docs: add comprehensive next steps plan

- Verification summary of all previous session work
- Prioritized action items (immediate, short-term, medium-term, long-term)
- Dependencies and success criteria
- Starting with immediate items: SLOS disclaimer, review count clarification, container status

docs/research/NEXT_STEPS_PLAN.md

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+# 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 found in this branch
+   - No Dockerfile or docker-compose for silver-autoproof
+   - Only package Dockerfiles in `.lake/packages/`
+   - **Action needed:** Document status or create container
+
+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.