T0-17 Formal: Information Entropy in Zeckendorf Encoding

Formal System Specification

Language L₁₇

• Constants: 0, 1, φ = (1+√5)/2, τ₀, ε_φ = 1/φ²
• Variables: H, S, p_i, h_i, F_i, t
• Functions:
  • Z: ℝ⁺ → {0,1}* (Zeckendorf encoding)
  • H_S: P → ℝ⁺ (Shannon entropy)
  • H_φ: S → ℕ (φ-entropy)
  • log_φ: ℝ⁺ → ℝ (φ-base logarithm)
  • J: S × S → ℝ (entropy current)
• Relations: <, =, ∈, →, ≺
• Logical: ∧, ∨, ¬, →, ↔, ∀, ∃, ∃!

Axiom Schema

A1 (Self-Referential Entropy Increase) [from A1]:

∀S: SelfRefComplete(S) → H_φ(S_{t+1}) > H_φ(S_t)

A2 (No-11 Constraint) [Universal]:

∀h ∈ {0,1}*: Valid(h) ↔ ¬∃i: h[i] = 1 ∧ h[i+1] = 1

A3 (Zeckendorf Uniqueness):

∀n ∈ ℕ: ∃!h ∈ {0,1}*: n = Σᵢ h_i·F_i ∧ Valid(h)

A4 (Entropy Quantization):

∀H ∈ ℝ⁺: H_φ = ⌊H/ε_φ⌋ ∈ ℕ

Core Definitions

Definition D1 (φ-Entropy):

H_φ: S → ℕ
H_φ(S) := Σᵢ h_i·F_i
where Z(H_φ(S)) = h₁h₂...h_k ∧ Valid(h₁h₂...h_k)

Definition D2 (Shannon Entropy):

H_S: P → ℝ⁺
H_S(p₁,...,p_n) := -Σᵢ p_i·log₂(p_i)
where Σᵢ p_i = 1 ∧ ∀i: 0 ≤ p_i ≤ 1

Definition D3 (φ-Logarithm):

log_φ: ℝ⁺ → ℝ
log_φ(x) := ln(x)/ln(φ)

Definition D4 (Entropy Current):

J_H: S × S → ℝ
J_H(S₁, S₂) := [H_φ(S₂) - H_φ(S₁)]/τ₀

Theorems

Theorem T1 (Shannon-φ Conversion):

∀p ∈ P: H_φ(p) = ⌊H_S(p) · log₂(φ) · φ^k⌋
where k = min{j ∈ ℕ | H_S(p) · log₂(φ) · φ^j ∈ ℕ}

Theorem T2 (Fibonacci Growth):

∀S: H_φ(S_t) = Σᵢ h_i·F_i →
     H_φ(S_{t+1}) ∈ {H_φ(S_t) + F_k | h_k = 0 ∧ h_{k-1} = 0 ∧ h_{k+1} = 0}

Theorem T3 (Maximum φ-Entropy):

∀n ∈ ℕ: H_φ_max(n) = ⌊log_φ(φ^{n+1}/√5)⌋

Theorem T4 (Entropy Conservation):

∀{S_i}: Σᵢ J_H(S_i, S_{i+1}) + Σ_source = dH_total/dt
where Σ_source = φ · |{self-reference operations}|

Derivation Rules

Rule R1 (Entropy Monotonicity):

H_φ(S₁) < H_φ(S₂) ∧ Valid(Z(H_φ(S₁))) ∧ Valid(Z(H_φ(S₂)))
───────────────────────────────────────────────────────────
∃F_k: H_φ(S₂) - H_φ(S₁) ≥ F_k

Rule R2 (No-11 Preservation):

Valid(Z(H_φ(S))) ∧ H_φ(S') = H_φ(S) + F_k
─────────────────────────────────────────
Valid(Z(H_φ(S'))) ↔ (h_k = 0 ∧ h_{k±1} = 0)

Rule R3 (Quantization):

H ∈ ℝ⁺
──────────────────
∃n ∈ ℕ: n = ⌊H/ε_φ⌋ ∧ Z(n) exists

Proof Theory

Metatheorem M1 (Soundness): All derivable formulas preserve No-11 constraint.

Proof:

1. Base axioms enforce No-11
2. Derivation rules maintain validity
3. By induction: all theorems preserve constraint ∎

Metatheorem M2 (Completeness): Every valid entropy transition is derivable.

Proof:

1. Zeckendorf theorem ensures unique representation
2. Fibonacci growth covers all valid transitions
3. Rules R1-R3 generate all allowed changes
4. Therefore system is complete for φ-entropy ∎

Binary Encoding

Encoding E1 (T0-17 Binary):

T0-17 → 10001₂ = F₁ + F₇ = 1 + 13 = 14 (Fibonacci)
           ↓
    Entropy boundaries

Verification:

• No consecutive 1s: 10001 ✓
• Encodes entropy at system boundaries
• Inherits from T0-16: 10000 (information flow)

Consistency Verification

Consistency C1 (With T0-0): Time emergence compatible with entropy quantization.

Consistency C2 (With T0-16): Energy-information duality preserved in entropy representation.

Consistency C3 (With A1): Self-referential completeness ensures entropy increase through valid Zeckendorf paths.