T20-1 φ-collapse-aware基础定理形式化规范
1. Zeckendorf编码基础类
1.1 Zeckendorf字符串类
import numpy as np
from typing import List, Tuple, Optional, Dict, Any
from dataclasses import dataclass
import math
class ZeckendorfString:
"""Zeckendorf编码字符串类，确保no-11约束"""
def __init__(self, value: int = 0):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci_cache = self._generate_fibonacci_sequence(50)
if isinstance(value, int):
self.representation = self._int_to_zeckendorf(value)
elif isinstance(value, str):
if self._is_valid_zeckendorf(value):
self.representation = value
else:
raise ValueError(f"Invalid Zeckendorf string: {value}")
else:
raise TypeError("Value must be int or str")
self.value = self._zeckendorf_to_int(self.representation)
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 _int_to_zeckendorf(self, n: int) -> str:
"""将整数转换为Zeckendorf表示"""
if n == 0:
return "0"
# 贪心算法构造Zeckendorf表示
result = []
remaining = n
# 从最大的Fibonacci数开始
for i in range(len(self.fibonacci_cache) - 1, -1, -1):
fib = self.fibonacci_cache[i]
if fib <= remaining:
result.append('1')
remaining -= fib
else:
result.append('0')
# 移除前导0
result_str = ''.join(result).lstrip('0')
return result_str if result_str else "0"
def _zeckendorf_to_int(self, zeck_str: str) -> int:
"""将Zeckendorf表示转换为整数"""
if zeck_str == "0":
return 0
total = 0
# 从左到右，对应从大到小的Fibonacci数
fib_index = len(zeck_str) - 1
for i, bit in enumerate(zeck_str):
if bit == '1':
if fib_index < len(self.fibonacci_cache):
total += self.fibonacci_cache[fib_index]
fib_index -= 1
return total
def _is_valid_zeckendorf(self, s: str) -> bool:
"""检查是否为有效的Zeckendorf表示（no-11约束）"""
if not all(c in '01' for c in s):
return False
return "11" not in s
def __str__(self) -> str:
return self.representation
def __eq__(self, other) -> bool:
if isinstance(other, ZeckendorfString):
return self.value == other.value
return False
def __hash__(self) -> int:
return hash(self.value)
def __add__(self, other: 'ZeckendorfString') -> 'ZeckendorfString':
"""Zeckendorf加法"""
return ZeckendorfString(self.value + other.value)
def __xor__(self, other: 'ZeckendorfString') -> 'ZeckendorfString':
"""Zeckendorf异或运算（特殊定义）"""
# 执行位级异或，然后转换回有效的Zeckendorf
max_len = max(len(self.representation), len(other.representation))
s1 = self.representation.zfill(max_len)
s2 = other.representation.zfill(max_len)
xor_result = ""
for i in range(max_len):
xor_result += str(int(s1[i]) ^ int(s2[i]))
# 转换回整数再转换为有效Zeckendorf
temp_value = 0
for i, bit in enumerate(xor_result):
if bit == '1':
# 简单的二进制权重
temp_value += 2 ** (max_len - 1 - i)
return ZeckendorfString(temp_value)
def compute_entropy(self) -> float:
"""计算Zeckendorf字符串的熵"""
if not self.representation or self.representation == "0":
return 0.0
ones = self.representation.count('1')
zeros = self.representation.count('0')
total = ones + zeros
if ones == 0 or zeros == 0:
return 0.0
p1 = ones / total
p0 = zeros / total
return -(p1 * np.log2(p1) + p0 * np.log2(p0)) * total
def phi_length(self) -> float:
"""计算φ-长度"""
phi_len = 0.0
for i, bit in enumerate(self.representation):
if bit == '1':
# φ权重基于位置
phi_len += 1.0 / (self.phi ** (i + 1))
return phi_len
2. ψ-collapse核心机制
2.1 collapse操作类
class PsiCollapse:
"""ψ = ψ(ψ) collapse操作实现"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def compute_phi_representation(self, state: ZeckendorfString) -> ZeckendorfString:
"""计算状态的φ-自指表示"""
# φ-自指：将每个Fibonacci位移位
original_value = state.value
# φ-变换：F_i -> F_{i+1}
phi_value = int(original_value * self.phi)
return ZeckendorfString(phi_value)
def psi_collapse_once(self, state: ZeckendorfString) -> ZeckendorfString:
"""执行一次ψ-collapse操作"""
# 计算φ-自指表示
phi_repr = self.compute_phi_representation(state)
# 组合原状态和φ-表示
collapsed = state + phi_repr
return collapsed
def psi_collapse_sequence(self, initial: ZeckendorfString,
max_iterations: int = 20) -> List[ZeckendorfString]:
"""执行collapse序列直到收敛或达到最大迭代次数"""
sequence = [initial]
current = initial
for i in range(max_iterations):
next_state = self.psi_collapse_once(current)
# 检查是否已经见过这个状态（周期检测）
if next_state in sequence:
# 找到周期
cycle_start = sequence.index(next_state)
sequence.append(next_state)
return sequence
sequence.append(next_state)
current = next_state
return sequence
def detect_cycle(self, sequence: List[ZeckendorfString]) -> Optional[Tuple[int, int]]:
"""检测序列中的周期"""
n = len(sequence)
# 检查不同的周期长度
for period in range(1, n // 2 + 1):
if self._is_periodic(sequence, period):
# 找到周期开始位置
for start in range(n - 2 * period + 1):
if self._check_period_at(sequence, start, period):
return (start, period)
return None
def _is_periodic(self, sequence: List[ZeckendorfString], period: int) -> bool:
"""检查序列是否具有给定周期"""
n = len(sequence)
if n < 2 * period:
return False
for i in range(period):
if (n - period + i < n and n - 2 * period + i >= 0 and
sequence[n - period + i] != sequence[n - 2 * period + i]):
return False
return True
def _check_period_at(self, sequence: List[ZeckendorfString],
start: int, period: int) -> bool:
"""检查在给定位置是否开始周期"""
n = len(sequence)
for i in range(period):
if (start + i < n and start + i + period < n and
sequence[start + i] != sequence[start + i + period]):
return False
return True
def is_self_referential(self, state: ZeckendorfString) -> bool:
"""检查状态是否满足ψ = ψ(ψ)"""
collapsed_once = self.psi_collapse_once(state)
collapsed_twice = self.psi_collapse_once(collapsed_once)
# 检查是否达到不动点或进入周期
return collapsed_once == collapsed_twice or collapsed_twice == state
2.2 φ-trace计算类
class PhiTrace:
"""φ-trace计算和分析"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
self.fibonacci_cache = self._generate_fibonacci_sequence(50)
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_trace(self, state: ZeckendorfString) -> int:
"""计算状态的φ-trace"""
trace_value = 0
representation = state.representation
for i, bit in enumerate(representation):
if bit == '1':
# trace权重基于Fibonacci索引
fib_index = len(representation) - 1 - i
if fib_index < len(self.fibonacci_cache):
trace_value += self.fibonacci_cache[fib_index] * (i + 1)
return trace_value
def trace_ratio(self, state1: ZeckendorfString,
state2: ZeckendorfString) -> float:
"""计算两个状态的trace比率"""
trace1 = self.compute_trace(state1)
trace2 = self.compute_trace(state2)
if trace1 == 0:
return float('inf') if trace2 > 0 else 1.0
return trace2 / trace1
def verify_phi_growth(self, sequence: List[ZeckendorfString],
tolerance: float = 0.2) -> bool:
"""验证序列中trace的φ-增长性质"""
if len(sequence) < 2:
return True
ratios = []
for i in range(1, len(sequence)):
ratio = self.trace_ratio(sequence[i-1], sequence[i])
if not math.isinf(ratio):
ratios.append(ratio)
if not ratios:
return True
# 检查比率是否接近φ
avg_ratio = np.mean(ratios)
return abs(avg_ratio - self.phi) < tolerance
def compute_trace_modulo(self, state: ZeckendorfString,
modulus: int) -> int:
"""计算模运算下的trace"""
trace = self.compute_trace(state)
return trace % modulus
def trace_invariant_check(self, original: ZeckendorfString,
collapsed: ZeckendorfString,
modulus: int) -> bool:
"""检查trace在collapse下的不变性"""
trace_orig = self.compute_trace_modulo(original, modulus)
trace_coll = self.compute_trace_modulo(collapsed, modulus)
# 检查φ-倍增关系
expected = (int(self.phi * trace_orig)) % modulus
return abs(trace_coll - expected) <= 1 # 允许一定误差
2.3 collapse深度分析类
class CollapseDepthAnalyzer:
"""collapse深度分析器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def compute_collapse_depth(self, initial: ZeckendorfString,
collapsed: ZeckendorfString) -> int:
"""计算collapse深度"""
entropy_initial = initial.compute_entropy()
entropy_collapsed = collapsed.compute_entropy()
entropy_diff = entropy_collapsed - entropy_initial
if entropy_diff <= 0:
return 0
depth = math.floor(math.log(entropy_diff + 1) / math.log(self.phi))
return max(0, depth)
def verify_entropy_increase(self, initial: ZeckendorfString,
collapsed: ZeckendorfString) -> bool:
"""验证collapse过程的熵增"""
entropy_initial = initial.compute_entropy()
entropy_collapsed = collapsed.compute_entropy()
# 严格熵增要求
min_increase = 1.0 / (self.phi ** self.compute_collapse_depth(initial, collapsed))
return entropy_collapsed > entropy_initial + min_increase
def analyze_depth_sequence(self, sequence: List[ZeckendorfString]) -> List[int]:
"""分析序列中每一步的collapse深度"""
depths = []
for i in range(1, len(sequence)):
depth = self.compute_collapse_depth(sequence[i-1], sequence[i])
depths.append(depth)
return depths
def compute_max_depth_bound(self, max_string_length: int) -> int:
"""计算最大collapse深度界限"""
max_entropy = max_string_length * math.log2(2) # 二进制串的最大熵
return math.floor(math.log(2**max_string_length + 1) / math.log(self.phi))
def verify_depth_bound(self, sequence: List[ZeckendorfString],
max_length: int) -> bool:
"""验证序列中所有深度都在界限内"""
max_bound = self.compute_max_depth_bound(max_length)
depths = self.analyze_depth_sequence(sequence)
return all(depth <= max_bound for depth in depths)
3. collapse-aware系统综合框架
3.1 系统状态类
@dataclass
class CollapseAwareState:
"""collapse-aware系统状态"""
def __init__(self, zeckendorf_str: ZeckendorfString):
self.state = zeckendorf_str
self.entropy = zeckendorf_str.compute_entropy()
self.trace = PhiTrace().compute_trace(zeckendorf_str)
self.phi_length = zeckendorf_str.phi_length()
self.timestamp = 0
def update_timestamp(self, t: int):
"""更新时间戳"""
self.timestamp = t
def compute_complexity(self) -> float:
"""计算状态复杂度"""
return self.entropy * self.phi_length * math.log(self.trace + 1)
def __eq__(self, other) -> bool:
if isinstance(other, CollapseAwareState):
return self.state == other.state
return False
def __hash__(self) -> int:
return hash(self.state)
3.2 主系统类
class CollapseAwareSystem:
"""完整的collapse-aware系统"""
def __init__(self, initial_value: int = 1):
self.phi = (1 + np.sqrt(5)) / 2
# 核心组件
self.psi_collapse = PsiCollapse()
self.phi_trace = PhiTrace()
self.depth_analyzer = CollapseDepthAnalyzer()
# 系统状态
self.initial_state = CollapseAwareState(ZeckendorfString(initial_value))
self.current_state = self.initial_state
self.history = [self.initial_state]
# 系统属性
self.total_entropy_increase = 0.0
self.total_trace_growth = 0
self.max_depth_reached = 0
def execute_collapse(self) -> CollapseAwareState:
"""执行一次collapse操作"""
# 执行ψ-collapse
new_zeck_state = self.psi_collapse.psi_collapse_once(self.current_state.state)
new_state = CollapseAwareState(new_zeck_state)
new_state.update_timestamp(len(self.history))
# 验证熵增
entropy_increase = new_state.entropy - self.current_state.entropy
if entropy_increase <= 0:
raise ValueError("Collapse failed: entropy did not increase")
# 更新系统统计
self.total_entropy_increase += entropy_increase
self.total_trace_growth += new_state.trace - self.current_state.trace
# 计算collapse深度
depth = self.depth_analyzer.compute_collapse_depth(
self.current_state.state, new_state.state)
self.max_depth_reached = max(self.max_depth_reached, depth)
# 更新状态
self.current_state = new_state
self.history.append(new_state)
return new_state
def execute_collapse_sequence(self, num_steps: int) -> List[CollapseAwareState]:
"""执行collapse序列"""
sequence = []
for i in range(num_steps):
try:
new_state = self.execute_collapse()
sequence.append(new_state)
# 检查是否进入周期
if self.detect_convergence():
break
except ValueError as e:
print(f"Collapse stopped at step {i}: {e}")
break
return sequence
def detect_convergence(self) -> bool:
"""检测系统是否收敛"""
if len(self.history) < 4:
return False
# 检查最近的状态是否重复
recent_states = [state.state for state in self.history[-4:]]
return len(set(recent_states)) < len(recent_states)
def verify_psi_property(self) -> bool:
"""验证ψ = ψ(ψ)性质"""
return self.psi_collapse.is_self_referential(self.current_state.state)
def analyze_phi_trace_growth(self) -> Dict[str, Any]:
"""分析φ-trace增长模式"""
if len(self.history) < 2:
return {'error': 'Insufficient history'}
traces = [state.trace for state in self.history]
# 计算增长比率
ratios = []
for i in range(1, len(traces)):
if traces[i-1] > 0:
ratios.append(traces[i] / traces[i-1])
return {
'traces': traces,
'growth_ratios': ratios,
'average_ratio': np.mean(ratios) if ratios else 0,
'phi_convergence': abs(np.mean(ratios) - self.phi) < 0.3 if ratios else False,
'phi_growth_verified': self.phi_trace.verify_phi_growth(
[state.state for state in self.history])
}
def system_diagnostics(self) -> Dict[str, Any]:
"""系统诊断信息"""
return {
'total_steps': len(self.history) - 1,
'current_entropy': self.current_state.entropy,
'total_entropy_increase': self.total_entropy_increase,
'current_trace': self.current_state.trace,
'total_trace_growth': self.total_trace_growth,
'max_depth_reached': self.max_depth_reached,
'psi_property_satisfied': self.verify_psi_property(),
'phi_trace_analysis': self.analyze_phi_trace_growth(),
'converged': self.detect_convergence(),
'current_state_value': self.current_state.state.value,
'current_state_repr': str(self.current_state.state)
}
def reset_system(self, new_initial_value: int = 1):
"""重置系统"""
self.initial_state = CollapseAwareState(ZeckendorfString(new_initial_value))
self.current_state = self.initial_state
self.history = [self.initial_state]
self.total_entropy_increase = 0.0
self.total_trace_growth = 0
self.max_depth_reached = 0
4. 验证和测试接口
4.1 系统验证器
class CollapseAwareValidator:
"""系统验证器"""
def __init__(self):
self.phi = (1 + np.sqrt(5)) / 2
def validate_zeckendorf_constraints(self, state: ZeckendorfString) -> bool:
"""验证Zeckendorf约束"""
return "11" not in state.representation
def validate_entropy_increase(self, system: CollapseAwareSystem) -> bool:
"""验证熵增性质"""
return system.total_entropy_increase > 0
def validate_phi_structure(self, system: CollapseAwareSystem) -> bool:
"""验证φ-结构保持"""
analysis = system.analyze_phi_trace_growth()
return analysis.get('phi_growth_verified', False)
def validate_self_reference(self, system: CollapseAwareSystem) -> bool:
"""验证自指性质"""
return system.verify_psi_property()
def comprehensive_validation(self, system: CollapseAwareSystem) -> Dict[str, bool]:
"""综合验证"""
return {
'zeckendorf_valid': all(self.validate_zeckendorf_constraints(state.state)
for state in system.history),
'entropy_increase': self.validate_entropy_increase(system),
'phi_structure': self.validate_phi_structure(system),
'self_reference': self.validate_self_reference(system),
'no_errors': len(system.history) > 1 # 至少执行了一次collapse
}
这个形式化规范提供了T20-1理论的完整实现，包括Zeckendorf编码、ψ-collapse机制、φ-trace计算和系统验证，确保了理论与实现的完全一致性。