T19-4 张力驱动collapse定理 - 形式化规范

形式化系统定义

基础结构

Definition F19.4.1 (Tension-Driven Collapse System):

CollapseSystem := ⟨S, T, Υ, C, D⟩
where:
  S := {s₁, s₂, ..., sₙ} is the set of system components  
  T := {T₁, T₂, ..., Tₙ} are the component tensions
  Υ := tension imbalance function
  C := collapse detection predicates
  D := collapse dynamics operators

Definition F19.4.2 (Tension Imbalance Measure):

Υ(T, t) := sqrt(Σᵢ(Tᵢ(t)/T̄(t) - φ⁻ⁱ)²)
where:
  T̄(t) := (1/n) * Σᵢ Tᵢ(t)
  φ := golden ratio (1 + √5)/2

collapse检测算法

Algorithm F19.4.1 (Collapse Trigger Detection):

def detect_collapse_trigger(
    tensions: List[float],
    phi: float = (1 + math.sqrt(5)) / 2,
    threshold_factor: float = math.sqrt((1 + math.sqrt(5)) / 2) * math.log(math.sqrt(5), 2)
) -> Tuple[bool, float, str]:
    """
    检测是否达到collapse触发条件
    
    Args:
        tensions: 组件张力列表
        phi: 黄金比例常数
        threshold_factor: 阈值系数，约等于0.883
        
    Returns:
        (is_triggered, imbalance_measure, collapse_type)
    """
    if len(tensions) == 0:
        return False, 0.0, "empty_system"
    
    n = len(tensions)
    avg_tension = sum(tensions) / n
    
    if avg_tension == 0:
        return False, 0.0, "zero_tension"
    
    # 计算张力不平衡度
    imbalance_sum = 0.0
    for i, tension in enumerate(tensions):
        normalized_tension = tension / avg_tension
        ideal_ratio = phi ** (-(i+1))  # φ^(-i), 1-indexed
        deviation = normalized_tension - ideal_ratio
        imbalance_sum += deviation * deviation
    
    imbalance_measure = math.sqrt(imbalance_sum)
    
    # 检查是否超过临界阈值
    critical_threshold = threshold_factor  # Υc ≈ 0.883
    is_triggered = imbalance_measure >= critical_threshold
    
    # 确定collapse类型
    collapse_type = classify_collapse_type(tensions, phi, avg_tension)
    
    return is_triggered, imbalance_measure, collapse_type

def classify_collapse_type(
    tensions: List[float], 
    phi: float, 
    avg_tension: float
) -> str:
    """
    分类collapse类型
    
    Returns:
        "type_i_bottleneck": 瓶颈主导型collapse
        "type_ii_cascade": 级联型collapse  
        "type_iii_oscillatory": 振荡型collapse
    """
    n = len(tensions)
    max_tension = max(tensions)
    high_tension_count = sum(1 for t in tensions if t > phi * avg_tension)
    
    # Type-I: 单一组件张力远超其他
    if max_tension > phi * phi * (sum(tensions) - max_tension):
        return "type_i_bottleneck"
    
    # Type-II: 多个组件同时超阈值
    if high_tension_count >= math.ceil(n / phi):
        return "type_ii_cascade"
    
    # Type-III: 默认为振荡型
    return "type_iii_oscillatory"

Algorithm F19.4.2 (Collapse Time Estimation):

def estimate_collapse_time(
    imbalance_measure: float,
    collapse_type: str,
    phi: float = (1 + math.sqrt(5)) / 2
) -> float:
    """
    估算collapse时间
    
    Args:
        imbalance_measure: 张力不平衡度 Υ(t)
        collapse_type: collapse类型
        phi: 黄金比例
        
    Returns:
        预估的collapse时间 τ
    """
    if imbalance_measure <= 0:
        return float('inf')  # 无collapse
    
    # 基础时间标度
    log2_phi = math.log2(phi)
    base_time_scale = 1.0 / log2_phi  # ≈ 1.44
    
    if collapse_type == "type_i_bottleneck":
        # τ ~ log(Υ)
        time_factor = max(0.1, math.log(imbalance_measure))
        return base_time_scale * time_factor
    
    elif collapse_type == "type_ii_cascade":
        # τ ~ sqrt(Υ)  
        time_factor = math.sqrt(imbalance_measure)
        return base_time_scale * time_factor
    
    else:  # type_iii_oscillatory
        # τ 具有随机性，使用平均值
        time_factor = imbalance_measure ** (1/phi)  # Υ^(1/φ)
        return base_time_scale * time_factor * 1.5  # 额外因子表示不确定性

collapse动力学算法

Algorithm F19.4.3 (Tension Redistribution Dynamics):

def evolve_collapse_dynamics(
    tensions: np.ndarray,
    dt: float = 0.01,
    phi: float = (1 + math.sqrt(5)) / 2,
    gamma: Optional[float] = None
) -> np.ndarray:
    """
    演化collapse过程的张力重分配
    
    实现方程: dTᵢ/dt = -γ(Tᵢ - Tᵢᵉᵠ) + ξᵢ(t)
    
    Args:
        tensions: 当前张力值
        dt: 时间步长
        phi: 黄金比例
        gamma: collapse速率常数，默认为φ²/log₂(φ)
        
    Returns:
        演化后的张力值
    """
    if gamma is None:
        log2_phi = math.log2(phi)
        gamma = (phi * phi) / log2_phi  # ≈ 3.803
    
    n = len(tensions)
    total_tension = np.sum(tensions)
    
    # 计算平衡张力 T_i^eq = (T_total/n) * φ^(-i)
    equilibrium_tensions = np.zeros(n)
    avg_tension = total_tension / n
    
    for i in range(n):
        equilibrium_tensions[i] = avg_tension * (phi ** (-(i+1)))
    
    # 归一化以保持总张力守恒
    eq_sum = np.sum(equilibrium_tensions)
    if eq_sum > 0:
        equilibrium_tensions *= total_tension / eq_sum
    
    # 动力学演化
    new_tensions = tensions.copy()
    
    for i in range(n):
        # 主要动力学项: -γ(Tᵢ - Tᵢᵉᵠ)
        dynamics_term = -gamma * (tensions[i] - equilibrium_tensions[i])
        
        # Zeckendorf量化噪声 ξᵢ(t)
        noise_amplitude = math.sqrt(dt) * math.log2(phi) * 0.1  # 小噪声
        quantization_noise = np.random.normal(0, noise_amplitude)
        
        # 更新张力
        dT_dt = dynamics_term + quantization_noise
        new_tensions[i] += dT_dt * dt
        
        # 确保张力非负
        new_tensions[i] = max(0, new_tensions[i])
    
    # 应用Zeckendorf量化
    for i in range(n):
        new_tensions[i] = zeckendorf_quantize_tension(new_tensions[i])
    
    return new_tensions

def zeckendorf_quantize_tension(tension: float) -> float:
    """
    将张力量化到Zeckendorf表示
    """
    if tension <= 0:
        return 0
    
    # 使用Fibonacci数量化
    phi = (1 + math.sqrt(5)) / 2
    fibonacci_cache = generate_fibonacci_cache(20)  # 足够的Fibonacci数
    
    # 贪心算法找到最接近的Fibonacci组合
    best_sum = 0
    remaining = tension
    
    for fib in reversed(fibonacci_cache):
        if fib <= remaining:
            best_sum += fib
            remaining -= fib
            
        if remaining < 0.01:  # 精度阈值
            break
    
    return best_sum

def generate_fibonacci_cache(n: int) -> List[int]:
    """生成Fibonacci数缓存"""
    if n <= 0:
        return []
    if n == 1:
        return [1]
    
    fib = [1, 1]
    for i in range(2, n):
        fib.append(fib[i-1] + fib[i-2])
    
    return fib

Algorithm F19.4.4 (Collapse Completion Detection):

def detect_collapse_completion(
    tensions_history: List[np.ndarray],
    window_size: int = 10,
    stability_threshold: float = 0.01
) -> Tuple[bool, Dict[str, float]]:
    """
    检测collapse是否完成，系统是否达到新的平衡态
    
    Args:
        tensions_history: 张力演化历史
        window_size: 稳定性检测窗口大小
        stability_threshold: 稳定性阈值
        
    Returns:
        (is_completed, stability_metrics)
    """
    if len(tensions_history) < window_size:
        return False, {"insufficient_data": True}
    
    # 分析最近窗口内的张力变化
    recent_tensions = tensions_history[-window_size:]
    
    # 计算张力变化的标准差
    tension_changes = []
    for i in range(1, len(recent_tensions)):
        change = np.linalg.norm(recent_tensions[i] - recent_tensions[i-1])
        tension_changes.append(change)
    
    if not tension_changes:
        return False, {"no_changes": True}
    
    avg_change = np.mean(tension_changes)
    std_change = np.std(tension_changes)
    
    # 检测稳定性
    is_stable = avg_change < stability_threshold and std_change < stability_threshold/2
    
    # 验证黄金比例分布
    final_tensions = recent_tensions[-1]
    phi_distribution_score = evaluate_phi_distribution(final_tensions)
    
    metrics = {
        "avg_change": avg_change,
        "std_change": std_change,
        "phi_distribution_score": phi_distribution_score,
        "is_stable": is_stable,
        "is_phi_distributed": phi_distribution_score > 0.8
    }
    
    is_completed = is_stable and phi_distribution_score > 0.7
    
    return is_completed, metrics

def evaluate_phi_distribution(tensions: np.ndarray) -> float:
    """
    评估张力分布与黄金比例分布的符合程度
    
    Returns:
        符合程度分数 (0-1)，1表示完美符合
    """
    if len(tensions) <= 1 or np.sum(tensions) == 0:
        return 0.0
    
    phi = (1 + math.sqrt(5)) / 2
    n = len(tensions)
    
    # 归一化张力
    total_tension = np.sum(tensions)
    normalized_tensions = tensions / total_tension
    
    # 计算理论黄金比例分布
    ideal_distribution = np.zeros(n)
    for i in range(n):
        ideal_distribution[i] = phi ** (-(i+1))
    
    # 归一化理论分布
    ideal_sum = np.sum(ideal_distribution)
    if ideal_sum > 0:
        ideal_distribution /= ideal_sum
    
    # 计算分布相似度 (1 - Jensen-Shannon散度)
    # 使用KL散度的对称化版本
    def kl_divergence(p, q):
        """计算KL散度，处理零值"""
        epsilon = 1e-10
        p_safe = p + epsilon
        q_safe = q + epsilon
        return np.sum(p_safe * np.log(p_safe / q_safe))
    
    # Jensen-Shannon散度
    m = 0.5 * (normalized_tensions + ideal_distribution)
    js_divergence = 0.5 * (kl_divergence(normalized_tensions, m) + 
                          kl_divergence(ideal_distribution, m))
    
    # 转换为相似度分数 (0-1)
    similarity_score = math.exp(-js_divergence)
    
    return min(1.0, similarity_score)

不可逆性验证算法

Algorithm F19.4.5 (Irreversibility Verification):

def verify_collapse_irreversibility(
    initial_tensions: np.ndarray,
    final_tensions: np.ndarray,
    phi: float = (1 + math.sqrt(5)) / 2
) -> Tuple[bool, float]:
    """
    验证collapse过程的不可逆性
    
    Args:
        initial_tensions: 初始张力分布
        final_tensions: 最终张力分布  
        phi: 黄金比例
        
    Returns:
        (is_irreversible, distance_measure)
    """
    # 计算状态间距离
    distance = np.linalg.norm(final_tensions - initial_tensions)
    
    # 最小不可逆距离 ΔT_min = log₂(φ)
    min_irreversible_distance = math.log2(phi)
    
    # 验证不可逆性条件
    is_irreversible = distance >= min_irreversible_distance
    
    return is_irreversible, distance

def compute_information_loss(
    initial_tensions: np.ndarray,
    final_tensions: np.ndarray
) -> float:
    """
    计算collapse过程中的信息损失
    
    Returns:
        信息损失量 (bits)
    """
    def tension_entropy(tensions):
        """计算张力分布的熵"""
        if np.sum(tensions) == 0:
            return 0
        
        # 归一化为概率分布
        probs = tensions / np.sum(tensions)
        
        # 计算Shannon熵
        entropy = 0
        for p in probs:
            if p > 0:
                entropy -= p * math.log2(p)
        
        return entropy
    
    initial_entropy = tension_entropy(initial_tensions)
    final_entropy = tension_entropy(final_tensions)
    
    # 信息损失 = 初始熵 - 最终熵
    information_loss = initial_entropy - final_entropy
    
    return max(0, information_loss)  # 确保非负

形式化验证条件

不变量

Invariant I19.4.1 (Tension Conservation during Collapse):

∀t ∈ collapse_period: Σᵢ Tᵢ(t) = T_total_initial

Invariant I19.4.2 (Non-Negative Tensions):

∀t ≥ 0, ∀i ∈ [1,n]: Tᵢ(t) ≥ 0

Invariant I19.4.3 (Zeckendorf Quantization):

∀i, ∀t: Tᵢ(t) ∈ ZeckendorfRepresentable

Invariant I19.4.4 (Collapse Threshold Consistency):

Υ(t) ≥ Υc ⟹ ∃τ > t: Collapse(τ)

测试谓词

Predicate P19.4.1 (Collapse Trigger Condition):

def collapse_trigger_satisfied(tensions, threshold):
    imbalance = compute_tension_imbalance(tensions)
    return imbalance >= threshold

Predicate P19.4.2 (Dynamics Correctness):

def dynamics_evolution_correct(old_tensions, new_tensions, dt, gamma):
    expected = evolve_collapse_dynamics(old_tensions, dt, gamma=gamma)
    tolerance = 0.01
    return np.allclose(new_tensions, expected, atol=tolerance)

Predicate P19.4.3 (Equilibrium Achievement):

def equilibrium_achieved(tensions, phi):
    phi_score = evaluate_phi_distribution(tensions)
    return phi_score > 0.8

复杂度分析

• Collapse Detection: O(n) where n = number of components
• Dynamics Evolution: O(n) per time step
• Completion Detection: O(k*n) where k = history window size
• Irreversibility Verification: O(n)

数值稳定性

• 张力演化使用显式Euler方法，时间步长需要满足稳定性条件
• Zeckendorf量化避免舍入误差累积
• 黄金比例φ使用高精度常数计算
• 总张力守恒通过重归一化保证

边界条件处理

1. 零张力系统: 特殊处理，避免除零错误
2. 单组件系统: collapse条件自动满足
3. 极大张力: 使用对数空间计算避免溢出
4. 数值精度: 设置最小张力阈值避免下溢

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注记: 本形式化规范提供了T19-4张力驱动collapse定理的完全可执行数学模型。所有算法都严格遵循Zeckendorf编码约束和φ-量化特性，确保理论与实现的一致性。collapse检测、动力学演化和平衡态验证都有明确的数值判据。