T0-14: Discrete-Continuous Transition - Formal Specification

Core Axiom

A1: Self-referential completeness → entropy increase

Primary Constraint

No-11: ∀b ∈ Binary: ¬(b[i] = 1 ∧ b[i+1] = 1)

Fundamental Definitions

D14.1: Zeckendorf Discrete Space

Z = {n ∈ ℕ | n = Σᵢ bᵢFᵢ, bᵢ ∈ {0,1}, bᵢ·bᵢ₊₁ = 0}

D14.2: φ-adic Real Extension

r ∈ ℝ⁺: r = lim_{n→∞} Σᵢ₌₋ₙ^∞ εᵢ·Fᵢ/φⁿ
where εᵢ ∈ {0,1}, εᵢ·εᵢ₊₁ = 0

D14.3: Bridging Function

B: Z → ℝ
B(z) = Σᵢ∈I(z) Fᵢ·φ⁻ᵈᵉᵖᵗʰ⁽ⁱ⁾

D14.4: Information Cost Function

I: ℝ⁺ → ℝ⁺
I(ε) = ⌈log_φ(1/ε)⌉·log₂(φ) + O(log log(1/ε))

D14.5: Observer Precision

P(O) = min{δ | O distinguishes states separated by δ}

Core Theorems

T14.1: Continuity Necessity

SelfRef(S) ∧ Complete(S) ∧ Finite(Observer) → Continuous(Perception)

T14.2: Convergence Rate

|r - rₙ| ≤ F₋ₙ/φⁿ ≈ φ⁻²ⁿ/√5

T14.3: Information Scaling

I(ε) ~ log_φ(1/ε)·log₂(φ)

T14.4: Entropy Cost

ΔH_{d→c} = log₂(φ)·depth + H(boundary)

T14.5: No-11 Continuity Constraint

|f(x+δ) - f(x)| ≤ K·φ⁻⌊log_φ(1/δ)⌋

T14.6: φ-Differentiability

f'(x) = lim_{n→∞} [f(x + Fₙ/φ²ⁿ) - f(x)]·φ²ⁿ/Fₙ

T14.7: Continuity Threshold

P(O) > Fₖ/φ²ᵏ → Continuous(O's perception)

T14.8: Measurement-Continuity Duality

Cost(continuous perception) = log₂(φ) bits/level

T14.9: Quantum-Classical Transition

|ψ_classical⟩ = lim_{n→∞} Σ_{z∈Zₙ} αz|z⟩
where Σ|αz|² = 1

T14.10: Decoherence Scale

Γ_decoherence = φ^{E/kT}

Derived Corollaries

C14.1: Gap Unbridgeability

∀n: ∄ valid encoding between Z(n) and Z(n+1) in discrete space

C14.2: Smoothness from Constraint

No-11 → bounded variation rate in continuous functions

C14.3: Density of Zeckendorf Rationals

∀r ∈ ℝ, ∀ε > 0: ∃z ∈ Z_rational: |r - z| < ε

C14.4: Perception Determines Reality

Continuous(Reality) ⟺ P(All Observers) > discrete_separation

Relations to Other T0 Theories

R14.1: Time Continuity (T0-0)

t_continuous = lim_{n→∞} Σᵢ Fᵢ·τᵢ/φⁿ

R14.2: Depth Continuity (T0-11)

C_continuous(x) = lim_{d→∞} depth(x,d)/log_φ(d)

R14.3: Observer Continuity (T0-12)

O_perception(x) = ⌊x·φ^precision⌋/φ^precision

R14.4: Boundary Continuity (T0-13)

Boundary_continuous(x) = ∫ B(x,width)·φ⁻|x-y| dy

Mathematical Structure

Complete System

DCS = (Z, ℝ, B, I, H, O) where:
- Z: discrete Zeckendorf space
- ℝ: continuous real space  
- B: Z → ℝ bridging function
- I: ε → ℝ⁺ information cost
- H: entropy measure
- O: observer resolution function

Master Equations

z ∈ Z (discrete state)
r = B(z) + ε (continuous approximation)
I(ε) = log_φ(1/ε)·log₂(φ) (information cost)
ΔH = log₂(φ)·depth(z) (entropy increase)
O(r) = continuous if P(O) > ε (perception)

Physical Applications

Spacetime Metric

ds² = lim_{n→∞} Σ g_μν^(n) dx^μ dx^ν

Quantum Fields

φ(x) = Σ_{k∈K_φ} aₖ e^{ikx}
where K_φ satisfies No-11

Computational Verification

Convergence Rate Test

def verify_convergence():
    for n in range(100):
        error_n = φ^(-2n)/√5
        assert |approximation_error| ≤ error_n

Information Cost Test

def verify_info_cost(ε):
    bits = ceil(log(1/ε, φ)) * log(φ, 2)
    assert actual_bits ≈ bits

Core Result

Final Theorem (T0-14 Core)

Discrete(Z) + No-11 + Finite(Measurement) = Continuous(Phenomena)

∀O: Zeckendorf(Universe) ∧ Finite(O) ∧ No-11(Encoding) → 
     Continuous(Perception(O))

This establishes that continuity necessarily emerges from discrete Zeckendorf structures through φ-convergent limits constrained by No-11 and finite observation.