T2-8-formal: φ-表示动态适应性定理的形式化规范

机器验证元数据

type: theorem  
verification: machine_ready
dependencies: ["A1-formal.md", "T2-6-formal.md", "T2-7-formal.md", "L1-5-formal.md"]
verification_points:
  - efficiency_lower_bound
  - local_recoding_correctness
  - no11_preservation
  - convergence_to_stability
  - adaptation_process_validity

核心定理

定理 T2-8（φ-表示的动态适应性）

DynamicAdaptability : Prop ≡
  ∀t : Time, S_t : SystemState, φ_t : PhiEncoding .
    Evolution(S_t, S_{t+1}) ∧ H(S_{t+1}) > H(S_t) →
      ∃φ_{t+1} : PhiEncoding .
        (Efficiency(φ_{t+1}) ≥ EfficiencyMin ∧ 
         AdaptProcess(φ_t → φ_{t+1}) ∈ Valid_11 ∧
         LocallyOptimal(φ_{t+1}, S_{t+1}))

where
  EfficiencyMin : Real = 1/(1 + c·log(t))  // c是系统常数
  Valid_11 : Set = {binary | no_consecutive_ones(binary)}
  LocallyOptimal : 局部最优性条件

形式化组件

1. 动态环境定义

DynamicEnvironment : Type ≡
  record {
    states : Time → SystemState
    entropy_law : ∀t . H(states(t+1)) > H(states(t))
    info_growth : ∀t . |Info(states(t+1))| > |Info(states(t))|
  }

2. 编码压力度量

EncodingPressure : Time → Real ≡
  λt . H(S_t) / log₂(|Available_Codes(φ_t)|)

Lemma_PressureMonotonic : Prop ≡
  ∀t₁ t₂ . t₁ < t₂ → EncodingPressure(t₁) ≤ EncodingPressure(t₂)

3. 局部重编码机制

LocalRecode : Info × PhiEncoding → PhiEncoding ≡
  λ(x, φ) . 
    if Efficiency(x, φ) ≥ θ then
      φ(x)  // 保持原编码
    else
      OptimalFibonacciEncode(x)  // 重新编码

where
  θ : Real = dynamic_threshold(φ, current_pressure)

4. 效率下界证明

EfficiencyLowerBound : Prop ≡
  ∃c > 0 . ∀t : Time, φ_t : PhiEncoding .
    IsAdaptive(φ_t) → Efficiency(φ_t) ≥ 1/(1 + c·log(t))

Proof_sketch:
  1. 需要重编码的元素数量 N_t = O(√|Info(S_t)|)
  2. Fibonacci性质保证稀疏性
  3. 重编码开销次线性
  4. 整体效率有下界

5. No-11约束保持

AdaptationPreservesNo11 : Prop ≡
  ∀φ : PhiEncoding, x : Info .
    Valid_11(φ(x)) → Valid_11(LocalRecode(x, φ))

Proof_components:
  - Initial_φ_valid: by T2-6
  - Fibonacci_decomposition_valid: by mathematical property
  - Composition_preserves_validity: by structural induction

6. 收敛性质

DynamicStability : Prop ≡
  ∀ε > 0 . ∃T : Time . ∀t > T .
    |dEfficiency(φ_t)/dt| < ε

ProcessEntropyGrowth : Prop ≡
  ∀t : Time . dH_process(S_t)/dt > 0

StructuralStability : Prop ≡
  lim_{t→∞} H_structure(φ_t) = H_max

算法规范

动态适应算法

AdaptivePhiEncode : Algorithm ≡
  Input: info_stream : Stream[Info], φ_current : PhiEncoding
  Output: φ_adapted : PhiEncoding
  
  Invariants:
    - ∀step . Valid_11(φ_current)
    - ∀step . Efficiency(φ_current) ≥ EfficiencyMin
  
  Process:
    1. threshold ← ComputeThreshold(φ_current)
    2. for each info ∈ info_stream:
         eff ← ComputeEfficiency(info, φ_current)
         if eff < threshold:
           new_code ← OptimalFibonacciEncode(info)
           φ_current.update(info → new_code)
         assert Valid_11(φ_current[info])
    3. return φ_current

数学性质验证

性质1：效率单调性

EfficiencyMonotonicity : Prop ≡
  ∀t_i < t < t_{i+1} . 
    NoRecoding(t_i, t_{i+1}) → 
    Efficiency(φ_t) ≥ Efficiency(φ_{t+1})

性质2：重编码频率递减

RecodingFrequencyDecay : Prop ≡
  lim_{t→∞} |{s < t : Recoding_at(s)}| / t = 0

性质3：渐近最优性

AsymptoticOptimality : Prop ≡
  lim_{t→∞} Efficiency(φ_t) / Efficiency_optimal = 1

验证检查点

1. 效率下界验证

verify_efficiency_bound(φ_sequence, time_range):
  for t in time_range:
    assert Efficiency(φ_sequence[t]) ≥ 1/(1 + c*log(t))

2. No-11保持验证

verify_no11_preservation(adaptation_trace):
  for step in adaptation_trace:
    for encoding in step.encodings:
      assert is_valid_no11(encoding)

3. 收敛性验证

verify_convergence(φ_sequence, window=1000):
  late_sequence = φ_sequence[-window:]
  efficiency_changes = [abs(eff[i+1] - eff[i]) for i in range(len(late_sequence)-1)]
  assert mean(efficiency_changes) < ε

与其他定理的联系

依赖关系

• T2-6: 提供no-11约束的基础
• T2-7: 证明φ-表示的必然性
• L1-5: Fibonacci结构的数学性质

支撑定理

• T2-9: 将使用动态适应性处理误差传播
• T2-10: 完备性依赖于动态适应能力

$$\boxed{\text{形式化规范：φ-表示通过局部重编码实现动态适应，保持效率下界和no-11约束}}$$