unified Calculus of Typed Spectral Certification, Composition, Solvability Consequences, and Independent Replay

Abstract

Indefinite Spectral Certificate Calculus (ISCC) is a method for analyzing symmetric or Hermitian systems with both positive and negative directions. It separates negative directions that belong to a known background structure from negative directions that appear only after reduction to the effective quotient space. ISCC uses established tools such as Sylvester’s law of inertia, congruence transformations, LDL^{T} factorization, exact arithmetic, and interval methods. These tools are organized into a certification procedure for background sectors, null sectors, quotient forms, and index jumps.