Applied Structural Analysis

Status: Technical reference Version: 0.0.3.3 Date: March 2026

1. Scope

This document describes domain-specific applications of the nodal equation $\partial\mathrm{EPI}/\partial t = \nu_f \, \Delta\mathrm{NFR}(t)$ 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.

2. Spectral Factorization

2.1 Statement

For a composite integer $n > 1$, the Paley–Jacobi graph $G(m)$ constructed from quadratic residues exhibits spectral coherence structure under the canonical decoder sequence $[\mathrm{UM}, \mathrm{RA}, \mathrm{IL}, \mathrm{THOL}]$. The resulting partitions expose periodicities corresponding to the prime factors of $n$.

2.2 Algorithm

1. Graph construction: Build Paley graph from modulus $m$ using quadratic residues. Maximum graph size: 4097 nodes (configurable via TNFR_PARTITION_TARGET_SIZE).
2. Spectral analysis: Compute Laplacian eigenvalues $\lambda_2, \ldots, \lambda_k$ and eigenvectors.
3. Tetrad projection: Translate eigendata into structural fields ($\Phi_s$, $|\nabla\phi|$, $K_\phi$, $\xi_C$).
4. 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.
5. 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).
6. Factor inference: Detect periodicities in stabilized partitions via structural consistency.
7. Certification: Verify factor candidates against 8 structural criteria (§2.3).

2.3 Verification Criteria

A factor candidate is TNFR-certified when $\geq 4$ of 8 structural criteria hold and $\geq 50\%$ of partition endorsements are positive:

2.4 Confidence Metric

Structural periodicity confidence:

2.5 Pure Mode

When TNFR_PURE_MODE=1, factor certification relies exclusively on structural confidence ($\geq 0.6$) without arithmetic divisibility checks. This isolates the TNFR-specific signal from classical number-theoretic shortcuts.

2.6 Output Artifacts

All artifacts include SHA256 hashes and deterministic seeds for reproducibility.

3. Implementation

3.1 Public API

from tnfr_factorization.api import factorize

result = factorize(221, pure=True, trace=True)
# result.candidate_factors -> [13, 17]
# result.tnfr_certified_factors -> [13, 17]
# result.telemetry -> {phi_s, phase_gradient, ...}
# result.certificate_path -> path to operator certificate

3.2 Integration with TNFR Core

• TNFRAdvancedFFTEngine for spectral computation
• TNFRSelfOptimizingEngine for operator sequence recommendation
• Operator grammar validation (U1–U6) via validate_sequence()
• Strategy registry for resource estimation
• ArithmeticTNFRParameters and ArithmeticStructuralTerms for number-theoretic integration

4. Test Coverage

Key verified properties:

• Nodal sequence [UM, RA, IL, THOL] surfaces factors via periodicity (e.g., $221 = 13 \times 17$ yields both certified factors).
• Certificate emission obeys U1–U6 grammar.
• Verification criteria are balanced (strictness score $\in [0.3, 0.8]$).
• False-positive resistance: min_partition_flags $\geq 3$, dnfr_gain_min $\geq 0.15$, periodicity_confidence_min $\geq 0.5$.
• Deterministic seeding ensures reproducibility across runs.

Implementation & Examples

Executable Demonstrations

Key Source Modules

Note: the attack uses the emergent prime-NFR nodal pulse (νf = log n; zeros as destructive interference).

• src/tnfr/riemann/nodal_pulse.py — canonical nodal-pulse foundation (νf = log n; zeros as destructive interference)
• src/tnfr/riemann/prime_ladder_hamiltonian.py — canonical νf prime-ladder Hamiltonian (P14)
• src/tnfr/riemann/von_mangoldt.py — prime-ladder von Mangoldt reproduction (P12)

6. References

• FUNDAMENTAL_THEORY.md — Nodal equation and structural fields
• PHYSICAL_REGIME_CORRESPONDENCES.md — Quantization mechanism
• UNIFIED_GRAMMAR_RULES.md — U1–U6 derivations
• STRUCTURAL_CONSERVATION_THEOREM.md — Conservation laws
• GLOSSARY.md — Operational definitions