Operator-Sequence Certificates for Spectral Factorization

Purpose

Provide TNFR-native evidence that the Paley graph for n reorganizes coherently through validated operator sequences, producing sub-EPIs that correspond to the recovered factors. Certificates must include per-partition TNFR telemetry, operator-strategy plans, and invariant reports so downstream agents can replay the full sequence.

Canonical pipeline

1. Input n
2. Construct Paley/Jacobi graph and annotate nodes for FFT
3. Partition via deterministic planner (configurable target size/overlap)
4. Execute [UM, RA, IL, THOL] on each partition (nodal decoder)
5. Plan AL/IL/RA/SHA strategies per partition from StrategyRegistry
6. Emit enriched certificate with:
   • partition before/after TNFR fields (Φ_s, |∇φ|, K_φ, ξ_C, ΔNFR)
   • nodal decoder outcomes (coherence ratios, inferred factors)
   • operator strategy plan snapshot + nodal decoding snapshot
   • invariant report citing U1–U6 + canonical invariants (#1–#6)
7. Optionally verify candidates via arithmetic gcd (supporting evidence)

Certificate contents (JSON)

• operators, canonical_operators: sequence recommended/applied
• telemetry: global TNFR metrics (Φ_s, |∇φ|, K_φ, ξ_C, ΔNFR)
• pressure_components: three-term ΔNFR decomposition — factorization, divisor, sigma pressures (§6)
• dual_lever_analysis: capacity vs pressure lever classification of the operator sequence (§8)
• conservation_proxies: Noether charge Q = Φ_s + K_φ, Lyapunov energy E = 0.5·(Φ_s² + |∇φ|² + K_φ²)
• arithmetic_epi, arithmetic_nu_f: nodal equation components for the target integer (§5)
• partition_states: map partition_id -> {before, after, node_count, boundary_count, candidate_factors}
• strategy_plan_snapshot: subset of operator_strategy_plan covering AL/IL/RA/SHA
• nodal_decoding_snapshot: per-partition decoder log (sequence, ΔNFR convergence, dynamic factors)
• invariant_report: structured proof of U1–U6 + canonical invariants (#1–#6)
• tnfr_verification_snapshot: deterministic certification data (per-factor endorsements, ΔNFR gains, coherence windows) that replaces the old gcd-only confirmation
• partition_manifest* references + per-partition certificate files

Certificates live in results/certificates/, and partition artifacts are stored under results/certificates/partitioned/. Threshold controls (TNFR_PARTITION_OUTPUT_DIR, TNFR_PARTITION_FILELIST_THRESHOLD) determine where the files land.

Verification update: gcd/divisibility checks remain available for diagnostics, but TNFR publication artifacts now treat the tnfr_verification_snapshot as the canonical proof of factor identity. A factor is "accepted" when its partitions satisfy the deterministic ΔNFR/coherence/Φ_s criteria captured in that snapshot.

Workflow summary

n → Paley/Jacobi graph → partitions + telemetry → nodal decoder [UM,RA,IL,THOL]
→ strategy plan selection → certificate emission (partition states + invariants)
→ (optional) arithmetic gcd verification

Operational guidance

• Always call SpectralPaleyFactorizer.analyze(..., trace_certificates=True) when you need reproducible evidence. Certificates should accompany benchmark runs (paley_gap_*, full_spectrum_factorization) and CI validations.
• Inspect partition_states to see how each block satisfied ΔNFR convergence and coherence matching. The nodal_state metadata is the authoritative view of the local EPI evolution.
• Use invariant_report to confirm the run satisfied TNFR grammar and canonical invariants without re-running the analysis. If any rule is not “tracked”, the result should be considered provisional.

Future extensions

• Attach raw operator logs (beyond the nodal decoder) when additional sequences are introduced.
• Include visual artifacts (plots of Φ_s, |∇φ|, K_φ, ξ_C per partition) referenced from the certificate metadata.
• Integrate certificate summaries into Zenodo release bundles so every dataset ships with structural evidence.