# VERIFICATION LOG
## 3-Agent Consensus Required. No exceptions.

**Rule:** Every formula must be computed independently by 3 agents. All 3 must
agree on every intermediate term and the final result to 6 decimal places. If
any agent disagrees, the formula is a SORRY. STOP. Investigate. All 3 must
agree before the formula is accepted.

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## VERIFICATION 001: Fisher Distance d_F(p,q)

**Formula:** d_F(p,q) = 2·arccos( Σᵢ₌₁⁸ √(pᵢqᵢ) )

**Verified by:** Alpha, Beta, Gamma
**Consensus:** 0.440258
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

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## VERIFICATION 002: √p Embedding Inner Product

**Formula:** ⟨φ(p),φ(q)⟩ = Σᵢ₌₁⁸ √(pᵢqᵢ)

**Verified by:** Alpha, Beta, Gamma
**Consensus:** 0.97586930
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

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## VERIFICATION 003: P1 — Pair-Averaging Map C(p)

**Formula:** C(p)_{2k-1} = C(p)_{2k} = (p_{2k-1} + p_{2k})/2 for k=1,2,3,4

**Input:** p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
**Verified by:** Alpha, Beta, Gamma
**Consensus:** C(p) = (0.20000000, 0.20000000, 0.10000000, 0.10000000, 0.14000000, 0.14000000, 0.06000000, 0.06000000)
**Sum check:** 1.00000000 ✓
**Nonnegativity:** All ≥ 0 ✓
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

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## VERIFICATION 004: P2 — Idempotence C(C(p)) = C(p)

**Formula:** C(C(p)) compared to C(p) component by component

**Input:** C(p) = (0.2, 0.2, 0.1, 0.1, 0.14, 0.14, 0.06, 0.06)
**Verified by:** Alpha, Beta, Gamma
**Consensus:** C(C(p)) = C(p) exactly, all 8 differences = 0.00000000
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

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## VERIFICATION 005: P3 — Contraction d_F(C(p),C(q)) ≤ d_F(p,q)

**Formula:** Compare Fisher distance before and after pair-averaging

**Inputs:** p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05), q = (0.2, 0.2, 0.1, 0.1, 0.15, 0.1, 0.1, 0.05)
**Verified by:** Alpha, Beta, Gamma
**Consensus:**
- d_F(C(p),C(q)) = 0.100441
- d_F(p,q) = 0.440258
- Inequality: 0.100441 < 0.440258 ✓ (strict)
- Contraction ratio: 0.228142
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

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## VERIFICATION 006: P4 — Information Loss I_loss(p)

**Formula:** I_loss(p) = Σₖ₌₁⁴ sₖ · KL(p_{2k-1}/sₖ, p_{2k}/sₖ ‖ ½, ½)

**Input:** p = (0.3, 0.1, 0.15, 0.05, 0.2, 0.08, 0.07, 0.05)
**Verified by:** Alpha, Beta, Gamma
**Consensus:**
- I_loss = 0.106727 nats = 0.153975 bits
- Pair 1 (0.3,0.1): 0.052325 nats
- Pair 2 (0.15,0.05): 0.026162 nats
- Pair 3 (0.2,0.08): 0.026566 nats
- Pair 4 (0.07,0.05): 0.001674 nats
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

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## STATUS BOARD

| # | Formula | Status | Consensus Value | Date |
|---|---------|--------|-----------------|------|
| 001 | d_F(p,q) = 2·arccos(Σ√(pᵢqᵢ)) | ✅ VERIFIED | 0.440258 | 2026-06-23 |
| 002 | ⟨φ(p),φ(q)⟩ = Σ√(pᵢqᵢ) | ✅ VERIFIED | 0.97586930 | 2026-06-23 |
| 003 | P1: C(p) pair-averaging | ✅ VERIFIED | (0.2,0.2,0.1,0.1,0.14,0.14,0.06,0.06) | 2026-06-23 |
| 004 | P2: Idempotence C(C(p))=C(p) | ✅ VERIFIED | All diffs = 0 | 2026-06-23 |
| 005 | P3: Contraction d_F(C(p),C(q))<d_F(p,q) | ✅ VERIFIED | 0.100441 < 0.440258 | 2026-06-23 |
| 006 | P4: Information loss I_loss(p) | ✅ VERIFIED | 0.106727 nats | 2026-06-23 |
| 007 | P5: Φ-corkscrew f(n) | ⬜ PENDING | — | — |

**Rule:** Status PENDING → 3-agent verification → VERIFIED or REJECTED.
No formula moves from PENDING to VERIFIED without all 3 agents agreeing.