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docs(traceability): every formula has a pedigree — trace back to verified basics
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TRACEABILITY_GRAPH.md
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TRACEABILITY_GRAPH.md
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# TRACEABILITY GRAPH
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## Every formula has a pedigree. Trace it back to verified basics.
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**Rule:** A formula is only as strong as its weakest dependency. If any node
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in the chain is a SORRY, everything downstream stops.
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---
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## THE FOUNDATION (given — established mathematics)
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```
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[F0] √a · √b = √(a·b) GIVEN (field property of ℝ≥₀)
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[F1] Chentsov's theorem (1972) GIVEN (Fisher metric uniqueness)
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[F2] S⁷ round metric GIVEN (standard Riemannian geometry)
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[F3] arccos: [-1,1] → [0,π] GIVEN (standard calculus)
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```
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Status: **GIVEN** — these are not proven here. They are established results.
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If any GIVEN is ever falsified, the entire graph collapses.
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---
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## THE CHAIN (verified in this project)
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```
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[F0] √a·√b = √(ab)
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│
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▼
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[F1] Chentsov ──→ [V1] g_p(u,v) = Σ uᵢvᵢ/pᵢ
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│
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▼
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[V2] φ(p) = (√p₁,...,√p₈) ∈ S⁷
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│
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▼
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[F0+F2] ──→ [V3] d_{S⁷}(a,b) = arccos(⟨a,b⟩)
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│
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▼
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[V2+V3] ──→ [V4] ⟨φ(p),φ(q)⟩ = Σ√(pᵢqᵢ) ← 3 AGENTS VERIFIED
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│ CONSENSUS: 0.97586930
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▼
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[V1+V3+V4] ──→ [V5] d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ)) ← 3 AGENTS VERIFIED
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CONSENSUS: 0.440258
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```
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**Legend:**
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- [F#] = Foundation node (GIVEN, not proven here)
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- [V#] = Verified node (proven + 3-agent consensus in this project)
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- Arrow = logical dependency
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---
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## VERIFIED NODES
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| Node | Formula | Depends On | Verified By | Consensus Value | Status |
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|------|---------|-----------|-------------|-----------------|--------|
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| V1 | g_p(u,v) = Σ uᵢvᵢ/pᵢ | F1 (Chentsov) | Given theorem | N/A | ✅ GIVEN |
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| V2 | φ(p) = (√p₁,...,√p₈) | V1 (metric def) | Direct calc | ‖φ(p)‖₂ = 1.0 | ✅ VERIFIED |
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| V3 | d_{S⁷}(a,b) = arccos(⟨a,b⟩) | F2 (round metric) | Given theorem | N/A | ✅ GIVEN |
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| **V4** | **⟨φ(p),φ(q)⟩ = Σ√(pᵢqᵢ)** | **V2 + F0** | **Alpha,Beta,Gamma** | **0.97586930** | **✅ 3-AGENT** |
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| **V5** | **d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ))** | **V1+V3+V4** | **Alpha,Beta,Gamma** | **0.440258** | **✅ 3-AGENT** |
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---
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## DOWNSTREAM (pending verification)
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These formulas depend on V5 being correct. They cannot be verified until V5 is
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firm, and they each need their own 3-agent verification.
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```
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[V5] d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ))
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│
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├──→ [P1] C(p) = ((p₁+p₂)/2, (p₁+p₂)/2, ..., (p₇+p₈)/2)
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│ Status: PENDING — needs 3-agent verification
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│ Test: Apply C to test vector, check output sums to 1
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│
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├──→ [P2] C(C(p)) = C(p) (idempotence)
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│ Status: PENDING — needs 3-agent verification
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│ Depends on: P1
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│
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├──→ [P3] d_F(C(p),C(q)) ≤ d_F(p,q) (contraction)
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│ Status: PENDING — needs 3-agent verification
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│ Depends on: P1 + V5
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│
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├──→ [P4] I_loss(p) = Σₖ sₖ·KL(p_{2k-1}/sₖ ‖ ½)
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│ Status: PENDING — needs 3-agent verification
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│ Depends on: P1 + V5
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│
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└──→ [P5] Φ-corkscrew: f(n) = (√n·cos(nψ), √n·sin(nψ))
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Status: PENDING — needs 3-agent verification
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Depends on: V5 (for the S⁷ embedding context)
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```
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**Rule:** P1 through P5 are SORRIES until 3-agent verified. No code for them.
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---
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## THE TRACEABILITY TEST
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For any claimed result, ask: **Trace it back. What's the oldest verified node?**
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**Example — d_F(p,q) = 0.440258:**
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```
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d_F(p,q) = 0.440258
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← V5: d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ))
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← V4: ⟨φ(p),φ(q)⟩ = Σ√(pᵢqᵢ) = 0.97586930
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← V2: φ(p) = (√p₁,...,√p₈)
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← V1: g_p(u,v) = Σ uᵢvᵢ/pᵢ
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← F1: Chentsov's theorem (1972)
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← F0: √a·√b = √(ab)
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← V3: d_{S⁷} = arccos(⟨a,b⟩)
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← F2: S⁷ round metric
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```
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If you doubt 0.440258, trace it. The weakest link is F1 (Chentsov's theorem,
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given, not proven here) or F0 (field property, given). If you accept those,
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the number follows inevitably.
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---
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## WHAT TRACING BUYS US
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1. **No hand-waving:** Every formula has a paper trail.
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2. **Targeted re-verification:** If a node is questioned, only that node and
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downstream nodes need re-checking. Upstream verified nodes stand.
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3. **Clear SORRY boundaries:** If P3 (contraction) fails, we know P1 and V5
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are still solid. The failure is isolated to the contraction claim.
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4. **Independent audit:** An outsider can verify any node independently by
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tracing back to F0-F3.
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---
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## CURRENT PROJECT STATUS
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```
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GIVEN: F0, F1, F2, F3 (4 foundation nodes)
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VERIFIED: V1, V2, V3, V4, V5 (5 verified nodes, 2 with 3-agent consensus)
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PENDING: P1, P2, P3, P4, P5 (5 formulas awaiting 3-agent verification)
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SORRY: S1-S10 (10 items from adversarial review awaiting resolution)
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```
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**Next action:** 3-agent verification of P1 (pair-averaging map C(p)).
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