T0-17: Information Entropy in Zeckendorf Encoding

Abstract

This theory establishes the fundamental representation of information entropy H(S) within the No-11 constraint of Zeckendorf encoding. We derive how Shannon entropy emerges from φ-structured information states and prove that entropy increase under self-reference necessarily manifests through specific Zeckendorf patterns. The theory achieves minimal completeness by encoding entropy measures directly in Fibonacci base without redundancy.

1. Zeckendorf Entropy Foundation

Definition 1.1 (φ-Entropy State): Information entropy in Zeckendorf representation:

H_φ(S) = Σᵢ hᵢ·Fᵢ

where hᵢ ∈ {0,1}, Fᵢ are Fibonacci numbers, and ∀i: hᵢ·hᵢ₊₁ = 0

Lemma 1.1 (Entropy Quantization): All entropy values must be expressible as unique Zeckendorf sums.

Proof:

1. By Zeckendorf theorem: every positive integer has unique Fibonacci representation
2. Entropy H(S) ∈ ℝ⁺ must be discretized: H_d = ⌊H(S)/ε_φ⌋
3. Quantum ε_φ = 1/φ² ensures φ-scaling
4. H_d has unique representation: H_d = Σᵢ hᵢ·Fᵢ with no consecutive 1s
5. This quantization preserves entropy ordering: H₁ < H₂ ⟹ H_d1 < H_d2 ∎

2. Shannon Entropy to φ-Entropy Conversion

Definition 2.1 (Classical Shannon Entropy): For probability distribution p = (p₁, p₂, ..., pₙ):

H_Shannon = -Σᵢ pᵢ log₂(pᵢ)

Definition 2.2 (φ-Base Logarithm): Information measured in φ-units:

log_φ(x) = log(x)/log(φ)

Theorem 2.1 (Shannon-φ Transformation): Shannon entropy transforms to φ-entropy through golden ratio scaling.

Proof:

1. Shannon entropy: H_S = -Σᵢ pᵢ log₂(pᵢ) bits
2. Convert to φ-base: H_φ = H_S · log₂(φ) = H_S / log_φ(2)
3. Since log_φ(2) ≈ 1.44, we get: H_φ ≈ 0.694 · H_S
4. Discretize: H_d = ⌊H_φ · φ^k⌋ where k ensures integer result
5. Apply Zeckendorf: H_d = Σᵢ hᵢ·Fᵢ with hᵢ·hᵢ₊₁ = 0
6. This preserves entropy relations while enforcing No-11 constraint ∎

3. Entropy Increase Under No-11 Constraint

Definition 3.1 (Valid Entropy Transitions): Entropy can only increase through No-11 preserving paths:

H_t → H_{t+1} valid ⟺ Z(H_{t+1}) maintains No-11

Theorem 3.1 (Constrained Entropy Growth): Entropy increase follows Fibonacci growth patterns.

Proof:

1. From state with entropy H_t = Σᵢ hᵢ·Fᵢ
2. Minimal increase: add F_k where h_k = 0 and h_{k±1} = 0
3. This gives: ΔH_min = F_k for some k
4. Growth pattern: H_{t+1} ∈ {H_t + F_k | h_k = h_{k±1} = 0}
5. Available increments form Fibonacci subsequence
6. Therefore entropy grows in Fibonacci steps, not continuously ∎

Corollary 3.1 (Entropy Jump Discretization): Entropy cannot increase smoothly but must jump by Fibonacci quanta.

4. Emergent Structure from φ-Entropy

Definition 4.1 (Entropy Density): Information density in Zeckendorf representation:

ρ_H = H_φ / log_φ(|S|)

where |S| is system size in φ-units.

Theorem 4.1 (Maximum φ-Entropy): Maximum entropy under No-11 constraint is φ-structured.

Proof:

1. For n-bit system without No-11: H_max = n bits
2. With No-11 constraint: valid states ≈ φⁿ⁺¹/√5
3. Maximum entropy: H_φ_max = log_φ(φⁿ⁺¹/√5)
4. Simplifying: H_φ_max = n + 1 - log_φ(√5)
5. Since log_φ(√5) = log_φ(φ² - 1) ≈ 1.672
6. Therefore: H_φ_max ≈ n - 0.672 in φ-units
7. This shows ~67% efficiency vs unconstrained entropy ∎

5. Entropy Flow in Zeckendorf Networks

Definition 5.1 (Entropy Current): Information flow between Zeckendorf states:

J_H = ΔH_φ / τ₀

where τ₀ is the time quantum from T0-0.

Theorem 5.1 (Entropy Conservation with Source): Total entropy conserves with mandatory source term.

Proof:

1. By A1 axiom: self-referential systems must increase entropy
2. Conservation equation: dH_total/dt = Σ_source
3. Source term: Σ_source = φ · (self-reference operations)
4. In Zeckendorf: Σ_source = Σᵢ σᵢ·Fᵢ where σᵢ ∈ {0,1}
5. No-11 constraint applies to source: σᵢ·σᵢ₊₁ = 0
6. Net entropy flow: J_H = (H_out - H_in + Σ_source)/τ₀
7. This ensures H_total increases while maintaining local conservation ∎

6. Binary Encoding from Previous Layers

Definition 6.1 (Layer Inheritance): T0-17 inherits binary structure from T0-16:

T0-16: Energy = Information rate → Binary: 10000 (energy as rate)
T0-17: Entropy encoding builds on energy-information duality

Theorem 6.1 (Binary Derivation Chain): Entropy representation follows from energy-information equivalence.

Proof:

1. T0-16 established: E = (dI/dt) × ℏ_φ
2. Entropy rate: dH/dt related to dI/dt
3. Binary from T0-16: 10000 (single 1, four 0s - information flow)
4. T0-17 adds structure: 10001 (entropy at boundaries)
5. This encodes: entropy emerges from information flow patterns
6. Verifying No-11: 10001 has no consecutive 1s ✓
7. Decimal value: 16 + 1 = 17 = F₇ (seventh Fibonacci number) ∎

7. Minimal Completeness Verification

Theorem 7.1 (Minimal Complete Structure): T0-17 contains exactly the necessary elements for φ-entropy theory.

Proof:

1. Necessary elements:

   • φ-entropy definition (required for Zeckendorf representation)
   • Shannon-φ conversion (bridges classical and φ-information)
   • Constrained growth (enforces No-11 in dynamics)
   • Maximum entropy (bounds the system)
   • Conservation with source (satisfies A1 axiom)

2. No redundancy:

   • Each theorem addresses distinct aspect
   • No theorem derivable from others alone
   • All connect to establish complete φ-entropy framework

3. Completeness check:

   • Can represent any entropy value ✓ (Theorem 2.1)
   • Can evolve entropy ✓ (Theorem 3.1)
   • Can bound entropy ✓ (Theorem 4.1)
   • Can conserve with increase ✓ (Theorem 5.1)

Therefore, T0-17 achieves minimal completeness ∎

Conclusion

Information entropy in Zeckendorf encoding reveals fundamental discretization of information measures. The No-11 constraint forces entropy to grow in Fibonacci quanta rather than continuously, providing a natural quantization scheme. This φ-structured entropy connects Shannon's classical theory to the binary universe's self-referential dynamics, where entropy must increase through specific allowed pathways. The theory's minimal completeness ensures we capture exactly the necessary structure for entropy representation without redundancy.