This document describes domain-specific applications of the nodal equation ∂ E P I / ∂ t

ν f   Δ N F R ( t ) ∂EPI/∂t=ν f ​ ΔNFR(t) that are implemented, tested, and computationally verified in this repository. The primary application is spectral factorization via Paley–Jacobi graphs, which has a dedicated implementation with 10 test modules, structural verification criteria, and reproducible certificates.

Demonstration-level examples (particle collisions, elemental structure) are listed in §5 with their actual verification status.