Research-Stack/6-Documentation/docs/semantics/missingproofs

Missing Proofs Registry

Purpose: Track theorems, lemmas, and formalizations that are defined, documented, or conjectured but not yet proven in Lean 4.

Status: Living document — update as formalization progresses.

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Gap Categories

Category A: AVMR Core Theorems (Models 102-110)

#	Theorem	Statement	Current Status	Blocker	Auto-Provable?
102	Quasi-Periodic Square-Shell	n = k² + a = (k+1)² - b with a+b = 2k+1	✅ PROVEN — ring + omega	—	YES ✅
103	Tip Coordinate Vector	Tip(n) = (ab, a-b) as ℝ² embedding	Partial — tipCoord defined	Real embedding properties	NO — needs injectivity
104	Axial Event Production	A_k, G_k, C_k, T_k partition shell	✅ PROVEN — axialGeneratorExhaustivity	—	YES ✅
121	Axial Generator Exhaustivity	{A_k, G_k, C_k, T_k} exhaust S_k	✅ PROVEN — ring + nlinarith	—	YES ✅
105	Resonance Hub	Tip(m²) = (0, -(2k+1))	✅ PROVEN — simp + ring + omega	—	YES ✅
106	Standing-Wave Rear Field	Ψ_i(n_i - d) = α_d · χ_i	Partial — buildFieldAt computes	Field convergence	PARTIAL — echo sum proven
107	Interaction Score	J(n) = ab·F_m + (a-b)·F_p + ⟨χ,F_c⟩	✅ Implemented	—	—
108	Left-Right Transduction	(n,k,a,b) ↦ (e,τ,...) ↦ (S,P,G)	Missing — pipeline not formalized	Multi-stage bind	NO — needs pipeline
109	Temporal Error-Coding	t(n) = nR + τ, 8-slot	Missing — microtime lattice	Temporal algebra	NO — needs Fin arithmetic
110	AVMR Commitment	Σ_{k+1}[i] = Φ(Σ_k[2i], Σ_k[2i+1])	Missing — vectorized Merkle	Aggregation invariant	NO — needs algebraic structure

Search space shrunk: 4 of 11 theorems proven automatically (Models 102, 104, 105, 121). 7 require manual work.

Category B: Core Semantics Debt

Module	Debt	Location	Priority
BraidCross.lean	Fix16.add associativity	Line 72	High — blocks algebra
BraidCross.lean	Fix16.add commutativity	Line 84	High — blocks algebra
Constitution.lean	Lawful constitution preservation	sorry placeholder	Medium — ENE core
Canon.lean	Float elimination (timestamp, trustLevel)	float64 field kind	Medium — core purity
Graph.lean	Float → Q16_16 port	Geometry module	Low — shim acceptable

Category C: Domain Intersection Gaps

Intersection	Missing Bridge	Impact
E ↔ M (Verification ↔ Lean)	BEA consensus → Formal proof	Weak verification formalism
H ↔ G (Algebra ↔ Energy)	Geometric algebra → Thermodynamics	Noether connection missing
J ↔ L (Dynamics ↔ Application)	Phase transitions → FEA	Engineering bridge

Category D: Unified Compression Extensions

Feature	Status	Needed For
Temporal lattice integration	Missing	Model 109 connection
AVMR commitment hook	Missing	Model 110 connection
Microvoxel constraint check	Partial	Model 21-22 validation
Meta-LUT formalization	Missing	Model 23-24
HotSwap selector proof	Missing	Model 25
Final Score Law	🔄 Formalizing	Models 119-120 — cost aggregation
Score parameter bounds	Missing	λ₁,λ₂,λ₃,λ₄ ∈ [0, 2.0]
Cost monotonicity	Missing	L(X) increases with complexity

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Proof Templates

When adding a proof, use this format:

theorem <name> : <statement> := by
  <tactics>
  <;> try native_decide
  <;> try omega
  <;> try nlinarith

Preferred witnesses:

• #eval examples with expected output
• native_decide for concrete cases
• omega / nlinarith for arithmetic
• simp with appropriate lemmas

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Dependencies

Before proving higher theorems, these foundations are needed:

1. Fix16 algebra (BraidCross) — blocks all algebraic manipulation
2. Shell state totality — blocks AVMR event classification proofs
3. Field convergence — blocks standing-wave theorems
4. Q16_16 metric — blocks distance/geometry proofs

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Recent Updates

• 2026-04-17: Created registry, populated from MATH_MODEL_MAP (181 models)
• 2026-04-17: Added Unified Compression gaps (Models 111-118)
• 2026-04-17: Added Final Score Law (Models 119-120) — complete cost aggregation
• 2026-04-17: Added Agent F1/F2/F3 Tier Proofs (Models 121-131) — explicit Φ operators
• 2026-04-17: Identified 3 critical domain intersections needing bridges
• 2026-04-17: Prover ran — Models 102, 104, 105, 121 proven automatically

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How to Resolve

1. Pick a theorem from Category A or B
2. Create branch: proof/<theorem-name>
3. Add theorem + witness to appropriate .lean file
4. Update this registry with ✅ status
5. Run lake build to verify
6. PR with theorem statement and proof summary

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Enforcement: Per AGENTS.md Section 4 — every def computing cost or invariant must have #eval or theorem witness before commit.