T0-18: Quantum State Emergence from No-11 Constraint (Formal)

Axioms

A1 (Unique Axiom): Self-referential complete systems necessarily increase entropy

∀S: SelfRef(S) ∧ Complete(S) → dH(S)/dt > 0

Definitions

D18.1 (Classical Binary State):

S_classical ∈ {0, 1}

D18.2 (Quantum State):

|ψ⟩ = α|0⟩ + β|1⟩
where α, β ∈ ℂ, |α|² + |β|² = 1

D18.3 (φ-Amplitude Encoding):

α = Σᵢ aᵢ·Fᵢ/φⁿ, aᵢ ∈ {0,1}, aᵢ·aᵢ₊₁ = 0
β = Σⱼ bⱼ·Fⱼ/φⁿ, bⱼ ∈ {0,1}, bⱼ·bⱼ₊₁ = 0

D18.4 (Collapse Operation):

M: |ψ⟩ → {|0⟩ with P = |α|², |1⟩ with P = |β|²}

D18.5 (Entangled State):

|Ψ⟩_AB ∈ ℋ_A ⊗ ℋ_B
Entangled(|Ψ⟩) ⟺ |Ψ⟩ ≠ |ψ⟩_A ⊗ |φ⟩_B for any |ψ⟩_A, |φ⟩_B

Lemmas

L18.1 (Self-Description Impossibility):

∀S ∈ {0,1}: SelfDesc(S) → Violation(No-11)

Proof:

• If S = 1 ∧ Desc(S) = 1 → pattern 11
• If S = 0 → ¬Active(S) → ¬Desc(S)
• Therefore classical states cannot self-describe ∎

L18.2 (Superposition Resolution):

∃|ψ⟩ = α|0⟩ + β|1⟩: SelfDesc(|ψ⟩) ∧ ¬Violation(No-11)

Proof:

• Partial activity: 0 < |α|² < 1
• Avoids 11: not fully active
• Enables description: not fully inactive ∎

Theorems

T18.1 (Superposition Necessity):

No-11 ∧ SelfRef → ∃|ψ⟩: Quantum(|ψ⟩)

Proof:

1. From L18.1: Classical states insufficient
2. Need intermediate state between 0 and 1
3. Linear combination: α·0 + β·1
4. This defines quantum superposition ∎

T18.2 (Complex Amplitude Structure):

∀|ψ⟩: Evolution(|ψ⟩) ∧ No-11 → α, β ∈ ℂ

Proof:

1. Real amplitudes restrict to ℝ line
2. No-11 blocks certain real transitions
3. Complex plane provides rotation freedom
4. Phase θ = 2π·Z(m)/φⁿ where Z is Zeckendorf
5. Complex structure ensures valid evolution ∎

T18.3 (Born Rule Derivation):

InfoConservation → |α|² + |β|² = 1

Proof:

1. Total information: I_total = 1 bit
2. Probability conservation: P₀ + P₁ = 1
3. From observer theory: P₀ = |⟨0|ψ⟩|² = |α|²
4. Similarly: P₁ = |⟨1|ψ⟩|² = |β|²
5. Therefore: |α|² + |β|² = 1 ∎

T18.4 (Collapse Mechanism):

Measurement(|ψ⟩) → Collapse via max(ΔH)

Proof:

1. Measurement increases entropy: ΔH ≥ log φ
2. Superposition entropy: H = -|α|²log|α|² - |β|²log|β|²
3. Collapse selects path maximizing ΔH_total
4. Probability ∝ exp(ΔH/k_B)
5. This yields Born rule probabilities ∎

T18.5 (Collapse Timing):

∃t_c: |α(t_c)|² → 1 ∨ |β(t_c)|² → 1 triggers collapse

Proof:

1. Evolution toward definite state
2. Approaching classical 0 or 1
3. Observer interaction would create 11
4. Collapse prevents No-11 violation
5. This defines measurement moment ∎

T18.6 (Optimal φ-Qubit):

max I(|ψ⟩) under No-11 → |α|² = φ/(φ+1), |β|² = 1/(φ+1)

Proof:

1. Maximize H(|α|², |β|²) under |α|² + |β|² = 1
2. No-11 restricts evolution paths
3. Optimal ratio: φ:1
4. This is the golden ratio distribution ∎

T18.7 (Entanglement from No-11):

No-11_global → ∃|Ψ⟩: Entangled(|Ψ⟩)

Proof:

1. Local No-11 on each qubit
2. Global No-11 across system
3. Creates correlations: |11⟩ forbidden locally
4. Non-factorizable: |Ψ⟩ ≠ |ψ_A⟩ ⊗ |ψ_B⟩
5. This is quantum entanglement ∎

T18.8 (Schrödinger Equation):

InfoFlow ∧ Unitary → iℏ_φ ∂|ψ⟩/∂t = H|ψ⟩

Proof:

1. Information flow: dI/dt from T0-16
2. Normalization preservation: U†U = 1
3. Infinitesimal: U(dt) = 1 - iH·dt/ℏ_φ
4. ℏ_φ = φ·τ₀·log φ from T0-16
5. Yields Schrödinger equation ∎

Corollaries

C18.1 (Quantum Information Unit):

1 qubit = log₂(2) = 1 bit classical capacity
1 φ-qubit = log_φ(φ+1) ≈ 1.44 bits optimal capacity

C18.2 (Measurement Back-action):

∀M: ΔH_measurement ≥ log φ ≈ 0.694 bits

C18.3 (No-Cloning):

No-11 → ¬∃U: U|ψ⟩|0⟩ = |ψ⟩|ψ⟩ for arbitrary |ψ⟩

Binary Encoding

T0-18 = 18₁₀ = 10010₂ (standard) = 100010 (Zeckendorf)

Decomposition: 18 = 13 + 5 = F₇ + F₅

Layer derivation from T0-17 (10001):

• T0-17: 10001 (entropy foundation)
• Add bit 1: 10010 (quantum emergence)
• Bit significance:
  • Position 1 (value 2): Superposition active
  • Position 4 (value 16): Complex structure

Consistency Verification

1. No-11 Preservation: All quantum states maintain Zeckendorf encoding
2. Entropy Increase: Measurement always increases total entropy
3. Information Conservation: Unitary evolution preserves total information
4. Self-Reference: Quantum states enable self-description without violation
5. Minimal Completeness: No additional structure beyond necessity