C20-2 ψₒ自指映射推论 - 形式化规范

依赖导入

import numpy as np
import math
from typing import List, Dict, Tuple, Optional, Any, Set
from dataclasses import dataclass
from enum import Enum
from collections import deque

# 从前置定理导入
from T20_1_formal import ZeckendorfString, PsiCollapse, CollapseAwareSystem
from T20_2_formal import TraceStructure, TraceLayerDecomposer, TraceComponent
from T20_3_formal import RealityShell, BoundaryFunction

1. 自指映射核心

1.1 自指状态表示

@dataclass
class SelfReferentialState:
    """自指状态的完整表示"""
    
    def __init__(self, z_value: int):
        self.phi = (1 + np.sqrt(5)) / 2
        self.value = ZeckendorfString(z_value)
        self.recursion_depth = 0
        self.entropy = self._compute_entropy()
        self.is_fixed_point = False
        self.trace_layers = []
        
    def _compute_entropy(self) -> float:
        """计算状态熵"""
        # 基础熵
        base_entropy = math.log(self.value.value + 1)
        
        # 递归深度贡献
        depth_entropy = self.recursion_depth * math.log(self.phi)
        
        # trace层贡献
        trace_entropy = sum(math.log(layer + 1) for layer in self.trace_layers) if self.trace_layers else 0
        
        return base_entropy + depth_entropy + trace_entropy / (len(self.trace_layers) + 1)
        
    def apply_self(self) -> 'SelfReferentialState':
        """应用自指操作 ψ → ψ(ψ)"""
        # 核心自指计算
        new_value = self._self_application()
        
        # 创建新状态
        new_state = SelfReferentialState(new_value)
        new_state.recursion_depth = self.recursion_depth + 1
        new_state.trace_layers = self.trace_layers.copy()
        new_state.trace_layers.append(self.value.value)
        
        # 检查是否是不动点
        new_state.is_fixed_point = (new_value == self.value.value)
        
        return new_state
        
    def _self_application(self) -> int:
        """计算 ψ(ψ) 的核心逻辑"""
        v = self.value.value
        
        # 自指操作的数学定义
        # ψ(ψ) = ψ作用于自身
        # 在Zeckendorf空间中的实现
        
        # 方法1: 迭代应用
        result = v
        for _ in range(min(v, 10)):  # 限制迭代次数
            # 每次应用产生新的Fibonacci分量
            fib_components = self._get_fibonacci_components(result)
            result = sum(fib_components) + len(fib_components)
            
            # 确保满足no-11约束
            z = ZeckendorfString(result)
            if '11' not in z.representation:
                break
            # 如果违反，调整到最近的合法值
            result = self._adjust_to_valid(result)
            
        return result
        
    def _get_fibonacci_components(self, n: int) -> List[int]:
        """获取n的Fibonacci分解"""
        z = ZeckendorfString(n)
        components = []
        
        # 提取每个1对应的Fibonacci数
        for i, bit in enumerate(reversed(z.representation)):
            if bit == '1':
                # 第i位对应F_{i+2}
                components.append(self._fibonacci(i + 2))
                
        return components
        
    def _fibonacci(self, n: int) -> int:
        """计算第n个Fibonacci数"""
        if n <= 0:
            return 0
        if n == 1:
            return 1
        
        a, b = 0, 1
        for _ in range(2, n + 1):
            a, b = b, a + b
        return b
        
    def _adjust_to_valid(self, value: int) -> int:
        """调整到满足no-11约束的最近值"""
        # 向下调整
        for v in range(value, 0, -1):
            z = ZeckendorfString(v)
            if '11' not in z.representation:
                return v
        return 1  # 默认返回1

1.2 自指映射算子

class SelfReferentialMapping:
    """ψₒ自指映射的完整实现"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.fixed_points = []
        self.mapping_history = []
        
    def apply(self, state: SelfReferentialState) -> SelfReferentialState:
        """应用自指映射"""
        # 应用自指操作
        new_state = state.apply_self()
        
        # 记录历史
        self.mapping_history.append({
            'input': state.value.value,
            'output': new_state.value.value,
            'depth': new_state.recursion_depth,
            'entropy': new_state.entropy
        })
        
        # 验证熵增
        if new_state.entropy < state.entropy - 1e-10:
            raise ValueError(f"违反熵增定律: {state.entropy} -> {new_state.entropy}")
            
        return new_state
        
    def find_fixed_point(self, max_iterations: int = 1000) -> Optional[SelfReferentialState]:
        """寻找自指映射的不动点"""
        # 候选起始点（Fibonacci数更可能收敛到不动点）
        fibonacci_numbers = [1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233]
        
        for fib in fibonacci_numbers:
            state = SelfReferentialState(fib)
            visited = set()
            
            for iteration in range(max_iterations):
                if state.value.value in visited:
                    # 发现循环，检查是否是不动点
                    prev_state = state
                    state = self.apply(state)
                    
                    if state.value.value == prev_state.value.value:
                        # 找到不动点
                        state.is_fixed_point = True
                        self.fixed_points.append(state)
                        return state
                    else:
                        # 是循环但不是不动点
                        break
                        
                visited.add(state.value.value)
                state = self.apply(state)
                
        return None
        
    def analyze_convergence(self, initial_state: SelfReferentialState, 
                          max_iterations: int = 100) -> Dict[str, Any]:
        """分析向不动点的收敛"""
        states = [initial_state]
        distances = []
        
        # 首先找到不动点
        fixed_point = self.find_fixed_point()
        if not fixed_point:
            return {'error': 'No fixed point found'}
            
        fp_value = fixed_point.value.value
        
        for i in range(max_iterations):
            current = states[-1]
            next_state = self.apply(current)
            states.append(next_state)
            
            # 计算到不动点的距离
            distance = abs(next_state.value.value - fp_value)
            distances.append(distance)
            
            # 检查是否收敛
            if distance == 0:
                break
                
        # 计算收敛速率
        convergence_rates = []
        for i in range(1, len(distances)):
            if distances[i-1] > 0:
                rate = distances[i] / distances[i-1]
                convergence_rates.append(rate)
                
        avg_rate = np.mean(convergence_rates) if convergence_rates else 0
        
        return {
            'fixed_point': fp_value,
            'iterations': len(states) - 1,
            'final_distance': distances[-1] if distances else float('inf'),
            'convergence_rate': avg_rate,
            'theoretical_rate': 1 / self.phi,  # 理论收敛速率
            'states': [s.value.value for s in states],
            'distances': distances
        }

2. 递归深度分析

2.1 递归深度追踪器

class RecursionDepthTracker:
    """追踪和分析递归深度"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.depth_entropy_map = {}
        
    def track_recursion(self, initial_state: SelfReferentialState, 
                       max_depth: int) -> Dict[str, Any]:
        """追踪递归过程中的熵变化"""
        mapping = SelfReferentialMapping()
        
        states = [initial_state]
        entropies = [initial_state.entropy]
        
        for depth in range(1, max_depth + 1):
            # 应用映射
            new_state = mapping.apply(states[-1])
            states.append(new_state)
            entropies.append(new_state.entropy)
            
            # 记录深度-熵关系
            self.depth_entropy_map[depth] = {
                'state': new_state.value.value,
                'entropy': new_state.entropy,
                'entropy_increase': new_state.entropy - initial_state.entropy
            }
            
        # 验证递归深度定理
        theoretical_increases = []
        actual_increases = []
        
        for d in range(1, max_depth + 1):
            # 理论预测：S(ψ_d) - S(ψ_0) = d * log(φ) + O(log(d))
            theoretical = d * math.log(self.phi)
            actual = entropies[d] - entropies[0]
            
            theoretical_increases.append(theoretical)
            actual_increases.append(actual)
            
        # 计算误差
        errors = [abs(a - t) for a, t in zip(actual_increases, theoretical_increases)]
        max_error = max(errors) if errors else 0
        
        # 验证O(log(d))误差界
        error_bound_satisfied = all(
            err <= math.log(d + 1) + 1 for d, err in enumerate(errors)
        )
        
        return {
            'depths': list(range(max_depth + 1)),
            'entropies': entropies,
            'entropy_increases': actual_increases,
            'theoretical_increases': theoretical_increases,
            'max_error': max_error,
            'error_bound_satisfied': error_bound_satisfied,
            'final_state': states[-1].value.value
        }

2.2 自指循环检测器

class SelfReferentialCycleDetector:
    """检测自指映射的周期性"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def detect_cycle(self, initial_state: SelfReferentialState, 
                    max_iterations: int = 1000) -> Dict[str, Any]:
        """检测自指循环"""
        mapping = SelfReferentialMapping()
        
        # Floyd循环检测算法
        slow = initial_state
        fast = initial_state
        
        # Phase 1: 检测是否存在循环
        cycle_found = False
        for _ in range(max_iterations):
            slow = mapping.apply(slow)
            fast = mapping.apply(mapping.apply(fast))
            
            if slow.value.value == fast.value.value:
                cycle_found = True
                break
                
        if not cycle_found:
            return {'cycle_found': False}
            
        # Phase 2: 找到循环起点
        mu = 0  # 循环起点
        slow = initial_state
        while slow.value.value != fast.value.value:
            slow = mapping.apply(slow)
            fast = mapping.apply(fast)
            mu += 1
            
        # Phase 3: 找到循环长度
        cycle_length = 1
        fast = mapping.apply(slow)
        while slow.value.value != fast.value.value:
            fast = mapping.apply(fast)
            cycle_length += 1
            
        # 收集循环中的所有状态
        cycle_states = []
        state = slow
        for _ in range(cycle_length):
            cycle_states.append(state.value.value)
            state = mapping.apply(state)
            
        # 验证周期与φ的关系
        # T_ψ 应该满足 φ^T ≡ 1 (mod 某个Fibonacci数)
        phi_powers = []
        for t in range(1, cycle_length + 1):
            phi_power = self.phi ** t
            phi_powers.append(phi_power)
            
        # 检查是否与Fibonacci数相关
        fibonacci_related = any(
            state in [1, 2, 3, 5, 8, 13, 21, 34, 55, 89] 
            for state in cycle_states
        )
        
        return {
            'cycle_found': True,
            'cycle_start': mu,
            'cycle_length': cycle_length,
            'cycle_states': cycle_states,
            'fibonacci_related': fibonacci_related,
            'phi_power_at_period': self.phi ** cycle_length
        }

3. 收敛性分析

3.1 收敛速率计算器

class ConvergenceAnalyzer:
    """分析自指映射的收敛性质"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compute_lipschitz_constant(self, mapping: SelfReferentialMapping,
                                  test_range: range) -> float:
        """计算映射的Lipschitz常数"""
        max_ratio = 0
        
        for i in test_range:
            for j in test_range:
                if i != j:
                    state_i = SelfReferentialState(i)
                    state_j = SelfReferentialState(j)
                    
                    mapped_i = mapping.apply(state_i)
                    mapped_j = mapping.apply(state_j)
                    
                    dist_before = abs(i - j)
                    dist_after = abs(mapped_i.value.value - mapped_j.value.value)
                    
                    if dist_before > 0:
                        ratio = dist_after / dist_before
                        max_ratio = max(max_ratio, ratio)
                        
        return max_ratio
        
    def verify_contraction(self, mapping: SelfReferentialMapping) -> bool:
        """验证映射是否是压缩映射"""
        # 在小范围内测试
        lipschitz = self.compute_lipschitz_constant(mapping, range(1, 100))
        
        # 理论预测：Lipschitz常数应该 ≤ 1/φ
        theoretical_bound = 1 / self.phi
        
        return lipschitz <= theoretical_bound + 0.1  # 允许小误差
        
    def analyze_basin_of_attraction(self, mapping: SelfReferentialMapping,
                                   fixed_point: int,
                                   test_range: range) -> Dict[str, Any]:
        """分析不动点的吸引域"""
        converging_states = []
        diverging_states = []
        
        for initial_value in test_range:
            state = SelfReferentialState(initial_value)
            
            # 迭代检查是否收敛到不动点
            converged = False
            for _ in range(100):
                state = mapping.apply(state)
                if abs(state.value.value - fixed_point) < 1:
                    converged = True
                    break
                    
            if converged:
                converging_states.append(initial_value)
            else:
                diverging_states.append(initial_value)
                
        basin_size = len(converging_states)
        total_tested = len(test_range)
        
        return {
            'fixed_point': fixed_point,
            'basin_size': basin_size,
            'basin_ratio': basin_size / total_tested,
            'converging_states': converging_states[:10],  # 前10个
            'diverging_states': diverging_states[:10]
        }

4. 完整自指系统

4.1 自指系统集成

class CompleteSelfReferentialSystem:
    """完整的ψₒ自指映射系统"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.mapping = SelfReferentialMapping()
        self.depth_tracker = RecursionDepthTracker()
        self.cycle_detector = SelfReferentialCycleDetector()
        self.convergence_analyzer = ConvergenceAnalyzer()
        
    def full_analysis(self, initial_value: int) -> Dict[str, Any]:
        """对给定初始值进行完整分析"""
        initial_state = SelfReferentialState(initial_value)
        
        results = {
            'initial_value': initial_value,
            'initial_entropy': initial_state.entropy
        }
        
        # 1. 寻找不动点
        fixed_point = self.mapping.find_fixed_point()
        if fixed_point:
            results['fixed_point'] = {
                'value': fixed_point.value.value,
                'entropy': fixed_point.entropy,
                'is_fibonacci': fixed_point.value.value in [1,2,3,5,8,13,21,34,55,89]
            }
            
        # 2. 分析递归深度
        depth_analysis = self.depth_tracker.track_recursion(initial_state, 20)
        results['recursion_depth'] = {
            'max_depth': 20,
            'final_entropy': depth_analysis['entropies'][-1],
            'entropy_increase': depth_analysis['entropies'][-1] - initial_state.entropy,
            'theory_satisfied': depth_analysis['error_bound_satisfied']
        }
        
        # 3. 检测周期
        cycle_info = self.cycle_detector.detect_cycle(initial_state)
        results['cycle'] = cycle_info
        
        # 4. 分析收敛性
        if fixed_point:
            convergence = self.mapping.analyze_convergence(initial_state)
            results['convergence'] = {
                'converged': convergence['final_distance'] == 0,
                'iterations': convergence['iterations'],
                'rate': convergence['convergence_rate']
            }
            
        # 5. 验证压缩性质
        is_contraction = self.convergence_analyzer.verify_contraction(self.mapping)
        results['is_contraction'] = is_contraction
        
        return results
        
    def validate_theorems(self) -> Dict[str, bool]:
        """验证所有理论预测"""
        validations = {}
        
        # 1. 不动点存在性
        fp = self.mapping.find_fixed_point()
        validations['fixed_point_exists'] = (fp is not None)
        if fp:
            validations['fixed_point_is_fibonacci'] = fp.value.value in [1,2,3,5,8,13,21,34,55,89]
            
        # 2. 递归深度定理
        test_state = SelfReferentialState(5)
        depth_result = self.depth_tracker.track_recursion(test_state, 10)
        validations['recursion_depth_theorem'] = depth_result['error_bound_satisfied']
        
        # 3. 周期性
        cycle = self.cycle_detector.detect_cycle(test_state)
        if cycle['cycle_found']:
            validations['cycle_fibonacci_related'] = cycle['fibonacci_related']
            
        # 4. 收敛速率
        validations['contraction_mapping'] = self.convergence_analyzer.verify_contraction(self.mapping)
        
        return validations

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注记: C20-2的形式化规范提供了ψₒ自指映射的完整实现，包括不动点计算、递归深度追踪、周期检测和收敛性分析。所有实现严格遵守Zeckendorf编码的no-11约束，并满足熵增定律。