# Next Steps Plan

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

---

## 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

---

## 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?

---

## 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

---

## 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)

---

## 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

---

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