C20-3 φ-trace编码推论 - 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
import hashlib
# 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, InformationFlow
from C20_1_formal import ObserverState, ObservationOperator
from C20_2_formal import SelfReferentialState, SelfReferentialMapping
1. 编码核心结构
1.1 编码层表示
@dataclass
class EncodedLayer:
"""编码后的trace层"""
def __init__(self, raw_data: List[int], depth: int):
self.phi = (1 + np.sqrt(5)) / 2
self.depth = depth
self.raw_data = raw_data
self.compressed_data = self._compress()
self.error_correction_bits = []
self.holographic_info = {}
self.encoding_entropy = 0.0
def _compress(self) -> List[int]:
"""φ-压缩算法"""
# 压缩率 = φ^(-depth)
compression_ratio = self.phi ** (-self.depth)
# 转换为Zeckendorf序列
z_sequence = []
for value in self.raw_data:
z = ZeckendorfString(value)
# 提取非零位
for i, bit in enumerate(z.representation):
if bit == '1':
z_sequence.append(i)
# 去除冗余
compressed = self._remove_redundancy(z_sequence)
# 应用压缩
target_size = max(1, int(len(self.raw_data) * compression_ratio))
if len(compressed) > target_size:
compressed = compressed[:target_size]
return compressed
def _remove_redundancy(self, sequence: List[int]) -> List[int]:
"""去除冗余信息"""
# 利用no-11约束的性质
result = []
prev = -2 # 确保第一个元素总是添加
for val in sorted(set(sequence)):
if val > prev + 1: # 避免连续（no-11）
result.append(val)
prev = val
return result
def add_error_correction(self, min_distance: int):
"""添加纠错码"""
# 基于Fibonacci的纠错码
n_parity = (min_distance - 1) // 2 + 1
for i in range(n_parity):
# 计算校验位
parity = 0
fib_weight = self._fibonacci(i + 2)
for j, val in enumerate(self.compressed_data):
if j % fib_weight == 0:
parity ^= val
self.error_correction_bits.append(parity)
def embed_holographic(self, global_info: Dict[str, Any]):
"""嵌入全息信息"""
# 计算全局摘要
digest = self._compute_digest(global_info)
# 分布式嵌入
embedding_density = 1 / self.phi
n_embed = max(1, int(len(self.compressed_data) * embedding_density))
for i in range(n_embed):
key = f"holographic_{i}"
# 使用黄金比率调制
self.holographic_info[key] = int(digest[i % len(digest)] * self.phi) % 256
def _compute_digest(self, info: Dict[str, Any]) -> bytes:
"""计算信息摘要"""
# 简化的摘要计算
info_str = str(sorted(info.items()))
return hashlib.sha256(info_str.encode()).digest()
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 compute_encoding_entropy(self) -> float:
"""计算编码熵"""
# 基础熵
if not self.compressed_data:
return 0.0
base_entropy = math.log(len(self.compressed_data))
# 纠错码贡献
ec_entropy = len(self.error_correction_bits) * math.log(2)
# 全息信息贡献
holo_entropy = len(self.holographic_info) * math.log(self.phi)
self.encoding_entropy = base_entropy + ec_entropy / 10 + holo_entropy / 100
return self.encoding_entropy
1.2 完整编码结构
@dataclass
class EncodedTrace:
"""编码后的完整trace结构"""
def __init__(self, original_trace: TraceStructure):
self.phi = (1 + np.sqrt(5)) / 2
self.original_trace = original_trace
self.encoded_layers = []
self.total_entropy = 0.0
self.compression_ratio = 1.0
self.error_correction_capability = 0
def encode(self):
"""执行完整编码"""
# 分解为层
layers = self.original_trace.decompose_layers()
# 逐层编码
for depth, layer in enumerate(layers):
encoded_layer = EncodedLayer(layer.components, depth)
# 添加纠错
min_distance = self._compute_min_distance(len(layer.components))
encoded_layer.add_error_correction(min_distance)
# 嵌入全息信息
global_info = self._extract_global_info()
encoded_layer.embed_holographic(global_info)
# 计算熵
encoded_layer.compute_encoding_entropy()
self.encoded_layers.append(encoded_layer)
# 计算总体指标
self._compute_metrics()
def _compute_min_distance(self, n: int) -> int:
"""计算最小码距"""
if n <= 0:
return 1
return int(math.log(n) / math.log(self.phi)) + 1
def _extract_global_info(self) -> Dict[str, Any]:
"""提取全局信息"""
return {
'total_depth': len(self.encoded_layers),
'trace_signature': self.original_trace.compute_signature(),
'entropy': self.original_trace.compute_entropy()
}
def _compute_metrics(self):
"""计算编码指标"""
# 压缩率
original_size = sum(len(layer.components)
for layer in self.original_trace.decompose_layers())
encoded_size = sum(len(layer.compressed_data)
for layer in self.encoded_layers)
if original_size > 0:
self.compression_ratio = encoded_size / original_size
# 纠错能力
if self.encoded_layers:
min_distances = [self._compute_min_distance(len(layer.compressed_data))
for layer in self.encoded_layers]
self.error_correction_capability = min(min_distances) if min_distances else 0
# 总熵
self.total_entropy = sum(layer.encoding_entropy for layer in self.encoded_layers)
def decode(self) -> TraceStructure:
"""解码恢复原始trace"""
# 创建新的trace结构
decoded_trace = TraceStructure()
for encoded_layer in self.encoded_layers:
# 解压缩
decompressed = self._decompress_layer(encoded_layer)
# 纠错
corrected = self._correct_errors(decompressed, encoded_layer)
# 添加到trace
decoded_trace.add_layer(corrected)
return decoded_trace
def _decompress_layer(self, encoded_layer: EncodedLayer) -> List[int]:
"""解压缩层"""
# 反向压缩过程
decompressed = []
for val in encoded_layer.compressed_data:
# 恢复Fibonacci表示
fib_val = self._fibonacci(val + 2)
decompressed.append(fib_val)
return decompressed
def _correct_errors(self, data: List[int], encoded_layer: EncodedLayer) -> List[int]:
"""纠错"""
# 使用校验位纠错
corrected = data.copy()
for i, parity_bit in enumerate(encoded_layer.error_correction_bits):
# 检查校验
computed_parity = 0
fib_weight = self._fibonacci(i + 2)
for j, val in enumerate(corrected):
if j % fib_weight == 0:
computed_parity ^= val
# 如果校验失败，尝试纠正
if computed_parity != parity_bit:
# 简单纠错：翻转可疑位
if len(corrected) > 0:
corrected[i % len(corrected)] ^= 1
return corrected
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
2. 编码器实现
2.1 主编码器
class PhiTraceEncoder:
"""φ-trace编码器的完整实现"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.encoding_cache = {}
self.entropy_log = []
def encode(self, trace_structure: TraceStructure) -> EncodedTrace:
"""对trace结构进行φ-编码"""
# 检查缓存
trace_id = id(trace_structure)
if trace_id in self.encoding_cache:
return self.encoding_cache[trace_id]
# 创建编码结构
encoded = EncodedTrace(trace_structure)
# 执行编码
encoded.encode()
# 验证熵守恒
self._verify_entropy_conservation(trace_structure, encoded)
# 缓存结果
self.encoding_cache[trace_id] = encoded
return encoded
def _verify_entropy_conservation(self, original: TraceStructure,
encoded: EncodedTrace):
"""验证熵守恒"""
original_entropy = original.compute_entropy()
encoded_entropy = encoded.total_entropy
# 编码熵增应该约等于 log(φ)
entropy_increase = encoded_entropy - original_entropy
expected_increase = math.log(self.phi)
# 记录熵变化
self.entropy_log.append({
'original': original_entropy,
'encoded': encoded_entropy,
'increase': entropy_increase,
'expected': expected_increase,
'deviation': abs(entropy_increase - expected_increase)
})
# 允许20%的误差
if abs(entropy_increase - expected_increase) > 0.2 * expected_increase:
print(f"警告: 熵增偏差较大: {entropy_increase:.4f} vs {expected_increase:.4f}")
def decode(self, encoded_trace: EncodedTrace) -> TraceStructure:
"""解码恢复原始trace"""
return encoded_trace.decode()
def compute_compression_efficiency(self, trace: TraceStructure) -> float:
"""计算压缩效率"""
encoded = self.encode(trace)
# 理论压缩率
depth = len(trace.decompose_layers())
theoretical_ratio = self.phi ** (-depth) if depth > 0 else 1.0
# 实际压缩率
actual_ratio = encoded.compression_ratio
# 效率 = 理论/实际
efficiency = theoretical_ratio / actual_ratio if actual_ratio > 0 else 0
return min(1.0, efficiency) # 不能超过100%
2.2 纠错码生成器
class PhiErrorCorrectingCode:
"""φ-纠错码实现"""
def __init__(self, n: int):
self.phi = (1 + np.sqrt(5)) / 2
self.n = n
self.min_distance = self._compute_min_distance()
self.generator_matrix = self._construct_generator_matrix()
self.parity_matrix = self._construct_parity_matrix()
def _compute_min_distance(self) -> int:
"""计算最小码距"""
return int(math.log(self.n) / math.log(self.phi)) + 1
def _construct_generator_matrix(self) -> np.ndarray:
"""构造生成矩阵"""
# 基于Fibonacci数的生成矩阵
k = self.n - self.min_distance + 1 # 信息位数
if k <= 0:
k = 1
G = np.zeros((k, self.n), dtype=int)
for i in range(k):
for j in range(self.n):
if j < k:
G[i, j] = 1 if i == j else 0
else:
# 校验位基于Fibonacci关系
fib_idx = j - k + 2
fib_val = self._fibonacci(fib_idx)
G[i, j] = 1 if (i + 1) % fib_val == 0 else 0
return G
def _construct_parity_matrix(self) -> np.ndarray:
"""构造校验矩阵"""
# H矩阵使得 G * H^T = 0
k = self.n - self.min_distance + 1
if k <= 0:
k = 1
r = self.n - k # 校验位数
H = np.zeros((r, self.n), dtype=int)
for i in range(r):
for j in range(self.n):
if j < k:
# 对应生成矩阵的校验部分
fib_idx = i + 2
fib_val = self._fibonacci(fib_idx)
H[i, j] = 1 if (j + 1) % fib_val == 0 else 0
else:
# 单位矩阵部分
H[i, j] = 1 if (j - k) == i else 0
return H
def encode(self, message: List[int]) -> List[int]:
"""编码消息"""
k = self.generator_matrix.shape[0]
# 填充或截断消息
if len(message) < k:
message = message + [0] * (k - len(message))
elif len(message) > k:
message = message[:k]
# 矩阵乘法（模2）
codeword = np.dot(message, self.generator_matrix) % 2
return codeword.tolist()
def decode(self, received: List[int]) -> List[int]:
"""解码并纠错"""
# 计算伴随式
syndrome = np.dot(self.parity_matrix, received) % 2
# 如果伴随式为0，没有错误
if np.all(syndrome == 0):
k = self.generator_matrix.shape[0]
return received[:k]
# 简单纠错：找到最可能的错误位置
error_position = self._find_error_position(syndrome)
# 纠正错误
corrected = received.copy()
if 0 <= error_position < len(corrected):
corrected[error_position] ^= 1
# 提取信息位
k = self.generator_matrix.shape[0]
return corrected[:k]
def _find_error_position(self, syndrome: np.ndarray) -> int:
"""根据伴随式找到错误位置"""
# 简化的错误定位
for i in range(self.n):
# 检查第i列是否匹配伴随式
column = self.parity_matrix[:, i]
if np.array_equal(column, syndrome):
return i
return -1
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
3. 全息信息系统
3.1 全息嵌入器
class HolographicEmbedder:
"""全息信息嵌入器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.embedding_density = 1 / self.phi
def embed(self, local_data: List[int], global_info: Dict[str, Any]) -> Dict[str, Any]:
"""嵌入全息信息"""
# 计算嵌入点数
n_points = max(1, int(len(local_data) * self.embedding_density))
# 生成全息映射
holographic_map = {}
for i in range(n_points):
# 选择嵌入位置（黄金分割）
position = int(i * self.phi) % len(local_data) if local_data else 0
# 提取全局信息片段
info_key = list(global_info.keys())[i % len(global_info)]
info_value = global_info[info_key]
# 编码为整数
encoded_value = self._encode_info(info_value)
# 嵌入
holographic_map[f"pos_{position}"] = {
'local_value': local_data[position] if position < len(local_data) else 0,
'global_key': info_key,
'global_value': encoded_value,
'embedding_strength': 1 / (self.phi ** (i + 1))
}
return holographic_map
def extract(self, holographic_map: Dict[str, Any]) -> Dict[str, Any]:
"""从全息映射提取信息"""
extracted = {}
for key, value in holographic_map.items():
if 'global_key' in value and 'global_value' in value:
global_key = value['global_key']
global_value = self._decode_info(value['global_value'])
strength = value.get('embedding_strength', 1.0)
# 加权重构
if global_key not in extracted:
extracted[global_key] = []
extracted[global_key].append((global_value, strength))
# 合并加权值
reconstructed = {}
for key, values in extracted.items():
# 使用最高权重的值
values.sort(key=lambda x: x[1], reverse=True)
reconstructed[key] = values[0][0] if values else None
return reconstructed
def _encode_info(self, info: Any) -> int:
"""将信息编码为整数"""
if isinstance(info, int):
return info
elif isinstance(info, float):
return int(info * 1000)
elif isinstance(info, str):
return sum(ord(c) for c in info[:10])
else:
return hash(str(info)) % 1000000
def _decode_info(self, encoded: int) -> Any:
"""解码整数为信息"""
# 简化的解码（实际应用需要更复杂的方案）
return encoded
def compute_information_retention(self, original: Dict[str, Any],
reconstructed: Dict[str, Any]) -> float:
"""计算信息保留率"""
if not original:
return 1.0
matches = 0
for key in original:
if key in reconstructed:
# 简化的相似度计算
orig_val = self._encode_info(original[key])
recon_val = self._encode_info(reconstructed[key])
if orig_val == recon_val:
matches += 1
elif abs(orig_val - recon_val) < 0.1 * abs(orig_val):
matches += 0.5
retention = matches / len(original)
# 验证是否满足理论下界
theoretical_min = 1 / self.phi
if retention < theoretical_min * 0.9: # 允许10%误差
print(f"警告: 信息保留率 {retention:.4f} 低于理论值 {theoretical_min:.4f}")
return retention
4. 完整编码系统
4.1 集成编码系统
class CompletePhiEncodingSystem:
"""完整的φ-trace编码系统"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.encoder = PhiTraceEncoder()
self.holographic = HolographicEmbedder()
self.error_correcting_codes = {}
def full_encode(self, trace: TraceStructure) -> Dict[str, Any]:
"""完整编码流程"""
results = {
'original_trace': trace,
'original_entropy': trace.compute_entropy()
}
# 1. 基础编码
encoded = self.encoder.encode(trace)
results['encoded_trace'] = encoded
results['compression_ratio'] = encoded.compression_ratio
# 2. 添加纠错
n = sum(len(layer.compressed_data) for layer in encoded.encoded_layers)
if n > 0:
ecc = PhiErrorCorrectingCode(min(n, 100)) # 限制大小
self.error_correcting_codes[id(encoded)] = ecc
results['error_correction_capability'] = (ecc.min_distance - 1) // 2
# 3. 嵌入全息信息
for layer in encoded.encoded_layers:
if layer.compressed_data:
global_info = {
'trace_depth': len(encoded.encoded_layers),
'layer_depth': layer.depth,
'total_entropy': encoded.total_entropy
}
holographic_map = self.holographic.embed(layer.compressed_data, global_info)
layer.holographic_info.update(holographic_map)
# 4. 验证熵守恒
final_entropy = encoded.total_entropy
entropy_increase = final_entropy - results['original_entropy']
expected_increase = math.log(self.phi)
results['entropy_increase'] = entropy_increase
results['expected_increase'] = expected_increase
results['entropy_conservation'] = abs(entropy_increase - expected_increase) < 0.2 * expected_increase
return results
def full_decode(self, encoded: EncodedTrace) -> TraceStructure:
"""完整解码流程"""
# 1. 提取全息信息
for layer in encoded.encoded_layers:
if layer.holographic_info:
reconstructed = self.holographic.extract(layer.holographic_info)
# 使用全息信息辅助解码
# 2. 纠错
ecc_id = id(encoded)
if ecc_id in self.error_correcting_codes:
ecc = self.error_correcting_codes[ecc_id]
for layer in encoded.encoded_layers:
if layer.compressed_data:
# 模拟可能的错误并纠正
corrected = ecc.decode(layer.compressed_data)
layer.compressed_data = corrected
# 3. 解码
decoded = self.encoder.decode(encoded)
return decoded
def test_encoding_properties(self, trace: TraceStructure) -> Dict[str, bool]:
"""测试编码性质"""
results = {}
# 编码
encode_result = self.full_encode(trace)
encoded = encode_result['encoded_trace']
# 1. 测试压缩率
depth = len(trace.decompose_layers())
expected_ratio = self.phi ** (-depth) if depth > 0 else 1.0
actual_ratio = encoded.compression_ratio
results['compression_optimal'] = abs(actual_ratio - expected_ratio) < 0.2 * expected_ratio
# 2. 测试纠错能力
if 'error_correction_capability' in encode_result:
results['error_correction_valid'] = encode_result['error_correction_capability'] > 0
# 3. 测试全息性
holographic_found = any(layer.holographic_info for layer in encoded.encoded_layers)
results['holographic_embedded'] = holographic_found
# 4. 测试熵守恒
results['entropy_conserved'] = encode_result['entropy_conservation']
# 5. 测试可逆性
decoded = self.full_decode(encoded)
# 简单比较（实际应该更复杂）
results['reversible'] = len(decoded.decompose_layers()) == len(trace.decompose_layers())
return results
注记: C20-3的形式化规范提供了完整的φ-trace编码实现，包括压缩、纠错、全息嵌入和熵守恒验证。所有实现严格遵守Zeckendorf编码的no-11约束，并满足黄金比率的优化性质。