T0-13 System Boundaries: Formal Specification

Core Data Types

BoundaryState = Zeckendorf  # Boundary configuration in Z-space
Position = int  # Fibonacci index for boundary position  
Thickness = float  # Boundary thickness in φ units
PermeabilityMatrix = List[List[float]]  # Information passage probabilities
FlowRate = float  # Information flow in φ bits/τ₀

Formal Definitions

D1: System Boundary

class Boundary:
    position: List[int]  # Fibonacci indices
    thickness: float  # In units of φ
    permeability: PermeabilityMatrix
    state: BoundaryState
    
    def __init__(self, system_size: int):
        # Boundary emerges from self-reference requirement
        self.position = self._compute_fibonacci_positions(system_size)
        self.thickness = self._quantize_thickness()
        self.permeability = self._initialize_permeability()
        self.state = self._encode_boundary()

D2: Boundary Emergence Function

def boundary_emergence(S: System) -> Boundary:
    """
    Computes emergent boundary from self-referential system
    
    Satisfies:
    1. Preserves system identity: I(S_in) ≠ I(S_out)
    2. Limits entropy flow: dH/dt < ∞
    3. Maintains No-11 constraint
    """
    entropy_pressure = compute_entropy_gradient(S)
    critical_thickness = phi ** ceil(log(entropy_pressure, phi))
    positions = fibonacci_sequence_up_to(S.size)
    
    # Remove consecutive positions (No-11)
    valid_positions = filter_non_consecutive(positions)
    
    return Boundary(
        position=valid_positions,
        thickness=critical_thickness,
        permeability=compute_permeability_matrix(S),
        state=zeckendorf_encode(valid_positions)
    )

D3: Information Flow Operator

def information_flow(B: Boundary, I_in: Information) -> Information:
    """
    Computes information passing through boundary
    
    Flow equation:
    I_out = Σᵢ P(B, Iᵢ) * filter(B, Iᵢ)
    
    Where P(B, I) = passage probability
    filter(B, I) = boundary filtering function
    """
    I_zeck = zeckendorf_encode(I_in)
    B_zeck = B.state
    
    # Check No-11 compatibility
    if creates_consecutive_ones(I_zeck, B_zeck):
        return Information(0)  # Blocked
    
    # Compute passage probability
    p_pass = compute_passage_probability(B.permeability, I_zeck)
    
    # Apply filtering with entropy cost
    filtered = apply_boundary_filter(I_in, B)
    entropy_cost = len(filtered) * log2(phi)
    
    return Information(
        data=filtered,
        entropy_added=entropy_cost,
        flow_rate=p_pass * phi / tau_0
    )

D4: Boundary Thickness Quantization

def quantize_thickness(raw_thickness: float) -> float:
    """
    Quantizes boundary thickness to powers of φ
    
    τ ∈ {φ⁰, φ¹, φ², ...}
    """
    if raw_thickness <= 0:
        return 1.0  # Minimum thickness
    
    n = floor(log(raw_thickness, phi))
    return phi ** n

D5: Openness Measure

def system_openness(S: System) -> int:
    """
    Computes discrete openness degree
    
    Ω(S) = Σᵢ P(Bᵢ) where i ∈ Fibonacci indices
    
    Returns Zeckendorf integer
    """
    boundaries = S.get_all_boundaries()
    total_permeability = 0
    
    for b in boundaries:
        if b.position in fibonacci_positions:
            p = sum(sum(row) for row in b.permeability)
            total_permeability += zeckendorf_round(p)
    
    return zeckendorf_encode(total_permeability)

Formal Axioms

A1: Boundary Necessity

axiom_boundary_necessity = """
∀S: self_referential(S) ∧ complete(S) →
    ∃B: boundary(B) ∧ separates(B, S, ¬S)
"""

A2: No-11 Constraint Preservation

axiom_no11_preservation = """
∀B: boundary(B) → 
    ∀i,j: adjacent(pos(B,i), pos(B,j)) →
        ¬(active(B,i) ∧ active(B,j))
"""

A3: Entropy Flow Requirement

axiom_entropy_flow = """
∀S: self_referential(S) →
    ∃Φ > 0: entropy_flow_rate(boundary(S)) = Φ
"""

Formal Theorems

T1: Boundary Position Quantization

def theorem_position_quantization():
    """
    Boundaries can only exist at Fibonacci-indexed positions
    
    Proof:
    1. Let p be boundary position
    2. Adjacent boundaries would create pattern 1p1
    3. If p=1, creates 111 (violates No-11)
    4. Therefore p must be at F_n positions
    5. QED
    """
    positions = []
    for n in range(2, MAX_DEPTH):
        f_n = fibonacci(n)
        # Check no adjacent positions
        if not any(abs(f_n - p) == 1 for p in positions):
            positions.append(f_n)
    return positions

T2: Thickness Quantization

def theorem_thickness_quantization():
    """
    Boundary thickness quantized in powers of φ
    
    Proof:
    1. Information resolution at depth d requires log₂(φᵈ) bits
    2. No-11 constraint forbids intermediate values
    3. Therefore τ ∈ {φ⁰, φ¹, φ², ...}
    """
    thicknesses = []
    for n in range(MAX_DEPTH):
        tau = phi ** n
        # Verify no consecutive representation
        z = zeckendorf_encode(int(tau * PRECISION))
        if is_valid_zeckendorf(z):
            thicknesses.append(tau)
    return thicknesses

T3: Flow Rate Quantization

def theorem_flow_quantization():
    """
    Information flow quantized in units of φ bits/τ₀
    
    Proof:
    1. Time quantized in τ₀ units (from T0-0)
    2. Information quantized in φ units
    3. Flow rate Φ = n·φ/m·τ₀, n,m ∈ ℕ
    4. Simplest quantum: Φ₀ = φ/τ₀
    """
    base_flow = phi / tau_0
    flows = []
    for n in fibonacci_sequence(MAX_FLOWS):
        flow = n * base_flow
        flows.append(flow)
    return flows

T4: Perfect Closure Impossibility

def theorem_no_perfect_closure():
    """
    No self-referential system can be perfectly closed
    
    Proof by contradiction:
    1. Assume perfectly closed: Ω(S) = 0
    2. From A1: S must increase entropy
    3. No outflow → H(S) → ∞
    4. Finite system cannot have infinite entropy
    5. Contradiction, therefore Ω(S) > 0
    """
    min_openness = 1  # Zeckendorf 1 = minimum nonzero
    return min_openness > 0

T5: Critical Transitions

def theorem_critical_transitions():
    """
    Phase transitions occur at φⁿ information density
    
    Proof:
    1. Hierarchy levels at φⁿ (from T0-11)
    2. Boundary must adapt to hierarchy
    3. Adaptation is discrete (No-11)
    4. Transitions at exactly φⁿ thresholds
    """
    critical_points = []
    for n in range(MAX_HIERARCHY):
        density = phi ** n
        # Verify discrete transition
        below = compute_permeability(density - epsilon)
        above = compute_permeability(density + epsilon)
        if below != above:
            critical_points.append(density)
    return critical_points

T6: Boundary Entropy Generation

def theorem_boundary_entropy():
    """
    Boundaries actively generate entropy
    
    Proof:
    1. Information filtering requires measurement
    2. Measurement increases entropy (from T0-12)
    3. Each filtered bit: ΔS ≥ k_B ln(2)
    4. Rate: dS/dt = Φ(B)·k_B ln(2)
    """
    def boundary_entropy_rate(B: Boundary) -> float:
        flow = information_flow_rate(B)
        entropy_per_bit = k_B * log(2)
        return flow * entropy_per_bit
    
    return boundary_entropy_rate

T7: Collapse Threshold

def theorem_collapse_threshold():
    """
    Boundary collapses when H_internal = φ·τ(B)
    
    Proof:
    1. Boundary capacity C = τ(B) bits
    2. Critical ratio H/C = φ
    3. At this ratio, structure fails
    4. System merges with environment
    """
    def collapse_condition(B: Boundary, H_internal: float) -> bool:
        threshold = phi * B.thickness
        return H_internal >= threshold
    
    return collapse_condition

T8: Network Topology Constraint

def theorem_network_topology():
    """
    Boundary networks have maximum degree φⁿ
    
    Proof:
    1. Each coupling requires avoiding 11
    2. Maximum non-interfering couplings = φⁿ
    3. Higher connectivity violates No-11
    4. Creates sparse network topology
    """
    def max_connections(hierarchy_level: int) -> int:
        return floor(phi ** hierarchy_level)
    
    return max_connections

Computational Functions

F1: Fibonacci Position Generator

def fibonacci_positions(max_n: int) -> List[int]:
    """Generate valid boundary positions"""
    positions = []
    a, b = 1, 2
    while a <= max_n:
        positions.append(a)
        a, b = b, a + b
        # Skip consecutive (No-11)
        if b - a == 1:
            a, b = b, a + b
    return positions

F2: Permeability Matrix Calculator

def compute_permeability_matrix(B: Boundary) -> PermeabilityMatrix:
    """
    Compute information passage probabilities
    """
    size = len(B.position)
    matrix = [[0.0] * size for _ in range(size)]
    
    for i, pos_i in enumerate(B.position):
        for j, pos_j in enumerate(B.position):
            if abs(pos_i - pos_j) > 1:  # No-11 constraint
                # Probability decreases with distance
                p = 1.0 / (phi ** abs(i - j))
                matrix[i][j] = p
    
    return matrix

F3: Entropy Pressure Calculator

def compute_entropy_pressure(S: System) -> float:
    """
    Compute outward entropy pressure on boundaries
    """
    internal_entropy = S.compute_entropy()
    volume = S.compute_volume()
    
    # Pressure proportional to entropy density
    pressure = internal_entropy / volume
    
    # Quantize to Fibonacci levels
    return quantize_to_fibonacci(pressure)

F4: Boundary Work Function

def boundary_maintenance_work(B: Boundary, time_steps: int) -> float:
    """
    Compute work needed to maintain boundary
    """
    base_work = phi * k_B * T * log(2)  # Per time quantum
    
    # Account for thickness and permeability
    thickness_factor = B.thickness
    permeability_factor = sum(sum(row) for row in B.permeability)
    
    total_work = base_work * thickness_factor * permeability_factor * time_steps
    
    return total_work

F5: Quantum Boundary State

def quantum_boundary_superposition(B1: Boundary, B2: Boundary) -> QuantumBoundary:
    """
    Create quantum superposition of boundaries
    """
    # Verify both satisfy No-11
    assert is_valid_zeckendorf(B1.state)
    assert is_valid_zeckendorf(B2.state)
    
    # Create superposition preserving constraint
    alpha = 1.0 / sqrt(2)
    beta = 1.0 / sqrt(2)
    
    return QuantumBoundary(
        state = alpha * B1.state + beta * B2.state,
        collapse_options = [B1, B2],
        preserves_no11 = True
    )

Validation Constraints

C1: No-11 Preservation

def validate_no11(B: Boundary) -> bool:
    """All boundary operations must preserve No-11"""
    binary = to_binary(B.state)
    return '11' not in binary

C2: Entropy Monotonicity

def validate_entropy_increase(B: Boundary, dt: float) -> bool:
    """Entropy must increase through boundary"""
    S_before = measure_entropy_before(B)
    S_after = measure_entropy_after(B, dt)
    return S_after > S_before

C3: Fibonacci Positioning

def validate_fibonacci_positions(B: Boundary) -> bool:
    """Positions must be at Fibonacci indices"""
    fib_set = set(fibonacci_sequence(MAX_POSITION))
    return all(p in fib_set for p in B.position)

C4: Thickness Quantization

def validate_thickness(B: Boundary) -> bool:
    """Thickness must be power of φ"""
    log_phi = log(B.thickness, phi)
    return abs(log_phi - round(log_phi)) < EPSILON

Error Bounds

E1: Position Uncertainty

POSITION_UNCERTAINTY = 1  # Minimum Fibonacci spacing

E2: Flow Rate Precision

FLOW_PRECISION = phi / tau_0  # Minimum flow quantum

E3: Thickness Resolution

THICKNESS_RESOLUTION = 1.0  # φ⁰ = minimum thickness

E4: Permeability Accuracy

PERMEABILITY_EPSILON = 1.0 / phi**10  # Numerical precision limit