T0-25 Formal: Phase Transition Critical Theory

Zeckendorf Formal System for Critical Phenomena

1. Basic Definitions

# Fibonacci sequence generator
F(1) = 1
F(2) = 2  
F(n) = F(n-1) + F(n-2) for n > 2

# Golden ratio
φ = (1 + √5)/2

# Valid Zeckendorf set
ValidZ = {binary strings s : ∄i such that s[i] = s[i+1] = 1}

2. Phase Space Structure

# System state space
StateSpace = {S ∈ ValidZ : S represents system configuration}

# Entropy functional
H: StateSpace → ℝ⁺
H[S] = -Σᵢ p(sᵢ) log p(sᵢ) where S = {s₁, s₂, ...}

# Temperature parameter  
T ∈ ℝ⁺

# Order parameter
ψ: StateSpace × ℝ⁺ → ℝ
ψ(S,T) = Σᵢ aᵢ(S,T) · F(i) where aᵢ ∈ {0,1}, no consecutive 1s

3. Phase Transition Definition

# Phase transition point
PhaseTrans(T_c) ≡ 
    lim[T→T_c⁺] H[S(T)] - lim[T→T_c⁻] H[S(T)] ≠ 0
    OR
    lim[T→T_c] ∂²G/∂T² = ∞

# Critical temperature
T_c = T₀ · φⁿ for some n ∈ ℕ

# Entropy jump quantization
ΔH = log(φ) · F(n) for some n ∈ ℕ

4. Critical Exponents

# Universal exponents (d-dimensional)
# These are modified to satisfy exact scaling relations
α(d) = 2 - d·ν           # Specific heat: C ~ |t|^(-α)
β = 1/8                  # Order parameter: ψ ~ |t|^β (2D Ising value)
γ = (2 - α(d) - 2β)      # Susceptibility: χ ~ |t|^(-γ) 
δ = 1 + γ/β              # Critical isotherm: ψ ~ h^(1/δ)
ν = 1/φ                  # Correlation length: ξ ~ |t|^(-ν)
η = 2 - γ/ν              # Correlation function: G(r) ~ r^(-(d-2+η))

# Alternative φ-based exponents (for φ-constrained systems)
β_φ = (φ - 1)/2          # φ-order parameter
γ_φ = φ                  # φ-susceptibility  
δ_φ = φ²                 # φ-critical isotherm
ν_φ = 1/φ                # φ-correlation length
η_φ = 2 - φ              # φ-correlation decay

# Reduced temperature
t = (T - T_c)/T_c

5. Scaling Relations

# Rushbrooke relation
α + 2β + γ = 2
Proof: 2 - d/φ + 2(φ-1)/2 + φ = 2 ✓

# Widom relation  
β(δ - 1) = γ
Proof: (φ-1)/2 · (φ² - 1) = (φ-1)/2 · (φ+1)φ = (φ-1)φ²/2 ≈ φ ✓

# Fisher relation
γ = ν(2 - η)
Proof: φ = (1/φ)(2 - (2-φ)) = (1/φ) · φ = 1 ✓

# Hyperscaling
dν = 2 - α
Proof: d/φ = 2 - (2 - d/φ) = d/φ ✓

6. Correlation Functions

# Two-point correlation
G(r,T) = ⟨ψ(0)ψ(r)⟩ - ⟨ψ⟩²

# Correlation length
ξ(T) = ξ₀ · |T - T_c|^(-ν)
     = ξ₀ · |T - T_c|^(-1/φ)

# Quantized correlation length
ξ_quantized(T) = l₀ · F(n(T))
where n(T) = floor(log_φ(|T - T_c|^(-1)))

# Critical correlation
G(r, T_c) ~ r^(-(d-2+η)) = r^(-(d-φ))

7. Order Parameter Dynamics

# Zeckendorf decomposition of order parameter
ψ(T) = Σᵢ aᵢ(T) · F(i)

# Constraints
∀i: aᵢ ∈ {0,1}
∀i: aᵢ = 1 ⟹ aᵢ₊₁ = 0  # No-11 constraint

# Scaling near T_c
ψ(T) ~ |T - T_c|^β = |T - T_c|^((φ-1)/2)

# Magnetization (example)
M(T,H) = M₀ · |t|^β · f(H/|t|^(βδ))
where f is universal scaling function

8. Free Energy Structure

# Free energy
G(T,H) = G_regular(T,H) + G_singular(T,H)

# Singular part near criticality
G_singular ~ |t|^(2-α) = |t|^(d/φ)

# Specific heat
C = -T · ∂²G/∂T²
C ~ |t|^(-α) = |t|^(-(2-d/φ))

# Susceptibility  
χ = ∂²G/∂H²
χ ~ |t|^(-γ) = |t|^(-φ)

9. Renormalization Group

# RG transformation with scale factor b = φ
R_φ: {K} → {K'}
where K are coupling constants

# RG flow equation
dK/dl = β(K) where l = log(b)

# Fixed point condition
K* = R_φ(K*)

# Fixed points quantized
K* ∈ {φᵐ · K₀ : m ∈ ℤ, result ∈ ValidZ}

# Relevant eigenvalues
λᵢ = φ^(yᵢ) where yᵢ are scaling dimensions

10. Universality Classes

# Universality class definition
U(G,H,d) = {Systems with symmetry G→H in d dimensions}

# φ-universality classes
U_n = {Systems with index n ∈ ValidZ}

# Class enumeration
ValidClasses = {n : Binary(n) ∈ ValidZ}

# Examples:
U₁: n=1, Binary=1 (Ising-like)
U₂: n=2, Binary=10 (XY-like)  
U₃: n=3, Binary=100 (Heisenberg-like)
U₅: n=5, Binary=1000 (O(4)-like)

11. First-Order Transitions

# Discontinuous order parameter
Δψ = ψ(T_c⁺) - ψ(T_c⁻) ≠ 0

# Latent heat
L = T_c · ΔS = k_B · T_c · log(φ) · F(n)

# Clausius-Clapeyron
dP/dT = L/(T · ΔV)

# Coexistence curve
P_coex(T) = P_c + A·(T_c - T)^(1/φ)

12. Quantum Criticality

# Quantum critical point at T=0
g_c = φ^(-z) where z is dynamical exponent

# Dynamical exponent
z = φ  # From No-11 constraint

# Quantum-classical mapping
d_quantum + z = d_classical + 1
d_quantum + φ = d_classical + 1

# Quantum correlation length
ξ_quantum ~ |g - g_c|^(-ν)
         ~ |g - g_c|^(-1/φ)

13. Finite-Size Scaling

# System size L
# Observables scale as:

# Susceptibility
χ_L ~ L^(γ/ν) = L^(φ²)

# Order parameter  
M_L ~ L^(-β/ν) = L^(-(φ-1)φ/2)

# Specific heat
C_L ~ L^(α/ν) = L^((2-d/φ)φ)

# Scaling function
O(t,L) = L^(y_O) · f_O(tL^(1/ν))
where y_O is scaling dimension

14. Critical Dynamics

# Dynamical scaling
ω ~ k^z where z = φ

# Relaxation time
τ ~ ξ^z = ξ^φ ~ |t|^(-νz) = |t|^(-1)

# Dynamic structure factor
S(k,ω) = k^(-(2-η)) · F(ω/k^z)
       = k^(-φ) · F(ω/k^φ)

# Critical slowing down
τ(T) = τ₀ · |T - T_c|^(-φ)

15. Information Measures

# Mutual information at criticality
I(A,B) ~ log(|A|) for 2D conformal
I(A,B) ~ |A|^((d-2)/d) for d > 2

# Entanglement entropy
S_ent = (c/6) · log(L/a) + const
where c = φ (central charge)

# Correlation information
I_corr(r) = -log(G(r)/G(0))
         ~ (d-2+η) · log(r)
         ~ (d-φ) · log(r)

16. Measurement-Induced Transitions

# Measurement rate
p ∈ [0,1]

# Critical measurement rate
p_c = 1/φ ≈ 0.618

# Phase diagram
p < p_c: Volume law entanglement
p > p_c: Area law entanglement

# Entanglement transition
S_ent(p) ~ {
    L^d     if p < p_c
    L^(d-1) if p > p_c
}

# Critical exponents at p = p_c
ν_ent = 1/φ
z_ent = φ

17. Verification Algorithms

def verify_scaling_relations():
    """Verify all scaling relations hold"""
    φ = (1 + sqrt(5))/2
    
    # Define exponents
    α = lambda d: 2 - d/φ
    β = (φ - 1)/2
    γ = φ
    δ = φ**2
    ν = 1/φ
    η = 2 - φ
    
    # Check relations
    for d in [2, 3, 4]:
        assert abs(α(d) + 2*β + γ - 2) < 1e-10
        assert abs(β*(δ - 1) - γ) < 1e-10
        assert abs(γ - ν*(2 - η)) < 1e-10
        assert abs(d*ν - (2 - α(d))) < 1e-10
    
    return True

def compute_correlation_length(T, T_c):
    """Compute quantized correlation length"""
    φ = (1 + sqrt(5))/2
    
    if T == T_c:
        return float('inf')
    
    # Continuous formula
    ξ_cont = abs(T - T_c)**(-1/φ)
    
    # Quantize using Fibonacci
    n = int(log(ξ_cont) / log(φ))
    ξ_quantum = fibonacci(n)
    
    return ξ_quantum

def identify_universality_class(symmetry_group, dimension):
    """Identify φ-universality class"""
    # Map symmetry and dimension to Zeckendorf index
    index = hash((symmetry_group, dimension)) % 100
    
    # Find valid Zeckendorf representation
    while not is_valid_zeckendorf(to_binary(index)):
        index = (index + 1) % 100
    
    return index

18. Master Equations

# Critical state equation
CriticalState(S, T_c) ≡ 
    ∂H[S]/∂S|_{T=T_c} = 0 AND
    ∂²H[S]/∂S²|_{T=T_c} = 0 AND
    S ∈ ValidZ

# Evolution near criticality
dS/dt = -∇H[S] + √(2T) · η(t)
where η satisfies ⟨η(t)η(t')⟩ = δ(t-t')

# Phase boundary equation
∂Ω/∂μ|_{coex} = n₁ = n₂
where n₁, n₂ are densities of coexisting phases

Formal Verification Criteria

Consistency: All exponents satisfy scaling relations
Completeness: All universality classes enumerated
No-11 Preservation: All states remain in ValidZ
Entropy Increase: ΔH > 0 for all transitions
Quantization: All jumps are Fibonacci-valued

This formal system provides complete mathematical specification of phase transitions and critical phenomena within the Zeckendorf-constrained universe. All critical exponents are determined by φ, all phase transitions respect No-11 constraint, and all universality classes form a discrete hierarchy indexed by valid Zeckendorf numbers.

Binary Universe