T0-22: Probability Measure Emergence from Zeckendorf Uncertainty

Formal Statement

From the path multiplicity of Zeckendorf encoding and the information incompleteness under No-11 constraint, probability measures emerge necessarily in self-referential systems.

Axioms

Axiom T0-22.1 (Path Uncertainty Principle): For any integer n with Zeckendorf representation Z(n), there exist multiple algorithmic paths π₁, π₂, ... leading to the same representation, with path count N(n) ~ φ^(log_φ n)/√5.

Axiom T0-22.2 (Observer Incompleteness): No finite observer can simultaneously determine both the complete Zeckendorf state and the specific path taken to reach that state.

Definitions

Definition D22.1 (Zeckendorf Probability Space): The triple (Ω_Z, Σ_φ, P_φ) where:

• Ω_Z = {all finite binary strings satisfying No-11}
• Σ_φ = σ-algebra generated by cylinder sets
• P_φ = φ-probability measure

Definition D22.2 (φ-Probability Measure): For cylinder set [z], the measure is:

P_φ([z]) = φ^(-H_φ(z)) / Z_φ

where H_φ(z) is the φ-entropy and Z_φ is the normalization constant.

Definition D22.3 (Path Amplitude): For path π from initial to final state:

A(π) = exp(iS[π]/ℏ_φ)

where S[π] is the path action.

Theorems

Theorem T22.1 (Measure Well-Definedness): P_φ satisfies the Kolmogorov axioms:

1. Non-negativity: P_φ(A) ≥ 0 for all A ∈ Σ_φ
2. Normalization: P_φ(Ω_Z) = 1
3. Countable additivity: P_φ(∪ᵢAᵢ) = Σᵢ P_φ(Aᵢ) for disjoint {Aᵢ}

Theorem T22.2 (Born Rule Derivation): For quantum state |ψ⟩ = Σₖ αₖ|k⟩, measurement probability:

P(outcome = k) = |αₖ|² = |Σ_π∈Ωₖ A(π)|²

where Ωₖ contains all paths leading to outcome k.

Theorem T22.3 (Maximum Entropy Distribution): Under No-11 constraint, the maximum entropy probability distribution has form:

p(z) = (1/Z_φ) · φ^(-λ·v(z))

where v(z) is the Zeckendorf value and λ is determined by normalization.

Theorem T22.4 (Continuum Limit): As refinement n → ∞, discrete φ-measure P_φ^(n) converges weakly to continuous measure μ_φ with density:

dμ_φ/dx = φ^(-H_φ^cont(x))

Lemmas

Lemma L22.1 (Path Counting): The number of distinct paths to Zeckendorf representation of n grows as:

N(n) = F_{k+1} where F_k ≤ n < F_{k+1}

Lemma L22.2 (Entropy Quantization): φ-entropy takes discrete values:

H_φ ∈ {log_φ(F_k) : k ∈ ℕ}

Lemma L22.3 (Measure Concentration): P_φ concentrates on low-entropy states:

P_φ({z : H_φ(z) ≤ h}) ≥ 1 - φ^(-h)

Corollaries

Corollary C22.1 (Quantum Probability Necessity): Quantum probabilities are necessary, not fundamental—they emerge from path uncertainty under No-11.

Corollary C22.2 (Classical Limit): As ℏ_φ → 0, quantum probabilities reduce to classical determinism:

lim_{ℏ_φ→0} P_φ^quantum = δ(classical_path)

Corollary C22.3 (Information-Probability Duality): Probability and information are dual:

P(state) · I(state) = constant

Proofs

Proof of Theorem T22.1

Non-negativity follows from φ^(-H_φ(z)) > 0. Normalization is ensured by Z_φ. For countable additivity, note that cylinder sets are either disjoint or nested, and the measure construction preserves additivity through the σ-algebra generation.

Proof of Theorem T22.2

Consider all Zeckendorf-compatible paths from initial to final state. Each path contributes amplitude A(π). Due to No-11 constraint, phase correlations between adjacent paths are destroyed, leading to:

P(k) = |Σ_π A(π)|² = |αₖ|²

This recovers the Born rule from path interference under constraint.

Proof of Theorem T22.3

Maximize entropy H = -Σ p(z)log p(z) subject to normalization Σp(z) = 1 and No-11 constraint (implicit in summation domain). Using Lagrange multipliers:

∂H/∂p(z) = -log p(z) - 1 - λv(z) = 0

Solving: p(z) ∝ exp(-λv(z)) = φ^(-λ'v(z)) (choosing φ as base).

Physical Implications

Implication 1 (Measurement Cost): Every measurement requires minimum information exchange of log φ bits.

Implication 2 (Fluctuation Spectrum): Thermodynamic fluctuations scale as:

⟨(ΔE)²⟩ = k_B T² · φ · C_v

Implication 3 (Cosmological Perturbations): Primordial density perturbations have spectral index:

n_s = 2 - log_φ(2) ≈ 0.96

Computational Verification

The φ-measure can be computed algorithmically:

1. Enumerate No-11 compatible states
2. Calculate φ-entropy for each state
3. Construct probability weights φ^(-H)
4. Normalize by partition function Z_φ

Consistency with T0 Framework

• T0-17: Provides φ-entropy definition
• T0-18: Explains quantum superposition probabilities
• T0-19: Collapse probabilities determined by measure
• T0-20: Measure defined on metric space
• T0-21: Mass density probability distribution

Minimal Completeness

Theory introduces only necessary concepts:

1. Path uncertainty (physically required)
2. φ-measure construction (mathematically required)
3. Entropy maximization (principle required)

No redundant structures—satisfies Occam's razor.