T20-2 ψₒ-trace结构定理形式化规范
1. 基础结构定义
1.1 trace层次组件类
import numpy as np
import math
from typing import List, Dict, Tuple, Optional, Any
from dataclasses import dataclass
from enum import Enum
@dataclass
class TraceComponent:
"""trace结构的单层组件"""
def __init__(self, layer: int, value: int, weight: float = 1.0):
self.phi = (1 + np.sqrt(5)) / 2
self.layer = layer
self.value = value
self.weight = weight
self.fibonacci_index = layer + 2 # 对应的Fibonacci索引
self.phi_power = self.phi ** layer
def fibonacci_bound(self) -> Tuple[int, int]:
"""返回该层对应的Fibonacci数界限"""
fib_cache = self._fibonacci_sequence(self.fibonacci_index + 2)
lower = fib_cache[self.fibonacci_index]
upper = fib_cache[self.fibonacci_index + 1]
return (lower, upper)
def _fibonacci_sequence(self, n: int) -> List[int]:
"""生成Fibonacci序列"""
if n < 2:
return [1, 2]
fib = [1, 2]
for i in range(2, n):
fib.append(fib[i-1] + fib[i-2])
return fib
def normalize_value(self) -> float:
"""归一化值到[0,1]区间"""
lower, upper = self.fibonacci_bound()
if upper == lower:
return 0.0
return (self.value - lower) / (upper - lower)
def phi_weight(self) -> float:
"""计算φ-权重"""
return self.weight / self.phi_power
def __eq__(self, other) -> bool:
if isinstance(other, TraceComponent):
return (self.layer == other.layer and
self.value == other.value)
return False
def __hash__(self) -> int:
return hash((self.layer, self.value))
@dataclass
class TraceStructure:
"""完整的trace结构"""
def __init__(self, components: Dict[int, TraceComponent]):
self.phi = (1 + np.sqrt(5)) / 2
self.components = components
self.max_layer = max(components.keys()) if components else 0
self.total_value = sum(comp.value for comp in components.values())
self.structural_core = self._compute_core()
self.entropy = self._compute_entropy()
def _compute_core(self) -> int:
"""计算结构核"""
if not self.components:
return 1
values = [comp.value for comp in self.components.values()]
return math.gcd(*values) if len(values) > 1 else values[0]
def _compute_entropy(self) -> float:
"""计算结构熵"""
if not self.components or self.total_value == 0:
return 0.0
entropy = 0.0
for comp in self.components.values():
if comp.value > 0:
p = comp.value / self.total_value
entropy -= p * math.log(p, self.phi)
return entropy
def get_layer_distribution(self) -> Dict[int, float]:
"""获取层次分布"""
if self.total_value == 0:
return {}
return {layer: comp.value / self.total_value
for layer, comp in self.components.items()}
def layer_complexity(self) -> float:
"""计算层次复杂度"""
if not self.components:
return 0.0
return len(self.components) * math.log(self.max_layer + 1, self.phi)
def structural_signature(self) -> str:
"""生成结构签名"""
if not self.components:
return "empty"
signature_parts = []
for layer in sorted(self.components.keys()):
comp = self.components[layer]
signature_parts.append(f"{layer}:{comp.value}")
return "-".join(signature_parts)
def __eq__(self, other) -> bool:
if isinstance(other, TraceStructure):
return self.components == other.components
return False
def __hash__(self) -> int:
return hash(self.structural_signature())
1.2 Zeckendorf扩展trace计算
class ZeckendorfTraceCalculator:
"""基于Zeckendorf编码的trace计算器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci_cache = self._generate_fibonacci_sequence(100)
def _generate_fibonacci_sequence(self, n: int) -> List[int]:
"""生成Fibonacci序列"""
if n < 2:
return [1, 2]
fib = [1, 2]
for i in range(2, n):
fib.append(fib[i-1] + fib[i-2])
return fib
def compute_full_trace(self, zeck_string: 'ZeckendorfString') -> int:
"""计算完整的ψₒ-trace值"""
if not zeck_string.representation or zeck_string.representation == "0":
return 0
trace_value = 0
representation = zeck_string.representation
for i, bit in enumerate(representation):
if bit == '1':
# 位置权重 * Fibonacci权重
position_weight = i + 1
fib_index = len(representation) - 1 - i
if fib_index < len(self.fibonacci_cache):
fib_weight = self.fibonacci_cache[fib_index]
trace_value += position_weight * fib_weight
return trace_value
def compute_weighted_trace(self, zeck_string: 'ZeckendorfString',
weights: List[float]) -> float:
"""计算加权trace值"""
if not zeck_string.representation:
return 0.0
trace_value = 0.0
representation = zeck_string.representation
for i, bit in enumerate(representation):
if bit == '1':
weight = weights[i] if i < len(weights) else 1.0
fib_index = len(representation) - 1 - i
if fib_index < len(self.fibonacci_cache):
fib_weight = self.fibonacci_cache[fib_index]
trace_value += weight * fib_weight
return trace_value
def trace_fibonacci_decomposition(self, trace_value: int) -> Dict[int, int]:
"""将trace值分解为Fibonacci组成"""
decomposition = {}
remaining = trace_value
for i in range(len(self.fibonacci_cache) - 1, -1, -1):
fib = self.fibonacci_cache[i]
if fib <= remaining:
count = remaining // fib
if count > 0:
decomposition[i] = count
remaining -= count * fib
return decomposition
def compute_trace_modular_structure(self, zeck_string: 'ZeckendorfString',
modulus: int) -> Dict[str, int]:
"""计算trace的模结构"""
trace_value = self.compute_full_trace(zeck_string)
structure = {
'full_trace': trace_value,
'mod_trace': trace_value % modulus,
'quotient': trace_value // modulus,
'residue_class': trace_value % modulus
}
# 计算模φ的特殊性质
phi_mod = int(self.phi * modulus) % modulus
structure['phi_residue'] = (trace_value * phi_mod) % modulus
return structure
2. 层次分解器
2.1 trace结构分解器
class TraceLayerDecomposer:
"""trace层次结构分解器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.trace_calculator = ZeckendorfTraceCalculator()
self.fibonacci_cache = self.trace_calculator.fibonacci_cache
def decompose_trace_structure(self, zeck_string: 'ZeckendorfString') -> TraceStructure:
"""分解trace为层次结构"""
trace_value = self.trace_calculator.compute_full_trace(zeck_string)
if trace_value == 0:
return TraceStructure({})
# 确定最大分解层数
max_layer = self._determine_max_layer(trace_value)
components = {}
# 按层分解
for k in range(max_layer + 1):
layer_component = self._extract_layer_component(
zeck_string, trace_value, k)
if layer_component.value > 0:
components[k] = layer_component
return TraceStructure(components)
def _determine_max_layer(self, trace_value: int) -> int:
"""确定最大分解层数"""
if trace_value <= 1:
return 0
return int(math.log(trace_value + 1) / math.log(self.phi))
def _extract_layer_component(self, zeck_string: 'ZeckendorfString',
full_trace: int, layer: int) -> TraceComponent:
"""提取指定层的trace组件"""
# 计算该层对应的Fibonacci界限
fib_lower = self.fibonacci_cache[layer + 2] if layer + 2 < len(self.fibonacci_cache) else 1
fib_upper = self.fibonacci_cache[layer + 3] if layer + 3 < len(self.fibonacci_cache) else fib_lower * 2
# 提取该层的贡献
layer_value = 0
representation = zeck_string.representation
for i, bit in enumerate(representation):
if bit == '1':
position_weight = i + 1
fib_index = len(representation) - 1 - i
if fib_index < len(self.fibonacci_cache):
fib_weight = self.fibonacci_cache[fib_index]
contribution = position_weight * fib_weight
# 检查是否属于当前层
if fib_lower <= contribution < fib_upper:
layer_value += contribution
# 计算层权重
weight = self._compute_layer_weight(layer, full_trace)
return TraceComponent(layer, layer_value, weight)
def _compute_layer_weight(self, layer: int, full_trace: int) -> float:
"""计算层权重"""
if full_trace == 0:
return 0.0
base_weight = 1.0 / (self.phi ** layer)
normalization = math.log(full_trace + 1, self.phi)
return base_weight * normalization
def validate_decomposition(self, original_trace: int,
structure: TraceStructure) -> bool:
"""验证分解的正确性"""
reconstructed = sum(
comp.value * (self.phi ** comp.layer)
for comp in structure.components.values()
)
# 允许小的数值误差
tolerance = max(1, original_trace * 0.01)
return abs(reconstructed - original_trace) <= tolerance
def compute_layer_interactions(self, structure: TraceStructure) -> Dict[Tuple[int, int], float]:
"""计算层间相互作用"""
interactions = {}
for layer1, comp1 in structure.components.items():
for layer2, comp2 in structure.components.items():
if layer1 < layer2:
# 计算层间耦合强度
coupling = self._compute_coupling_strength(comp1, comp2)
if coupling > 0:
interactions[(layer1, layer2)] = coupling
return interactions
def _compute_coupling_strength(self, comp1: TraceComponent,
comp2: TraceComponent) -> float:
"""计算两层间的耦合强度"""
layer_diff = abs(comp2.layer - comp1.layer)
value_ratio = min(comp1.value, comp2.value) / max(comp1.value, comp2.value)
# φ-衰减的耦合强度
coupling = value_ratio / (self.phi ** layer_diff)
return coupling
2.2 结构核提取器
class TraceCoreExtractor:
"""trace结构核提取器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def extract_structural_core(self, structure: TraceStructure) -> Dict[str, Any]:
"""提取结构核信息"""
if not structure.components:
return {'core_value': 1, 'core_layers': [], 'core_type': 'empty'}
core_info = {
'core_value': structure.structural_core,
'core_layers': self._identify_core_layers(structure),
'core_type': self._classify_core_type(structure),
'core_stability': self._compute_core_stability(structure),
'phi_invariant': self._check_phi_invariance(structure)
}
return core_info
def _identify_core_layers(self, structure: TraceStructure) -> List[int]:
"""识别核心贡献层"""
core_value = structure.structural_core
core_layers = []
for layer, comp in structure.components.items():
if comp.value % core_value == 0:
core_layers.append(layer)
return sorted(core_layers)
def _classify_core_type(self, structure: TraceStructure) -> str:
"""分类核类型"""
core_value = structure.structural_core
if core_value == 1:
return 'trivial'
elif self._is_fibonacci_number(core_value):
return 'fibonacci'
elif self._is_phi_related(core_value):
return 'phi_related'
else:
return 'composite'
def _is_fibonacci_number(self, n: int) -> bool:
"""检查是否为Fibonacci数"""
fib_cache = [1, 2]
while fib_cache[-1] < n:
fib_cache.append(fib_cache[-1] + fib_cache[-2])
return n in fib_cache
def _is_phi_related(self, n: int) -> bool:
"""检查是否与φ相关"""
# 检查是否为φ的整数倍的近似
phi_multiples = [int(self.phi * k) for k in range(1, 20)]
return n in phi_multiples
def _compute_core_stability(self, structure: TraceStructure) -> float:
"""计算核稳定性"""
if not structure.components:
return 1.0
core_value = structure.structural_core
total_deviation = 0.0
for comp in structure.components.values():
deviation = abs(comp.value - core_value * (comp.value // core_value))
total_deviation += deviation
if structure.total_value == 0:
return 1.0
stability = 1.0 - (total_deviation / structure.total_value)
return max(0.0, stability)
def _check_phi_invariance(self, structure: TraceStructure) -> bool:
"""检查φ-不变性"""
core_value = structure.structural_core
# 检查核值是否满足φ-不变性质
phi_scaled = int(core_value * self.phi)
phi_inverse_scaled = int(core_value / self.phi)
# 检查在某个模意义下的不变性
modulus = max(10, core_value)
return (phi_scaled % modulus) == (core_value % modulus)
def compute_core_evolution(self, initial_core: int, num_steps: int) -> List[int]:
"""计算核在collapse下的演化"""
evolution = [initial_core]
current_core = initial_core
for step in range(num_steps):
# φ-变换下的核演化
next_core = int(current_core * self.phi) % (current_core * 10 + 1)
evolution.append(next_core)
current_core = next_core
return evolution
def verify_core_invariance(self, original_structure: TraceStructure,
collapsed_structure: TraceStructure,
modulus: int) -> bool:
"""验证核在collapse下的不变性"""
original_core = original_structure.structural_core
collapsed_core = collapsed_structure.structural_core
# 检查φ-倍数关系
expected_core = (int(original_core * self.phi)) % modulus
actual_core = collapsed_core % modulus
return abs(expected_core - actual_core) <= 1
3. 螺旋演化追踪器
3.1 演化轨迹分析器
class SpiralEvolutionTracker:
"""螺旋演化追踪器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.decomposer = TraceLayerDecomposer()
def track_evolution_sequence(self, initial_state: 'ZeckendorfString',
collapse_func: callable,
num_steps: int) -> List[Dict[str, Any]]:
"""追踪演化序列"""
evolution_data = []
current_state = initial_state
for step in range(num_steps + 1):
# 分析当前状态的trace结构
structure = self.decomposer.decompose_trace_structure(current_state)
# 记录演化数据
step_data = {
'step': step,
'state': current_state,
'structure': structure,
'spiral_phase': self._compute_spiral_phase(structure, step),
'evolution_vector': self._compute_evolution_vector(structure, step),
'convergence_measure': self._compute_convergence_measure(evolution_data, structure)
}
evolution_data.append(step_data)
# 执行collapse到下一状态
if step < num_steps:
try:
current_state = collapse_func(current_state)
except:
break
return evolution_data
def _compute_spiral_phase(self, structure: TraceStructure, step: int) -> complex:
"""计算螺旋相位"""
if not structure.components:
return complex(0, 0)
# 基于层次分布计算相位
phase_contributions = []
for layer, comp in structure.components.items():
layer_phase = (comp.value / structure.total_value) * (2 * math.pi * layer / len(structure.components))
phase_contributions.append(layer_phase)
total_phase = sum(phase_contributions)
radius = structure.total_value * (self.phi ** step)
return complex(radius * math.cos(total_phase), radius * math.sin(total_phase))
def _compute_evolution_vector(self, structure: TraceStructure, step: int) -> np.ndarray:
"""计算演化向量"""
if not structure.components:
return np.array([0.0])
# 构造多维演化向量
max_layer = structure.max_layer
vector = np.zeros(max_layer + 1)
for layer, comp in structure.components.items():
# φ^step缩放的分量
vector[layer] = comp.value * (self.phi ** step)
return vector
def _compute_convergence_measure(self, evolution_data: List[Dict],
current_structure: TraceStructure) -> float:
"""计算收敛度量"""
if len(evolution_data) < 2:
return 0.0
# 比较最近几步的结构相似性
recent_steps = min(5, len(evolution_data))
similarities = []
for i in range(1, recent_steps + 1):
if len(evolution_data) >= i:
prev_structure = evolution_data[-i]['structure']
similarity = self._compute_structure_similarity(
current_structure, prev_structure)
similarities.append(similarity)
return np.mean(similarities) if similarities else 0.0
def _compute_structure_similarity(self, struct1: TraceStructure,
struct2: TraceStructure) -> float:
"""计算结构相似度"""
if not struct1.components and not struct2.components:
return 1.0
if not struct1.components or not struct2.components:
return 0.0
# 计算层次分布的相似度
dist1 = struct1.get_layer_distribution()
dist2 = struct2.get_layer_distribution()
all_layers = set(dist1.keys()) | set(dist2.keys())
if not all_layers:
return 1.0
similarity = 0.0
for layer in all_layers:
p1 = dist1.get(layer, 0.0)
p2 = dist2.get(layer, 0.0)
similarity += min(p1, p2)
return similarity
def analyze_spiral_properties(self, evolution_data: List[Dict]) -> Dict[str, Any]:
"""分析螺旋性质"""
if len(evolution_data) < 3:
return {'error': 'Insufficient data'}
# 提取螺旋相位序列
phases = [data['spiral_phase'] for data in evolution_data]
# 计算螺旋参数
growth_rate = self._compute_growth_rate(phases)
angular_velocity = self._compute_angular_velocity(phases)
pitch = self._compute_spiral_pitch(phases)
# 检查φ-增长特征
phi_growth = self._verify_phi_growth(evolution_data)
return {
'growth_rate': growth_rate,
'angular_velocity': angular_velocity,
'spiral_pitch': pitch,
'phi_growth_verified': phi_growth,
'spiral_type': self._classify_spiral_type(growth_rate, angular_velocity),
'convergence_detected': self._detect_convergence_pattern(evolution_data)
}
def _compute_growth_rate(self, phases: List[complex]) -> float:
"""计算增长率"""
if len(phases) < 2:
return 0.0
radii = [abs(phase) for phase in phases]
growth_rates = []
for i in range(1, len(radii)):
if radii[i-1] > 0:
rate = radii[i] / radii[i-1]
growth_rates.append(rate)
return np.mean(growth_rates) if growth_rates else 0.0
def _compute_angular_velocity(self, phases: List[complex]) -> float:
"""计算角速度"""
if len(phases) < 2:
return 0.0
angles = [math.atan2(phase.imag, phase.real) for phase in phases]
angular_differences = []
for i in range(1, len(angles)):
diff = angles[i] - angles[i-1]
# 处理角度跳跃
while diff > math.pi:
diff -= 2 * math.pi
while diff < -math.pi:
diff += 2 * math.pi
angular_differences.append(diff)
return np.mean(angular_differences) if angular_differences else 0.0
def _compute_spiral_pitch(self, phases: List[complex]) -> float:
"""计算螺旋螺距"""
growth_rate = self._compute_growth_rate(phases)
angular_velocity = self._compute_angular_velocity(phases)
if angular_velocity == 0:
return float('inf')
return growth_rate / angular_velocity
def _verify_phi_growth(self, evolution_data: List[Dict]) -> bool:
"""验证φ-增长特征"""
if len(evolution_data) < 3:
return False
total_values = [data['structure'].total_value for data in evolution_data]
growth_ratios = []
for i in range(1, len(total_values)):
if total_values[i-1] > 0:
ratio = total_values[i] / total_values[i-1]
growth_ratios.append(ratio)
if not growth_ratios:
return False
avg_ratio = np.mean(growth_ratios)
return abs(avg_ratio - self.phi) < 0.3
def _classify_spiral_type(self, growth_rate: float, angular_velocity: float) -> str:
"""分类螺旋类型"""
if abs(growth_rate - self.phi) < 0.1:
return 'golden_spiral'
elif growth_rate > 1.1:
return 'expanding_spiral'
elif growth_rate < 0.9:
return 'contracting_spiral'
else:
return 'stable_spiral'
def _detect_convergence_pattern(self, evolution_data: List[Dict]) -> bool:
"""检测收敛模式"""
if len(evolution_data) < 5:
return False
convergence_measures = [data['convergence_measure'] for data in evolution_data[-5:]]
# 检查收敛度量是否递增
is_converging = all(convergence_measures[i] >= convergence_measures[i-1] - 0.1
for i in range(1, len(convergence_measures)))
# 检查是否达到高收敛度
high_convergence = convergence_measures[-1] > 0.8
return is_converging and high_convergence
4. 结构熵分析器
4.1 熵计算和分析
class StructuralEntropyAnalyzer:
"""结构熵分析器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def compute_structural_entropy(self, structure: TraceStructure) -> Dict[str, float]:
"""计算多种结构熵"""
if not structure.components:
return {
'shannon_entropy': 0.0,
'phi_entropy': 0.0,
'layer_entropy': 0.0,
'complexity_entropy': 0.0
}
distribution = structure.get_layer_distribution()
return {
'shannon_entropy': self._shannon_entropy(distribution),
'phi_entropy': self._phi_entropy(distribution),
'layer_entropy': self._layer_entropy(structure),
'complexity_entropy': self._complexity_entropy(structure)
}
def _shannon_entropy(self, distribution: Dict[int, float]) -> float:
"""计算Shannon熵"""
entropy = 0.0
for prob in distribution.values():
if prob > 0:
entropy -= prob * math.log2(prob)
return entropy
def _phi_entropy(self, distribution: Dict[int, float]) -> float:
"""计算φ-熵"""
entropy = 0.0
for prob in distribution.values():
if prob > 0:
entropy -= prob * math.log(prob, self.phi)
return entropy
def _layer_entropy(self, structure: TraceStructure) -> float:
"""计算层次熵"""
if not structure.components:
return 0.0
layer_complexities = []
for layer, comp in structure.components.items():
# 层复杂度基于层级和值
complexity = comp.value * math.log(layer + 2, self.phi)
layer_complexities.append(complexity)
total_complexity = sum(layer_complexities)
if total_complexity == 0:
return 0.0
entropy = 0.0
for complexity in layer_complexities:
if complexity > 0:
prob = complexity / total_complexity
entropy -= prob * math.log(prob, self.phi)
return entropy
def _complexity_entropy(self, structure: TraceStructure) -> float:
"""计算复杂度熵"""
base_entropy = structure.entropy
layer_factor = math.log(len(structure.components) + 1, self.phi)
core_factor = math.log(structure.structural_core + 1, self.phi)
return base_entropy * layer_factor * core_factor
def analyze_entropy_evolution(self, evolution_data: List[Dict]) -> Dict[str, Any]:
"""分析熵演化"""
if len(evolution_data) < 2:
return {'error': 'Insufficient data'}
entropy_sequence = []
for data in evolution_data:
structure = data['structure']
entropies = self.compute_structural_entropy(structure)
entropy_sequence.append(entropies)
return {
'entropy_sequence': entropy_sequence,
'entropy_increase_verified': self._verify_entropy_increase(entropy_sequence),
'entropy_growth_rate': self._compute_entropy_growth_rate(entropy_sequence),
'entropy_convergence': self._check_entropy_convergence(entropy_sequence),
'phi_entropy_pattern': self._analyze_phi_entropy_pattern(entropy_sequence)
}
def _verify_entropy_increase(self, entropy_sequence: List[Dict[str, float]]) -> Dict[str, bool]:
"""验证熵增"""
results = {}
for entropy_type in ['shannon_entropy', 'phi_entropy', 'layer_entropy', 'complexity_entropy']:
values = [entropies[entropy_type] for entropies in entropy_sequence]
# 检查是否非递减（允许小的数值误差）
is_increasing = all(values[i] >= values[i-1] - 1e-6
for i in range(1, len(values)))
results[entropy_type] = is_increasing
return results
def _compute_entropy_growth_rate(self, entropy_sequence: List[Dict[str, float]]) -> Dict[str, float]:
"""计算熵增长率"""
growth_rates = {}
for entropy_type in ['shannon_entropy', 'phi_entropy', 'layer_entropy', 'complexity_entropy']:
values = [entropies[entropy_type] for entropies in entropy_sequence]
if len(values) < 2:
growth_rates[entropy_type] = 0.0
continue
rates = []
for i in range(1, len(values)):
if values[i-1] > 0:
rate = (values[i] - values[i-1]) / values[i-1]
rates.append(rate)
growth_rates[entropy_type] = np.mean(rates) if rates else 0.0
return growth_rates
def _check_entropy_convergence(self, entropy_sequence: List[Dict[str, float]]) -> Dict[str, bool]:
"""检查熵收敛"""
if len(entropy_sequence) < 5:
return {entropy_type: False for entropy_type in ['shannon_entropy', 'phi_entropy', 'layer_entropy', 'complexity_entropy']}
convergence = {}
for entropy_type in ['shannon_entropy', 'phi_entropy', 'layer_entropy', 'complexity_entropy']:
values = [entropies[entropy_type] for entropies in entropy_sequence[-5:]]
# 检查最近几步的变化是否很小
max_change = max(abs(values[i] - values[i-1]) for i in range(1, len(values)))
convergence[entropy_type] = max_change < 0.01
return convergence
def _analyze_phi_entropy_pattern(self, entropy_sequence: List[Dict[str, float]]) -> Dict[str, Any]:
"""分析φ-熵模式"""
phi_entropies = [entropies['phi_entropy'] for entropies in entropy_sequence]
if len(phi_entropies) < 3:
return {'pattern': 'insufficient_data'}
# 检查是否符合φ-增长模式
growth_ratios = []
for i in range(1, len(phi_entropies)):
if phi_entropies[i-1] > 0:
ratio = phi_entropies[i] / phi_entropies[i-1]
growth_ratios.append(ratio)
if not growth_ratios:
return {'pattern': 'no_growth'}
avg_ratio = np.mean(growth_ratios)
if abs(avg_ratio - self.phi) < 0.2:
pattern = 'phi_growth'
elif avg_ratio > 1.1:
pattern = 'super_phi_growth'
elif avg_ratio < 0.9:
pattern = 'sub_phi_growth'
else:
pattern = 'linear_growth'
return {
'pattern': pattern,
'average_ratio': avg_ratio,
'phi_deviation': abs(avg_ratio - self.phi)
}
def compute_entropy_increase_bound(self, initial_structure: TraceStructure,
collapse_depth: int) -> float:
"""计算熵增下界"""
# 根据T20-2理论，最小熵增为1/φ^{collapse_depth}
min_increase = 1.0 / (self.phi ** collapse_depth)
return min_increase
def verify_entropy_increase_law(self, initial_structure: TraceStructure,
collapsed_structure: TraceStructure,
collapse_depth: int) -> bool:
"""验证熵增律"""
initial_entropies = self.compute_structural_entropy(initial_structure)
collapsed_entropies = self.compute_structural_entropy(collapsed_structure)
min_increase = self.compute_entropy_increase_bound(initial_structure, collapse_depth)
# 检查至少一种熵是否满足增长要求
for entropy_type in ['shannon_entropy', 'phi_entropy', 'layer_entropy', 'complexity_entropy']:
actual_increase = collapsed_entropies[entropy_type] - initial_entropies[entropy_type]
if actual_increase >= min_increase:
return True
return False
5. 综合ψₒ-trace系统
5.1 主系统类
class PsiTraceSystem:
"""完整的ψₒ-trace结构系统"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
# 核心组件
self.trace_calculator = ZeckendorfTraceCalculator()
self.decomposer = TraceLayerDecomposer()
self.core_extractor = TraceCoreExtractor()
self.evolution_tracker = SpiralEvolutionTracker()
self.entropy_analyzer = StructuralEntropyAnalyzer()
# 系统状态
self.current_structure = None
self.evolution_history = []
def analyze_complete_structure(self, zeck_string: 'ZeckendorfString') -> Dict[str, Any]:
"""完整的结构分析"""
# 基础trace计算
trace_value = self.trace_calculator.compute_full_trace(zeck_string)
# 层次结构分解
structure = self.decomposer.decompose_trace_structure(zeck_string)
# 结构核分析
core_info = self.core_extractor.extract_structural_core(structure)
# 结构熵分析
entropy_info = self.entropy_analyzer.compute_structural_entropy(structure)
return {
'trace_value': trace_value,
'structure': structure,
'core_info': core_info,
'entropy_info': entropy_info,
'structural_signature': structure.structural_signature(),
'complexity_measure': structure.layer_complexity()
}
def simulate_collapse_evolution(self, initial_state: 'ZeckendorfString',
collapse_func: callable,
num_steps: int) -> Dict[str, Any]:
"""模拟collapse演化过程"""
# 追踪演化序列
evolution_data = self.evolution_tracker.track_evolution_sequence(
initial_state, collapse_func, num_steps)
# 分析螺旋性质
spiral_analysis = self.evolution_tracker.analyze_spiral_properties(evolution_data)
# 分析熵演化
entropy_evolution = self.entropy_analyzer.analyze_entropy_evolution(evolution_data)
# 验证理论性质
theory_verification = self._verify_theoretical_properties(evolution_data)
return {
'evolution_data': evolution_data,
'spiral_analysis': spiral_analysis,
'entropy_evolution': entropy_evolution,
'theory_verification': theory_verification,
'convergence_detected': spiral_analysis.get('convergence_detected', False)
}
def _verify_theoretical_properties(self, evolution_data: List[Dict]) -> Dict[str, bool]:
"""验证理论性质"""
if len(evolution_data) < 2:
return {'insufficient_data': True}
verification = {
'layer_decomposition_valid': True,
'core_invariance_verified': True,
'spiral_evolution_confirmed': True,
'entropy_increase_satisfied': True
}
# 验证层次分解有效性
for data in evolution_data:
structure = data['structure']
if structure.components:
# 检查分解是否覆盖了总trace值
total_reconstructed = sum(
comp.value * (self.phi ** comp.layer)
for comp in structure.components.values()
)
if total_reconstructed == 0:
verification['layer_decomposition_valid'] = False
# 验证核不变性
if len(evolution_data) >= 2:
for i in range(1, len(evolution_data)):
prev_structure = evolution_data[i-1]['structure']
curr_structure = evolution_data[i]['structure']
if not self.core_extractor.verify_core_invariance(
prev_structure, curr_structure, 100):
verification['core_invariance_verified'] = False
break
# 验证螺旋演化
phases = [data['spiral_phase'] for data in evolution_data]
if len(phases) >= 3:
growth_rate = self.evolution_tracker._compute_growth_rate(phases)
if abs(growth_rate - self.phi) > 0.5:
verification['spiral_evolution_confirmed'] = False
# 验证熵增
entropies = []
for data in evolution_data:
structure = data['structure']
entropy_info = self.entropy_analyzer.compute_structural_entropy(structure)
entropies.append(entropy_info)
entropy_increase = self.entropy_analyzer._verify_entropy_increase(entropies)
if not any(entropy_increase.values()):
verification['entropy_increase_satisfied'] = False
return verification
def compare_structures(self, struct1: TraceStructure,
struct2: TraceStructure) -> Dict[str, Any]:
"""比较两个trace结构"""
comparison = {
'structural_similarity': self.evolution_tracker._compute_structure_similarity(struct1, struct2),
'core_relationship': self._analyze_core_relationship(struct1, struct2),
'entropy_difference': self._compute_entropy_difference(struct1, struct2),
'layer_mapping': self._compute_layer_mapping(struct1, struct2)
}
return comparison
def _analyze_core_relationship(self, struct1: TraceStructure,
struct2: TraceStructure) -> Dict[str, Any]:
"""分析结构核关系"""
core1 = struct1.structural_core
core2 = struct2.structural_core
if core1 == 0 or core2 == 0:
return {'relationship': 'undefined'}
ratio = core2 / core1
gcd_value = math.gcd(core1, core2)
relationship = {
'ratio': ratio,
'gcd': gcd_value,
'phi_related': abs(ratio - self.phi) < 0.1,
'integer_multiple': abs(ratio - round(ratio)) < 0.01
}
return relationship
def _compute_entropy_difference(self, struct1: TraceStructure,
struct2: TraceStructure) -> Dict[str, float]:
"""计算熵差异"""
entropy1 = self.entropy_analyzer.compute_structural_entropy(struct1)
entropy2 = self.entropy_analyzer.compute_structural_entropy(struct2)
differences = {}
for entropy_type in entropy1.keys():
differences[entropy_type] = entropy2[entropy_type] - entropy1[entropy_type]
return differences
def _compute_layer_mapping(self, struct1: TraceStructure,
struct2: TraceStructure) -> Dict[int, int]:
"""计算层间映射"""
mapping = {}
for layer1, comp1 in struct1.components.items():
best_match = None
best_similarity = 0.0
for layer2, comp2 in struct2.components.items():
# 基于值和权重的相似度
value_similarity = min(comp1.value, comp2.value) / max(comp1.value, comp2.value)
weight_similarity = min(comp1.weight, comp2.weight) / max(comp1.weight, comp2.weight)
similarity = (value_similarity + weight_similarity) / 2
if similarity > best_similarity:
best_similarity = similarity
best_match = layer2
if best_match is not None and best_similarity > 0.5:
mapping[layer1] = best_match
return mapping
def generate_structure_report(self, zeck_string: 'ZeckendorfString') -> str:
"""生成结构分析报告"""
analysis = self.analyze_complete_structure(zeck_string)
report = f"""
ψₒ-trace结构分析报告
====================
输入状态: {zeck_string.representation} (值: {zeck_string.value})
Trace值: {analysis['trace_value']}
层次结构分解:
-----------
"""
structure = analysis['structure']
for layer, comp in structure.components.items():
report += f" 层{layer}: 值={comp.value}, 权重={comp.weight:.3f}, φ权重={comp.phi_weight():.3f}\n"
report += f"""
结构核信息:
---------
核值: {analysis['core_info']['core_value']}
核类型: {analysis['core_info']['core_type']}
核稳定性: {analysis['core_info']['core_stability']:.3f}
φ不变性: {analysis['core_info']['phi_invariant']}
结构熵:
------
Shannon熵: {analysis['entropy_info']['shannon_entropy']:.3f}
φ熵: {analysis['entropy_info']['phi_entropy']:.3f}
层次熵: {analysis['entropy_info']['layer_entropy']:.3f}
复杂度熵: {analysis['entropy_info']['complexity_entropy']:.3f}
复杂度度量: {analysis['complexity_measure']:.3f}
结构签名: {analysis['structural_signature']}
"""
return report
这个形式化规范提供了T20-2理论的完整实现，包括层次分解、结构核提取、螺旋演化追踪和结构熵分析，确保了理论与实现的完全一致性。