T2-9-formal: φ-表示误差传播控制定理的形式化规范

机器验证元数据

type: theorem  
verification: machine_ready
dependencies: ["A1-formal.md", "T2-6-formal.md", "T2-7-formal.md", "T2-8-formal.md"]
verification_points:
  - single_bit_error_bound
  - error_decay_verification
  - multiple_error_subadditivity
  - error_detection_capability
  - propagation_control_mechanism

核心定理

定理 T2-9（φ-表示的误差传播控制）

ErrorPropagationControl : Prop ≡
  ∀ε₀ : Error, d : Distance .
    P(|ε_d| > δ) ≤ α · φ^(-d) · |ε₀|

where
  φ : Real = (1 + √5) / 2  // 黄金比率
  α : Real = system_constant
  δ : Real = error_threshold

形式化组件

1. 误差模型定义

Error : Type ≡
  record {
    position : ℕ
    magnitude : Real
    type : ErrorType
  }

ErrorType : Type ≡ 
  | SingleBit    // 单比特翻转
  | Burst        // 突发误差
  | Systematic   // 系统性误差

2. 误差影响度量

ErrorImpact : PhiCode × Error → Real ≡
  λ(code, error) .
    |decode(code) - decode(corrupt(code, error))|

where
  corrupt : PhiCode × Error → PhiCode
  decode : PhiCode → ℕ

3. 传播距离定义

PropagationDistance : Error × Error → ℕ ≡
  λ(e₁, e₂) . |position(e₂) - position(e₁)|

4. 误差衰减定理

ErrorDecay : Prop ≡
  ∀code : PhiCode, e : Error, d : ℕ .
    Valid_11(code) →
    ImpactAt(code, e, d) ≤ ImpactAt(code, e, 0) · φ^(-d)

where
  ImpactAt(code, e, d) = magnitude of error e after propagating distance d

5. 多重误差次可加性

MultipleErrorSubadditivity : Prop ≡
  ∀code : PhiCode, E : Set[Error] .
    Valid_11(code) →
    ImpactTotal(code, E) < ∑_{e ∈ E} ImpactSingle(code, e)

Proof_sketch:
  1. No-11约束限制了某些误差组合
  2. Fibonacci数的非线性增长
  3. 某些误差相互抵消

6. 误差检测能力

ErrorDetection : Prop ≡
  ∃detector : PhiCode → Option[Error] .
    ∀code : PhiCode, e : Error .
      corrupt(code, e) violates no-11 →
      detector(corrupt(code, e)) = Some(e)

DetectionProbability : Real ≡
  |{e : Error | detectable(e)}| / |{e : Error | possible(e)}|
  ≥ 1 - φ^(-1)
  ≈ 0.382

算法规范

误差传播分析算法

ErrorPropagationAnalysis : Algorithm ≡
  Input: code : PhiCode, errors : List[Error]
  Output: PropagationReport
  
  Invariants:
    - ∀e ∈ errors . Valid_11(corrupt(code, e)) ∨ detectable(e)
    - Decay rate ≤ φ^(-1)
  
  Process:
    1. original ← decode(code)
    2. for each e in errors:
         corrupted ← corrupt(code, e)
         if Valid_11(corrupted):
           impact[e] ← |decode(corrupted) - original|
         else:
           detected.add(e)
    3. combined ← corrupt_multiple(code, errors)
    4. verify SubAdditivity:
         assert |decode(combined) - original| < sum(impact.values())
    5. compute decay_rate from impact distribution
    6. return PropagationReport(impact, detected, decay_rate)

误差界限计算

ComputeErrorBound : Algorithm ≡
  Input: n : ℕ, error_prob : Real
  Output: bound : Real
  
  Process:
    1. max_fib_index ← ⌊log_φ(n)⌋
    2. single_bit_max ← F[max_fib_index]
    3. expected_errors ← n × error_prob
    4. bound ← single_bit_max × (1 - φ^(-expected_errors))
    5. return bound

数学性质验证

性质1：单比特误差界限

SingleBitErrorBound : Prop ≡
  ∀n : ℕ, i < ⌊log_φ(n)⌋ .
    ErrorImpact(encode(n), SingleBit(i)) ≤ F_i

性质2：误差期望值

ExpectedError : Prop ≡
  ∀n : ℕ, p : Probability .
    E[|Error|] = O(√n × p)
    
Compare with binary:
  Binary: E[|Error|] = O(n × p)
  Improvement factor: O(√n)

性质3：误差局部性

ErrorLocality : Prop ≡
  ∀code : PhiCode, e : Error .
    ∃radius : ℕ .
      ∀i . distance(i, position(e)) > radius →
        bit(code, i) = bit(corrupt(code, e), i)

验证检查点

1. 单比特误差界限验证

verify_single_bit_bound(test_size):
  for n in range(1, test_size):
    code = encode(n)
    for i in range(len(code)):
      if code[i] == 1:
        corrupted = flip_bit(code, i)
        if is_valid_no11(corrupted):
          impact = abs(decode(corrupted) - n)
          assert impact == fib[i]

2. 误差衰减验证

verify_error_decay(code, error_pos):
  impacts = []
  for d in range(len(code)):
    if error_pos + d < len(code):
      impact = measure_impact_at_distance(code, error_pos, d)
      impacts.append(impact)
  
  # 验证指数衰减
  for i in range(1, len(impacts)):
    if impacts[i] > 0 and impacts[i-1] > 0:
      decay_rate = impacts[i] / impacts[i-1]
      assert decay_rate <= 1/φ + ε

3. 多重误差次可加性验证

verify_subadditivity(code, error_positions):
  # 单独误差影响
  individual_impacts = []
  for pos in error_positions:
    impact = measure_single_error(code, pos)
    individual_impacts.append(impact)
  
  # 组合误差影响
  combined_impact = measure_combined_errors(code, error_positions)
  
  # 验证次可加性
  assert combined_impact < sum(individual_impacts)

4. 误差检测能力验证

verify_detection_capability(test_size):
  detectable_count = 0
  total_count = 0
  
  for n in range(1, test_size):
    code = encode(n)
    for i in range(len(code)):
      total_count += 1
      corrupted = flip_bit(code, i)
      if not is_valid_no11(corrupted):
        detectable_count += 1
  
  detection_rate = detectable_count / total_count
  assert detection_rate >= 1 - 1/φ - ε

与其他定理的联系

依赖关系

• T2-6: 提供no-11约束的数学基础
• T2-7: 确立φ-表示的必然性
• T2-8: 动态适应性需要误差控制

支撑定理

• T2-10: 完备性在有误差情况下仍然保持
• T2-11: 最大熵增率不受误差显著影响

实用函数

def compute_error_resilience(n: int) -> float:
    """计算数n的φ-表示的误差韧性"""
    code = encode(n)
    total_bits = len(code)
    detectable_errors = count_detectable_errors(code)
    
    resilience = detectable_errors / total_bits
    return resilience

def estimate_propagation_bound(initial_error: float, distance: int) -> float:
    """估计误差传播界限"""
    φ = (1 + sqrt(5)) / 2
    α = 2.0  # 系统常数，经验值
    
    bound = α * (φ ** (-distance)) * initial_error
    return bound

def analyze_error_cascade(code: List[int], error_positions: List[int]) -> Dict:
    """分析误差级联效应"""
    cascade_effects = {}
    
    for primary_pos in error_positions:
        affected_positions = find_affected_positions(code, primary_pos)
        cascade_effects[primary_pos] = {
            'direct_impact': compute_direct_impact(code, primary_pos),
            'cascade_range': len(affected_positions),
            'total_impact': compute_total_impact(code, primary_pos, affected_positions)
        }
    
    return cascade_effects

$$\boxed{\text{形式化规范：φ-表示通过Fibonacci结构实现误差传播的指数衰减控制}}$$