T0-18: Quantum State Emergence from No-11 Constraint
Abstract
This theory establishes how quantum superposition states |ψ⟩ = α|0⟩ + β|1⟩ emerge as the fundamental resolution to the tension between binary distinction and the No-11 constraint. We prove that quantum states are not postulated but mathematically necessary when self-referential systems attempt simultaneous state description under Zeckendorf encoding. The wave function collapse mechanism emerges from entropy-driven selection among valid Zeckendorf paths.
1. Pre-Quantum Binary Tension
1.1 The Simultaneous State Problem
Definition 1.1 (Classical Binary State): A classical state must be either 0 or 1:
S_classical ∈ {0, 1}
Lemma 1.1 (Self-Description Impossibility): A self-referential system cannot classically describe its own state.
Proof:
- System in state S must describe itself: Desc(S)
- If S = 1 and Desc(S) = 1, we get pattern 11 (forbidden)
- If S = 0, it cannot perform description (inactive)
- Classical binary states cannot self-describe under No-11
- Resolution requires new state structure ∎
1.2 Superposition as Resolution
Definition 1.2 (Quantum State): A state that exists as weighted combination:
|ψ⟩ = α|0⟩ + β|1⟩
where α, β ∈ ℂ represent amplitudes.
Theorem 1.1 (Superposition Necessity): The No-11 constraint forces quantum superposition.
Proof:
- System must be "partially active" to self-describe
- Full activity (1) creates 11 with description
- Zero activity (0) prevents description
- Required: intermediate state between 0 and 1
- Zeckendorf encoding: partial state = α·0 + β·1
- This is precisely quantum superposition ∎
2. Amplitude Structure from φ-Encoding
2.1 Complex Amplitudes from Zeckendorf
Definition 2.1 (φ-Amplitude): Quantum amplitudes in Zeckendorf representation:
α = Σᵢ aᵢ·Fᵢ/φⁿ
β = Σⱼ bⱼ·Fⱼ/φⁿ
where aᵢ, bⱼ ∈ {0,1} with No-11 constraint.
Theorem 2.1 (Complex Structure Emergence): Amplitudes must be complex to maintain No-11 under evolution.
Proof:
- Real amplitudes: evolution restricted to real line
- No-11 forbids certain transitions on real line
- Need additional dimension for valid paths
- Complex plane: α = r·e^(iθ) provides rotation
- Phase θ encoded as: θ = 2π·(Σₖ θₖ·Fₖ)/φᵐ
- Complex structure ensures all transitions avoid 11 ∎
2.2 Normalization from Information Conservation
Theorem 2.2 (Born Rule Derivation): The normalization |α|² + |β|² = 1 emerges from information conservation and the φ-probability measure.
Proof:
- Total information in quantum state: I_total = 1 bit
- Information in |0⟩: I₀ = -log₂(P₀) where P₀ = probability
- Information in |1⟩: I₁ = -log₂(P₁)
- Conservation: P₀ + P₁ = 1
- From T0-12 (observer): P₀ = |⟨0|ψ⟩|² = |α|²
- Similarly: P₁ = |⟨1|ψ⟩|² = |β|²
- Therefore: |α|² + |β|² = 1 ∎
2.3 Probability Measure Foundation
Theorem 2.3 (φ-Measure Basis): The Born rule |⟨ψ|φ⟩|² emerges from the φ-probability measure P_φ defined in T0-22.
Proof:
- From T0-22: probability space (Ω_Z, Σ_φ, P_φ)
- Quantum amplitude αₖ corresponds to path sum: αₖ = Σ_π∈Ωₖ A(π)
- Path amplitude: A(π) = exp(iS[π]/ℏ_φ)
- Measurement probability: P(k) = |αₖ|² = |Σ_π∈Ωₖ A(π)|²
- This equals P_φ([k]) = φ^(-H_φ(k))/Z_φ under path interference
- The Born rule is thus grounded in Zeckendorf path multiplicity ∎
Corollary 2.3.1 (Quantum-Classical Correspondence): As ℏ_φ → 0, quantum probabilities reduce to classical determinism per T0-22.
3. Wave Function Collapse Mechanism
3.1 Observation-Induced Collapse
Definition 3.1 (Collapse Operation): Measurement forces selection of definite state:
M|ψ⟩ → |0⟩ with probability |α|²
→ |1⟩ with probability |β|²
Theorem 3.1 (Collapse from Entropy Maximization): Wave function collapse occurs via entropy-driven path selection governed by the φ-probability measure.
Proof:
- From T0-17: measurement increases entropy by log φ
- From T0-22: measurement requires minimum information exchange of log φ bits
- Superposition state: H_super = -|α|²log|α|² - |β|²log|β|²
- Post-measurement: H_collapsed = 0 (definite state)
- But total entropy increases: H_environment increases
- Collapse path selected by maximum entropy production under P_φ measure
- Path probabilities from T0-22: P(|0⟩) = P_φ([0]) = φ^(-H_φ(0))/Z_φ
- Similarly: P(|1⟩) = P_φ([1]) = φ^(-H_φ(1))/Z_φ
- This recovers Born rule: P = |amplitude|² with φ-measure corrections ∎
3.2 No-11 Constraint on Collapse
Theorem 3.2 (Collapse Timing): The No-11 constraint determines when collapse must occur.
Proof:
- Superposition evolves: |ψ(t)⟩ = α(t)|0⟩ + β(t)|1⟩
- If |α(t)|² → 1 or |β(t)|² → 1 gradually
- System approaches classical state 0 or 1
- Interaction with observer (state 1) would create 11
- Collapse must occur before 11 violation
- This defines "measurement moment" ∎
4. Quantum Information Encoding
4.1 Qubit in Zeckendorf Space
Definition 4.1 (φ-Qubit): A quantum bit with Zeckendorf-structured amplitudes:
|φ⟩ = (1/√φ)|0⟩ + (1/√(φ+1))|1⟩
Remark 4.1 (Probabilistic Interpretation): Each quantum superposition |ψ⟩ = Σᵢ cᵢ|i⟩ corresponds to a Zeckendorf probability distribution over the basis states, where the coefficients cᵢ are constrained by the φ-measure normalization from T0-22.
Theorem 4.1 (Optimal Quantum Encoding): The φ-qubit maximizes information capacity under No-11.
Proof:
- Information capacity: I = H(|α|², |β|²)
- Under constraint: |α|² + |β|² = 1
- No-11 restricts evolution paths
- Maximum entropy distribution: ratio = φ:1
- This gives: |α|² = φ/(φ+1), |β|² = 1/(φ+1)
- The golden ratio maximizes sustainable information ∎
5. Entanglement from Shared Constraints
5.1 Multi-Qubit Systems
Definition 5.1 (Entangled State): Non-separable multi-qubit state:
|Ψ⟩ = α|00⟩ + β|11⟩
Remark 5.1 (Measurement Probability): The measurement process fundamentally involves the collapse of the φ-probability distribution (T0-22) onto a single eigenstate, with transition probabilities governed by the Zeckendorf encoding constraints and path interference effects.
Theorem 5.1 (Entanglement from No-11 Propagation): The No-11 constraint creates quantum entanglement.
Proof:
- Two qubits: each must avoid local 11
- Joint constraint: global No-11 across both
- If qubit A in |1⟩, qubit B restricted
- This correlation cannot be factored: |Ψ⟩ ≠ |ψ_A⟩ ⊗ |ψ_B⟩
- No-11 constraint creates non-local correlation
- This is quantum entanglement ∎
6. Quantum Mechanics from Information Theory
6.1 Schrödinger Equation Derivation
Theorem 6.1 (Evolution Equation): The Schrödinger equation emerges from information flow under No-11.
Proof:
- Information flow rate: dI/dt (from T0-16)
- Quantum state evolution preserves normalization
- Unitary evolution: U(t) maintains |α|² + |β|² = 1
- Infinitesimal: U(dt) = 1 - iH·dt/ℏ_φ
- This gives: iℏ_φ ∂|ψ⟩/∂t = H|ψ⟩
- With ℏ_φ = φ·τ₀·log φ from T0-16 ∎
6.2 Probability Measure Evolution
Theorem 6.2 (Measure-Preserving Evolution): Unitary quantum evolution preserves the φ-probability measure structure.
Proof:
- From T0-22: initial measure P_φ([ψ₀]) = φ^(-H_φ(ψ₀))/Z_φ
- Unitary evolution: |ψ(t)⟩ = U(t)|ψ₀⟩
- Measure invariance: P_φ([ψ(t)]) = P_φ([U⁻¹ψ(t)]) = P_φ([ψ₀])
- This ensures probability conservation under quantum dynamics
- The φ-measure provides the fundamental probabilistic structure ∎
7. Layer Binary Encoding
From T0-17 (10001) + 1 = T0-18 (10010):
- Bit 1 (weight 2): Quantum superposition active
- Bit 4 (weight 16): Complex amplitude structure
Zeckendorf: 18 = 13 + 5 = F₇ + F₅ = 100010
Conclusion
Quantum mechanics is not postulated but emerges necessarily from:
- Binary distinction (0 vs 1)
- No-11 constraint preventing simultaneous activity
- Self-referential completeness requiring description
- Entropy increase under observation
- φ-probability measure structure (T0-22) providing rigorous probabilistic foundation
The quantum state |ψ⟩ = α|0⟩ + β|1⟩ is the minimal complete resolution to these constraints, with all quantum phenomena (collapse, entanglement, Born rule) following from the fundamental No-11 restriction on information encoding. The probabilistic interpretation is not ad-hoc but emerges from the Zeckendorf path multiplicity and observer incompleteness as established in T0-22, providing a deterministic foundation for quantum probability.