T1-4 熵增方向唯一性形式化规范

1. 基础数学对象

1.1 时间方向

class TimeDirection:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def entropy_gradient(self, state: 'SystemState') -> float:
        """计算熵梯度，确定时间方向"""
        
    def is_forward(self, t1: float, t2: float, H1: float, H2: float) -> bool:
        """判断t1到t2是否为正向时间"""
        return H2 > H1 if t2 > t1 else H1 > H2
        
    def verify_uniqueness(self, trajectory: List[Tuple[float, float]]) -> bool:
        """验证时间方向的唯一性"""

1.2 递归展开结构

class RecursiveUnfolding:
    def __init__(self):
        self.depth_history = {}
        self.phi = (1 + np.sqrt(5)) / 2
        
    def unfold(self, state: 'SystemState') -> 'SystemState':
        """递归展开操作"""
        new_state = state.copy()
        new_state.add_description(self.describe(state))
        new_state.increment_depth()
        return new_state
        
    def is_reversible(self, state: 'SystemState') -> bool:
        """检查展开是否可逆（应该总是False）"""
        unfolded = self.unfold(state)
        # 尝试逆操作
        return self.can_reverse(unfolded, state)
        
    def can_reverse(self, unfolded: 'SystemState', 
                   original: 'SystemState') -> bool:
        """尝试逆向展开"""
        # 由于信息累积，这应该不可能
        return False

1.3 Zeckendorf方向性

class ZeckendorfDirectionality:
    def __init__(self):
        self.fib_cache = {0: 0, 1: 1, 2: 1}
        
    def zeckendorf_representation(self, n: int) -> List[int]:
        """计算n的Zeckendorf表示"""
        if n == 0:
            return [0]
            
        # 生成足够的Fibonacci数
        fibs = self._generate_fibonacci_list(n)
        
        # 贪心算法
        result = []
        remaining = n
        
        for i in range(len(fibs) - 1, -1, -1):
            if fibs[i] <= remaining:
                result.append(1)
                remaining -= fibs[i]
            else:
                result.append(0)
                
        # 移除前导零
        while result and result[0] == 0:
            result.pop(0)
            
        return result
        
    def evolution_rule(self, z_n: List[int]) -> List[int]:
        """Zeckendorf表示的演化规则 n -> n+1"""
        # 实现具体的演化规则
        
    def is_evolution_reversible(self, z_n: List[int], 
                               z_n_plus_1: List[int]) -> bool:
        """检查演化是否可逆"""
        # 由于规则的不对称性，应该返回False
        
    def verify_irreversibility(self, max_n: int = 100) -> bool:
        """验证演化的不可逆性"""

2. 时间反演不可能性

2.1 时间反演算子

class TimeReversalOperator:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def attempt_reversal(self, state: 'SystemState') -> Optional['SystemState']:
        """尝试时间反演（应该失败）"""
        # 检查是否可能构造反演状态
        return None
        
    def verify_no_reversal(self, trajectory: List['SystemState']) -> bool:
        """验证不存在时间反演"""
        for i in range(len(trajectory) - 1):
            if self._can_reverse_step(trajectory[i], trajectory[i+1]):
                return False
        return True
        
    def _can_reverse_step(self, state1: 'SystemState', 
                         state2: 'SystemState') -> bool:
        """检查单步是否可逆"""
        # 熵增使得反演不可能
        return state2.entropy() <= state1.entropy()

2.2 不可逆性证明

class IrreversibilityProof:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def prove_recursive_irreversibility(self, depth: int) -> bool:
        """证明递归展开的不可逆性"""
        # 构造深度为depth的递归结构
        # 证明不能反向构造
        
    def prove_entropy_irreversibility(self, states: List['SystemState']) -> bool:
        """证明熵增的不可逆性"""
        # 验证熵严格递增
        for i in range(len(states) - 1):
            if states[i+1].entropy() <= states[i].entropy():
                return False
        return True
        
    def prove_no11_irreversibility(self, sequence: List[int]) -> bool:
        """证明no-11约束导致的不可逆性"""
        # 检查约束的累积效应

3. 熵梯度结构

3.1 熵梯度场

class EntropyGradientField:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def gradient_at(self, state: 'SystemState') -> 'Vector':
        """计算状态空间中的熵梯度"""
        
    def curl(self, region: 'StateSpaceRegion') -> float:
        """计算熵梯度的旋度（应该为0）"""
        
    def verify_irrotational(self, sample_points: int = 1000) -> bool:
        """验证熵梯度场是无旋的"""
        
    def flow_lines(self, initial: 'SystemState', 
                   steps: int) -> List['SystemState']:
        """沿熵梯度的流线"""

3.2 方向性度量

class DirectionalityMeasure:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def arrow_of_time_strength(self, trajectory: List['SystemState']) -> float:
        """时间箭头的强度"""
        total_entropy_increase = 0
        for i in range(len(trajectory) - 1):
            dH = trajectory[i+1].entropy() - trajectory[i].entropy()
            total_entropy_increase += dH
            
        return total_entropy_increase / (len(trajectory) - 1)
        
    def reversibility_index(self, process: 'Process') -> float:
        """过程的可逆性指数（0=完全不可逆，1=完全可逆）"""
        # 基于熵产生计算
        
    def causal_asymmetry(self, state1: 'SystemState', 
                        state2: 'SystemState') -> float:
        """因果不对称性度量"""

4. 物理验证

4.1 CPT对称性破坏

class CPTViolation:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def cpt_violation_amplitude(self, complexity: int) -> float:
        """CPT破坏振幅"""
        return 1.0 / (self.phi ** complexity)
        
    def measure_time_asymmetry(self, process: 'QuantumProcess') -> float:
        """测量过程的时间不对称性"""
        
    def predict_violation(self, system: 'PhysicalSystem') -> float:
        """预测CPT破坏的大小"""

4.2 因果结构

class CausalStructure:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def causal_order(self, events: List['Event']) -> List['Event']:
        """确定事件的因果顺序"""
        # 基于熵增排序
        
    def verify_no_closed_timelike_curves(self, 
                                       spacetime: 'Spacetime') -> bool:
        """验证无闭合类时曲线"""
        
    def future_past_asymmetry(self, event: 'Event') -> float:
        """未来-过去不对称性"""

5. 信息论结构

5.1 信息累积

class InformationAccumulation:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def information_content(self, state: 'SystemState') -> float:
        """状态的信息含量"""
        
    def verify_no_information_destruction(self, 
                                        process: 'Process') -> bool:
        """验证信息不灭"""
        initial_info = self.information_content(process.initial_state)
        final_info = self.information_content(process.final_state)
        return final_info >= initial_info
        
    def memory_direction(self, memory: 'Memory') -> str:
        """记忆的方向（应该总是指向过去）"""
        return "past"

5.2 计算不可逆性

class ComputationalIrreversibility:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def landauer_limit(self, temperature: float) -> float:
        """Landauer极限"""
        k_B = 1.38e-23  # Boltzmann常数
        return k_B * temperature * np.log(2)
        
    def erasure_cost(self, bits: int, temperature: float) -> float:
        """擦除信息的能量代价"""
        return bits * self.landauer_limit(temperature)
        
    def verify_computational_arrow(self, 
                                 computation: 'Computation') -> bool:
        """验证计算的时间箭头"""

6. 宇宙学应用

6.1 宇宙演化方向

class CosmologicalDirection:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def universe_entropy(self, time: float) -> float:
        """宇宙熵随时间的演化"""
        # 包括物质、辐射、黑洞等贡献
        
    def verify_no_big_crunch_reversal(self) -> bool:
        """验证即使大挤压也不能完全时间反演"""
        
    def initial_condition_entropy(self) -> float:
        """初始条件的熵（应该很低）"""

6.2 黑洞不可逆性

class BlackHoleIrreversibility:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def bekenstein_hawking_entropy(self, mass: float) -> float:
        """Bekenstein-Hawking熵"""
        # S = A/4 in Planck units
        
    def formation_entropy_jump(self, initial_mass: float, 
                             final_mass: float) -> float:
        """黑洞形成的熵跃变"""
        
    def verify_no_white_holes(self) -> bool:
        """验证白洞不存在（时间反演的黑洞）"""

7. 实验预测

7.1 量子测量方向性

class QuantumMeasurementDirection:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def collapse_entropy_increase(self, initial_state: 'QuantumState',
                                measured_state: 'QuantumState') -> float:
        """波函数坍缩的熵增"""
        
    def retrocausation_bound(self, system_size: int) -> float:
        """逆因果的上界"""
        return 1.0 / (self.phi ** system_size)

7.2 统计力学验证

class StatisticalVerification:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def boltzmann_h_theorem(self, distribution: 'Distribution') -> float:
        """Boltzmann H定理的验证"""
        
    def fluctuation_theorem_asymmetry(self, 
                                    trajectory: 'Trajectory') -> float:
        """涨落定理的不对称性"""

8. 数学结构

8.1 单向半群

class UnidirectionalSemigroup:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compose(self, t1: 'TimeEvolution', 
                t2: 'TimeEvolution') -> 'TimeEvolution':
        """时间演化的复合（只能正向）"""
        
    def has_inverse(self, t: 'TimeEvolution') -> bool:
        """检查是否有逆（应该返回False）"""
        return False
        
    def verify_semigroup_properties(self) -> bool:
        """验证半群性质"""

8.2 熵偏序

class EntropyPartialOrder:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def compare(self, state1: 'SystemState', 
                state2: 'SystemState') -> Optional[str]:
        """比较两个状态的时间顺序"""
        H1 = state1.entropy()
        H2 = state2.entropy()
        
        if H1 < H2:
            return "before"
        elif H1 > H2:
            return "after"
        else:
            return None  # 不可比较
            
    def verify_partial_order_properties(self) -> bool:
        """验证偏序性质"""

9. 关键常数

# 基础常数
PHI = (1 + np.sqrt(5)) / 2  # 黄金分割率

# 物理常数
K_B = 1.38064852e-23  # Boltzmann常数 (J/K)
HBAR = 1.054571817e-34  # 约化Planck常数 (J·s)
C = 299792458  # 光速 (m/s)

# 方向性参数
ARROW_STRENGTH = 1.0  # 时间箭头强度
CPT_VIOLATION_SCALE = PHI ** (-10)  # CPT破坏尺度
MEMORY_EFFICIENCY = 1.0 / PHI  # 记忆效率因子

10. 错误处理

class DirectionalityError(Exception):
    """方向性错误基类"""
    
class TimeReversalError(DirectionalityError):
    """时间反演错误"""
    
class EntropyDecreaseError(DirectionalityError):
    """熵减少错误"""
    
class CausalityViolationError(DirectionalityError):
    """因果性违反错误"""