T10-3 自相似性定理 - 形式化描述
1. 形式化框架
1.1 自相似性系统
class SelfSimilaritySystem:
"""自相似性定理的数学模型"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci = [1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597]
self.similarity_threshold = 0.9 # 相似度阈值
def apply_scale_transform(self, state: str, scale_factor: int) -> str:
"""应用φ-尺度变换 T_λ where λ = φ^k"""
if not state or scale_factor == 0:
return state
# φ尺度变换的离散实现
if scale_factor > 0:
# 放大变换：通过φ-展开增加细节
result = state
for _ in range(scale_factor):
result = self.phi_expansion(result)
else:
# 缩小变换：通过φ-收缩减少细节
result = state
for _ in range(-scale_factor):
result = self.phi_contraction(result)
return self.enforce_no11_constraint(result)
def phi_expansion(self, state: str) -> str:
"""φ-展开变换：增加自相似细节"""
if not state:
return "10"
expanded = ""
for i, bit in enumerate(state):
if bit == '1':
# 用Fibonacci模式展开
if i < len(self.fibonacci) - 1:
pattern = self.generate_fibonacci_pattern(i)
expanded += pattern
else:
expanded += "10"
else:
expanded += "0"
return expanded
def phi_contraction(self, state: str) -> str:
"""φ-收缩变换：减少细节保持结构"""
if len(state) <= 2:
return state
# 提取主要结构
contracted = ""
i = 0
while i < len(state):
if i + 1 < len(state) and state[i:i+2] == "10":
contracted += "1"
i += 2
else:
if state[i] == '1' and (not contracted or contracted[-1] != '1'):
contracted += state[i]
elif state[i] == '0':
contracted += state[i]
i += 1
return contracted if contracted else "10"
def generate_fibonacci_pattern(self, index: int) -> str:
"""生成Fibonacci模式"""
if index == 0:
return "1"
elif index == 1:
return "10"
else:
# 使用Fibonacci数生成模式
fib_num = self.fibonacci[min(index, len(self.fibonacci)-1)]
pattern = bin(fib_num)[2:]
return self.enforce_no11_constraint(pattern)
def calculate_hausdorff_dimension(self, periodic_orbit: List[str]) -> float:
"""计算周期轨道的Hausdorff维数 D_H = log(p)/log(φ)"""
if not periodic_orbit:
return 0
period_length = len(periodic_orbit)
if period_length == 0:
return 0
# Hausdorff维数公式
dimension = np.log(period_length) / np.log(self.phi)
return dimension
def verify_scale_invariance(self, state: str, scale_factor: int,
collapse_operator) -> Dict[str, Any]:
"""验证尺度不变性：T_λ[Ξ^n[S]] ~ Ξ^[n/λ][S]"""
# 左边：先collapse再尺度变换
left_sequence = []
current = state
for n in range(10): # 测试10步
current = collapse_operator(current)
scaled = self.apply_scale_transform(current, scale_factor)
left_sequence.append(scaled)
# 右边：先尺度变换再collapse
right_sequence = []
scaled_state = self.apply_scale_transform(state, scale_factor)
current = scaled_state
# 计算对应的collapse步数
for n in range(10):
steps = max(1, n // (self.phi ** abs(scale_factor)))
temp = current
for _ in range(int(steps)):
temp = collapse_operator(temp)
right_sequence.append(temp)
# 计算相似度
similarities = []
for l, r in zip(left_sequence, right_sequence):
sim = self.calculate_structural_similarity(l, r)
similarities.append(sim)
avg_similarity = np.mean(similarities)
return {
'scale_factor': scale_factor,
'left_sequence': left_sequence[:5], # 前5个
'right_sequence': right_sequence[:5],
'similarities': similarities,
'average_similarity': avg_similarity,
'scale_invariant': avg_similarity > self.similarity_threshold
}
def calculate_structural_similarity(self, state1: str, state2: str) -> float:
"""计算结构相似度"""
if not state1 and not state2:
return 1.0
if not state1 or not state2:
return 0.0
# 提取结构特征
features1 = self.extract_structural_features(state1)
features2 = self.extract_structural_features(state2)
# 计算特征相似度
similarity = 0
total_weight = 0
for feature in features1:
if feature in features2:
weight = features1[feature]['weight']
sim = 1 - abs(features1[feature]['value'] - features2[feature]['value']) / \
max(features1[feature]['value'], features2[feature]['value'], 1)
similarity += weight * sim
total_weight += weight
return similarity / total_weight if total_weight > 0 else 0
def extract_structural_features(self, state: str) -> Dict[str, Dict[str, float]]:
"""提取结构特征"""
features = {}
if not state:
return features
# 特征1：密度分布
density = state.count('1') / len(state)
features['density'] = {'value': density, 'weight': 1.0}
# 特征2：模式分布
patterns = {'10': 0, '01': 0, '00': 0}
for i in range(len(state) - 1):
pattern = state[i:i+2]
if pattern in patterns:
patterns[pattern] += 1
total_patterns = sum(patterns.values())
if total_patterns > 0:
for pattern, count in patterns.items():
features[f'pattern_{pattern}'] = {
'value': count / total_patterns,
'weight': 0.5
}
# 特征3：φ-结构
phi_weight = self.calculate_phi_weight(state)
features['phi_weight'] = {'value': phi_weight, 'weight': 2.0}
return features
def calculate_phi_weight(self, state: str) -> float:
"""计算φ-权重"""
if not state:
return 0
weight = 0
for i, bit in enumerate(state):
if bit == '1':
weight += 1 / (self.phi ** i)
return weight
def verify_recursive_isomorphism(self, periodic_orbit: List[str]) -> Dict[str, Any]:
"""验证递归结构同构：Structure(S*) ≅ Structure(Ξ^k[S*])"""
if not periodic_orbit:
return {'isomorphic': False, 'reason': 'Empty orbit'}
# 计算轨道中每个状态的结构特征
structures = []
for state in periodic_orbit:
structure = self.extract_structural_features(state)
structures.append(structure)
# 验证结构的循环同构
isomorphisms = []
for i in range(len(periodic_orbit)):
j = (i + 1) % len(periodic_orbit)
similarity = self.compare_structures(structures[i], structures[j])
isomorphisms.append({
'state_i': i,
'state_j': j,
'similarity': similarity,
'isomorphic': similarity > self.similarity_threshold
})
all_isomorphic = all(iso['isomorphic'] for iso in isomorphisms)
avg_similarity = np.mean([iso['similarity'] for iso in isomorphisms])
return {
'isomorphic': all_isomorphic,
'isomorphisms': isomorphisms,
'average_similarity': avg_similarity,
'period_length': len(periodic_orbit)
}
def compare_structures(self, struct1: Dict, struct2: Dict) -> float:
"""比较两个结构的相似度"""
if not struct1 and not struct2:
return 1.0
if not struct1 or not struct2:
return 0.0
all_features = set(struct1.keys()) | set(struct2.keys())
similarity = 0
total_weight = 0
for feature in all_features:
if feature in struct1 and feature in struct2:
val1 = struct1[feature]['value']
val2 = struct2[feature]['value']
weight = (struct1[feature]['weight'] + struct2[feature]['weight']) / 2
if max(val1, val2) > 0:
feat_sim = 1 - abs(val1 - val2) / max(val1, val2)
else:
feat_sim = 1.0
similarity += weight * feat_sim
total_weight += weight
else:
# 特征缺失惩罚
weight = struct1.get(feature, struct2.get(feature))['weight']
total_weight += weight
return similarity / total_weight if total_weight > 0 else 0
def enforce_no11_constraint(self, state: str) -> str:
"""强制no-11约束"""
result = ""
i = 0
while i < len(state):
if i < len(state) - 1 and state[i] == '1' and state[i+1] == '1':
result += "10"
i += 2
else:
result += state[i]
i += 1
return result
1.2 分形维数分析器
class FractalDimensionAnalyzer:
"""分形维数计算和分析"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def calculate_box_dimension(self, trajectory: List[str], max_scale: int = 10) -> float:
"""计算盒维数（数值逼近Hausdorff维数）"""
if not trajectory:
return 0
# 将轨道嵌入到度量空间
embedded_points = self.embed_trajectory(trajectory)
# 不同尺度的盒子计数
scales = []
counts = []
for scale in range(1, max_scale + 1):
epsilon = 1 / (self.phi ** scale)
count = self.count_boxes(embedded_points, epsilon)
if count > 0:
scales.append(np.log(1/epsilon))
counts.append(np.log(count))
if len(scales) < 2:
return 0
# 线性拟合 log(N) = D * log(1/ε) + C
dimension = np.polyfit(scales, counts, 1)[0]
return dimension
def embed_trajectory(self, trajectory: List[str]) -> List[np.ndarray]:
"""将轨道嵌入到度量空间"""
points = []
for state in trajectory:
# 使用φ-坐标嵌入
coords = []
for i, bit in enumerate(state[:10]): # 限制维度
if bit == '1':
coords.append(1 / (self.phi ** i))
else:
coords.append(0)
if coords:
points.append(np.array(coords))
return points
def count_boxes(self, points: List[np.ndarray], epsilon: float) -> int:
"""计算覆盖点集所需的ε-盒子数"""
if not points:
return 0
# 简化的盒计数算法
covered = set()
for point in points:
# 将点映射到盒子索引
box_index = tuple(int(coord / epsilon) for coord in point)
covered.add(box_index)
return len(covered)
def calculate_correlation_dimension(self, trajectory: List[str],
max_pairs: int = 1000) -> float:
"""计算关联维数"""
if len(trajectory) < 2:
return 0
embedded = self.embed_trajectory(trajectory)
n = len(embedded)
if n < 2:
return 0
# 计算点对距离
distances = []
pairs = min(max_pairs, n * (n - 1) // 2)
for _ in range(pairs):
i = np.random.randint(0, n)
j = np.random.randint(0, n)
if i != j:
dist = np.linalg.norm(embedded[i] - embedded[j])
if dist > 0:
distances.append(dist)
if not distances:
return 0
# 计算关联积分
distances.sort()
correlations = []
radii = []
for r in np.logspace(np.log10(min(distances)), np.log10(max(distances)), 20):
c = sum(1 for d in distances if d < r) / len(distances)
if c > 0:
correlations.append(np.log(c))
radii.append(np.log(r))
if len(radii) < 2:
return 0
# 线性拟合
dimension = np.polyfit(radii, correlations, 1)[0]
return dimension
def verify_critical_dimension(self) -> Dict[str, float]:
"""验证临界维数 D_c = log_φ(F_∞)"""
# Fibonacci数列的渐近增长率
golden_ratio = self.phi
# 理论临界维数
theoretical_dc = np.log(golden_ratio) / np.log(self.phi) # = 1
# 数值估计（使用大Fibonacci数）
large_fib = [987, 1597, 2584, 4181, 6765]
growth_rates = []
for i in range(len(large_fib) - 1):
rate = large_fib[i+1] / large_fib[i]
growth_rates.append(rate)
avg_growth = np.mean(growth_rates)
numerical_dc = np.log(avg_growth) / np.log(self.phi)
return {
'theoretical_critical_dimension': theoretical_dc,
'numerical_critical_dimension': numerical_dc,
'golden_ratio_growth': avg_growth,
'relative_error': abs(numerical_dc - theoretical_dc) / theoretical_dc
}
1.3 多尺度周期性分析器
class MultiScalePeriodicityAnalyzer:
"""多尺度周期性分析"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.similarity_system = SelfSimilaritySystem()
def analyze_nested_periods(self, trajectory: List[str],
max_depth: int = 3) -> Dict[str, Any]:
"""分析嵌套周期结构"""
if not trajectory:
return {'nested_periods': [], 'depth': 0}
nested_periods = []
current_level = trajectory
for depth in range(max_depth):
# 检测当前尺度的周期
period_info = self.detect_period_at_scale(current_level, depth)
if period_info['period'] > 0:
nested_periods.append(period_info)
# 提取更细尺度的结构
current_level = self.extract_finer_scale(current_level, period_info['period'])
if not current_level:
break
else:
break
# 验证φ-关系
phi_relations = []
for i in range(len(nested_periods) - 1):
p1 = nested_periods[i]['period']
p2 = nested_periods[i+1]['period']
if p2 > 0:
ratio = p1 / p2
phi_power = np.log(ratio) / np.log(self.phi)
phi_relations.append({
'scale_i': i,
'scale_j': i + 1,
'period_ratio': ratio,
'phi_power': phi_power,
'is_phi_multiple': abs(phi_power - round(phi_power)) < 0.1
})
return {
'nested_periods': nested_periods,
'depth': len(nested_periods),
'phi_relations': phi_relations,
'follows_phi_scaling': all(rel['is_phi_multiple'] for rel in phi_relations)
}
def detect_period_at_scale(self, sequence: List[str], scale: int) -> Dict[str, Any]:
"""在特定尺度检测周期"""
if not sequence:
return {'period': 0, 'scale': scale}
# 应用尺度变换
scaled_sequence = []
for state in sequence:
scaled = self.similarity_system.apply_scale_transform(state, -scale)
scaled_sequence.append(scaled)
# 检测周期
n = len(scaled_sequence)
for period in range(1, n // 2 + 1):
is_periodic = True
for i in range(n - period):
if scaled_sequence[i] != scaled_sequence[i + period]:
is_periodic = False
break
if is_periodic:
return {
'period': period,
'scale': scale,
'sequence_sample': scaled_sequence[:min(5, period)]
}
return {'period': 0, 'scale': scale}
def extract_finer_scale(self, sequence: List[str], period: int) -> List[str]:
"""提取更细尺度的结构"""
if not sequence or period == 0:
return []
# 取一个周期的子序列
sub_sequence = sequence[:period]
# 展开到更细尺度
finer_sequence = []
for state in sub_sequence:
expanded = self.similarity_system.phi_expansion(state)
finer_sequence.append(expanded)
return finer_sequence
def verify_scale_hierarchy(self, base_pattern: str, scales: List[int]) -> Dict[str, Any]:
"""验证尺度层级结构"""
patterns = {}
# 生成不同尺度的模式
current = base_pattern
patterns[0] = current
for scale in scales:
if scale > 0:
current = self.similarity_system.apply_scale_transform(current, scale)
else:
current = self.similarity_system.apply_scale_transform(current, scale)
patterns[scale] = current
# 验证自相似关系
similarities = []
for i, scale1 in enumerate(scales[:-1]):
scale2 = scales[i+1]
pattern1 = patterns[scale1]
pattern2 = patterns[scale2]
# 调整到相同尺度比较
scale_diff = scale2 - scale1
adjusted1 = self.similarity_system.apply_scale_transform(pattern1, scale_diff)
similarity = self.similarity_system.calculate_structural_similarity(adjusted1, pattern2)
similarities.append({
'scale_from': scale1,
'scale_to': scale2,
'similarity': similarity,
'self_similar': similarity > self.similarity_system.similarity_threshold
})
return {
'patterns': patterns,
'similarities': similarities,
'average_similarity': np.mean([s['similarity'] for s in similarities]),
'maintains_hierarchy': all(s['self_similar'] for s in similarities)
}
2. 自相似性验证系统
2.1 完整验证框架
class SelfSimilarityVerifier:
"""T10-3自相似性定理的综合验证"""
def __init__(self):
self.similarity_system = SelfSimilaritySystem()
self.dimension_analyzer = FractalDimensionAnalyzer()
self.periodicity_analyzer = MultiScalePeriodicityAnalyzer()
self.phi = (1 + np.sqrt(5)) / 2
def verify_scale_invariance_property(self, test_states: List[str],
collapse_operator) -> Dict[str, Any]:
"""验证尺度不变性质"""
results = []
scale_factors = [-2, -1, 1, 2] # φ^k的不同k值
for state in test_states:
state_results = []
for k in scale_factors:
verification = self.similarity_system.verify_scale_invariance(
state, k, collapse_operator
)
state_results.append(verification)
results.append({
'initial_state': state,
'scale_results': state_results,
'average_invariance': np.mean([r['average_similarity'] for r in state_results])
})
overall_invariance = np.mean([r['average_invariance'] for r in results])
return {
'individual_results': results,
'overall_scale_invariance': overall_invariance,
'passes_threshold': overall_invariance > 0.7 # 放宽阈值因为是离散系统
}
def verify_fractal_dimension_formula(self, periodic_orbits: List[List[str]]) -> Dict[str, Any]:
"""验证分形维数公式 D_H = log(p)/log(φ)"""
results = []
for orbit in periodic_orbits:
if not orbit:
continue
# 理论维数
theoretical_dim = self.similarity_system.calculate_hausdorff_dimension(orbit)
# 数值维数（盒维数）
numerical_dim = self.dimension_analyzer.calculate_box_dimension(orbit)
# 关联维数
correlation_dim = self.dimension_analyzer.calculate_correlation_dimension(orbit)
results.append({
'period_length': len(orbit),
'theoretical_dimension': theoretical_dim,
'box_dimension': numerical_dim,
'correlation_dimension': correlation_dim,
'relative_error': abs(numerical_dim - theoretical_dim) / theoretical_dim if theoretical_dim > 0 else float('inf')
})
avg_error = np.mean([r['relative_error'] for r in results if r['relative_error'] < float('inf')])
return {
'dimension_results': results,
'average_relative_error': avg_error,
'formula_verified': avg_error < 0.3 # 30%误差容限
}
def verify_recursive_isomorphism_property(self, periodic_orbits: List[List[str]]) -> Dict[str, Any]:
"""验证递归结构同构性"""
results = []
for orbit in periodic_orbits:
if not orbit:
continue
isomorphism_result = self.similarity_system.verify_recursive_isomorphism(orbit)
results.append(isomorphism_result)
success_rate = sum(1 for r in results if r['isomorphic']) / len(results) if results else 0
avg_similarity = np.mean([r['average_similarity'] for r in results]) if results else 0
return {
'orbit_results': results,
'isomorphism_success_rate': success_rate,
'average_structural_similarity': avg_similarity,
'property_verified': success_rate > 0.7
}
def verify_multi_scale_periodicity(self, test_trajectories: List[List[str]]) -> Dict[str, Any]:
"""验证多尺度周期性"""
results = []
for trajectory in test_trajectories:
if not trajectory:
continue
nested_analysis = self.periodicity_analyzer.analyze_nested_periods(trajectory)
results.append(nested_analysis)
# 统计φ-缩放关系
phi_scaling_rate = sum(1 for r in results if r['follows_phi_scaling']) / len(results) if results else 0
avg_depth = np.mean([r['depth'] for r in results]) if results else 0
return {
'trajectory_results': results,
'phi_scaling_rate': phi_scaling_rate,
'average_nesting_depth': avg_depth,
'multi_scale_verified': phi_scaling_rate > 0.6 and avg_depth > 1
}
def generate_test_orbits(self, collapse_operator, num_orbits: int = 5) -> List[List[str]]:
"""生成测试用的周期轨道"""
test_states = ["10", "101", "1010", "10100", "101001"]
orbits = []
for i in range(min(num_orbits, len(test_states))):
state = test_states[i]
trajectory = []
current = state
# 生成轨迹直到进入周期
seen = {}
step = 0
while current not in seen and step < 50:
seen[current] = step
trajectory.append(current)
current = collapse_operator(current)
step += 1
# 提取周期轨道
if current in seen:
period_start = seen[current]
periodic_orbit = trajectory[period_start:]
if periodic_orbit:
orbits.append(periodic_orbit)
return orbits
def run_comprehensive_verification(self, collapse_operator) -> Dict[str, Any]:
"""运行综合验证"""
print("生成测试轨道...")
test_orbits = self.generate_test_orbits(collapse_operator)
test_states = ["10", "101", "1010", "10101"]
print("验证尺度不变性...")
scale_invariance = self.verify_scale_invariance_property(test_states, collapse_operator)
print("验证分形维数...")
fractal_dimension = self.verify_fractal_dimension_formula(test_orbits)
print("验证递归同构...")
recursive_isomorphism = self.verify_recursive_isomorphism_property(test_orbits)
print("验证多尺度周期性...")
multi_scale = self.verify_multi_scale_periodicity(test_orbits)
# 综合评分
scores = [
scale_invariance['passes_threshold'],
fractal_dimension['formula_verified'],
recursive_isomorphism['property_verified'],
multi_scale['multi_scale_verified']
]
overall_score = sum(scores) / len(scores)
return {
'scale_invariance': scale_invariance,
'fractal_dimension': fractal_dimension,
'recursive_isomorphism': recursive_isomorphism,
'multi_scale_periodicity': multi_scale,
'overall_verification': {
'individual_scores': scores,
'overall_score': overall_score,
'theorem_verified': overall_score > 0.6
}
}
def generate_verification_report(self, results: Dict[str, Any]) -> str:
"""生成验证报告"""
report = "# T10-3 自相似性定理验证报告\n\n"
overall = results['overall_verification']
report += f"## 总体验证结果\n"
report += f"- 综合得分: {overall['overall_score']:.3f}\n"
report += f"- 定理验证: {'通过' if overall['theorem_verified'] else '未通过'}\n\n"
# 尺度不变性
scale = results['scale_invariance']
report += f"### 尺度不变性\n"
report += f"- 整体不变性: {scale['overall_scale_invariance']:.3f}\n"
report += f"- 通过阈值: {'是' if scale['passes_threshold'] else '否'}\n\n"
# 分形维数
fractal = results['fractal_dimension']
report += f"### 分形维数公式\n"
report += f"- 平均相对误差: {fractal['average_relative_error']:.3f}\n"
report += f"- 公式验证: {'通过' if fractal['formula_verified'] else '未通过'}\n\n"
# 递归同构
iso = results['recursive_isomorphism']
report += f"### 递归结构同构\n"
report += f"- 同构成功率: {iso['isomorphism_success_rate']:.3f}\n"
report += f"- 平均结构相似度: {iso['average_structural_similarity']:.3f}\n"
report += f"- 性质验证: {'通过' if iso['property_verified'] else '未通过'}\n\n"
# 多尺度周期性
multi = results['multi_scale_periodicity']
report += f"### 多尺度周期性\n"
report += f"- φ-缩放率: {multi['phi_scaling_rate']:.3f}\n"
report += f"- 平均嵌套深度: {multi['average_nesting_depth']:.2f}\n"
report += f"- 验证结果: {'通过' if multi['multi_scale_verified'] else '未通过'}\n\n"
# 结论
if overall['overall_score'] > 0.75:
report += "## 结论\nT10-3自相似性定理得到强有力的验证支持。"
elif overall['overall_score'] > 0.5:
report += "## 结论\nT10-3自相似性定理得到部分验证支持。"
else:
report += "## 结论\nT10-3自相似性定理需要进一步完善。"
return report
2.2 Fibonacci自相似验证
class FibonacciSelfSimilarityVerifier:
"""Fibonacci串的自相似性验证"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def generate_fibonacci_string(self, n: int) -> str:
"""生成第n个Fibonacci串"""
if n == 0:
return "0"
elif n == 1:
return "01"
else:
s1 = "0"
s2 = "01"
for _ in range(2, n + 1):
s1, s2 = s2, s2 + s1
return s2
def verify_fibonacci_self_similarity(self, max_n: int = 10) -> Dict[str, Any]:
"""验证Fibonacci串的自相似性"""
strings = [self.generate_fibonacci_string(i) for i in range(max_n)]
# 验证递归结构
recursive_verified = []
for i in range(2, max_n):
s_n = strings[i]
s_n1 = strings[i-1]
s_n2 = strings[i-2]
# 验证 S_n = S_{n-1} + S_{n-2}
constructed = s_n1 + s_n2
verified = (s_n == constructed)
recursive_verified.append(verified)
# 验证自相似模式
pattern_analysis = []
for i in range(3, max_n):
string = strings[i]
patterns = self.extract_self_similar_patterns(string)
pattern_analysis.append({
'n': i,
'string_length': len(string),
'unique_patterns': len(patterns),
'pattern_distribution': patterns
})
# 计算分形维数
dimension_analysis = []
for i in range(5, max_n):
string = strings[i]
dim = self.calculate_string_dimension(string)
theoretical_dim = np.log(self.phi) / np.log(2) # 约0.694
dimension_analysis.append({
'n': i,
'calculated_dimension': dim,
'theoretical_dimension': theoretical_dim,
'relative_error': abs(dim - theoretical_dim) / theoretical_dim
})
avg_dimension_error = np.mean([d['relative_error'] for d in dimension_analysis])
return {
'recursive_structure_verified': all(recursive_verified),
'pattern_analysis': pattern_analysis,
'dimension_analysis': dimension_analysis,
'average_dimension_error': avg_dimension_error,
'self_similarity_confirmed': avg_dimension_error < 0.2
}
def extract_self_similar_patterns(self, string: str) -> Dict[str, int]:
"""提取自相似模式"""
patterns = {}
# 不同长度的模式
for length in [2, 3, 5, 8]: # Fibonacci数
for i in range(len(string) - length + 1):
pattern = string[i:i+length]
patterns[pattern] = patterns.get(pattern, 0) + 1
# 只保留重复出现的模式
return {p: c for p, c in patterns.items() if c > 1}
def calculate_string_dimension(self, string: str) -> float:
"""计算字符串的分形维数"""
if len(string) < 2:
return 0
# 使用滑动窗口方法
scales = []
counts = []
for window_size in range(1, min(len(string) // 2, 20)):
patterns = set()
for i in range(len(string) - window_size + 1):
patterns.add(string[i:i+window_size])
if len(patterns) > 0:
scales.append(np.log(window_size))
counts.append(np.log(len(patterns)))
if len(scales) < 2:
return 0
# 线性拟合
dimension = np.polyfit(scales, counts, 1)[0]
return dimension
3. 总结
本形式化框架提供了：
- 自相似性系统：实现尺度变换和结构分析
- 分形维数分析：计算Hausdorff维数、盒维数和关联维数
- 多尺度周期分析：验证嵌套周期的φ-关系
- 综合验证框架：全面测试定理的各个方面
- Fibonacci验证：特殊案例的深入分析
这为T10-3自相似性定理提供了严格的数学基础和可验证的实现。