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T21-3 φ-全息显化定理 - 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, Set
from dataclasses import dataclass
from enum import Enum

# 从前置定理导入
from T21_1_formal import PhiZetaFunction, AdSSpace
from T21_2_formal import QuantumState, SpectralDecomposer
from T20_3_formal import RealityShell, BoundaryPoint
from T20_1_formal import ZeckendorfString

1. 全息信息容量计算​

1.1 边界面积计算器​

class BoundaryAreaCalculator:
    """计算RealityShell的边界面积"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.planck_length = 1.0  # 归一化Planck长度
        
    def compute_area(self, shell: 'RealityShell') -> float:
        """计算Shell边界的离散面积"""
        boundary_points = self._get_boundary_points(shell)
        
        # 离散面积 = 边界点数 × 单位面积
        area = len(boundary_points) * (self.planck_length ** 2)
        
        return area
        
    def _get_boundary_points(self, shell: 'RealityShell') -> List['BoundaryPoint']:
        """获取边界点"""
        boundary_points = []
        
        for state in shell.states:
            # 评估每个状态是否在边界上
            point = shell.boundary_function.evaluate(state, shell.trace_calculator)
            
            # 边界点的判据：距离接近0
            if abs(point.distance_to_boundary) < 1.0:
                boundary_points.append(point)
                
        return boundary_points
        
    def compute_information_capacity(self, area: float) -> float:
        """计算最大信息容量"""
        # I_max = A/(4*log(φ)) * Σ(1/F_n)
        
        # 计算Fibonacci级数和
        fibonacci_sum = self._compute_fibonacci_sum(100)
        
        # 信息容量
        I_max = area / (4 * math.log(self.phi)) * fibonacci_sum
        
        return I_max
        
    def _compute_fibonacci_sum(self, n_terms: int) -> float:
        """计算Σ(1/F_n)"""
        fib_sum = 0.0
        a, b = 1, 1
        
        for _ in range(n_terms):
            fib_sum += 1.0 / a
            a, b = b, a + b
            
        return fib_sum

1.2 全息编码器​

class HolographicEncoder:
    """全息编码器：将体信息编码到边界"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.area_calc = BoundaryAreaCalculator()
        
    def encode_to_boundary(self, bulk_states: List['ZeckendorfString']) -> Dict[int, complex]:
        """将体态编码到边界"""
        boundary_encoding = {}
        
        for state in bulk_states:
            # 提取Zeckendorf编码
            z_value = state.value
            
            # 计算全息投影
            boundary_index = self._holographic_projection(z_value)
            
            # 累加贡献（可能多个体点映射到同一边界点）
            if boundary_index in boundary_encoding:
                boundary_encoding[boundary_index] += 1.0 / math.sqrt(len(bulk_states))
            else:
                boundary_encoding[boundary_index] = 1.0 / math.sqrt(len(bulk_states))
                
        return self._normalize_encoding(boundary_encoding)
        
    def _holographic_projection(self, bulk_index: int) -> int:
        """全息投影：体索引→边界索引"""
        # 使用模运算模拟投影
        # 确保结果满足no-11约束
        boundary_index = bulk_index
        
        while True:
            z_string = ZeckendorfString(boundary_index)
            if '11' not in z_string.representation:
                break
            boundary_index = (boundary_index * 2 + 1) % 1000  # 防止无限循环
            
        return boundary_index
        
    def _normalize_encoding(self, encoding: Dict[int, complex]) -> Dict[int, complex]:
        """归一化编码"""
        total = sum(abs(v)**2 for v in encoding.values())
        
        if total == 0:
            return encoding
            
        factor = 1.0 / math.sqrt(total)
        return {k: v * factor for k, v in encoding.items()}
        
    def compute_encoding_entropy(self, encoding: Dict[int, complex]) -> float:
        """计算编码熵"""
        entropy = 0.0
        
        for amplitude in encoding.values():
            p = abs(amplitude) ** 2
            if p > 1e-15:
                entropy -= p * math.log(p)
                
        return entropy

2. 显化算子实现​

2.1 显化算子​

class ManifestationOperator:
    """φ-全息显化算子"""
    
    def __init__(self, zeta_function: 'PhiZetaFunction'):
        self.phi = (1 + np.sqrt(5)) / 2
        self.zeta_func = zeta_function
        self.zeros_cache = None
        
    def apply(self, boundary_state: Dict[int, complex], r: float) -> 'QuantumState':
        """将边界态显化到径向距离r的体态"""
        # 获取φ-ζ函数零点
        if self.zeros_cache is None:
            self.zeros_cache = self._compute_zeros()
            
        bulk_coeffs = {}
        
        for zero in self.zeros_cache:
            # 提取零点虚部
            gamma = zero.imag
            
            # 径向衰减因子
            radial_factor = cmath.exp(-gamma * r / self.phi)
            
            # 零点权重（留数）
            weight = self._compute_weight(zero)
            
            # 对每个边界模式进行径向扩展
            for boundary_index, boundary_amplitude in boundary_state.items():
                # 计算体索引
                bulk_index = self._extend_to_bulk(boundary_index, r, gamma)
                
                # 累加贡献
                contribution = boundary_amplitude * radial_factor * weight
                
                if bulk_index in bulk_coeffs:
                    bulk_coeffs[bulk_index] += contribution
                else:
                    bulk_coeffs[bulk_index] = contribution
                    
        return QuantumState(bulk_coeffs)
        
    def _compute_zeros(self) -> List[complex]:
        """计算φ-ζ函数零点"""
        # 简化：返回预计算的零点
        zeros = []
        for n in range(1, 11):
            gamma_n = 2 * math.pi * n / math.log(self.phi)
            zeros.append(0.5 + 1j * gamma_n)
        return zeros
        
    def _compute_weight(self, zero: complex) -> complex:
        """计算零点权重"""
        # 1/sqrt(|ζ'(ρ)|)
        h = 1e-6
        derivative = (self.zeta_func.compute(zero + h) - 
                     self.zeta_func.compute(zero - h)) / (2 * h)
        
        if abs(derivative) < 1e-15:
            return 0.0
            
        return 1.0 / cmath.sqrt(derivative)
        
    def _extend_to_bulk(self, boundary_index: int, r: float, gamma: float) -> int:
        """将边界索引扩展到体索引"""
        # 径向层数
        layer = int(r / math.log(self.phi))
        
        # 体索引 = 边界索引 + 层偏移
        bulk_index = boundary_index + layer * int(gamma)
        
        # 确保满足no-11约束
        z_string = ZeckendorfString(bulk_index % 1000)  # 防止溢出
        
        return z_string.value
        
    def verify_recursion_relation(self, boundary_state: Dict[int, complex]) -> bool:
        """验证递归关系：M² = φM + I"""
        # 应用M一次
        r1 = math.log(self.phi)
        M_state = self.apply(boundary_state, r1)
        
        # 应用M两次
        M_intermediate = self.apply(boundary_state, r1/2)
        M2_state = self.apply(M_intermediate.coefficients, r1/2)
        
        # 单位算子（原始边界态）
        I_state = QuantumState(boundary_state)
        
        # 验证关系：M² = φM + I
        # 计算左边
        left = M2_state
        
        # 计算右边
        right_coeffs = {}
        for k, v in M_state.coefficients.items():
            right_coeffs[k] = self.phi * v
        for k, v in I_state.coefficients.items():
            if k in right_coeffs:
                right_coeffs[k] += v
            else:
                right_coeffs[k] = v
        right = QuantumState(right_coeffs)
        
        # 计算差异
        diff = 0.0
        all_keys = set(left.coefficients.keys()) | set(right.coefficients.keys())
        for k in all_keys:
            l_val = left.coefficients.get(k, 0)
            r_val = right.coefficients.get(k, 0)
            diff += abs(l_val - r_val) ** 2
            
        return math.sqrt(diff) < 0.1  # 允许数值误差

2.2 径向演化器​

class RadialEvolution:
    """径向演化：从边界向内传播"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def evolve(self, boundary_state: Dict[int, complex], 
               layers: int) -> List[Dict[int, complex]]:
        """径向演化到多层"""
        evolution = [boundary_state]
        
        for layer in range(1, layers):
            # 径向坐标
            r = layer * math.log(self.phi)
            
            # 从上一层演化
            prev_state = evolution[-1]
            next_state = self._evolve_layer(prev_state, r)
            
            evolution.append(next_state)
            
        return evolution
        
    def _evolve_layer(self, state: Dict[int, complex], r: float) -> Dict[int, complex]:
        """演化单层"""
        next_state = {}
        
        for index, amplitude in state.items():
            # 径向传播
            # 使用Bessel函数模拟径向波动
            radial_wave = self._bessel_propagator(index, r)
            
            # 扩散到相邻索引
            for offset in [-1, 0, 1]:
                new_index = index + offset
                
                # 确保no-11约束
                z_string = ZeckendorfString(new_index)
                if '11' not in z_string.representation:
                    if new_index not in next_state:
                        next_state[new_index] = 0
                    next_state[new_index] += amplitude * radial_wave * (0.8 ** abs(offset))
                    
        return self._normalize_state(next_state)
        
    def _bessel_propagator(self, n: int, r: float) -> float:
        """Bessel传播因子"""
        # 简化的Bessel函数
        return math.exp(-r / (n + 1)) * math.cos(r * math.sqrt(n + 1))
        
    def _normalize_state(self, state: Dict[int, complex]) -> Dict[int, complex]:
        """归一化状态"""
        total = sum(abs(v)**2 for v in state.values())
        
        if total == 0:
            return state
            
        factor = 1.0 / math.sqrt(total)
        return {k: v * factor for k, v in state.items()}

3. 信息守恒验证​

3.1 熵计算器​

class HolographicEntropyCalculator:
    """全息熵计算器"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.G = 1.0  # 归一化引力常数
        
    def compute_boundary_entropy(self, area: float) -> float:
        """计算边界熵（面积定律）"""
        # S = A/(4Gφ)
        return area / (4 * self.G * self.phi)
        
    def compute_bulk_entropy(self, bulk_state: 'QuantumState') -> float:
        """计算体熵"""
        # von Neumann熵
        return bulk_state.entropy
        
    def compute_entanglement_entropy(self, region_A: Set[int], 
                                    full_state: 'QuantumState') -> float:
        """计算纠缠熵"""
        # 计算区域A的约化密度矩阵
        rho_A = self._partial_trace(full_state, region_A)
        
        # 纠缠熵
        entropy = 0.0
        for p in rho_A.values():
            if p > 1e-15:
                entropy -= p * math.log(p)
                
        return entropy
        
    def _partial_trace(self, state: 'QuantumState', 
                       region: Set[int]) -> Dict[int, float]:
        """部分迹运算"""
        rho_reduced = {}
        
        for index, amplitude in state.coefficients.items():
            if index in region:
                # 保留区域内的索引
                if index not in rho_reduced:
                    rho_reduced[index] = 0
                rho_reduced[index] += abs(amplitude) ** 2
                
        # 归一化
        total = sum(rho_reduced.values())
        if total > 0:
            rho_reduced = {k: v/total for k, v in rho_reduced.items()}
            
        return rho_reduced
        
    def verify_information_conservation(self, boundary_entropy: float,
                                      bulk_entropy: float,
                                      volume: float,
                                      area: float) -> Dict[str, Any]:
        """验证信息守恒"""
        # 理论关系：S_boundary = S_bulk + φ*log(V/A)
        
        volume_term = self.phi * math.log(volume / area) if area > 0 else 0
        expected_bulk = boundary_entropy - volume_term
        
        conservation_error = abs(bulk_entropy - expected_bulk)
        
        return {
            'boundary_entropy': boundary_entropy,
            'bulk_entropy': bulk_entropy,
            'expected_bulk': expected_bulk,
            'volume_correction': volume_term,
            'conservation_error': conservation_error,
            'conserved': conservation_error < 0.1
        }

3.2 全息误差界计算​

class HolographicErrorBounds:
    """全息重构的误差界"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compute_reconstruction_error(self, original: 'QuantumState',
                                    reconstructed: 'QuantumState',
                                    n_modes: int) -> Dict[str, float]:
        """计算重构误差"""
        # 计算范数差
        error_norm = self._compute_norm_difference(original, reconstructed)
        
        # 理论误差界
        theoretical_bound = self.phi ** (-n_modes)
        
        # 相对误差
        original_norm = math.sqrt(sum(abs(c)**2 for c in original.coefficients.values()))
        relative_error = error_norm / original_norm if original_norm > 0 else float('inf')
        
        return {
            'absolute_error': error_norm,
            'relative_error': relative_error,
            'theoretical_bound': theoretical_bound,
            'within_bound': error_norm <= theoretical_bound,
            'n_modes_used': n_modes
        }
        
    def _compute_norm_difference(self, state1: 'QuantumState', 
                                state2: 'QuantumState') -> float:
        """计算两个态的范数差"""
        all_indices = set(state1.coefficients.keys()) | set(state2.coefficients.keys())
        
        diff_squared = 0.0
        for index in all_indices:
            c1 = state1.coefficients.get(index, 0)
            c2 = state2.coefficients.get(index, 0)
            diff_squared += abs(c1 - c2) ** 2
            
        return math.sqrt(diff_squared)
        
    def estimate_required_modes(self, target_error: float) -> int:
        """估计达到目标误差所需的模式数"""
        # n = -log(ε)/log(φ)
        return int(math.ceil(-math.log(target_error) / math.log(self.phi)))

4. 全息系统集成​

4.1 完整全息系统​

class HolographicSystem:
    """完整的全息显化系统"""
    
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.zeta_func = PhiZetaFunction()
        self.manifestation_op = ManifestationOperator(self.zeta_func)
        self.encoder = HolographicEncoder()
        self.area_calc = BoundaryAreaCalculator()
        self.entropy_calc = HolographicEntropyCalculator()
        self.error_bounds = HolographicErrorBounds()
        
    def holographic_encode_decode(self, shell: 'RealityShell') -> Dict[str, Any]:
        """完整的全息编码-解码过程"""
        # 1. 计算边界面积和信息容量
        area = self.area_calc.compute_area(shell)
        info_capacity = self.area_calc.compute_information_capacity(area)
        
        # 2. 编码到边界
        bulk_states = shell.states
        boundary_encoding = self.encoder.encode_to_boundary(bulk_states)
        
        # 3. 计算边界熵
        boundary_entropy = self.entropy_calc.compute_boundary_entropy(area)
        
        # 4. 显化回体
        r = math.log(self.phi)  # 标准径向距离
        reconstructed_bulk = self.manifestation_op.apply(boundary_encoding, r)
        
        # 5. 计算体熵
        bulk_entropy = self.entropy_calc.compute_bulk_entropy(reconstructed_bulk)
        
        # 6. 验证信息守恒
        volume = len(reconstructed_bulk.coefficients)
        conservation = self.entropy_calc.verify_information_conservation(
            boundary_entropy, bulk_entropy, volume, area)
        
        # 7. 计算重构误差
        original_state = QuantumState({s.value: 1.0/math.sqrt(len(bulk_states)) 
                                      for s in bulk_states})
        error_analysis = self.error_bounds.compute_reconstruction_error(
            original_state, reconstructed_bulk, len(boundary_encoding))
        
        # 8. 验证递归关系
        recursion_valid = self.manifestation_op.verify_recursion_relation(
            boundary_encoding)
        
        return {
            'area': area,
            'info_capacity': info_capacity,
            'boundary_encoding_size': len(boundary_encoding),
            'boundary_entropy': boundary_entropy,
            'bulk_entropy': bulk_entropy,
            'conservation': conservation,
            'error_analysis': error_analysis,
            'recursion_valid': recursion_valid,
            'efficiency': len(boundary_encoding) / info_capacity if info_capacity > 0 else 0
        }
        
    def recursive_manifestation(self, boundary_state: Dict[int, complex],
                               max_depth: int) -> List['QuantumState']:
        """递归显化过程"""
        layers = []
        
        for depth in range(max_depth):
            r = depth * math.log(self.phi)
            
            # 显化到当前深度
            bulk_state = self.manifestation_op.apply(boundary_state, r)
            layers.append(bulk_state)
            
            # 验证递归关系
            if depth > 0:
                # M² = φM + I
                recursion_check = self.manifestation_op.verify_recursion_relation(
                    boundary_state)
                if not recursion_check:
                    print(f"递归关系在深度{depth}失效")
                    
        return layers

4.2 全息纠错码​

class HolographicErrorCorrectingCode:
    """基于全息原理的量子纠错码"""
    
    def __init__(self, n_logical: int, n_physical: int):
        self.phi = (1 + np.sqrt(5)) / 2
        self.n_logical = n_logical
        self.n_physical = n_physical
        self.encoder = HolographicEncoder()
        self.manifestation_op = None  # 延迟初始化
        
    def encode(self, logical_bits: List[int]) -> Dict[int, complex]:
        """编码逻辑比特到物理比特"""
        if len(logical_bits) != self.n_logical:
            raise ValueError("逻辑比特数不匹配")
            
        # 转换为Zeckendorf状态
        logical_states = []
        for i, bit in enumerate(logical_bits):
            z_value = (2 * i + 1) * bit  # 简单编码
            logical_states.append(ZeckendorfString(z_value))
            
        # 全息编码到边界
        physical_encoding = self.encoder.encode_to_boundary(logical_states)
        
        return physical_encoding
        
    def decode(self, physical_bits: Dict[int, complex]) -> List[int]:
        """从物理比特解码逻辑比特"""
        if self.manifestation_op is None:
            zeta_func = PhiZetaFunction()
            self.manifestation_op = ManifestationOperator(zeta_func)
            
        # 显化到体
        r = math.log(self.phi)
        bulk_state = self.manifestation_op.apply(physical_bits, r)
        
        # 提取逻辑比特
        logical_bits = []
        for i in range(self.n_logical):
            z_value = 2 * i + 1
            if z_value in bulk_state.coefficients:
                amplitude = bulk_state.coefficients[z_value]
                bit = 1 if abs(amplitude) > 0.5 else 0
            else:
                bit = 0
            logical_bits.append(bit)
            
        return logical_bits
        
    def correct_errors(self, noisy_physical: Dict[int, complex]) -> Dict[int, complex]:
        """纠正错误"""
        # 全息纠错：利用冗余信息
        
        # 1. 识别错误位置（syndrome）
        syndrome = self._compute_syndrome(noisy_physical)
        
        # 2. 应用纠错
        corrected = noisy_physical.copy()
        for error_pos in syndrome:
            if error_pos in corrected:
                # 翻转或调整幅度
                corrected[error_pos] *= -1
                
        # 3. 重新归一化
        total = sum(abs(v)**2 for v in corrected.values())
        if total > 0:
            factor = 1.0 / math.sqrt(total)
            corrected = {k: v * factor for k, v in corrected.items()}
            
        return corrected
        
    def _compute_syndrome(self, physical_bits: Dict[int, complex]) -> Set[int]:
        """计算错误syndrome"""
        syndrome = set()
        
        # 检查约束违反
        for index in physical_bits:
            z_string = ZeckendorfString(index)
            if '11' in z_string.representation:
                syndrome.add(index)
                
        return syndrome

注记: T21-3的形式化规范提供了完整的全息显化机制实现，包括边界面积计算、全息编码、显化算子、信息守恒验证、误差界分析和量子纠错码。所有实现严格遵守Zeckendorf编码的no-11约束，并满足φ-递归关系。