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T21-2 φ-谱共识定理 - 形式化规范

依赖导入​

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

# 从前置定理导入
from T21_1_formal import PhiZetaFunction, ZeroDistributionCalculator
from T20_2_formal import TraceStructure, TraceLayerDecomposer
from T20_3_formal import RealityShell, InformationFlow
from T20_1_formal import ZeckendorfString, PsiCollapse

1. 量子态和φ-本征态​

1.1 量子态表示​

@dataclass
class QuantumState:
    """量子态的Zeckendorf表示"""
    
    def __init__(self, coefficients: Dict[int, complex]):
        self.phi = (1 + np.sqrt(5)) / 2
        self.coefficients = self._normalize_coefficients(coefficients)
        self.dimension = max(coefficients.keys()) + 1 if coefficients else 0
        self.entropy = self._compute_entropy()
        
    def _normalize_coefficients(self, coeffs: Dict[int, complex]) -> Dict[int, complex]:
        """归一化系数，保证模方和为1"""
        total = sum(abs(c)**2 for c in coeffs.values())
        if total == 0:
            return {}
        factor = 1.0 / math.sqrt(total)
        return {k: c * factor for k, c in coeffs.items()}
        
    def _compute_entropy(self) -> float:
        """计算von Neumann熵"""
        entropy = 0.0
        for c in self.coefficients.values():
            p = abs(c) ** 2
            if p > 1e-15:
                entropy -= p * math.log(p)
        return entropy
        
    def inner_product(self, other: 'QuantumState') -> complex:
        """计算内积"""
        result = 0.0 + 0.0j
        for n in self.coefficients:
            if n in other.coefficients:
                result += np.conj(self.coefficients[n]) * other.coefficients[n]
        return result
        
    def tensor_product(self, other: 'QuantumState') -> 'QuantumState':
        """张量积"""
        new_coeffs = {}
        for n1, c1 in self.coefficients.items():
            for n2, c2 in other.coefficients.items():
                # 使用Zeckendorf编码组合索引
                combined_index = self._combine_indices(n1, n2)
                new_coeffs[combined_index] = c1 * c2
        return QuantumState(new_coeffs)
        
    def _combine_indices(self, n1: int, n2: int) -> int:
        """组合两个索引，保持no-11约束"""
        # 使用Fibonacci数的性质组合
        z1 = ZeckendorfString(n1)
        z2 = ZeckendorfString(n2)
        combined = z1 + z2
        return combined.value

1.2 φ-本征态生成器​

class PhiEigenstateGenerator:
    """φ-本征态生成器"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.cache = {}
        
    def generate_eigenstate(self, n: int) -> QuantumState:
        """生成第n个φ-本征态"""
        if n in self.cache:
            return self.cache[n]
            
        if n == 0:
            # 基态
            coeffs = {0: 1.0 + 0.0j}
        else:
            # 递归生成
            prev_state = self.generate_eigenstate(n - 1)
            coeffs = self._apply_creation_operator(prev_state, n)
            
        eigenstate = QuantumState(coeffs)
        self.cache[n] = eigenstate
        return eigenstate
        
    def _apply_creation_operator(self, state: QuantumState, n: int) -> Dict[int, complex]:
        """应用产生算子"""
        new_coeffs = {}
        
        for k, c in state.coefficients.items():
            # φ-递归关系
            new_index = int(k * self.phi) + n
            
            # 确保满足no-11约束
            z_string = ZeckendorfString(new_index)
            if self._check_no_11(z_string.representation):
                amplitude = c * (self.phi ** (-n/2))
                new_coeffs[z_string.value] = amplitude
                
        return new_coeffs
        
    def _check_no_11(self, binary_str: str) -> bool:
        """检查no-11约束"""
        return '11' not in binary_str
        
    def compute_completeness(self, max_n: int) -> float:
        """计算前max_n个本征态的完备性"""
        # 构造投影算子之和
        total_projection = 0.0
        
        for n in range(max_n):
            eigenstate = self.generate_eigenstate(n)
            # |φ_n⟩⟨φ_n|的迹
            projection = sum(abs(c)**2 for c in eigenstate.coefficients.values())
            total_projection += projection
            
        return total_projection

2. 共识算子实现​

2.1 共识算子类​

class ConsensusOperator:
    """φ-谱共识算子"""
    
    def __init__(self, zeta_function: PhiZetaFunction):
        self.phi = (1 + np.sqrt(5)) / 2
        self.zeta_func = zeta_function
        self.zeros_cache = None
        self.eigenstate_gen = PhiEigenstateGenerator()
        
    def apply(self, state1: QuantumState, state2: QuantumState, 
              t: float) -> QuantumState:
        """应用共识算子"""
        # 获取φ-ζ函数零点
        if self.zeros_cache is None:
            self.zeros_cache = self._compute_zeros()
            
        # 初始化共识态
        consensus_coeffs = {}
        
        for rho in self.zeros_cache:
            # 提取零点虚部
            gamma = rho.imag
            
            # 时间演化因子
            evolution = cmath.exp(1j * gamma * t)
            
            # 计算留数（零点导数的倒数）
            residue = self._compute_residue(rho)
            
            # 贡献到共识态
            contribution = self._zero_contribution(state1, state2, 
                                                  evolution, residue)
            
            for k, v in contribution.items():
                if k in consensus_coeffs:
                    consensus_coeffs[k] += v
                else:
                    consensus_coeffs[k] = v
                    
        return QuantumState(consensus_coeffs)
        
    def _compute_zeros(self, max_zeros: int = 10) -> List[complex]:
        """计算φ-ζ函数的前几个零点"""
        zeros = self.zeta_func.find_zeros_in_critical_strip(0.1, 50.0, t_step=0.5)
        return zeros[:max_zeros]
        
    def _compute_residue(self, zero: complex) -> complex:
        """计算零点处的留数"""
        h = 1e-6
        
        # 数值微分计算导数
        f_plus = self.zeta_func.compute(zero + h)
        f_minus = self.zeta_func.compute(zero - h)
        derivative = (f_plus - f_minus) / (2 * h)
        
        if abs(derivative) < 1e-15:
            return 0.0 + 0.0j
            
        return 1.0 / derivative
        
    def _zero_contribution(self, state1: QuantumState, state2: QuantumState,
                          evolution: complex, residue: complex) -> Dict[int, complex]:
        """计算单个零点的贡献"""
        contribution = {}
        
        # 张量积的系数
        for n1, c1 in state1.coefficients.items():
            for n2, c2 in state2.coefficients.items():
                # 组合索引
                z1 = ZeckendorfString(n1)
                z2 = ZeckendorfString(n2)
                combined = z1 + z2
                
                # 贡献幅度
                amplitude = c1 * c2 * evolution * residue
                
                if combined.value in contribution:
                    contribution[combined.value] += amplitude
                else:
                    contribution[combined.value] = amplitude
                    
        return contribution

2.2 谱分解器​

class SpectralDecomposer:
    """频谱分解器"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.eigenstate_gen = PhiEigenstateGenerator()
        
    def decompose(self, state: QuantumState, max_components: int = 20) -> Dict[int, complex]:
        """将态分解到φ-本征态基"""
        decomposition = {}
        
        for n in range(max_components):
            # 生成第n个本征态
            eigenstate = self.eigenstate_gen.generate_eigenstate(n)
            
            # 计算投影系数
            coefficient = state.inner_product(eigenstate)
            
            # φ-调制
            coefficient *= self.phi ** (-n/2)
            
            # 存储非零系数
            if abs(coefficient) > 1e-10:
                decomposition[n] = coefficient
                
        return decomposition
        
    def reconstruct(self, decomposition: Dict[int, complex]) -> QuantumState:
        """从谱分解重构量子态"""
        reconstructed_coeffs = {}
        
        for n, c_n in decomposition.items():
            eigenstate = self.eigenstate_gen.generate_eigenstate(n)
            
            # 反φ-调制
            amplitude = c_n * (self.phi ** (n/2))
            
            # 累加贡献
            for k, v in eigenstate.coefficients.items():
                if k in reconstructed_coeffs:
                    reconstructed_coeffs[k] += amplitude * v
                else:
                    reconstructed_coeffs[k] = amplitude * v
                    
        return QuantumState(reconstructed_coeffs)
        
    def compute_spectral_entropy(self, decomposition: Dict[int, complex]) -> float:
        """计算谱熵"""
        entropy = 0.0
        
        # 归一化
        total = sum(abs(c)**2 for c in decomposition.values())
        
        for c in decomposition.values():
            p = abs(c)**2 / total
            if p > 1e-15:
                entropy -= p * math.log(p)
                
        return entropy

3. Fourier变换和谱共识​

3.1 φ-调制Fourier变换​

class PhiModulatedFourier:
    """φ-调制的Fourier变换"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.omega_phi = 2 * math.pi / math.log(self.phi)
        
    def transform(self, trace_structure: TraceStructure) -> np.ndarray:
        """对trace结构进行Fourier变换"""
        # 提取trace值序列
        trace_values = []
        max_layer = max(trace_structure.components.keys())
        
        for n in range(1, max_layer + 1):
            if n in trace_structure.components:
                trace_values.append(trace_structure.components[n].value)
            else:
                trace_values.append(0)
                
        # 应用FFT
        spectrum = np.fft.fft(trace_values)
        
        # φ-调制
        for k in range(len(spectrum)):
            spectrum[k] *= self.phi ** (-k / len(spectrum))
            
        return spectrum
        
    def inverse_transform(self, spectrum: np.ndarray) -> List[float]:
        """逆Fourier变换"""
        # 反φ-调制
        demodulated = spectrum.copy()
        for k in range(len(demodulated)):
            demodulated[k] *= self.phi ** (k / len(demodulated))
            
        # 应用逆FFT
        trace_values = np.fft.ifft(demodulated).real
        
        return list(trace_values)
        
    def compute_power_spectrum(self, trace_structure: TraceStructure) -> np.ndarray:
        """计算功率谱"""
        spectrum = self.transform(trace_structure)
        power = np.abs(spectrum) ** 2
        
        # 验证标度律：P(ω) ~ ω^(-2+1/φ)
        exponent = -2 + 1/self.phi
        
        return power
        
    def verify_scaling_invariance(self, spectrum: np.ndarray) -> bool:
        """验证φ-标度不变性"""
        n = len(spectrum)
        
        # 比较 F[φω] 和 φ^(-1)F[ω]
        for k in range(1, n//2):
            k_scaled = int(k * self.phi) % n
            
            if k_scaled < n:
                left = spectrum[k_scaled]
                right = spectrum[k] / self.phi
                
                # 允许小误差
                if abs(left - right) > 0.1 * abs(right):
                    return False
                    
        return True

3.2 谱共识验证器​

class SpectralConsensusVerifier:
    """谱共识验证器"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.fourier = PhiModulatedFourier()
        
    def verify_consensus(self, shell1: RealityShell, shell2: RealityShell) -> Dict[str, Any]:
        """验证两个Shell的谱共识"""
        # 计算trace结构
        trace1 = self._extract_trace_structure(shell1)
        trace2 = self._extract_trace_structure(shell2)
        
        # Fourier变换
        spectrum1 = self.fourier.transform(trace1)
        spectrum2 = self.fourier.transform(trace2)
        
        # 计算共识度量
        consensus_metric = self._compute_consensus_metric(spectrum1, spectrum2)
        
        # 检查共识条件
        omega_phi = self.fourier.omega_phi
        product = spectrum1 * np.conj(spectrum2)
        
        # 理论预测
        expected = np.zeros_like(product, dtype=complex)
        expected[int(omega_phi * len(product) / (2 * np.pi))] = self.phi ** (1j * omega_phi)
        
        # 计算偏差
        deviation = np.linalg.norm(product - expected) / np.linalg.norm(expected)
        
        return {
            'consensus_achieved': deviation < 0.1,
            'consensus_metric': consensus_metric,
            'deviation': deviation,
            'dominant_frequency': self._find_dominant_frequency(product),
            'phase_coherence': self._compute_phase_coherence(spectrum1, spectrum2)
        }
        
    def _extract_trace_structure(self, shell: RealityShell) -> TraceStructure:
        """从Shell提取trace结构"""
        components = {}
        
        for i, state in enumerate(shell.states):
            trace_value = shell.trace_calculator.compute_full_trace(state)
            components[i+1] = type('TraceComponent', (), {'value': trace_value})()
            
        return type('TraceStructure', (), {'components': components})()
        
    def _compute_consensus_metric(self, spectrum1: np.ndarray, 
                                 spectrum2: np.ndarray) -> float:
        """计算共识度量"""
        # 归一化
        s1_norm = spectrum1 / np.linalg.norm(spectrum1)
        s2_norm = spectrum2 / np.linalg.norm(spectrum2)
        
        # 谱相似度
        similarity = abs(np.vdot(s1_norm, s2_norm))
        
        return similarity
        
    def _find_dominant_frequency(self, spectrum: np.ndarray) -> float:
        """找到主导频率"""
        power = np.abs(spectrum) ** 2
        dominant_index = np.argmax(power)
        
        # 转换为频率
        frequency = 2 * np.pi * dominant_index / len(spectrum)
        
        return frequency
        
    def _compute_phase_coherence(self, spectrum1: np.ndarray, 
                                spectrum2: np.ndarray) -> float:
        """计算相位相干性"""
        # 计算相位差
        phase1 = np.angle(spectrum1)
        phase2 = np.angle(spectrum2)
        phase_diff = phase1 - phase2
        
        # 相干性度量
        coherence = abs(np.mean(np.exp(1j * phase_diff)))
        
        return coherence

4. 熵增监测系统​

4.1 熵计算器​

class EntropyCalculator:
    """熵计算和监测"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compute_von_neumann_entropy(self, state: QuantumState) -> float:
        """计算von Neumann熵"""
        # 已在QuantumState中实现
        return state.entropy
        
    def compute_entanglement_entropy(self, state1: QuantumState, 
                                    state2: QuantumState) -> float:
        """计算纠缠熵"""
        # 计算约化密度矩阵的熵
        combined = state1.tensor_product(state2)
        
        # 部分迹运算（简化实现）
        reduced_coeffs = {}
        for k, v in combined.coefficients.items():
            # 取模运算模拟部分迹
            reduced_index = k % len(state1.coefficients)
            if reduced_index in reduced_coeffs:
                reduced_coeffs[reduced_index] += abs(v) ** 2
            else:
                reduced_coeffs[reduced_index] = abs(v) ** 2
                
        # 计算熵
        entropy = 0.0
        total = sum(reduced_coeffs.values())
        
        for p in reduced_coeffs.values():
            p_norm = p / total
            if p_norm > 1e-15:
                entropy -= p_norm * math.log(p_norm)
                
        return entropy
        
    def verify_entropy_increase(self, initial_state: QuantumState,
                              final_state: QuantumState,
                              dt: float) -> Dict[str, Any]:
        """验证熵增定律"""
        S_initial = initial_state.entropy
        S_final = final_state.entropy
        
        dS = S_final - S_initial
        
        # 理论预测（简化）
        overlap = abs(initial_state.inner_product(initial_state)) ** 2
        expected_dS = self.phi * overlap * dt
        
        return {
            'initial_entropy': S_initial,
            'final_entropy': S_final,
            'entropy_increase': dS,
            'expected_increase': expected_dS,
            'violation': dS < 0,  # 是否违反熵增
            'relative_error': abs(dS - expected_dS) / expected_dS if expected_dS > 0 else 0
        }

4.2 共识过程监测器​

class ConsensusProcessMonitor:
    """共识过程监测器"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.entropy_calc = EntropyCalculator()
        self.history = []
        
    def record_step(self, state1: QuantumState, state2: QuantumState,
                   consensus_state: QuantumState, t: float):
        """记录共识步骤"""
        step_data = {
            'time': t,
            'entropy1': state1.entropy,
            'entropy2': state2.entropy,
            'entropy_consensus': consensus_state.entropy,
            'total_entropy': state1.entropy + state2.entropy,
            'entanglement_entropy': self.entropy_calc.compute_entanglement_entropy(
                state1, state2),
            'overlap': abs(state1.inner_product(state2)) ** 2
        }
        
        self.history.append(step_data)
        
    def verify_monotonic_increase(self) -> bool:
        """验证熵的单调增加"""
        if len(self.history) < 2:
            return True
            
        for i in range(1, len(self.history)):
            if self.history[i]['entropy_consensus'] <= self.history[i-1]['entropy_consensus']:
                return False
                
        return True
        
    def compute_entropy_production_rate(self) -> float:
        """计算熵产生率"""
        if len(self.history) < 2:
            return 0.0
            
        total_dS = 0.0
        total_dt = 0.0
        
        for i in range(1, len(self.history)):
            dS = self.history[i]['entropy_consensus'] - self.history[i-1]['entropy_consensus']
            dt = self.history[i]['time'] - self.history[i-1]['time']
            
            total_dS += dS
            total_dt += dt
            
        return total_dS / total_dt if total_dt > 0 else 0.0
        
    def get_consensus_timeline(self) -> Dict[str, List[float]]:
        """获取共识时间线"""
        timeline = {
            'times': [],
            'entropies': [],
            'overlaps': [],
            'entanglement': []
        }
        
        for step in self.history:
            timeline['times'].append(step['time'])
            timeline['entropies'].append(step['entropy_consensus'])
            timeline['overlaps'].append(step['overlap'])
            timeline['entanglement'].append(step['entanglement_entropy'])
            
        return timeline

5. 完整共识系统​

5.1 谱共识系统​

class SpectralConsensusSystem:
    """完整的谱共识系统"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.zeta_func = PhiZetaFunction()
        self.consensus_op = ConsensusOperator(self.zeta_func)
        self.decomposer = SpectralDecomposer()
        self.verifier = SpectralConsensusVerifier()
        self.monitor = ConsensusProcessMonitor()
        
    def achieve_consensus(self, shell1: RealityShell, shell2: RealityShell,
                         max_iterations: int = 100) -> Dict[str, Any]:
        """达成两个Shell的共识"""
        # 提取量子态
        state1 = self._shell_to_quantum_state(shell1)
        state2 = self._shell_to_quantum_state(shell2)
        
        # 量子化时间步长
        dt = math.log(self.phi)
        t = 0.0
        
        consensus_state = None
        
        for iteration in range(max_iterations):
            # 应用共识算子
            consensus_state = self.consensus_op.apply(state1, state2, t)
            
            # 记录监测数据
            self.monitor.record_step(state1, state2, consensus_state, t)
            
            # 验证共识条件
            verification = self.verifier.verify_consensus(shell1, shell2)
            
            if verification['consensus_achieved']:
                break
                
            # 更新时间
            t += dt
            
            # 更新状态（部分反馈）
            state1 = self._update_state(state1, consensus_state, 0.1)
            state2 = self._update_state(state2, consensus_state, 0.1)
            
        # 验证熵增
        entropy_verified = self.monitor.verify_monotonic_increase()
        
        return {
            'consensus_achieved': verification['consensus_achieved'],
            'iterations': iteration + 1,
            'final_time': t,
            'final_state': consensus_state,
            'entropy_verified': entropy_verified,
            'entropy_production_rate': self.monitor.compute_entropy_production_rate(),
            'timeline': self.monitor.get_consensus_timeline()
        }
        
    def _shell_to_quantum_state(self, shell: RealityShell) -> QuantumState:
        """将Shell转换为量子态"""
        coeffs = {}
        
        for i, state in enumerate(shell.states):
            # 使用Zeckendorf值作为基态索引
            coeffs[state.value] = complex(1.0 / math.sqrt(len(shell.states)), 0)
            
        return QuantumState(coeffs)
        
    def _update_state(self, state: QuantumState, consensus: QuantumState,
                     mixing: float) -> QuantumState:
        """更新状态（部分混合）"""
        new_coeffs = {}
        
        # 线性混合
        for k in set(state.coefficients.keys()) | set(consensus.coefficients.keys()):
            c1 = state.coefficients.get(k, 0)
            c2 = consensus.coefficients.get(k, 0)
            new_coeffs[k] = (1 - mixing) * c1 + mixing * c2
            
        return QuantumState(new_coeffs)

注记: T21-2的形式化规范提供了完整的φ-谱共识机制实现，包括量子态表示、φ-本征态生成、共识算子、Fourier变换、熵监测等核心组件。所有实现严格遵守Zeckendorf编码的no-11约束，并与φ-ζ函数零点结构紧密结合。