Research-Stack/6-Documentation/docs/semantics/missingproofs/SUBAGENT_ASSIGNMENTS.md

Subagent Assignments — Remaining AVMR Theorems

Generated: 2026-04-19
Total Remaining: 10 theorems across 4 difficulty tiers
Priority: P0 (blocking hardware extraction) → P3 (research enrichment)

---

Tier 1: Algebraic Geometry (P1 — 3 theorems)

Location: 0-Core-Formalism/lean/Semantics/Semantics/AVMR.lean
Domain: Number theory, hyperbola intersection, shell geometry
Estimated Effort: 2-3 days each

Theorem 122: tipCoordinateMassResonance

Statement: an * bn = am * bm for resonance condition
Gap: Requires solving hyperbola intersection equation
Mathematical Approach:

• Express a_n, b_n in terms of n, k_n = ⌊√n⌋
• Mass resonance occurs when ab_i = ab_j
• This is a Diophantine equation on hyperbola branches
• Use nlinarith with proper bounds on a, b

Subagent Assignment:
→ Agent F2 (Algebra Specialist)
Required: Number theory, nonlinear arithmetic, nlinarith mastery
Deliverable: Complete proof with explicit bounds extraction

---

Theorem 123: tipCoordinateMirrorResonance

Statement: (an : Int) - (bn : Int) = -((am : Int) - (bm : Int))
Gap: Shell coupling between different k values
Mathematical Approach:

• Mirror resonance: (a-b)_i = -(a-b)_j
• Requires relating shells k_i and k_j
• Use omega for integer constraints
• May need case analysis on |k_i - k_j|

Subagent Assignment:
→ Agent F2 (Algebra Specialist)
Required: Integer arithmetic, case analysis, shell geometry
Deliverable: Proof with coupling condition explicit

---

Theorem 124: fortyFiveLineFactorRevelation

Statement: For even n, 45° line contains all divisors d|n in {a_m, b_m}
Gap: Fermat factorization mapping
Mathematical Approach:

• Fermat's factorization: n = x² - y² = (x+y)(x-y)
• Map to shell coordinates: a = x - k, b = k + 1 - x
• Requires constructing m for each divisor
• Use Nat.divisors and explicit construction

Subagent Assignment:
→ Agent F3 (Number Theory Specialist)
Required: Fermat factorization, divisor theory, explicit constructions
Deliverable: Existence proof with constructive witness

---

Tier 2: Information Theory (P2 — 4 theorems)

Location: 0-Core-Formalism/lean/Semantics/Semantics/AVMR.lean
Domain: Float computation, enumeration, summation
Estimated Effort: 1-2 days each

Theorem 159: rscuSumSynonymous

Statement: Σ_{c ∈ aa} RSCU(c) = d (degeneracy)
Gap: Enumeration over all codons for an amino acid
Mathematical Approach:

• Use Fin 64 to enumerate all codons
• Filter by geneticCode c = aa
• Compute sum and prove equals codonDegeneracy aa
• Can use native_decide for each amino acid case

Subagent Assignment:
→ Agent F1 (Enumeration Specialist)
Required: Fintype, enumeration tactics, summation
Deliverable: Proof by exhaustive check or symbolic sum

---

Theorem 160: kraftInequality

Statement: Σ_{c: Codon} 2^(-L(c)) ≤ 1
Gap: Computation over all 64 codons
Mathematical Approach:

• Enumerate all 64 codons
• Compute optimalCodeLength for each
• Verify sum ≤ 1 via native_decide
• Or prove symbolically using Shannon's source coding

Subagent Assignment:
→ Agent F1 (Enumeration Specialist)
Required: Float computation, summation, native_decide
Deliverable: Numerical verification proof

---

Theorem 163: speciesBetterThanGeneric

Statement: minRedundancyCodeSize s n < (n.toFloat * 6.0) / 8.0 (average case bound)
Gap: Float inequality with entropy
Mathematical Approach: -- minRedundancyCodeSize = n * speciesEntropy s / 8 -- speciesEntropy s < 6.0 (proven) -- Direct Float inequality: n * H_s / 8 < n * 6 / 8 Note: Changed from per-codon bound (false for rare codons) to average case Subagent Assignment:
→ Agent F1 (Enumeration Specialist)
Required: Float inequalities, summation bounds
Deliverable: Direct corollary of speciesEntropy theorem

---

Theorem 162: caiBounds

Statement: 0.0 ≤ cai s gene ∧ cai s gene ≤ 1.0
Gap: Bounds on geometric mean of RSCU
Mathematical Approach:

• RSCU > 0 by definition
• RSCU ≤ max frequency / min expected
• Use Float bounds and product inequalities
• May need sorry for Float inequality lemmas

Subagent Assignment:
→ Agent F2 (Algebra Specialist)
Required: Float arithmetic, geometric mean bounds
Deliverable: Tight bounds proof

---

Tier 3: Continuous Dynamics (P0 — 1 theorem)

Location: 0-Core-Formalism/lean/Semantics/Semantics/AVMR.lean
Domain: ODE/SDE, continuous extension
Estimated Effort: 5-7 days (research problem)

Theorem 131: missingLinkODE

Statement: Continuous limit: d/dt(a,b) = (1,-1) + ε·∇J
Gap: Requires continuous extension of discrete braid dynamics
Mathematical Approach:

• Between axial events: linear drift
• At events: gradient ascent on J
• Needs piecewise ODE formulation
• May require new Lean 4 tactics for ODEs

Subagent Assignment:
→ Agent F3 (Analysis Specialist) + Human Architect
Required: ODE theory, differential inclusions, continuous dynamics
Deliverable: Formalized ODE system with existence proof
Note: This is a genuine research gap, not just formalization

---

Tier 4: Documentation (P3 — 2 theorems)

Location: docs/semantics/missingproofs/AVMR_Theorems.lean
Purpose: Template completeness

Theorem 157: codonFrequencySum

Statement: Σ codonFrequency = 1000.0
Status: Marked PROVEN in template
Action: Verify or implement native_decide proof
Subagent: Agent F1 (quick verification)

Theorem 160 (template): kraftInequality_Missing

Status: Duplicated in template
Action: Sync with main file, mark appropriately
Subagent: Agent F1 (documentation sync)

---

Summary Matrix

Tier	Priority	Count	Assignee	Domain	Effort
1	P1	3	Agent F2 + F3	Algebraic Geometry	6-9 days
2	P2	4	Agent F1	Information Theory	4-8 days
3	P0	1	Agent F3 + Human	ODE/Analysis	5-7 days
4	P3	2	Agent F1	Documentation	1 day
Total	—	10	—	—	16-25 days

---

Cross-Dependencies

missingLinkODE (P0)
    └── Depends on: tipCoordinateMassResonance (P1)
    └── Blocks: Hardware extraction of continuous dynamics

rscuSumSynonymous (P2)
    └── Enables: caiBounds proof
    └── Enables: kraftInequality proof

speciesBetterThanGeneric (P2)
    └── Depends on: speciesEntropyLessThanUniform (✅ PROVEN)
    └── Enables: Compression engine formalization

---

Success Criteria

Per Theorem:

• lake build passes with zero errors
• #eval example with expected output
• No sorry or admit in proof term
• Reference in MATH_MODEL_MAP marked ✅ PROVEN

Global:

• Layer M proven count ≥ 30 (currently 23)
• P0 theorems resolved
• Hardware extraction unblocked

---

Communication Protocol

1. Claim: Comment on this file with theorem ID + ETA
2. Progress: Daily updates in #lean-semantics channel
3. Blockers: Escalate to Human Architect after 2 days stuck
4. Completion: PR with proof + #eval witness + map update

Blackboard Sessions Required For:

• Theorem 131 (missingLinkODE)
• Theorem 124 (Fermat factorization)

---

Document ID: SUBAGENT_ASSIGNMENTS_2026-04-19
Authority: AGENTS.md §9 — Subagent delegation