# 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.

---

## 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

---

## 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

---

## 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

---

## 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

---

## 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

---

## 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

---

## 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) | ✅ VERIFIED | f(20121)=(-137.80079576,-33.64432624), dist=264.41810784 | 2026-06-23 |
| 008 | F("a+b=c") byte-frequency map | ✅ VERIFIED | (0,0.2,0,0.2,0,0,0.6,0) | 2026-06-23 |
| 009 | √p embedding ‖φ(F("a+b=c"))‖₂ | ✅ VERIFIED | 1.00000000 | 2026-06-23 |
| 010 | τ("a+b=c") parse-tree feature | ✅ VERIFIED | (0.6, 0.2, 0.2, 0) | 2026-06-23 |
| 011 | G2: Φ collision breaking (a+b=c vs p/q=r) | ✅ VERIFIED | F equal, τ different, Φ different | 2026-06-23 |

---

## VERIFICATION 007: P5 — Φ-Corkscrew f(n)

**Formula:** f(n) = (√n·cos(nψ), √n·sin(nψ)), ψ = 2π/φ²

**Test values:** n₁ = 20121, n₂ = 20122
**Verified by:** Alpha, Beta, Gamma
**Consensus:**
- φ = 1.61803399
- ψ = 2.39996323
- f(20121) = (-137.80079576, -33.64432624)
- f(20122) = (124.33952368, -68.27651757)
- Distance = 264.41810784
- Different: YES (consecutive integers produce different points)
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

**Note:** Injectivity follows from ψ/2π = 1/φ² being irrational. If f(m)=f(n)
with m>n, then (m-n)ψ ∈ 2πℤ, requiring ψ/2π ∈ ℚ. But φ² = φ+1 is irrational,
so 1/φ² is irrational. Contradiction. Therefore f is injective.

---

## VERIFICATION 008: Byte-Frequency Map F("a+b=c")

**Formula:** F(E)ᵢ = countᵢ(E) / Σⱼ countⱼ(E)

**Input:** E = "a+b=c" (5 characters)
**Byte classes:** control(0), punct-low(1), digits(2), punct-mid(3), upper(4), punct-high(5), lower(6), extended(7)
**Verified by:** Alpha, Beta, Gamma
**Consensus:**
- ASCII: a=97→6, +=43→1, b=98→6, ==61→3, c=99→6
- Counts: (0, 1, 0, 1, 0, 0, 3, 0), total = 5
- F("a+b=c") = (0.000000, 0.200000, 0.000000, 0.200000, 0.000000, 0.000000, 0.600000, 0.000000)
- Sum = 1.000000 ✓, all ≥ 0 ✓
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

---

## VERIFICATION 009: √p Embedding of F("a+b=c")

**Formula:** φ(p) = (√p₁, ..., √p₈), verify ‖φ(p)‖₂ = 1

**Input:** p = F("a+b=c") = (0, 0.2, 0, 0.2, 0, 0, 0.6, 0)
**Verified by:** Alpha, Beta, Gamma
**Consensus:**
- φ(p) = (0, 0.44721360, 0, 0.44721360, 0, 0, 0.77459667, 0)
- ‖φ(p)‖₂² = 1.00000000
- ‖φ(p)‖₂ = 1.00000000 ✓
**Date:** 2026-06-23
**Status:** ✅ VERIFIED

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