Skip to main content

C20-1 collapse-aware观测推论 - 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 datetime import datetime

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

1. 观测者模型​

1.1 观测者状态​

@dataclass
class ObserverState:
    """观测者的量子态表示"""
    
    def __init__(self, z_state: ZeckendorfString):
        self.phi = (1 + np.sqrt(5)) / 2
        self.state = z_state
        self.memory = []  # 观测记忆
        self.entropy = self._compute_entropy()
        self.observation_count = 0
        
    def _compute_entropy(self) -> float:
        """计算观测者熵"""
        # 基础熵
        base_entropy = math.log(self.state.value + 1)
        
        # 记忆贡献
        memory_entropy = 0.0
        if self.memory:
            for mem in self.memory:
                memory_entropy += mem['information_content'] / len(self.memory)
                
        return base_entropy + memory_entropy
        
    def update_state(self, observation_result: Dict[str, Any]):
        """根据观测结果更新状态"""
        # 提取信息
        info = observation_result['information']
        
        # 更新Zeckendorf状态
        new_value = self.state.value + int(info * self.phi)
        self.state = ZeckendorfString(new_value)
        
        # 记录到记忆
        self.memory.append({
            'timestamp': observation_result['timestamp'],
            'information_content': info,
            'system_state': observation_result['system_state']
        })
        
        # 限制记忆大小
        if len(self.memory) > 100:
            self.memory = self.memory[-100:]
            
        # 更新熵
        self.entropy = self._compute_entropy()
        self.observation_count += 1
        
    def get_observation_capacity(self) -> float:
        """获取观测能力"""
        # 基于状态复杂度和熵
        return self.phi ** (self.entropy / math.log(self.phi))

1.2 观测算子​

class ObservationOperator:
    """观测算子：实现观测者对系统的观测"""
    
    def __init__(self, observer_state: ObserverState):
        self.phi = (1 + np.sqrt(5)) / 2
        self.observer = observer_state
        self.observation_history = []
        self.precision_limit = math.log(self.phi)
        
    def observe(self, system: CollapseAwareSystem) -> Dict[str, Any]:
        """执行观测，返回观测结果和反作用"""
        # 记录初始状态
        initial_system_state = system.current_state.value
        initial_observer_entropy = self.observer.entropy
        initial_system_entropy = system.compute_total_entropy()
        
        # 执行观测（导致collapse）
        observation = self._extract_information(system)
        
        # 系统因观测而collapse
        system.execute_collapse()
        
        # 计算反作用
        final_system_state = system.current_state.value
        final_observer_entropy = self.observer.entropy
        final_system_entropy = system.compute_total_entropy()
        
        backaction = self._compute_backaction(
            initial_system_state, final_system_state,
            initial_observer_entropy, final_observer_entropy
        )
        
        # 验证φ-比例关系
        self._verify_phi_proportion(backaction)
        
        # 验证熵增
        total_entropy_increase = (final_system_entropy + final_observer_entropy) - \
                               (initial_system_entropy + initial_observer_entropy)
        
        if total_entropy_increase < -1e-10:  # 允许数值误差
            raise ValueError(f"违反熵增定律: ΔS = {total_entropy_increase}")
            
        result = {
            'observation': observation,
            'backaction': backaction,
            'entropy_increase': total_entropy_increase,
            'timestamp': datetime.now(),
            'observer_state': self.observer.state.value,
            'system_state': system.current_state.value
        }
        
        # 更新观测者状态
        self.observer.update_state(result)
        
        # 记录历史
        self.observation_history.append(result)
        
        return result
        
    def _extract_information(self, system: CollapseAwareSystem) -> Dict[str, Any]:
        """从系统提取信息"""
        # 计算可观测量
        trace = system.compute_phi_trace()
        
        # 观测精度受限
        precision = min(self.observer.get_observation_capacity(), 
                       self.precision_limit)
        
        # 添加观测噪声
        noise = np.random.normal(0, 1/precision)
        observed_trace = trace + noise
        
        # 提取的信息量
        information = math.log(abs(observed_trace) + 1) / math.log(self.phi)
        
        return {
            'trace': observed_trace,
            'information': information,
            'precision': precision,
            'raw_trace': trace
        }
        
    def _compute_backaction(self, initial_sys: int, final_sys: int,
                           initial_obs: float, final_obs: float) -> Dict[str, float]:
        """计算观测反作用"""
        # 系统状态改变
        sys_change = abs(final_sys - initial_sys)
        
        # 观测者熵改变
        obs_change = abs(final_obs - initial_obs)
        
        # φ-比例关系
        expected_ratio = 1 / self.phi
        actual_ratio = obs_change / (sys_change + 1)  # 避免除零
        
        return {
            'system_change': sys_change,
            'observer_change': obs_change,
            'expected_ratio': expected_ratio,
            'actual_ratio': actual_ratio,
            'deviation': abs(actual_ratio - expected_ratio)
        }
        
    def _verify_phi_proportion(self, backaction: Dict[str, float]):
        """验证φ-比例关系"""
        deviation = backaction['deviation']
        
        # 允许10%的误差
        if deviation > 0.1 * backaction['expected_ratio']:
            print(f"警告: φ-比例关系偏差较大: {deviation:.4f}")

2. 观测精度界限​

2.1 精度计算器​

class ObservationPrecisionCalculator:
    """观测精度计算和界限验证"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.min_info_unit = math.log2(self.phi)
        self.min_time_unit = math.log(self.phi)
        
    def compute_precision_bound(self, info_content: float, 
                               observation_time: float) -> Dict[str, Any]:
        """计算观测精度界限"""
        # 不确定性乘积
        uncertainty_product = info_content * observation_time
        
        # 理论下界
        lower_bound = self.min_info_unit * self.min_time_unit
        
        # 验证界限
        satisfies_bound = uncertainty_product >= lower_bound - 1e-10
        
        # 相对精度
        relative_precision = uncertainty_product / lower_bound if lower_bound > 0 else float('inf')
        
        return {
            'info_content': info_content,
            'observation_time': observation_time,
            'uncertainty_product': uncertainty_product,
            'lower_bound': lower_bound,
            'satisfies_bound': satisfies_bound,
            'relative_precision': relative_precision
        }
        
    def optimal_observation_strategy(self, target_info: float) -> Dict[str, float]:
        """计算最优观测策略"""
        # 给定目标信息量，计算最小观测时间
        min_time = self.min_time_unit * target_info / self.min_info_unit
        
        # 实际可行时间（考虑实际约束）
        practical_time = max(min_time, self.min_time_unit)
        
        # 对应的信息精度
        achievable_info = self.min_info_unit * practical_time / self.min_time_unit
        
        return {
            'target_info': target_info,
            'min_time': min_time,
            'practical_time': practical_time,
            'achievable_info': achievable_info,
            'efficiency': target_info / achievable_info if achievable_info > 0 else 0
        }

2.2 连续观测模型​

class ContinuousObservationModel:
    """连续观测模型（量子Zeno效应）"""
    
    def __init__(self, observer: ObserverState):
        self.phi = (1 + np.sqrt(5)) / 2
        self.observer = observer
        self.observation_op = ObservationOperator(observer)
        self.zeno_threshold = 1 / math.log(self.phi)
        
    def continuous_observe(self, system: CollapseAwareSystem, 
                          frequency: float, 
                          duration: float) -> Dict[str, Any]:
        """执行连续观测"""
        # 观测间隔
        interval = 1 / frequency if frequency > 0 else float('inf')
        
        # 观测次数
        n_observations = int(duration / interval)
        
        # 记录系统演化
        evolution = []
        zeno_frozen = False
        
        for i in range(n_observations):
            # 执行观测
            result = self.observation_op.observe(system)
            
            # 记录状态
            evolution.append({
                'time': i * interval,
                'system_state': system.current_state.value,
                'observer_entropy': self.observer.entropy,
                'backaction': result['backaction']['system_change']
            })
            
            # 检查Zeno效应
            if result['backaction']['system_change'] < self.zeno_threshold:
                zeno_frozen = True
                break
                
        return {
            'frequency': frequency,
            'duration': duration,
            'n_observations': len(evolution),
            'evolution': evolution,
            'zeno_frozen': zeno_frozen,
            'final_state': system.current_state.value
        }
        
    def verify_zeno_effect(self, frequency: float) -> bool:
        """验证量子Zeno效应"""
        # 理论预测：当频率超过临界值时系统冻结
        critical_frequency = 1 / math.log(self.phi)
        
        return frequency > critical_frequency

3. 观测者纠缠​

3.1 纠缠观测系统​

class EntangledObservationSystem:
    """纠缠观测系统：多个观测者观测纠缠态"""
    
    def __init__(self, observers: List[ObserverState]):
        self.phi = (1 + np.sqrt(5)) / 2
        self.observers = observers
        self.observation_ops = [ObservationOperator(obs) for obs in observers]
        self.entanglement_matrix = self._initialize_entanglement()
        
    def _initialize_entanglement(self) -> np.ndarray:
        """初始化纠缠矩阵"""
        n = len(self.observers)
        matrix = np.zeros((n, n))
        
        for i in range(n):
            for j in range(n):
                if i != j:
                    # φ-关联强度
                    matrix[i, j] = 1 / (self.phi ** abs(i - j))
                    
        return matrix
        
    def entangled_observation(self, entangled_systems: List[CollapseAwareSystem]) -> Dict[str, Any]:
        """执行纠缠观测"""
        if len(entangled_systems) != len(self.observers):
            raise ValueError("系统数量必须与观测者数量匹配")
            
        results = []
        correlations = np.zeros((len(self.observers), len(self.observers)))
        
        # 每个观测者观测对应系统
        for i, (obs_op, system) in enumerate(zip(self.observation_ops, entangled_systems)):
            result = obs_op.observe(system)
            results.append(result)
            
            # 通过纠缠影响其他系统
            for j, other_system in enumerate(entangled_systems):
                if i != j:
                    # 纠缠导致的状态改变
                    influence = self.entanglement_matrix[i, j] * result['backaction']['system_change']
                    other_system.current_state = ZeckendorfString(
                        int(other_system.current_state.value + influence)
                    )
                    
        # 计算观测关联
        for i in range(len(results)):
            for j in range(len(results)):
                if i != j:
                    corr = self._compute_correlation(results[i], results[j])
                    correlations[i, j] = corr
                    
        return {
            'individual_results': results,
            'correlation_matrix': correlations,
            'average_correlation': np.mean(np.abs(correlations)),
            'max_correlation': np.max(np.abs(correlations))
        }
        
    def _compute_correlation(self, result1: Dict, result2: Dict) -> float:
        """计算两个观测结果的关联"""
        # 使用信息量的关联
        info1 = result1['observation']['information']
        info2 = result2['observation']['information']
        
        # 归一化关联
        if info1 * info2 == 0:
            return 0
            
        correlation = 2 * min(info1, info2) / (info1 + info2)
        
        # φ-调制
        return correlation / self.phi

4. 完整观测系统​

4.1 观测系统集成​

class CompleteObservationSystem:
    """完整的collapse-aware观测系统"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.precision_calc = ObservationPrecisionCalculator()
        self.observers = []
        self.systems = []
        self.observation_log = []
        
    def add_observer(self, z_value: int) -> ObserverState:
        """添加观测者"""
        observer = ObserverState(ZeckendorfString(z_value))
        self.observers.append(observer)
        return observer
        
    def add_system(self, z_value: int) -> CollapseAwareSystem:
        """添加被观测系统"""
        system = CollapseAwareSystem(ZeckendorfString(z_value))
        self.systems.append(system)
        return system
        
    def perform_observation(self, observer_idx: int, system_idx: int) -> Dict[str, Any]:
        """执行单次观测"""
        if observer_idx >= len(self.observers) or system_idx >= len(self.systems):
            raise IndexError("观测者或系统索引超出范围")
            
        observer = self.observers[observer_idx]
        system = self.systems[system_idx]
        
        # 创建观测算子
        obs_op = ObservationOperator(observer)
        
        # 记录观测开始时间
        start_time = datetime.now()
        
        # 执行观测
        result = obs_op.observe(system)
        
        # 记录观测结束时间
        end_time = datetime.now()
        observation_time = (end_time - start_time).total_seconds()
        
        # 验证精度界限
        precision_check = self.precision_calc.compute_precision_bound(
            result['observation']['information'],
            observation_time
        )
        
        # 完整记录
        full_result = {
            'observer_idx': observer_idx,
            'system_idx': system_idx,
            'observation_result': result,
            'precision_check': precision_check,
            'timestamp': start_time
        }
        
        self.observation_log.append(full_result)
        
        return full_result
        
    def analyze_observation_history(self) -> Dict[str, Any]:
        """分析观测历史"""
        if not self.observation_log:
            return {'error': 'No observations recorded'}
            
        # 统计分析
        total_observations = len(self.observation_log)
        total_entropy_increase = sum(log['observation_result']['entropy_increase'] 
                                    for log in self.observation_log)
        
        # 精度统计
        precision_violations = sum(1 for log in self.observation_log 
                                 if not log['precision_check']['satisfies_bound'])
        
        # 反作用统计
        avg_backaction = np.mean([log['observation_result']['backaction']['system_change'] 
                                 for log in self.observation_log])
        
        return {
            'total_observations': total_observations,
            'total_entropy_increase': total_entropy_increase,
            'avg_entropy_per_observation': total_entropy_increase / total_observations,
            'precision_violations': precision_violations,
            'violation_rate': precision_violations / total_observations,
            'avg_backaction': avg_backaction,
            'observation_log': self.observation_log
        }

注记: C20-1的形式化规范提供了完整的观测者模型实现，包括观测算子、精度界限、连续观测和纠缠观测。所有实现严格遵守Zeckendorf编码的no-11约束，并满足熵增定律和φ-比例关系。