T26-1 瓶颈张力积累定理 - 形式化规范

依赖关系

• 前置: A1 (唯一公理：自指完备系统必然熵增)
• 前置: C7-4 (木桶原理系统瓶颈推论)
• 前置: D1-3 (no-11约束)
• 前置: D1-8 (φ-表示系统)

形式化系统定义

基础结构

Definition F26.1.1 (Zeckendorf System with Bottleneck Tension):

ZeckendorfBottleneckSystem := ⟨S, C, H, T, Ω, ≺⟩
where:
  S := {s₁, s₂, ..., sₙ} is the set of system components
  C := {C₁, C₂, ..., Cₙ} are the component capacities  
  H := {H₁, H₂, ..., Hₙ} are the component entropy levels
  T := {T₁, T₂, ..., Tₙ} are the component tensions
  Ω := {Ω₁, Ω₂, ..., Ωₙ} are the self-reference coefficients
  ≺ is the partial order on components

Definition F26.1.2 (Zeckendorf Component Encoding):

∀i ∈ [1,n]: sᵢ = Σⱼ∈Kᵢ Fⱼ
where:
  Kᵢ ⊆ ℕ is the index set for component i
  ∀j,k ∈ Kᵢ: |j-k| ≥ 2 (no-11 constraint)
  Fⱼ is the j-th Fibonacci number

张力计算算法

Algorithm F26.1.1 (Tension Computation):

def compute_component_tension(
    component_id: int,
    H_actual: float,
    H_required: float, 
    capacity: float,
    omega: float
) -> float:
    """
    Compute tension for a component in Zeckendorf system
    
    Args:
        component_id: Component identifier
        H_actual: Current entropy level of component
        H_required: Required entropy level for system evolution  
        capacity: Zeckendorf-encoded entropy capacity
        omega: Self-reference coefficient
        
    Returns:
        Component tension T_i(t)
    """
    # Core tension formula from T26-1
    entropy_deficit = H_required - H_actual
    normalized_deficit = entropy_deficit / capacity
    tension = normalized_deficit * omega
    
    # Ensure Zeckendorf constraints
    return zeckendorf_quantize(tension)

def zeckendorf_quantize(value: float) -> float:
    """Quantize value to nearest Zeckendorf-representable number"""
    if value <= 0:
        return 0
    
    # Find best Fibonacci combination without consecutive terms
    phi = (1 + math.sqrt(5)) / 2
    fibonacci_cache = generate_fibonacci_cache(int(math.log(value * math.sqrt(5))/math.log(phi)) + 5)
    
    best_sum = 0
    used_indices = set()
    remaining = value
    
    # Greedy algorithm with no-11 constraint
    for i in range(len(fibonacci_cache) - 1, -1, -1):
        if fibonacci_cache[i] <= remaining and (i+1) not in used_indices and (i-1) not in used_indices:
            best_sum += fibonacci_cache[i]
            used_indices.add(i)
            remaining -= fibonacci_cache[i]
            
    return best_sum

Algorithm F26.1.2 (Self-Reference Coefficient):

def compute_self_reference_coefficient(
    component_id: int,
    connection_matrix: np.ndarray,
    component_state: int,
    max_fibonacci: int
) -> float:
    """
    Compute self-reference coefficient Ω_i(t)
    
    Args:
        component_id: Component identifier  
        connection_matrix: Adjacency matrix of component connections
        component_state: Current Zeckendorf-encoded state
        max_fibonacci: Maximum Fibonacci number in component encoding
        
    Returns:
        Self-reference coefficient Ω_i(t)
    """
    n_self_connections = connection_matrix[component_id, component_id]
    n_total_connections = np.sum(connection_matrix[component_id, :])
    
    if n_total_connections == 0:
        return 0
        
    connection_ratio = n_self_connections / n_total_connections
    state_factor = math.log2(1 + component_state / max_fibonacci)
    
    return connection_ratio * state_factor

瓶颈识别算法

Algorithm F26.1.3 (Bottleneck Identification):

def identify_bottleneck_component(
    entropy_levels: List[float],
    capacities: List[float]
) -> int:
    """
    Identify the bottleneck component with highest saturation
    
    Args:
        entropy_levels: Current entropy levels H_i(t)
        capacities: Component capacities C_i
        
    Returns:
        Index of bottleneck component j*
    """
    saturations = [H / C for H, C in zip(entropy_levels, capacities)]
    return int(np.argmax(saturations))

张力不均匀性验证算法

Algorithm F26.1.4 (Tension Inequality Verification):

def verify_tension_inequality(
    tensions: List[float],
    bottleneck_id: int,
    phi: float = (1 + math.sqrt(5)) / 2
) -> Tuple[bool, float, float]:
    """
    Verify T26-1 tension inequality theorem
    
    Args:
        tensions: List of component tensions
        bottleneck_id: Index of bottleneck component
        phi: Golden ratio constant
        
    Returns:
        (is_valid, bottleneck_ratio, min_ratio) where:
        - is_valid: Whether T_j* >= φ * T_avg and ∃i: T_i <= T_avg/φ
        - bottleneck_ratio: T_j* / T_avg  
        - min_ratio: min(T_i) / T_avg
    """
    if len(tensions) == 0:
        return False, 0, 0
        
    avg_tension = sum(tensions) / len(tensions)
    if avg_tension == 0:
        return True, 0, 0  # Trivial case
        
    bottleneck_tension = tensions[bottleneck_id]
    min_tension = min(tensions)
    
    bottleneck_ratio = bottleneck_tension / avg_tension
    min_ratio = min_tension / avg_tension
    
    # Check T_j* >= φ * T_avg
    condition1 = bottleneck_ratio >= phi - 1e-10  # Small tolerance for numerical errors
    
    # Check ∃i: T_i <= T_avg/φ  
    condition2 = min_ratio <= (1/phi) + 1e-10
    
    is_valid = condition1 and condition2
    return is_valid, bottleneck_ratio, min_ratio

张力动力学算法

Algorithm F26.1.5 (Tension Dynamics Evolution):

def evolve_tension_dynamics(
    current_tension: float,
    entropy_saturation: float,  # H_j*/C_j*
    accumulation_rate: float,   # λ parameter
    max_tension: float,         # T_max = φ * log₂(φ)
    dt: float,
    phi: float = (1 + math.sqrt(5)) / 2
) -> float:
    """
    Evolve bottleneck tension according to T26-1 dynamics
    
    dT_j*/dt = λ * (H_j*/C_j*)^φ * (1 - T_j*/T_max)
    
    Args:
        current_tension: Current tension T_j*(t)
        entropy_saturation: Saturation ratio H_j*/C_j*
        accumulation_rate: Rate constant λ
        max_tension: Maximum theoretical tension T_max
        dt: Time step
        phi: Golden ratio
        
    Returns:
        Updated tension T_j*(t + dt)
    """
    if max_tension == 0:
        return current_tension
        
    saturation_term = entropy_saturation ** phi
    capacity_term = 1 - (current_tension / max_tension)
    
    dtension_dt = accumulation_rate * saturation_term * capacity_term
    
    new_tension = current_tension + dtension_dt * dt
    
    # Ensure tension doesn't exceed theoretical maximum
    new_tension = min(new_tension, max_tension)
    
    # Quantize to Zeckendorf representation
    return zeckendorf_quantize(new_tension)

张力传播算法

Algorithm F26.1.6 (Tension Propagation):

def propagate_tension(
    tensions: np.ndarray,
    component_states: np.ndarray,
    adjacency_matrix: np.ndarray,
    dt: float,
    phi: float = (1 + math.sqrt(5)) / 2
) -> np.ndarray:
    """
    Propagate tension according to Zeckendorf diffusion equation
    
    ∂T_i/∂t = D_eff * Σⱼ~ᵢ [F_gcd(s_i,s_j) / √(s_i*s_j)] * (T_j - T_i)
    
    Args:
        tensions: Current tension values T_i(t)  
        component_states: Zeckendorf-encoded states s_i
        adjacency_matrix: Component connection matrix
        dt: Time step
        phi: Golden ratio
        
    Returns:
        Updated tensions T_i(t + dt)
    """
    n = len(tensions)
    new_tensions = tensions.copy()
    
    # Effective diffusion coefficient from T26-1
    D_eff = math.log2(phi) / phi  # ≈ 0.694
    
    for i in range(n):
        dtension_dt = 0
        
        # Sum over connected components
        for j in range(n):
            if i != j and adjacency_matrix[i, j] > 0:
                # Compute coupling coefficient
                gcd_val = math.gcd(component_states[i], component_states[j])
                if gcd_val == 0:
                    continue
                    
                # Find Fibonacci number for GCD
                F_gcd = fibonacci_of_value(gcd_val)
                if F_gcd == 0:
                    continue
                    
                # Coupling strength
                coupling = F_gcd / math.sqrt(component_states[i] * component_states[j])
                
                # Tension difference
                tension_diff = tensions[j] - tensions[i]
                
                dtension_dt += coupling * tension_diff
                
        dtension_dt *= D_eff
        new_tensions[i] += dtension_dt * dt
        
        # Apply Zeckendorf quantization
        new_tensions[i] = zeckendorf_quantize(max(0, new_tensions[i]))
        
    return new_tensions

def fibonacci_of_value(value: int) -> int:
    """Find the largest Fibonacci number <= value"""
    if value <= 0:
        return 0
    if value == 1:
        return 1
        
    fib_prev, fib_curr = 1, 1
    while fib_curr <= value:
        fib_prev, fib_curr = fib_curr, fib_prev + fib_curr
        
    return fib_prev

形式化验证条件

不变量

Invariant I26.1.1 (Tension Non-Negativity):

∀t ≥ 0, ∀i ∈ [1,n]: T_i(t) ≥ 0

Invariant I26.1.2 (Zeckendorf Constraint):

∀i ∈ [1,n]: T_i(t) ∈ ZeckendorfRepresentable

Invariant I26.1.3 (Bottleneck Tension Dominance):

∀t ≥ 0: T_j*(t) ≥ φ * (1/n) * Σᵢ T_i(t)
where j* = argmax_i(H_i(t)/C_i)

Invariant I26.1.4 (Tension Conservation):

∀t ≥ 0: Σᵢ T_i(t) ≤ T_max_total
where T_max_total = n * φ * log₂(φ)

Invariant I26.1.5 (Entropy Increase Consistency):

∀t ≥ 0: H_system(t+1) - H_system(t) > 0 ⟹ ∃i: T_i(t) > 0

测试谓词

Predicate P26.1.1 (Tension Inequality):

def tension_inequality_holds(tensions, bottleneck_id, phi):
    """验证张力不均匀分布定理"""
    avg = sum(tensions) / len(tensions)
    return (tensions[bottleneck_id] >= phi * avg and 
            min(tensions) <= avg / phi)

Predicate P26.1.2 (Entropy Increase Consistency):

def entropy_increase_consistent(old_entropy, new_entropy, tensions):
    """验证熵增与张力的一致性"""
    total_entropy_increase = sum(new_entropy) - sum(old_entropy) 
    return total_entropy_increase > 0 or max(tensions) > 0

Predicate P26.1.3 (Zeckendorf Validity):

def all_tensions_zeckendorf_valid(tensions):
    """验证所有张力值满足Zeckendorf约束"""
    return all(is_zeckendorf_representable(t) for t in tensions)

Predicate P26.1.4 (Self-Reference Coefficient Bounds):

def omega_bounds_valid(omega_values):
    """验证自指系数的界限"""
    return all(0 <= omega <= math.log2(math.sqrt(5)) for omega in omega_values)

Predicate P26.1.5 (Tension Dynamics Stability):

def dynamics_stable(old_tensions, new_tensions, max_tension):
    """验证张力动力学的稳定性"""
    return all(0 <= new_t <= max_tension for new_t in new_tensions)

辅助函数

Function F26.1.1 (Generate Fibonacci Cache):

def generate_fibonacci_cache(max_index: int) -> List[int]:
    """生成Fibonacci数缓存"""
    cache = [1, 2]
    while len(cache) <= max_index:
        cache.append(cache[-1] + cache[-2])
    return cache

Function F26.1.2 (Is Zeckendorf Representable):

def is_zeckendorf_representable(value: float, tolerance: float = 1e-10) -> bool:
    """检查值是否可以用Zeckendorf表示"""
    if value < 0:
        return False
    quantized = zeckendorf_quantize(value)
    return abs(value - quantized) < tolerance

Function F26.1.3 (Compute System Entropy):

def compute_system_entropy(component_states: List[str]) -> float:
    """计算系统总熵"""
    total_entropy = 0
    for state in component_states:
        # 计算单个组件的Zeckendorf熵
        entropy = zeckendorf_entropy(state)
        total_entropy += entropy
    return total_entropy

def zeckendorf_entropy(state: str) -> float:
    """计算Zeckendorf状态的熵"""
    if not state or state == "0":
        return 0
    ones = state.count('1')
    length = len(state)
    if ones == 0:
        return 0
    p = ones / length
    phi = (1 + math.sqrt(5)) / 2
    return -p * math.log2(p) - (1-p) * math.log2(1-p) * length * math.log2(phi)

复杂度分析

• Tension Computation: O(log n) per component
• Bottleneck Identification: O(n)
• Tension Propagation: O(n²) per time step
• Zeckendorf Quantization: O(log F_max) where F_max is largest Fibonacci number
• Self-Reference Coefficient: O(n) per component
• System Entropy Calculation: O(n * L) where L is average state length

数值稳定性

精度要求

• 所有浮点运算使用双精度
• 张力值精度: ε_tension < 10⁻¹²
• 熵计算精度: ε_entropy < 10⁻¹⁴
• 黄金比例φ使用高精度常数

误差控制

• Zeckendorf量化避免累积误差
• 张力传播使用数值稳定的扩散格式
• 动力学演化采用显式Euler方法与自适应步长

边界条件处理

1. 零张力处理: 确保非负性
2. 饱和张力: 限制在理论最大值内
3. 空组件: 返回零张力和零熵
4. 数值溢出: 使用对数空间计算

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注记: 本形式化规范提供了T26-1定理的完全可执行数学模型，确保理论与实现的严格一致性。所有算法都保持Zeckendorf编码约束和φ-量化特性，严格遵循唯一公理：自指完备系统必然熵增。