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T13-2 形式化规范：自适应压缩算法

核心命题​

命题 T13-2：在φ编码宇宙中，自适应压缩通过识别递归模式实现深度优化。

形式化陈述​

∀S : PhiEncoded . ∃C : Compressor .
  depth(C(S)) ≤ depth(S) - log_φ(PatternRegularity(S)) ∧
  Unfold(C(S)) = S

其中：

  • S是任意φ编码的数据（已满足no-11约束）
  • depth衡量递归展开深度
  • PatternRegularity衡量S中的规律性
  • Unfold是递归展开操作

形式化组件​

1. 压缩系统结构​

CompressionSystem ≡ record {
  encode : Sequence → BitString
  decode : BitString → Sequence
  window_size : ℕ → ℕ
  mode_selector : Window → Mode
  compress_rate : Sequence → ℝ
}

AdaptivePhiCompression : CompressionSystem ≡ record {
  encode = adaptive_phi_encode
  decode = adaptive_phi_decode
  window_size = λn . ⌈log₂ n⌉
  mode_selector = complexity_based_selector
  compress_rate = λS . |encode(S)| / |S|
}

2. φ-熵定义​

PhiEntropy : Distribution → ℝ ≡
  λP . -∑_{w ∈ ValidPhi} P(w) × log₂(P(w))

ValidPhi : Set[String] ≡
  {w : String | ¬contains_11(w)}

StandardEntropy : Distribution → ℝ ≡
  λP . -∑_{w ∈ Σ*} P(w) × log₂(P(w))

EntropyRelation : 
  PhiEntropy(P) = StandardEntropy(P) - log₂(φ) + o(1)

3. 编码模式定义​

Mode ≡ Sparse | Dense | Recursive

SparseEncoding ≡ record {
  encode : Window → List[(Symbol, ZeckendorfRep)]
  decode : List[(Symbol, ZeckendorfRep)] → Window
  condition : λw . LocalComplexity(w) < 0.3
}

DenseEncoding ≡ record {
  encode : Window → (PhiReal, PhiReal)
  decode : (PhiReal, PhiReal) → Window
  condition : λw . 0.3 ≤ LocalComplexity(w) < 0.7
}

RecursiveEncoding ≡ record {
  encode : Window → RecursiveStructure
  decode : RecursiveStructure → Window
  condition : λw . LocalComplexity(w) ≥ 0.7 ∧ HasSelfSimilarity(w)
}

4. 局部复杂度度量​

LocalComplexity : Window → [0,1] ≡
  λw . PhiKolmogorov(w) / |w|

PhiKolmogorov : String → ℕ ≡
  λs . min{|p| : Program p ∧ Execute(p) = s ∧ ValidPhi(p)}

WindowComplexityProfile : Sequence → (ℕ → [0,1]) ≡
  λS w . Average[LocalComplexity(S[i:i+w]) for i in range(|S|-w)]

5. 自适应选择算法​

AdaptiveModeSelector : Window → Mode ≡
  λw . if LocalComplexity(w) < 0.3 then Sparse
       else if LocalComplexity(w) < 0.7 then Dense
       else if DetectSelfSimilarity(w) then Recursive
       else Dense

OptimalWindowSize : ℕ → ℕ ≡
  λn . argmin_{w} (CompressionOverhead(w) + ExpectedEncodingLength(w))
  where
    CompressionOverhead(w) = log₂(#modes) / w
    ExpectedEncodingLength(w) = w × AveragePhiEntropy

核心定理​

定理1：压缩率界限​

theorem CompressionRateBound:
  ∀S : Sequence . ∀A : AdaptivePhiCompression .
    CompressRate(A, S) ≤ PhiEntropy(S) + O(log log |S| / log |S|)
    
proof:
  设 w = OptimalWindowSize(|S|) = O(log |S|)
  每个窗口的编码包含：
  1. 模式标识：log₂(3) 位
  2. 模式数据：≤ w × PhiEntropy(S) + o(w) 位
  
  总压缩率 = ∑(模式标识 + 模式数据) / |S|
           = |S|/w × (log₂(3) + w × PhiEntropy(S) + o(w)) / |S|
           = log₂(3)/w + PhiEntropy(S) + o(1)
           = O(log log |S| / log |S|) + PhiEntropy(S)
  ∎

定理2：模式选择最优性​

theorem ModeSelectionOptimality:
  ∀w : Window . 
    SelectedMode(w) = argmin_{m ∈ Mode} EncodingLength(m, w)
    
proof:
  对于每种模式，编码长度为：
  - Sparse: O(k × log n)，k为游程数
  - Dense: O(n × H(w))，H为熵
  - Recursive: O(|pattern| + t × log n)，t为变换数
  
  局部复杂度与最优模式的对应关系：
  - Low complexity → Few runs → Sparse optimal
  - Medium complexity → High entropy → Dense optimal  
  - High complexity + self-similarity → Recursive optimal
  ∎

定理3：递归压缩不动点​

theorem RecursiveCompressionFixedPoint:
  ∃C* : CompressionAlgorithm .
    Compress(C*) = C* ∧
    |C*| = log₂|S| / (1 - log₂φ)
    
proof:
  设压缩函数为 f(x) = αx + β
  不动点条件：f(x*) = x*
  
  解得：x* = β/(1-α)
  
  其中 α = 1/φ (压缩比)
       β = log₂|S| (元数据)
       
  因此 |C*| = log₂|S| / (1 - 1/φ) = log₂|S| / (1 - log₂φ)
  ∎

算法规范​

算法1：主压缩流程​

def adaptive_phi_compress(data: Sequence) -> BitString:
    """自适应φ压缩主算法"""
    # 前置条件
    assert len(data) > 0
    
    # 确定最优窗口大小
    w = optimal_window_size(len(data))
    
    compressed = []
    for i in range(0, len(data), w):
        window = data[i:i+w]
        
        # 计算局部特征
        complexity = local_complexity(window)
        mode = select_mode(complexity)
        
        # 根据模式编码
        if mode == Mode.SPARSE:
            encoded = sparse_encode(window)
        elif mode == Mode.DENSE:
            encoded = dense_encode(window)
        else:  # RECURSIVE
            encoded = recursive_encode(window)
        
        compressed.append((mode, encoded))
    
    # 后置条件
    assert can_decode(compressed, data)
    return pack_compressed(compressed)

算法2：稀疏编码实现​

def sparse_encode(window: Window) -> SparseCoding:
    """游程编码的φ变体"""
    runs = []
    i = 0
    
    while i < len(window):
        # 找到游程
        symbol = window[i]
        length = 1
        while i + length < len(window) and window[i + length] == symbol:
            length += 1
        
        # 使用Zeckendorf编码长度
        zeck_length = to_zeckendorf(length)
        runs.append((symbol, zeck_length))
        
        i += length
    
    # 验证no-11约束
    for _, zeck in runs:
        assert is_valid_phi(zeck)
    
    return runs

算法3：密集编码实现​

def dense_encode(window: Window) -> DenseCoding:
    """算术编码的φ变体"""
    # 统计符号频率
    freq = compute_frequency(window)
    
    # 构建φ进制累积概率表
    cumulative = build_phi_cumulative(freq)
    
    # 算术编码过程
    low, high = PhiReal(0), PhiReal(1)
    
    for symbol in window:
        range_width = high - low
        high = low + range_width * cumulative[symbol + 1]
        low = low + range_width * cumulative[symbol]
    
    # 选择区间内的φ进制数
    code = select_phi_number(low, high)
    
    return (code, len(window))

算法4：递归编码实现​

def recursive_encode(window: Window) -> RecursiveCoding:
    """自相似结构的递归编码"""
    # 模式检测
    pattern, occurrences = detect_pattern(window)
    
    if pattern and len(occurrences) > 2:
        # 递归压缩模式
        compressed_pattern = adaptive_phi_compress(pattern)
        
        # 编码变换序列
        transforms = []
        for occ in occurrences:
            trans = compute_transform(pattern, occ)
            transforms.append(encode_transform(trans))
        
        # 编码残差
        residual = compute_residual(window, pattern, occurrences)
        compressed_residual = adaptive_phi_compress(residual)
        
        return RecursiveCoding(
            pattern=compressed_pattern,
            transforms=transforms,
            residual=compressed_residual
        )
    else:
        # 退化到密集编码
        return RecursiveCoding(fallback=dense_encode(window))

验证条件​

1. 压缩正确性​

  • decode(encode(S)) = S
  • |encode(S)| < |S| （对于可压缩数据）
  • 保持no-11约束

2. 渐近最优性​

  • 压缩率趋近φ-熵
  • 额外开销 = O(log log n / log n)
  • 对所有数据类型有效

3. 模式选择有效性​

  • 每种模式在其适用范围内最优
  • 模式切换开销最小
  • 自适应阈值收敛

4. 计算效率​

  • 编码时间：O(n log n)
  • 解码时间：O(n)
  • 空间复杂度：O(√n)

实现注意事项​

  1. 数值精度：φ进制运算需要高精度
  2. 模式检测：使用后缀数组加速
  3. 并行化：窗口间可并行处理
  4. 流式处理：支持在线压缩
  5. 错误恢复：添加同步标记