T22-1 φ-网络节点涌现定理 - 形式化规范

依赖导入

import numpy as np
import math
from typing import List, Dict, Set, Tuple, Optional
from dataclasses import dataclass
from collections import defaultdict
import networkx as nx

# 从前置理论导入
from T2_7_formal import PhiRepresentation
from T20_1_formal import CollapseAwareSystem
from D1_8_formal import ZeckendorfString
from L1_5_formal import FibonacciSequence

1. 网络节点定义

1.1 基础节点结构

@dataclass
class NetworkNode:
    """网络节点的φ-表示"""
    
    def __init__(self, node_id: int):
        self.id = node_id
        self.z_representation = ZeckendorfString(node_id)
        self.state = 0  # 节点状态
        self.degree = 0  # 节点度数
        self.phi = (1 + np.sqrt(5)) / 2
        self.entropy = 0.0
        
    def is_valid(self) -> bool:
        """验证节点满足no-11约束"""
        return '11' not in self.z_representation.representation
        
    def can_connect_to(self, other: 'NetworkNode') -> bool:
        """判断是否可以连接到另一节点"""
        # 连接后不能违反no-11约束
        combined = self.z_representation.value + other.z_representation.value
        z_combined = ZeckendorfString(combined)
        return '11' not in z_combined.representation

1.2 边结构

@dataclass
class NetworkEdge:
    """网络边的φ-表示"""
    
    def __init__(self, node1: NetworkNode, node2: NetworkNode):
        self.node1 = node1
        self.node2 = node2
        self.weight = 1.0
        self.phi = (1 + np.sqrt(5)) / 2
        
    def entropy_contribution(self) -> float:
        """计算边的熵贡献"""
        # 边的熵贡献与节点状态差相关
        state_diff = abs(self.node1.state - self.node2.state)
        return math.log(1 + state_diff) / self.phi

2. φ-网络结构

2.1 网络类定义

class PhiNetwork:
    """φ-网络的完整实现"""
    
    def __init__(self, n_initial: int = 3):
        self.phi = (1 + np.sqrt(5)) / 2
        self.nodes: Dict[int, NetworkNode] = {}
        self.edges: Set[NetworkEdge] = set()
        self.time = 0
        self.entropy_history = []
        
        # 初始化节点
        self._initialize_nodes(n_initial)
        
    def _initialize_nodes(self, n: int):
        """初始化n个节点"""
        fib_sequence = FibonacciSequence()
        
        for i in range(n):
            # 使用Fibonacci数作为节点ID
            node_id = fib_sequence.get(i + 2)  # 从F_2=1开始
            node = NetworkNode(node_id)
            
            if node.is_valid():
                self.nodes[node_id] = node
                
    def add_node(self) -> NetworkNode:
        """添加新节点（熵增驱动）"""
        # 找到下一个有效的Fibonacci数
        max_id = max(self.nodes.keys()) if self.nodes else 1
        
        fib_sequence = FibonacciSequence()
        new_id = max_id
        
        # 寻找下一个满足no-11的ID
        while True:
            new_id = fib_sequence.next_after(new_id)
            node = NetworkNode(new_id)
            if node.is_valid():
                self.nodes[new_id] = node
                return node
                
    def add_edge(self, node1_id: int, node2_id: int) -> bool:
        """添加边（如果满足约束）"""
        if node1_id not in self.nodes or node2_id not in self.nodes:
            return False
            
        node1 = self.nodes[node1_id]
        node2 = self.nodes[node2_id]
        
        if not node1.can_connect_to(node2):
            return False
            
        edge = NetworkEdge(node1, node2)
        self.edges.add(edge)
        
        # 更新度数
        node1.degree += 1
        node2.degree += 1
        
        return True

2.2 熵增演化

class EntropyDrivenEvolution:
    """熵增驱动的网络演化"""
    
    def __init__(self, network: PhiNetwork):
        self.network = network
        self.phi = (1 + np.sqrt(5)) / 2
        
    def evolve_step(self):
        """单步演化"""
        # 计算当前熵
        current_entropy = self.compute_network_entropy()
        
        # 决定是添加节点还是边
        if np.random.random() < 1 / self.phi:
            self._add_node_by_entropy()
        else:
            self._add_edge_by_entropy()
            
        # 验证熵增
        new_entropy = self.compute_network_entropy()
        entropy_increase = new_entropy - current_entropy
        
        # 记录熵历史
        self.network.entropy_history.append(new_entropy)
        
        # 验证熵增满足理论预测
        expected_increase = math.log(self.phi)
        assert entropy_increase >= expected_increase * 0.9  # 允许10%误差
        
    def _add_node_by_entropy(self):
        """根据熵增需求添加节点"""
        new_node = self.network.add_node()
        
        # 根据熵增概率连接到现有节点
        for node_id, node in self.network.nodes.items():
            if node_id != new_node.id:
                # 计算连接的熵增
                delta_s = self._compute_entropy_increase(new_node.id, node_id)
                
                # 连接概率
                p_connect = delta_s / (self.phi * self._max_entropy())
                
                if np.random.random() < p_connect:
                    self.network.add_edge(new_node.id, node_id)
                    
    def _add_edge_by_entropy(self):
        """根据熵增概率添加边"""
        node_ids = list(self.network.nodes.keys())
        
        for i, id1 in enumerate(node_ids):
            for id2 in node_ids[i+1:]:
                # 检查是否已连接
                if self._are_connected(id1, id2):
                    continue
                    
                # 计算熵增
                delta_s = self._compute_entropy_increase(id1, id2)
                
                # 连接概率
                p_connect = delta_s / (self.phi * self._max_entropy())
                
                if np.random.random() < p_connect:
                    self.network.add_edge(id1, id2)
                    break
                    
    def _compute_entropy_increase(self, id1: int, id2: int) -> float:
        """计算添加边的熵增"""
        node1 = self.network.nodes[id1]
        node2 = self.network.nodes[id2]
        
        # 基础熵增
        base_entropy = math.log(2)  # 新连接的信息
        
        # 度数贡献
        degree_entropy = math.log(1 + node1.degree) + math.log(1 + node2.degree)
        
        # Zeckendorf编码的熵贡献
        z_entropy = len(node1.z_representation.representation) + \
                   len(node2.z_representation.representation)
        
        return (base_entropy + degree_entropy / 10 + z_entropy / 100) / self.phi
        
    def _max_entropy(self) -> float:
        """计算最大可能熵"""
        n = len(self.network.nodes)
        if n <= 1:
            return 1.0
        return n * math.log(n)
        
    def _are_connected(self, id1: int, id2: int) -> bool:
        """检查两节点是否已连接"""
        for edge in self.network.edges:
            if (edge.node1.id == id1 and edge.node2.id == id2) or \
               (edge.node1.id == id2 and edge.node2.id == id1):
                return True
        return False
        
    def compute_network_entropy(self) -> float:
        """计算网络总熵"""
        # 节点熵
        node_entropy = sum(node.entropy for node in self.network.nodes.values())
        
        # 边熵
        edge_entropy = sum(edge.entropy_contribution() for edge in self.network.edges)
        
        # 结构熵
        n = len(self.network.nodes)
        m = len(self.network.edges)
        
        if n > 0:
            structure_entropy = math.log(1 + n) + math.log(1 + m)
        else:
            structure_entropy = 0
            
        # 总熵包含log(φ)项
        total_entropy = node_entropy + edge_entropy + structure_entropy + math.log(self.phi)
        
        return total_entropy

3. 度分布验证

3.1 φ-度分布类

class PhiDegreeDistribution:
    """验证网络度分布遵循φ-表示"""
    
    def __init__(self, network: PhiNetwork):
        self.network = network
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compute_degree_distribution(self) -> Dict[int, int]:
        """计算度分布"""
        distribution = defaultdict(int)
        
        for node in self.network.nodes.values():
            distribution[node.degree] += 1
            
        return dict(distribution)
        
    def verify_fibonacci_clustering(self) -> bool:
        """验证度数聚集在Fibonacci数附近"""
        distribution = self.compute_degree_distribution()
        degrees = list(distribution.keys())
        
        if not degrees:
            return True
            
        fib_sequence = FibonacciSequence()
        fibonacci_numbers = [fib_sequence.get(i) for i in range(2, 20)]
        
        # 检查每个度数
        for degree in degrees:
            # 度数必须可Zeckendorf表示
            z = ZeckendorfString(degree)
            if '11' in z.representation:
                return False
                
            # 检查是否接近Fibonacci数
            min_distance = min(abs(degree - fib) for fib in fibonacci_numbers)
            
            # 允许一定偏差
            if min_distance > degree * 0.2:  # 20%偏差
                return False
                
        return True
        
    def compute_scaling_exponent(self) -> float:
        """计算度分布的标度指数"""
        distribution = self.compute_degree_distribution()
        
        if len(distribution) < 2:
            return 0.0
            
        # 对数-对数拟合
        degrees = []
        counts = []
        
        for degree, count in distribution.items():
            if degree > 0 and count > 0:
                degrees.append(math.log(degree))
                counts.append(math.log(count))
                
        if len(degrees) < 2:
            return 0.0
            
        # 线性回归
        n = len(degrees)
        sum_x = sum(degrees)
        sum_y = sum(counts)
        sum_xx = sum(x*x for x in degrees)
        sum_xy = sum(x*y for x, y in zip(degrees, counts))
        
        denominator = n * sum_xx - sum_x * sum_x
        if abs(denominator) < 1e-10:
            return 0.0
            
        slope = (n * sum_xy - sum_x * sum_y) / denominator
        
        # 理论预测：标度指数应接近-φ
        return -slope

4. 网络熵守恒

4.1 熵守恒验证

class NetworkEntropyConservation:
    """验证网络熵守恒定律"""
    
    def __init__(self, network: PhiNetwork):
        self.network = network
        self.phi = (1 + np.sqrt(5)) / 2
        
    def decompose_entropy(self) -> Dict[str, float]:
        """分解网络熵为各组分"""
        # 节点熵
        node_entropy = self._compute_node_entropy()
        
        # 边熵
        edge_entropy = self._compute_edge_entropy()
        
        # 结构熵（log φ项）
        structure_entropy = math.log(self.phi)
        
        # 总熵
        total_entropy = node_entropy + edge_entropy + structure_entropy
        
        return {
            'node_entropy': node_entropy,
            'edge_entropy': edge_entropy,
            'structure_entropy': structure_entropy,
            'total_entropy': total_entropy
        }
        
    def _compute_node_entropy(self) -> float:
        """计算节点熵"""
        entropy = 0.0
        
        for node in self.network.nodes.values():
            if node.state > 0:
                entropy += math.log(node.state)
                
            # Zeckendorf表示的熵贡献
            z_length = len(node.z_representation.representation)
            entropy += z_length * math.log(2) / 10
            
        return entropy
        
    def _compute_edge_entropy(self) -> float:
        """计算边熵"""
        entropy = 0.0
        
        # 度分布熵
        degree_dist = defaultdict(int)
        for node in self.network.nodes.values():
            degree_dist[node.degree] += 1
            
        total_nodes = len(self.network.nodes)
        if total_nodes > 0:
            for degree, count in degree_dist.items():
                if count > 0:
                    p = count / total_nodes
                    entropy -= p * math.log(p)
                    
        # 边权熵
        for edge in self.network.edges:
            entropy += edge.entropy_contribution()
            
        return entropy
        
    def verify_conservation(self, tolerance: float = 0.1) -> bool:
        """验证熵守恒"""
        components = self.decompose_entropy()
        
        # 验证分解公式
        expected_total = (components['node_entropy'] + 
                         components['edge_entropy'] + 
                         components['structure_entropy'])
        
        actual_total = components['total_entropy']
        
        # 检查守恒（允许数值误差）
        return abs(actual_total - expected_total) < tolerance

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注记: 本形式化规范提供了T22-1定理的完整数学实现，包括网络节点、边、熵增演化和度分布验证的所有必要组件。所有实现严格遵守no-11约束和Zeckendorf编码要求。