TRACEABILITY GRAPH

Every formula has a pedigree. Trace it back to verified basics.

Rule: A formula is only as strong as its weakest dependency. If any node in the chain is a SORRY, everything downstream stops.

THE FOUNDATION (given — established mathematics)

[F0] √a · √b = √(a·b)          GIVEN   (field property of ℝ≥₀)
[F1] Chentsov's theorem (1972)   GIVEN   (Fisher metric uniqueness)
[F2] S⁷ round metric             GIVEN   (standard Riemannian geometry)
[F3] arccos: [-1,1] → [0,π]     GIVEN   (standard calculus)

Status: GIVEN — these are not proven here. They are established results. If any GIVEN is ever falsified, the entire graph collapses.

THE CHAIN (verified in this project)

                    [F0] √a·√b = √(ab)
                           │
                           ▼
[F1] Chentsov (invariant, not unique) ──→ [V1] g_p(u,v) = Σ uᵢvᵢ/pᵢ
                           │
                           ▼
                    [V2] φ(p) = (√p₁,...,√p₈) ∈ S⁷
                           │
                           ▼
              [F0+F2] ──→ [V3] d_{S⁷}(a,b) = arccos(⟨a,b⟩)
                           │
                           ▼
              [V2+V3] ──→ [V4] ⟨φ(p),φ(q)⟩ = Σ√(pᵢqᵢ)    ← 3 AGENTS VERIFIED
                           │                                    CONSENSUS: 0.97586930
                           ▼
        [V1+V3+V4] ──→   [V5] d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ)) ← 3 AGENTS VERIFIED
                                                                CONSENSUS: 0.440258

Legend:

[F#] = Foundation node (GIVEN, not proven here)
[V#] = Verified node (proven + 3-agent consensus in this project)
Arrow = logical dependency

VERIFIED NODES

Node	Formula	Depends On	Verified By	Consensus Value	Status
V1	g_p(u,v) = Σ uᵢvᵢ/pᵢ	F1 (Chentsov invariant)	Proven: invariance under coarse-graining (Verification 005)	N/A	✅ INVARIANT (uniqueness open)
V2	φ(p) = (√p₁,...,√p₈)	V1 (metric def)	Direct calc	‖φ(p)‖₂ = 1.0	✅ VERIFIED
V3	d_{S⁷}(a,b) = arccos(⟨a,b⟩)	F2 (round metric)	Given theorem	N/A	✅ GIVEN
V4	⟨φ(p),φ(q)⟩ = Σ√(pᵢqᵢ)	V2 + F0	Alpha,Beta,Gamma	0.97586930	✅ 3-AGENT
V5	d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ))	V1+V3+V4	Alpha,Beta,Gamma	0.440258	✅ 3-AGENT

DOWNSTREAM (pending verification)

These formulas depend on V5 being correct. They cannot be verified until V5 is firm, and they each need their own 3-agent verification.

[V5] d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ))
        │
        ├──→ [P1] C(p) = ((p₁+p₂)/2, (p₁+p₂)/2, ..., (p₇+p₈)/2)
        │           Status: PENDING — needs 3-agent verification
        │           Test: Apply C to test vector, check output sums to 1
        │
        ├──→ [P2] C(C(p)) = C(p)  (idempotence)
        │           Status: PENDING — needs 3-agent verification
        │           Depends on: P1
        │
        ├──→ [P3] d_F(C(p),C(q)) ≤ d_F(p,q)  (contraction)
        │           Status: PENDING — needs 3-agent verification
        │           Depends on: P1 + V5
        │
        ├──→ [P4] I_loss(p) = Σₖ sₖ·KL(p_{2k-1}/sₖ ‖ ½)
        │           Status: PENDING — needs 3-agent verification
        │           Depends on: P1 + V5
        │
        └──→ [P5] Φ-corkscrew: f(n) = (√n·cos(nψ), √n·sin(nψ))
                    Status: PENDING — needs 3-agent verification
                    Depends on: V5 (for the S⁷ embedding context)

Rule: P1 through P5 are SORRIES until 3-agent verified. No code for them.

THE TRACEABILITY TEST

For any claimed result, ask: Trace it back. What's the oldest verified node?

Example — d_F(p,q) = 0.440258:

d_F(p,q) = 0.440258
    ← V5: d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ))
        ← V4: ⟨φ(p),φ(q)⟩ = Σ√(pᵢqᵢ) = 0.97586930
            ← V2: φ(p) = (√p₁,...,√p₈)
                ← V1: g_p(u,v) = Σ uᵢvᵢ/pᵢ
                    ← F1: Chentsov's theorem (1972)
            ← F0: √a·√b = √(ab)
        ← V3: d_{S⁷} = arccos(⟨a,b⟩)
            ← F2: S⁷ round metric

If you doubt 0.440258, trace it. The weakest link is F1 (Chentsov's theorem, given, not proven here) or F0 (field property, given). If you accept those, the number follows inevitably.

WHAT TRACING BUYS US

No hand-waving: Every formula has a paper trail.
Targeted re-verification: If a node is questioned, only that node and downstream nodes need re-checking. Upstream verified nodes stand.
Clear SORRY boundaries: If P3 (contraction) fails, we know P1 and V5 are still solid. The failure is isolated to the contraction claim.
Independent audit: An outsider can verify any node independently by tracing back to F0-F3.

CURRENT PROJECT STATUS

GIVEN:    F0, F1, F2, F3          (4 foundation nodes)
VERIFIED: V1, V2, V3, V4, V5     (5 verified nodes, 2 with 3-agent consensus)
PENDING:  P1, P2, P3, P4, P5     (5 formulas awaiting 3-agent verification)
SORRY:    S1-S10                  (10 items from adversarial review awaiting resolution)

Next action: 3-agent verification of P1 (pair-averaging map C(p)).

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