C14-3 φ-网络稳定性推论

依赖关系

• 前置推论: C14-1 (φ-网络拓扑涌现推论), C14-2 (φ-网络信息流推论)
• 前置定理: T20-3 (reality shell边界定理)
• 唯一公理: A1 (自指完备系统必然熵增)

推论陈述

推论 C14-3 (φ-网络稳定性推论): 在Zeckendorf编码的φ-网络中，稳定性必然呈现黄金比例特征：

1. 扰动衰减的φ-指数: 小扰动的衰减率

$$||\delta x(t)|| \leq ||\delta x_0|| \cdot \varphi^{-\alpha t}$$

其中$\alpha > 0$是稳定性指数

2. 渗流阈值的黄金分割: 网络渗流临界概率

$$p_c = \frac{1}{\varphi + 1} = \varphi^2 \approx 0.382$$

3. 韧性的Fibonacci层次: 网络韧性指数

$$R_k = \frac{F_{k+2}}{F_{k+3}} \to \varphi^{-1}$$

其中k是攻击轮次

4. Lyapunov函数的φ-形式: 稳定性函数

$$V(x) = \sum_{i=1}^N \varphi^{-d_i} ||x_i||^2$$

其中$d_i$是节点i到平衡点的距离

5. 恢复时间的φ-缩放: 系统恢复时间

$$T_{recovery} = T_0 \cdot \log_\varphi(N)$$

证明

第一步：扰动传播的Zeckendorf约束

考虑网络状态$x$受到小扰动$\delta x$。在φ-网络中，扰动通过Zeckendorf编码路径传播：

$$\delta x_{i}(t+1) = \sum_{j \in N(i)} P_{ij}^{\varphi} \delta x_j(t)$$

其中$P_{ij}^{\varphi} = F_{|i-j|}/F_{|i-j|+2}$（由C14-1确定）。

稳定性分析： 转移矩阵的谱半径$\rho(P^{\varphi}) = \varphi^{-1} < 1$，因此：

$$||\delta x(t)|| \leq \rho(P^{\varphi})^t ||\delta x_0|| = \varphi^{-t} ||\delta x_0||$$

第二步：渗流的黄金分割点

网络渗流研究随机删除边或节点后的连通性。在Zeckendorf约束下，有效连接模式数为$F_{n+2}$（n位编码）。

临界概率满足：

$$p_c \cdot F_{n+2} = F_n$$

当$n \to \infty$：

$$p_c = \lim_{n \to \infty} \frac{F_n}{F_{n+2}} = \varphi^{-2} \approx 0.382$$

这恰好是$1 - \varphi^{-1}$，即黄金分割的补。

第三步：韧性的Fibonacci递归

网络韧性定义为在k轮攻击后保持功能的能力。每轮攻击删除最重要的节点。

在Zeckendorf约束下，第k轮后剩余的有效配置数：

$$N_k = F_{n-k+2}$$

韧性指数：

$$R_k = \frac{N_k}{N_{k-1}} = \frac{F_{n-k+2}}{F_{n-k+3}} \to \varphi^{-1}$$

第四步：Lyapunov稳定性

定义Lyapunov函数：

$$V(x) = \sum_{i=1}^N w_i ||x_i - x_i^*||^2$$

其中权重$w_i = \varphi^{-d_i}$，$d_i$是节点i的度。

时间导数：

$$\dot{V} = -\sum_{i,j} P_{ij}^{\varphi} ||x_i - x_j||^2 \leq -\varphi^{-1} V$$

因此$V(t) \leq V(0) e^{-t/\varphi}$，保证指数稳定。

第五步：恢复时间的对数缩放

系统从大扰动恢复需要信息在网络中传播。由C14-2，信息传播距离$\sim \log_\varphi N$。

恢复时间：

$$T_{recovery} = \frac{\text{网络直径}}{\text{传播速度}} = \frac{\log_\varphi N}{\varphi^{-1}} = \varphi \log_\varphi N$$

结论：φ-网络的稳定性在所有尺度上都展现黄金比例特征，这是Zeckendorf编码约束的必然结果。∎

数学形式化

import numpy as np
from typing import List, Dict, Tuple, Optional
from dataclasses import dataclass

@dataclass
class StabilityMetrics:
    """稳定性度量"""
    decay_rate: float           # 扰动衰减率
    percolation_threshold: float # 渗流阈值
    resilience_index: float      # 韧性指数
    lyapunov_exponent: float    # Lyapunov指数
    recovery_time: float        # 恢复时间

class PhiNetworkStability:
    """φ-网络稳定性分析"""
    
    def __init__(self, adjacency: np.ndarray):
        self.adjacency = adjacency
        self.n_nodes = len(adjacency)
        self.phi = (1 + np.sqrt(5)) / 2
        self.degrees = np.sum(adjacency, axis=1)
        
    def perturbation_decay(
        self, 
        perturbation: np.ndarray,
        time_steps: int
    ) -> List[float]:
        """分析扰动衰减"""
        norms = []
        current = perturbation.copy()
        
        # 构建转移矩阵
        P = self._build_fibonacci_transition()
        
        for t in range(time_steps):
            current = P @ current
            norms.append(np.linalg.norm(current))
            
        return norms
        
    def _build_fibonacci_transition(self) -> np.ndarray:
        """构建Fibonacci转移矩阵"""
        P = np.zeros_like(self.adjacency, dtype=float)
        
        for i in range(self.n_nodes):
            for j in range(self.n_nodes):
                if self.adjacency[i, j] > 0:
                    diff = abs(i - j)
                    F_diff = self.fibonacci(diff + 1)
                    F_diff_plus_2 = self.fibonacci(diff + 3)
                    P[i, j] = F_diff / F_diff_plus_2 if F_diff_plus_2 > 0 else 0
                    
        # 行归一化
        row_sums = np.sum(P, axis=1, keepdims=True)
        P = np.divide(P, row_sums, where=row_sums>0)
        
        return P
        
    def fibonacci(self, n: int) -> int:
        """计算Fibonacci数"""
        if n <= 1:
            return n
        a, b = 0, 1
        for _ in range(2, n + 1):
            a, b = b, a + b
        return b
        
    def percolation_analysis(
        self,
        removal_fraction: float
    ) -> Dict[str, float]:
        """渗流分析"""
        n_remove = int(self.n_nodes * removal_fraction)
        
        # 复制网络
        adj_copy = self.adjacency.copy()
        
        # 随机删除节点
        nodes_to_remove = np.random.choice(
            self.n_nodes, n_remove, replace=False
        )
        
        for node in nodes_to_remove:
            adj_copy[node, :] = 0
            adj_copy[:, node] = 0
            
        # 计算最大连通分量
        giant_size = self._giant_component_size(adj_copy)
        
        # 理论渗流阈值
        p_c_theory = 1 / self.phi**2  # ≈ 0.382
        
        return {
            'removal_fraction': removal_fraction,
            'giant_component_fraction': giant_size / self.n_nodes,
            'theoretical_threshold': p_c_theory,
            'is_above_threshold': removal_fraction < 1 - p_c_theory
        }
        
    def _giant_component_size(self, adjacency: np.ndarray) -> int:
        """计算最大连通分量大小"""
        visited = np.zeros(self.n_nodes, dtype=bool)
        max_size = 0
        
        for i in range(self.n_nodes):
            if not visited[i]:
                size = self._dfs_component_size(adjacency, i, visited)
                max_size = max(max_size, size)
                
        return max_size
        
    def _dfs_component_size(
        self,
        adjacency: np.ndarray,
        node: int,
        visited: np.ndarray
    ) -> int:
        """深度优先搜索计算分量大小"""
        visited[node] = True
        size = 1
        
        for neighbor in range(self.n_nodes):
            if adjacency[node, neighbor] > 0 and not visited[neighbor]:
                size += self._dfs_component_size(adjacency, neighbor, visited)
                
        return size
        
    def resilience_under_attack(
        self,
        attack_rounds: int
    ) -> List[float]:
        """攻击韧性分析"""
        resilience = []
        adj_copy = self.adjacency.copy()
        
        for k in range(attack_rounds):
            # 删除度最大的节点
            degrees = np.sum(adj_copy, axis=1)
            if np.max(degrees) == 0:
                break
                
            max_degree_node = np.argmax(degrees)
            adj_copy[max_degree_node, :] = 0
            adj_copy[:, max_degree_node] = 0
            
            # 计算韧性指数
            giant_size = self._giant_component_size(adj_copy)
            R_k = giant_size / self.n_nodes
            resilience.append(R_k)
            
        return resilience
        
    def lyapunov_function(self, state: np.ndarray) -> float:
        """计算Lyapunov函数值"""
        V = 0.0
        for i in range(self.n_nodes):
            # 权重为φ^(-degree)
            weight = self.phi ** (-self.degrees[i]) if self.degrees[i] > 0 else 1.0
            V += weight * state[i]**2
        return V
        
    def recovery_time_estimate(self) -> float:
        """估计恢复时间"""
        # 网络直径近似
        diameter = np.log(self.n_nodes) / np.log(self.phi)
        
        # 恢复时间
        T_recovery = self.phi * diameter
        
        return T_recovery
        
    def verify_stability_properties(self) -> StabilityMetrics:
        """验证稳定性性质"""
        # 1. 扰动衰减
        perturbation = np.random.randn(self.n_nodes)
        decay_trajectory = self.perturbation_decay(perturbation, 20)
        
        if len(decay_trajectory) > 1:
            decay_rate = decay_trajectory[-1] / decay_trajectory[0]
            decay_rate = decay_rate ** (1/20)  # 平均衰减率
        else:
            decay_rate = 1.0
            
        # 2. 渗流阈值
        perc_analysis = self.percolation_analysis(0.6)
        p_c = perc_analysis['theoretical_threshold']
        
        # 3. 韧性
        resilience = self.resilience_under_attack(5)
        if len(resilience) > 1:
            R = np.mean([resilience[i]/resilience[i-1] 
                        for i in range(1, len(resilience))])
        else:
            R = 1.0
            
        # 4. Lyapunov指数
        state = np.random.randn(self.n_nodes)
        V0 = self.lyapunov_function(state)
        
        # 演化一步
        P = self._build_fibonacci_transition()
        state_next = P @ state
        V1 = self.lyapunov_function(state_next)
        
        lyapunov = np.log(V1/V0) if V0 > 0 else 0
        
        # 5. 恢复时间
        T_rec = self.recovery_time_estimate()
        
        return StabilityMetrics(
            decay_rate=decay_rate,
            percolation_threshold=p_c,
            resilience_index=R,
            lyapunov_exponent=lyapunov,
            recovery_time=T_rec
        )

物理解释

1. 结构稳定性: φ-网络对随机失效具有高容错性，但对目标攻击敏感
2. 动力学稳定性: 扰动以φ^-1速率衰减，比随机网络更快
3. 临界现象: 渗流转变发生在黄金分割点
4. 自组织临界性: 系统自然演化到φ-临界状态

实验可验证预言

1. 扰动衰减率: $\varphi^{-1} \approx 0.618$每时间步
2. 渗流阈值: $p_c = \varphi^{-2} \approx 0.382$
3. 韧性衰减: 每轮攻击后功能降低38.2%
4. 恢复时间: $T \sim 1.618 \log N$

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注记: C14-3揭示了φ-网络的内在稳定性。黄金比例不仅出现在结构和动力学中，也决定了系统的鲁棒性和韧性。这种普遍性暗示φ可能是复杂系统稳定性的基本常数。