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T1-3 熵增速率形式化规范

1. 基础数学对象​

1.1 φ数系统​

class PhiNumber:
    def __init__(self, value: float):
        self.phi = (1 + np.sqrt(5)) / 2
        self.value = float(value)
        
    def to_fibonacci_basis(self) -> List[int]:
        """转换为Fibonacci基表示（Zeckendorf表示）"""
        
    def verify_no_11(self) -> bool:
        """验证表示中无连续的11"""

1.2 熵函数​

class EntropyFunction:
    def __init__(self, system: 'SelfReferentialSystem'):
        self.system = system
        self.phi = (1 + np.sqrt(5)) / 2
        self.k0 = 1.0  # 基准速率常数
        
    def compute_entropy(self, time: int) -> 'PhiNumber':
        """计算时刻t的系统熵"""
        
    def entropy_rate(self, time: int) -> 'PhiNumber':
        """计算熵增速率 dH/dt"""
        
    def recursive_depth(self, time: int) -> int:
        """计算递归深度d(t)"""

1.3 递归深度​

class RecursiveDepth:
    def __init__(self):
        self.depths = {0: 0, 1: 1}  # 初始条件
        
    def compute_depth(self, time: int) -> int:
        """计算时刻t的递归深度（Fibonacci增长）"""
        if time in self.depths:
            return self.depths[time]
        
        # Fibonacci递归：d(t+1) = d(t) + d(t-1)
        depth = self.compute_depth(time-1) + self.compute_depth(time-2)
        self.depths[time] = depth
        return depth
        
    def verify_fibonacci_growth(self) -> bool:
        """验证递归深度的Fibonacci增长性"""

2. 约束因子​

2.1 no-11约束调制​

class No11Modulation:
    def __init__(self, epsilon: float = 0.01, T: float = 1.0):
        self.epsilon = epsilon  # 约束强度
        self.T = T  # 特征时间尺度
        
    def theta_factor(self, time: float) -> float:
        """计算no-11约束因子Θ(t)"""
        theta = 1.0
        
        # Fibonacci频率的调制
        fib_prev, fib_curr = 1, 1
        for n in range(1, 20):  # 前20个Fibonacci数
            theta -= self.epsilon * np.sin(2 * np.pi * fib_curr * time / self.T) / fib_curr
            fib_prev, fib_curr = fib_curr, fib_prev + fib_curr
            
        return theta
        
    def verify_bounds(self) -> bool:
        """验证|Θ(t) - 1| ≤ ε"""

2.2 Fibonacci序列生成器​

class FibonacciGenerator:
    def __init__(self):
        self.cache = {0: 0, 1: 1, 2: 1}
        
    def fibonacci(self, n: int) -> int:
        """生成第n个Fibonacci数"""
        if n in self.cache:
            return self.cache[n]
            
        fib = self.fibonacci(n-1) + self.fibonacci(n-2)
        self.cache[n] = fib
        return fib
        
    def fibonacci_sequence(self, max_n: int) -> List[int]:
        """生成Fibonacci序列"""
        
    def is_fibonacci(self, num: int) -> bool:
        """判断是否为Fibonacci数"""

3. 熵增速率核心​

3.1 熵增速率计算器​

class EntropyRateCalculator:
    def __init__(self, k0: float = 1.0):
        self.k0 = k0
        self.phi = (1 + np.sqrt(5)) / 2
        self.depth_calc = RecursiveDepth()
        self.modulation = No11Modulation()
        
    def compute_rate(self, time: float) -> 'PhiNumber':
        """计算熵增速率 dH/dt = k0 * φ^d(t) * Θ(t)"""
        d_t = self.depth_calc.compute_depth(int(time))
        theta_t = self.modulation.theta_factor(time)
        
        rate = self.k0 * (self.phi ** d_t) * theta_t
        return PhiNumber(rate)
        
    def verify_bounds(self, time: float) -> bool:
        """验证速率在理论界限内"""
        
    def asymptotic_behavior(self, time: float) -> 'PhiNumber':
        """计算渐近行为"""

3.2 熵演化器​

class EntropyEvolution:
    def __init__(self, H0: 'PhiNumber'):
        self.H0 = H0  # 初始熵
        self.rate_calc = EntropyRateCalculator()
        self.history = [(0, H0)]
        
    def evolve(self, time_steps: int, dt: float = 0.01) -> List[Tuple[float, 'PhiNumber']]:
        """演化系统熵"""
        
    def integrate_entropy(self, t_start: float, t_end: float) -> 'PhiNumber':
        """积分计算累积熵变"""
        
    def find_phase_transitions(self) -> List[float]:
        """找出相变点（熵增速率突变）"""

4. 物理量计算​

4.1 时间涌现​

class EmergentTime:
    def __init__(self, entropy_evolution: 'EntropyEvolution'):
        self.entropy_evolution = entropy_evolution
        
    def proper_time_rate(self, coordinate_time: float) -> 'PhiNumber':
        """计算固有时流逝率 dτ/dt = 1/(dH/dt)"""
        
    def time_dilation_factor(self, t1: float, t2: float) -> 'PhiNumber':
        """计算两个时刻间的时间膨胀因子"""

4.2 信息处理速率​

class InformationRate:
    def __init__(self, entropy_rate: 'EntropyRateCalculator'):
        self.entropy_rate = entropy_rate
        
    def max_processing_rate(self, time: float) -> 'PhiNumber':
        """最大信息处理速率 dI/dt ≤ dH/dt"""
        
    def channel_capacity(self, time: float) -> 'PhiNumber':
        """信道容量"""

4.3 复杂度演化​

class ComplexityEvolution:
    def __init__(self, alpha: float = 0.5):
        self.alpha = alpha  # 复杂度-熵转换因子
        self.entropy_rate = EntropyRateCalculator()
        
    def complexity_rate(self, time: float) -> 'PhiNumber':
        """复杂度增长率 dC/dt = α * dH/dt"""
        
    def total_complexity(self, time: float) -> 'PhiNumber':
        """累积复杂度"""

5. 数学结构​

5.1 微分方程求解器​

class EntropyDifferentialEquation:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def second_order_eq(self, t: float, H: float, dH_dt: float) -> float:
        """二阶微分方程：d²H/dt² = dH/dt * (log φ + dΘ/dt/Θ)"""
        
    def solve_ivp(self, t_span: Tuple[float, float], 
                  initial_conditions: Tuple[float, float]) -> 'Solution':
        """求解初值问题"""

5.2 变分原理​

class EntropyVariational:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def lagrangian(self, H: float, dH_dt: float) -> float:
        """拉格朗日量 L(H, dH/dt)"""
        
    def action(self, path: List[Tuple[float, float]], dt: float) -> float:
        """作用量 S = ∫[H*dH/dt - L]dt"""
        
    def find_extremal_path(self, boundary_conditions: Tuple) -> List[Tuple[float, float]]:
        """寻找使作用量极值的路径"""

6. 实验预测​

6.1 黑洞熵计算器​

class BlackHoleEntropy:
    def __init__(self, mass: float):
        self.mass = mass
        self.entropy_rate = EntropyRateCalculator()
        
    def bekenstein_hawking_rate(self, time: float) -> 'PhiNumber':
        """黑洞熵增速率"""
        
    def hawking_temperature(self) -> 'PhiNumber':
        """Hawking温度"""

6.2 量子退相干​

class QuantumDecoherence:
    def __init__(self, n_qubits: int, temperature: float):
        self.n_qubits = n_qubits
        self.temperature = temperature
        self.phi = (1 + np.sqrt(5)) / 2
        
    def decoherence_rate_bound(self) -> 'PhiNumber':
        """退相干率下界 γ ≥ k_B*T/ħ * φ^(-n)"""

6.3 生物复杂度​

class BiologicalComplexity:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        
    def complexity_growth_rate(self, generation: int) -> 'PhiNumber':
        """生物复杂度增长率 dC_bio/dt ∝ φ^generation"""

7. 数值特征​

7.1 渐近分析​

class AsymptoticAnalysis:
    def __init__(self):
        self.phi = (1 + np.sqrt(5)) / 2
        self.tau_phi = np.log(self.phi) / np.log(1 + 1/self.phi)
        
    def long_time_behavior(self, time: float) -> 'PhiNumber':
        """长时渐近行为 dH/dt ~ k0 * φ^(t/τ_φ)"""
        
    def oscillation_periods(self) -> List[float]:
        """主要振荡周期"""
        
    def phase_transition_points(self, T: float) -> List[float]:
        """相变点 t_c = T*F_n/(2π)"""

8. 验证函数​

8.1 理论一致性检查​

def verify_fibonacci_growth(depth_sequence: List[int]) -> bool:
    """验证递归深度的Fibonacci增长"""
    
def verify_entropy_bounds(rate_history: List[float], epsilon: float) -> bool:
    """验证熵增速率的上下界"""
    
def verify_no_11_constraint(evolution: List['PhiNumber']) -> bool:
    """验证演化过程满足no-11约束"""

8.2 数值稳定性​

def check_numerical_stability(evolution: 'EntropyEvolution') -> bool:
    """检查数值计算的稳定性"""
    
def estimate_truncation_error(n_terms: int) -> float:
    """估计级数截断误差"""

9. 关键常数​

# 基础常数
PHI = (1 + np.sqrt(5)) / 2  # 黄金分割率
TAU_PHI = np.log(PHI) / np.log(1 + 1/PHI)  # 特征时间 ≈ 1.44

# 物理常数（归一化单位）
K_B = 1.0  # Boltzmann常数
HBAR = 1.0  # 约化Planck常数
C = 1.0  # 光速
G = 1.0  # 引力常数

# 模型参数
K0_DEFAULT = 1.0  # 默认基准速率
EPSILON_DEFAULT = 0.01  # 默认约束强度
T_DEFAULT = 1.0  # 默认特征时间尺度

10. 错误处理​

class EntropyRateError(Exception):
    """熵增速率计算错误基类"""
    
class NegativeDepthError(EntropyRateError):
    """负递归深度错误"""
    
class ConstraintViolationError(EntropyRateError):
    """约束违反错误"""
    
class ConvergenceError(EntropyRateError):
    """数值收敛错误"""