SilverSight/docs/research/NEXT_STEPS_PLAN.md

Next Steps Plan

Date: 2026-07-04
Status: VERIFICATION COMPLETE — Planning next phase

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Verification Summary

✅ Confirmed from Previous Session

1. CRTSidon.lean — 0 sorries, both theorems proven

   • sidon_preserved: Componentwise CRT Sidon preservation
   • sidon_preserved_mod: Modular CRT Sidon preservation
   • Supporting lemmas: mod_eq_dvd, dvd_mod_eq, mod_eq_of_coprime

2. de Grey Graph — 1581-vertex construction verified

   • hn_hoffman_bound.py: Computes χ ≥ 3 via spectral method
   • Eigenvalue computation confirmed

3. Sidon-Sofa Coloring — v2 and v3 scripts present

   • v2: Coarse q-sweep (5 values), n ∈ {8, 13, 21}
   • v3: Fine q-sweep (100 values), n ∈ {8, 13, 21, 34}

4. StrandCapacityBound Triviality — Flagged in AGENTS.md

   • Known limitation documented

5. CITATION.cff — Prior art references added

   • Elsasser (1946) toroidal/poloidal decomposition
   • Wikipedia toroidal/poloidal coordinates

6. Quandela API Key — Encrypted and stored

   • secrets/quandela_api_key.enc.yaml (sops/age encrypted)
   • .sops.yaml configuration present

7. Golden Centering Constant — Fixed to 40504

   • formal/SilverSight/GoldenSpiral.lean: phiInvQ16 = 40504
   • formal/CoreFormalism/BraidEigensolid.lean: Consistent value

8. Adversarial Review — 18 findings documented

   • 4 Critical, 5 High, 6 Medium, 3 Low
   • Commit 54fd2283: Repaired sofa coloring scripts + Hoffman bound
   • SIDON_SOFA_COLORING_REVIEW.md: Full review receipt

9. Photonic Sidon Search — All 18 tests passing

   • photonic_sidon_search.py: Exact integer verification
   • Encoders imported: encoder_q16.py, padic_encoder.py
   • Test evidence: .openresearch/artifacts/photonic_sidon_evidence.jsonl

⚠️ Uncertain / Not Found

1. SLOS Classical-Sim Disclaimer — Not explicitly found

   • File header mentions "exact integer verification"
   • No explicit disclaimer about classical simulation limitations
   • Action needed: Add disclaimer to script header

2. Container (silver-autoproof:latest) — NOT REQUIRED

   • MCP autoproof infrastructure already exists and is functional
   • Python MCP server: scripts/mcp_autoproof.py (282 lines)
   • Python worker: scripts/mcp_worker.py (127 lines)
   • Rust backend: scripts/mcp_backend/ (thread-safe state management)
   • Uses neon-64gb API (http://100.92.88.64:8766/generate) for phi4 LLM
   • File-based locking for thread safety
   • Supports stdio and HTTP modes
   • Action needed: Document infrastructure in README or separate doc

3. Adversarial Review Count Discrepancy

   • Session summary mentions "14 issues"
   • Review document shows 18 findings (4+5+6+3)
   • Action needed: Clarify count (may be different categorization)

4. TOROIDAL_POLOIDAL_REFINEMENT.md — Planning document only

   • 6 refinement actions (R1-R6) all marked TODO or BLOCKED
   • No implementation yet
   • Action needed: Prioritize and implement

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Next Steps (Prioritized)

Immediate (Low Effort, High Impact)

1. Add SLOS Disclaimer (15 min)

   • Add to photonic_sidon_search.py header
   • Clarify classical vs quantum simulation
   • Document empirical nature of Omega-Sidon correlation

2. Clarify Adversarial Review Count (10 min)

   • Check if "14 issues" refers to unique issues vs total findings
   • Update session summary or review document for consistency

3. Document Container Status (20 min)

   • If silver-autoproof exists in another branch/repo, document location
   • If not, create Dockerfile based on current autoproof infrastructure
   • Update README with container usage instructions

Short-term (Medium Effort, Research Value)

4. Run Sidon-Sofa Coloring v3 (2-4 hours)

   • Generate artifacts for fine q-sweep + n=34
   • Analyze q=1 phase boundary behavior
   • Compare with v2 results

5. Verify Photonic Search Reproducibility (30 min)

   • Confirm all 18 tests still pass
   • Check test evidence matches committed version

6. Address TOROIDAL_POLOIDAL_REFINEMENT.md TODOs (4-8 hours)

   • R1 (High): Redefine modulus selection as q-profile design
   • R2 (High): Add cross-pair q-ratio coprimality check
   • R4 (Medium): Document 1.9× optimum as A2 sweet spot
   • R5 (High): Formalize discrete Elsasser decomposition

Medium-term (High Effort, Strategic Value)

7. Formalize Discrete Elsasser Decomposition (1-2 days)

   • Extend CRTSidon.lean with toroidal/poloidal coordinate system
   • Prove equivalence between CRT embedding and Elsasser decomposition
   • Connect to plasma physics literature (Elsasser 1946)

8. Implement q-Profile Design (2-3 days)

   • Refactor modulus selection to use q-profile (q = L₂/L₁)
   • Systematic sweep: q ∈ {0.5, 0.75, 1.0, 1.33, 2.0}
   • Analyze Sidon preservation as function of q
   • Identify optimal q-regime (conjectured: q < 1, poloidal-dominated)

9. Create silver-autoproof Container (1 day)

   • Dockerfile based on scripts/mcp_autoproof.py
   • Include Lean 4 toolchain + Mathlib
   • Expose MCP server endpoint for remote proof filling
   • Update CI/CD to use container for automated proof checking

Long-term (Research Directions)

10. Extend to Larger Sidon Sets (ongoing)

    • Current: n ≤ 34 (v3 script)
    • Target: n = 50, 100, 200
    • Requires GPU acceleration or distributed computation

11. Quantum Photonic Simulation (exploratory)

    • Replace classical SLOS with quantum circuit simulation
    • Test if quantum Omega shows stronger Sidon correlation
    • Requires Quandela API access (key already encrypted)

12. Connect to Hadwiger-Nelson Problem (theoretical)

    • Use Sidon-Sofa coloring results to bound χ(ℝ²)
    • Current lower bound: χ ≥ 5 (de Grey 2018)
    • Can Sidon structure improve this?

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Immediate Action Items

• Add SLOS disclaimer to photonic_sidon_search.py
• Clarify adversarial review count (14 vs 18)
• Document silver-autoproof container status
• Run sidon_sofa_coloring_v3.py and analyze results
• Verify photonic_sidon_search.py reproducibility
• Prioritize TOROIDAL_POLOIDAL_REFINEMENT.md TODOs

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Dependencies

• Perceval tokens: Required for R3 (SLOS K=4, K=5 tests)
• GPU access: Required for large-scale Sidon-Sofa coloring (n > 34)
• Quandela API: Required for quantum photonic simulation (optional)

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Success Criteria

• All immediate action items completed
• Sidon-Sofa Coloring v3 artifacts generated and analyzed
• Photonic Sidon Search reproducibility confirmed
• At least 2 refinement TODOs implemented (R1, R2, or R5)
• Container status documented or container created

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Next Session Start: Begin with immediate action items, then proceed to short-term tasks based on results.