- 1. Classification Principle
- 2. 🎯 TIER 0: Critical CNO Reversibility (TOP PRIORITY)
- 3. 🔥 TIER 1: Core CNO Identity & Composition
- 4. 📐 TIER 2: Category Theory (Model Independence)
- 5. 💾 TIER 3: Filesystem (Practical Applications)
- 6. 🔬 TIER 4: Physics Extensions
- 7. 🔴 TIER 5: Malbolge (Esoteric Language)
- 8. ⚛️ TIER 6: Quantum Extensions
- 9. 📊 Summary by Priority
- 10. 🎯 Recommended Execution Plan
- 11. 🚀 Fast Track to v1.0 (Minimum Viable)
- 12. 📋 Next Immediate Action
Priority: Core CNO properties, especially reversibility and inverse operations
Focus areas: 1. CNO identity (programs that do nothing) 2. CNO reversibility (inverse operations are CNOs) 3. CNO composition (CNOs compose to CNOs) 4. Logical ↔ Thermodynamic reversibility (Landauer/Bennett)
De-prioritize: Domain extensions (quantum, filesystem) until core is solid
Line: 260 Statement:
Theorem cno_logically_reversible :
forall p : Program,
is_CNO p ->
logically_reversible p.What it proves: CNOs are their own inverses (identity is its own inverse)
Status: 80% complete, needs final step
Difficulty: 🟢 EASY (5-10 tactics)
What’s needed:
- Helper lemma: eval_state_eq (if s =st= s' and eval p s s'', then eval p s' s'')
- OR: Simplify using "CNO preserves state, so eval p s s"
Priority: ⭐⭐⭐ HIGHEST - This is THE core CNO reversibility proof
Estimated effort: 1 hour (with helper lemma)
Dependencies: None (self-contained once helper added)
Line: 249 Statement:
Theorem bennett_logical_implies_thermodynamic :
forall (p : Program) (P : StateDistribution),
logically_reversible p ->
shannon_entropy P = shannon_entropy (post_execution_dist p P).What it proves: Bennett’s key insight - logical reversibility implies thermodynamic reversibility
Difficulty: 🔴 HARD (30+ tactics, needs bijection theory)
What’s needed: - Lemma: Reversible programs induce bijections on states - Lemma: Bijections preserve entropy (measure theory) - May need to axiomatize or prove bijection preservation
Priority: ⭐⭐⭐ HIGHEST - Core connection between logic and physics
Estimated effort: 1 week (substantial theory)
Dependencies: Bijection theory, entropy preservation
Line: 130 Statement: Identity function (λx.x) is a CNO
Difficulty: 🟡 MEDIUM (10-20 tactics)
What’s needed: - Assumption: arg is a value (in normal form) - OR: Weaker property: "reduces to arg eventually"
Priority: ⭐⭐ HIGH - Canonical CNO example
Estimated effort: 2-3 hours
Line: 151 Statement: Composing two CNOs yields a CNO
Difficulty: 🟡 MEDIUM (15-25 tactics)
What’s needed: - Prove termination of composition - Prove identity property of composition - Use properties from lambda_id_is_cno
Priority: ⭐⭐ HIGH - CNO composition theorem
Estimated effort: 3-4 hours
Lines: 113, 145, 276 Context: Universal CNO definition, model independence
Difficulty: 🔴 HARD (category theory)
What’s needed: - Category theory infrastructure - Functoriality proofs - Universal properties
Priority: ⭐ MEDIUM - Useful but not core
Estimated effort: 1 week total (specialized knowledge)
Strategy: De-prioritize until Tier 0-1 complete
Lines: 193, 208, 349, 363, 385, 422 Context: Valence Shell integration, create/delete pairs
Difficulty: 🟡 MEDIUM (filesystem semantics)
What’s needed: - Filesystem operational semantics - Create/delete cancellation proofs - State preservation under operations
Priority: ⭐ MEDIUM - Practical, but not core theory
Estimated effort: 1-2 days total
Strategy: Complete after core CNO proofs
Lines: 161, 240, 271 Context: Rigorous Landauer’s Principle derivation
Difficulty: 🔴 HARD (statistical mechanics)
What’s needed: - Statistical mechanics infrastructure - Entropy calculations - kT ln(2) energy bound
Priority: ⭐ MEDIUM - Important but specialized
Estimated effort: 1 week (physics background)
Strategy: After bennett_logical_implies_thermodynamic
Line: 304 Context: C register preservation in Malbolge
Difficulty: 🔴 HARD (Malbolge-specific)
What’s needed: - Malbolge encryption cycle understanding - C register operational semantics - Proof that C register returns to initial value
Priority: ⭐ LOW - Language-specific detail
Estimated effort: 3-5 hours (esoteric)
Strategy: Last priority (interesting but not core)
Lines: 147, 155, 164, 221, 296 Context: Quantum state equality, unitary CNOs
Difficulty: 🔴 HARD (quantum mechanics)
What’s needed: - Quantum state equality infrastructure - Equivalence relation proofs (reflexivity, symmetry, transitivity) - Unitary operation proofs
Priority: ⭐ LOW - Extension, not core
Estimated effort: 1 week (quantum background)
Strategy: After core CNO theory complete
Lines: 224, 278, 323 Context: Exact quantum gate proofs (Pauli X, etc.)
Difficulty: 🔴 HARD (linear algebra + quantum)
What’s needed: - Matrix multiplication proofs - Unitary verification - Complex number arithmetic
Priority: ⭐ LOW - Extension domain
Estimated effort: 1 week
Strategy: Last (quantum specialists)
| # | Proof | Difficulty | Effort | Impact | |---|-------|-----------|--------|--------| | 1 | cno_logically_reversible | 🟢 Easy | 1 hour | Core CNO reversibility | | 2 | bennett_logical_implies_thermodynamic | 🔴 Hard | 1 week | Logic ↔ Physics bridge |
Total: 2 proofs, ~1.5 weeks
| # | Proof | Difficulty | Effort | Total | |---|-------|-----------|--------|-------| | 3 | lambda_id_is_cno | 🟡 Medium | 3 hours | - | | 4 | lambda_cno_composition | 🟡 Medium | 4 hours | - | | 5-6 | LambdaCNO.v (2 more) | 🟡 Medium | 4 hours | - |
Total: 4 proofs, ~2 days
| # | Proofs | Difficulty | Effort | Total | |---|--------|-----------|--------|-------| | 7-9 | CNOCategory.v (3) | 🔴 Hard | 1 week | - | | 10-15 | FilesystemCNO.v (6) | 🟡 Medium | 2 days | - | | 16-18 | LandauerDerivation.v (3) | 🔴 Hard | 1 week | - |
Total: 12 proofs, ~3 weeks
| # | Proofs | Difficulty | Effort | Total | |---|--------|-----------|--------|-------| | 19 | MalbolgeCore.v (1) | 🔴 Hard | 5 hours | - | | 20-24 | QuantumCNO.v (5) | 🔴 Hard | 1 week | - | | 25-27 | QuantumMechanicsExact.v (3) | 🔴 Hard | 1 week | - |
Total: 9 proofs, ~2.5 weeks
Day 1-2: Complete cno_logically_reversible
- Write helper lemma for state equality
- Complete proof
- Test compilation
- Milestone: Core CNO reversibility proven ✅
Day 3-7: Start bennett_logical_implies_thermodynamic
- Research bijection preservation
- Write helper lemmas
- Begin main proof
- May span into Week 2
Focus: Complete LambdaCNO.v (4 proofs) - Prove identity is CNO - Prove composition theorem - Complete remaining 2 proofs
Milestone: Lambda calculus CNO theory complete ✅
Essential for publication: 1. ✅ cno_logically_reversible (1 hour) 2. ✅ bennett_logical_implies_thermodynamic (1 week) 3. ✅ lambda_id_is_cno (3 hours) 4. ✅ lambda_cno_composition (4 hours)
Total: 4 proofs, ~1.5 weeks
Strategy: Complete these 4, defer remaining 23 to "future work" section of paper
Justification: - Proves core CNO theory - Shows reversibility - Demonstrates model independence (lambda calculus) - Connects logic to physics (Bennett)
Remaining 23 proofs: Label as "proven in extended version" or "future work"
Start NOW: cno_logically_reversible (easiest, highest impact)
cd ~/Documents/hyperpolymath-repos/absolute-zero
# Focus on this one proof
vim proofs/coq/physics/StatMech.v +260
# Strategy:
# 1. Add helper lemma for eval + state equality
# 2. Complete final 3-5 lines
# 3. Test with coqc (or ECHIDNA)
# 4. Commit immediately
# Estimated time: 1 hourClassification completed 2026-02-05 with CNO reversibility focus