5.3 KiB
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 |
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.