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114 lines
5.1 KiB
Markdown
114 lines
5.1 KiB
Markdown
# Missing Proofs Registry
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**Purpose:** Track theorems, lemmas, and formalizations that are defined, documented, or conjectured but not yet proven in Lean 4.
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**Status:** Living document — update as formalization progresses.
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---
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## Gap Categories
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### Category A: AVMR Core Theorems (Models 102-110)
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| # | Theorem | Statement | Current Status | Blocker | Auto-Provable? |
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|---|---------|-----------|----------------|---------|----------------|
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| 102 | Quasi-Periodic Square-Shell | `n = k² + a = (k+1)² - b` with `a+b = 2k+1` | ✅ **PROVEN** — `ring + omega` | — | **YES** ✅ |
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| 103 | Tip Coordinate Vector | `Tip(n) = (ab, a-b)` as ℝ² embedding | Partial — `tipCoord` defined | Real embedding properties | **NO** — needs injectivity |
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| 104 | Axial Event Production | `A_k, G_k, C_k, T_k` partition shell | ✅ **PROVEN** — `axialGeneratorExhaustivity` | — | **YES** ✅ |
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| 121 | Axial Generator Exhaustivity | {A_k, G_k, C_k, T_k} exhaust S_k | ✅ **PROVEN** — `ring + nlinarith` | — | **YES** ✅ |
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| 105 | Resonance Hub | `Tip(m²) = (0, -(2k+1))` | ✅ **PROVEN** — `simp + ring + omega` | — | **YES** ✅ |
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| 106 | Standing-Wave Rear Field | `Ψ_i(n_i - d) = α_d · χ_i` | Partial — `buildFieldAt` computes | Field convergence | **PARTIAL** — echo sum proven |
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| 107 | Interaction Score | `J(n) = ab·F_m + (a-b)·F_p + ⟨χ,F_c⟩` | ✅ **Implemented** | — | — |
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| 108 | Left-Right Transduction | `(n,k,a,b) ↦ (e,τ,...) ↦ (S,P,G)` | Missing — pipeline not formalized | Multi-stage bind | **NO** — needs pipeline |
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| 109 | Temporal Error-Coding | `t(n) = nR + τ`, 8-slot | Missing — microtime lattice | Temporal algebra | **NO** — needs Fin arithmetic |
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| 110 | AVMR Commitment | `Σ_{k+1}[i] = Φ(Σ_k[2i], Σ_k[2i+1])` | Missing — vectorized Merkle | Aggregation invariant | **NO** — needs algebraic structure |
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**Search space shrunk:** 4 of 11 theorems proven automatically (Models 102, 104, 105, 121). 7 require manual work.
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### Category B: Core Semantics Debt
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| Module | Debt | Location | Priority |
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|--------|------|----------|----------|
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| `BraidCross.lean` | Fix16.add associativity | Line 72 | High — blocks algebra |
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| `BraidCross.lean` | Fix16.add commutativity | Line 84 | High — blocks algebra |
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| `Constitution.lean` | Lawful constitution preservation | `sorry` placeholder | Medium — ENE core |
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| `Canon.lean` | Float elimination (timestamp, trustLevel) | `float64` field kind | Medium — core purity |
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| `Graph.lean` | Float → Q16_16 port | Geometry module | Low — shim acceptable |
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### Category C: Domain Intersection Gaps
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| Intersection | Missing Bridge | Impact |
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|--------------|----------------|--------|
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| E ↔ M (Verification ↔ Lean) | BEA consensus → Formal proof | Weak verification formalism |
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| H ↔ G (Algebra ↔ Energy) | Geometric algebra → Thermodynamics | Noether connection missing |
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| J ↔ L (Dynamics ↔ Application) | Phase transitions → FEA | Engineering bridge |
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### Category D: Unified Compression Extensions
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| Feature | Status | Needed For |
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|---------|--------|------------|
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| Temporal lattice integration | Missing | Model 109 connection |
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| AVMR commitment hook | Missing | Model 110 connection |
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| Microvoxel constraint check | Partial | Model 21-22 validation |
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| Meta-LUT formalization | Missing | Model 23-24 |
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| HotSwap selector proof | Missing | Model 25 |
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| **Final Score Law** | 🔄 Formalizing | Models 119-120 — cost aggregation |
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| Score parameter bounds | Missing | λ₁,λ₂,λ₃,λ₄ ∈ [0, 2.0] |
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| Cost monotonicity | Missing | L(X) increases with complexity |
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---
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## Proof Templates
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When adding a proof, use this format:
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```lean
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theorem <name> : <statement> := by
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<tactics>
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<;> try native_decide
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<;> try omega
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<;> try nlinarith
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```
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Preferred witnesses:
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- `#eval` examples with expected output
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- `native_decide` for concrete cases
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- `omega` / `nlinarith` for arithmetic
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- `simp` with appropriate lemmas
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---
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## Dependencies
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Before proving higher theorems, these foundations are needed:
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1. **Fix16 algebra** (BraidCross) — blocks all algebraic manipulation
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2. **Shell state totality** — blocks AVMR event classification proofs
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3. **Field convergence** — blocks standing-wave theorems
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4. **Q16_16 metric** — blocks distance/geometry proofs
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---
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## Recent Updates
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- **2026-04-17:** Created registry, populated from MATH_MODEL_MAP (181 models)
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- **2026-04-17:** Added Unified Compression gaps (Models 111-118)
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- **2026-04-17:** Added Final Score Law (Models 119-120) — complete cost aggregation
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- **2026-04-17:** Added Agent F1/F2/F3 Tier Proofs (Models 121-131) — explicit Φ operators
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- **2026-04-17:** Identified 3 critical domain intersections needing bridges
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- **2026-04-17:** Prover ran — Models 102, 104, 105, 121 proven automatically
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---
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## How to Resolve
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1. Pick a theorem from Category A or B
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2. Create branch: `proof/<theorem-name>`
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3. Add theorem + witness to appropriate `.lean` file
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4. Update this registry with ✅ status
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5. Run `lake build` to verify
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6. PR with theorem statement and proof summary
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---
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**Enforcement:** Per AGENTS.md Section 4 — every `def` computing cost or invariant must have `#eval` or theorem witness before commit.
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