TNFR Factorization Roadmap

Phase 0 — Research Capture (complete)

• Summarize Paley spectral gap findings and tie them to TNFR coherence metrics.
• Define deliverables and repository scaffolding (README, docs, package skeleton).

Phase 1 — Spectral Instrumentation

• Implement Paley graph builder with safe fallbacks when (n) is not prime power.
• Integrate Laplacian computation (scipy/networkx) plus TNFR telemetry adapters.
• Produce baseline plots of (\lambda_2) vs (n) (prime vs composite).

Phase 2 — TNFR Coherence Mapping

• Convert eigenvectors to TNFR structural fields (Φ_s, |∇φ|, K_φ, ξ_C, Si).
• Detect coherence clusters using canonical operator sequences [UM, RA, IL].
• Record telemetry episodes showing how ΔNFR behaves when clusters align with factors.

Phase 3 — Factor Recovery Logic (in progress)

• Map cluster periodicity to candidate factors using modular arithmetic.
• Design dissonance-driven search (OZ, ZHIR) to refine ambiguous cluster sizes.
• Output factorization certificates: TNFR metrics + gcd validations.

Phase 3 Milestone — TNFR Certification (November 2025)

• Completeness proof: Formalize that the nodal decoder [UM, RA, IL, THOL] converges to a unique partition-periodicity pattern for every Paley/Jacobi configuration (see theory/TNFR_NODAL_FACTOR_COMPLETENESS.md).
• Deterministic verification: Replace the gcd "proof" step with TNFR-native invariants (ΔNFR attenuation, Φ_s confinement, coherence ratios). Certificates now include tnfr_verification_snapshot blocks showing which factors are certified solely via nodal evidence.
• Operational tooling: SpectralAnalysisResult exposes tnfr_certified_factors, and the full-spectrum benchmark records verification metadata for each run.

Status: gcd checks remain available as optional backstops, but publication artifacts (certificates, playbook) now treat TNFR verification as the canonical proof of factor identity.

Phase 3.1 — Theory Alignment (March 2026)

• Arithmetic tetrad recalibration: _coherence_score() now uses arithmetic-specific thresholds (Φ_s<0.7452, |∇φ|<0.2591, K_φ<3.2275) derived from §7.5 of TNFR_NUMBER_THEORY.md.
• Pressure decomposition: Three-term ΔNFR breakdown (factorization, divisor, sigma pressures per §6) exposed in certificates and telemetry.
• Dual-lever analysis: Operator sequences classified as capacity-dominated, pressure-dominated, or balanced per §8.
• Conservation proxies: Noether charge Q = Φ_s + K_φ and Lyapunov energy E = 0.5·‖tetrad‖² included in api.py telemetry.
• Enriched telemetry: factorize() result telemetry expanded from 6 to ~15 fields covering the full TNFR number-theoretic stack.

Status: Complete. All code changes landed in spectral_paley.py, api.py, and partitioning.py.

1. Input n
2. Construct Paley/Jacobi graph for lifted modulus
3. Partition graph deterministically with telemetry per block
4. Execute nodal sequence [UM, RA, IL, THOL] per partition
5. Record operator-strategy plan + nodal decoding results
6. Emit certificates with per-partition before/after TNFR fields + invariant report
7. Verify factors optionally via arithmetic (gcd)

Phase 4 — Benchmarks and Validation

• Build benchmark suite mirroring primality-test/benchmarks/ (random semiprimes, RSA-style composites, smooth numbers).
• Compare against classical algorithms (trial division, Pollard-ρ, QS) for reference.
• Document success rates, runtimes, and TNFR telemetry plots.
• Range coverage (new): full_spectrum_factorization.py now sweeps contiguous ranges (150–170 by default, plus CLI-configurable spans) and emits certificates per n, establishing empirical coverage beyond curated samples.

Phase 5 — Packaging & Publication

• Finalize API (tnfr_factorization.factorize(n)), CLI, and notebooks.
• Prepare Zenodo metadata, release notes, and integration guide for main TNFR repo.
• Share results with TNFR-Riemann program to explore cross-links with spectral operators.