T0-23 Formal: Causal Cone Structure

Zeckendorf Foundation Elements

Layer 0: Binary Encoding Base

Z₀ = {0, 1}  # Binary states
F = {1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, ...}  # Fibonacci sequence

Layer 1: No-11 Constraint Application

Binary representation: 10101000 (valid) Forbidden pattern: 11 (consecutive ones)

Zeckendorf encoding of causal constraints:

• Information state: I ∈ {0, 1}
• Simultaneous states: I(x,t) ⊗ I(y,t)
• No-11 requirement: I(x,t) · I(y,t) = 0 if x ≠ y at same t

Layer-by-Layer Theoretical Construction

Layer 2: Information Propagation Speed

From Layer 1 binary: 10101000 → extract pattern Pattern indicates: finite propagation speed required

Maximum Speed Derivation:

c_φ = l₀/τ₀

where:

• l₀ = minimum spatial quantum (from T0-15)
• τ₀ = minimum time quantum (from T0-0)

Zeckendorf representation of c:

c = F₉ + F₇ + F₄ = 34 + 13 + 3 = 50 (normalized units)
Binary: 100101000

Layer 3: Lightcone Geometry

From Layer 2 binary: 100101000 → extract structure

Future Lightcone:

L⁺(E) = {(x,t) : |x - x₀| ≤ c_φ(t - t₀), t > t₀}

Zeckendorf encoding of cone angle:

θ = arctan(1/c_φ) = F₅/F₈ = 5/21
Binary: 10010

Past Lightcone:

L⁻(E) = {(x,t) : |x - x₀| ≤ c_φ(t₀ - t), t < t₀}

Layer 4: Causal Classification

From Layer 3 binary: 10010 → derive intervals

Interval Types (Zeckendorf encoded):

1. Timelike: ds² < 0
   • Binary: 10000 (F₆ = 8)
2. Lightlike: ds² = 0
   • Binary: 00000 (null)
3. Spacelike: ds² > 0
   • Binary: 01010 (F₄ + F₂ = 3 + 1 = 4)

Layer 5: Metric Structure

From Layer 4 binary patterns → construct metric

φ-Minkowski Metric:

ds²_φ = -c²_φdt² + φ^(-2n)(dx² + dy² + dz²)

Recursive depth n in Zeckendorf:

n = F₆ + F₃ = 8 + 2 = 10
Binary: 100100

Entropy Analysis

Initial State Entropy

H₀ = 0 (single reference point)

After Causal Structure Emergence

H₁ = log(F₈) = log(21) ≈ 3.04 bits

With Full Lightcone

H₂ = log(F₁₀) = log(55) ≈ 5.78 bits

Entropy Increase Verification

ΔH = H₂ - H₀ = 5.78 bits > 0 ✓

Satisfies A1 axiom: self-referential system increases entropy.

Minimal Completeness Verification

Required Elements (all present):

1. ✓ Speed of light c_φ definition
2. ✓ Future/past lightcone structure
3. ✓ Causal classification (timelike/lightlike/spacelike)
4. ✓ Metric with φ-scaling
5. ✓ No-11 constraint enforcement

Excluded Elements (not needed):

• Detailed quantum corrections
• Black hole thermodynamics
• Cosmological implications
• Higher-order relativistic effects

Formal System Summary

CausalCone = {
    Speed: c_φ = l₀/τ₀,
    Cones: (L⁺, L⁻),
    Intervals: {timelike, lightlike, spacelike},
    Metric: ds²_φ,
    Constraint: No-11
}

All values Zeckendorf-encoded with binary verification:

• No consecutive 1s found ✓
• Entropy strictly increasing ✓
• Minimal complete structure achieved ✓

Key Predictions (Zeckendorf Values)

1. Lightcone discretization:

   Δr = l_P × φⁿ where n = F₅ = 5
   Binary: 10010
   

2. Critical information density:

   ρ_crit = 1/(l³_P × φ³) where φ³ ≈ F₇/F₄ = 13/3
   Binary: 101000/100
   

3. Causal uncertainty scale:

   ΔL = ℏ_φ/Δp where ℏ_φ = F₈ = 21
   Binary: 1000100
   

Final verification: All theoretical constructs maintain No-11 constraint and exhibit monotonic entropy increase, confirming minimal completeness under A1 axiom.