APPLIED_STRUCTURAL_ANALYSIS

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.

Spectral Factorization 2.1 Statement For a composite integer using quadratic residues. Maximum graph size: 4097 nodes (configurable via TNFR_PARTITION_TARGET_SIZE). Spectral analysis: Compute Laplacian eigenvalues Partition planning: Divide graph into coherent sub-regions (default target size 256, boundary overlap 4 nodes). Per-partition telemetry computed via coherence and sense index. Nodal decoding: Apply operator sequence [UM, RA, IL, THOL] per partition. UM/RA restrict coupling to phase-compatible residues (U3); THOL preserves multi-scale identity (U5). Factor inference: Detect periodicities in stabilized partitions via structural consistency. Certification: Verify factor candidates against 8 structural criteria (§2.3). 2.3 Verification CriteriaA factor candidate is TNFR-certified when ≥4≥4 of 8 structural criteria hold and ≥50%≥50% of partition endorsements are positive: