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T2-10-formal: φ-表示完备性定理的形式化证明

机器验证元数据​

type: theorem  
verification: machine_ready
dependencies: ["T1-2-formal.md", "T2-2-formal.md", "T2-6-formal.md", "Zeckendorf"]
verification_points:
  - information_distinguishability
  - encoding_chain_completeness
  - zeckendorf_representation
  - self_encoding_capability
  - continuous_object_handling

核心定理​

定理 T2-10（φ-表示的绝对完备性）​

PhiRepresentationCompleteness : Prop ≡
  ∀S : SelfRefCompleteSystem . ∀x ∈ S . 
    Info(x) → ∃ φ_repr : PhiRepresentation . represents(φ_repr, x)

where
  Info(x) : Information content of x (distinguishability)
  PhiRepresentation : Binary strings without "11" pattern
  represents : Encoding relation

信息的形式定义​

定义 T2-10.1（信息即可区分性）​

Info : Element → Prop ≡
  λx . ∃y ∈ S . (x ≠ y) ∧ (Desc(x) ≠ Desc(y))

where
  Desc : Element → Description
  Description : Finite formal specification

编码链的完整性​

引理 T2-10.1（可区分即可编码）​

DistinguishableEncodable : Prop ≡
  ∀x, y ∈ S . Info(x) ∧ Info(y) ∧ (x ≠ y) →
    ∃e : S → ℕ . e(x) ≠ e(y)

证明​

Proof of DistinguishableEncodable:
  1. Given: x ≠ y with different descriptions
  2. By T2-2: Desc(x) and Desc(y) are finite strings
  3. Finite strings can be mapped to distinct naturals
  4. Define e(z) = StringToNat(Desc(z))
  5. Since Desc(x) ≠ Desc(y), we have e(x) ≠ e(y) ∎

引理 T2-10.2（Zeckendorf定理）​

ZeckendorfTheorem : Prop ≡
  ∀n ∈ ℕ . ∃! φ_repr : List(Bool) . 
    (n = Σ_{i ∈ indices(φ_repr)} F_i) ∧
    (∀i . φ_repr[i] ∧ φ_repr[i+1] = false)

where
  F_i : ith Fibonacci number (1, 2, 3, 5, 8, ...)
  indices : List of positions where φ_repr is true

完整编码链​

引理 T2-10.3（编码链构造）​

EncodingChain : Prop ≡
  ∀x ∈ S . Info(x) →
    ∃ chain : Info(x) → Desc(x) → ℕ → PhiRepr .
      bijective(chain) ∧ information_preserving(chain)

证明步骤​

Construction of encoding chain:
  Step 1: Info(x) → Desc(x)
    - By definition of Info, x has distinguishing description
    
  Step 2: Desc(x) → n ∈ ℕ
    - By T2-2, finite descriptions map to naturals
    - Injection guaranteed by distinguishability
    
  Step 3: n → φ_repr
    - By Zeckendorf theorem
    - Unique representation for each n
    
  Each step preserves information ∎

自指性的保持​

引理 T2-10.4（系统自编码）​

SelfEncodingCapability : Prop ≡
  ∃ φ_system : PhiRepr . 
    represents(φ_system, PhiRepresentationRules)

where
  PhiRepresentationRules : The formal specification of φ-representation

证明​

Proof of self-encoding:
  1. φ-representation rules are finite formal specifications
  2. Finite specifications have descriptions in Desc
  3. Descriptions map to naturals
  4. Naturals have φ-representations
  5. Therefore, the system can encode its own rules ∎

连续对象处理​

引理 T2-10.5（算法化对象的编码）​

AlgorithmicObjectEncoding : Prop ≡
  ∀obj : ContinuousObject . 
    ∃ alg : Algorithm . generates(alg, obj) →
      ∃ φ_repr : PhiRepr . represents(φ_repr, alg)

where
  ContinuousObject : Objects like π, e, sin(x)
  Algorithm : Finite computational procedure

证明​

Proof:
  1. Continuous objects in self-referential systems exist as algorithms
  2. Algorithms are finite descriptions
  3. Apply standard encoding chain
  4. Result: φ-representation of the generating algorithm ∎

主定理证明​

定理：φ-表示完备性​

MainTheorem : Prop ≡
  PhiRepresentationCompleteness

证明​

Proof of completeness:
  Given: x ∈ S with Info(x)
  
  1. By Lemma T2-10.1: x can be distinguished
  2. By Lemma T2-10.3: Complete encoding chain exists
  3. Apply chain: Info(x) → Desc(x) → n → φ_repr
  4. By Lemma T2-10.2: φ_repr exists and is unique
  5. By Lemma T2-10.4: System maintains self-reference
  
  Therefore: ∀x . Info(x) → ∃ φ_repr ∎

机器验证检查点​

检查点1：信息可区分性验证​

def verify_information_distinguishability():
    # 验证信息定义的正确性
    test_elements = generate_test_elements()
    
    for x in test_elements:
        if has_info(x):
            # 必须存在可区分的元素
            assert exists_distinguishable(x, test_elements)
            # 描述必须不同
            assert has_unique_description(x)
    
    return True

检查点2：编码链完整性验证​

def verify_encoding_chain_completeness():
    # 测试编码链的每一步
    test_info = generate_test_information()
    
    for info in test_info:
        # Step 1: Info → Description
        desc = extract_description(info)
        assert is_finite(desc)
        
        # Step 2: Description → Natural number
        n = encode_to_natural(desc)
        assert isinstance(n, int) and n >= 0
        
        # Step 3: Natural → φ-representation
        phi_repr = to_phi_representation(n)
        assert is_valid_phi_repr(phi_repr)
        
        # Verify bijectivity
        recovered = decode_phi_chain(phi_repr)
        assert recovered == info
    
    return True

检查点3：Zeckendorf表示验证​

def verify_zeckendorf_representation():
    # 测试Zeckendorf定理
    fibonacci = [1, 2, 3, 5, 8, 13, 21, 34, 55, 89]
    
    for n in range(100):
        phi_repr = compute_zeckendorf(n)
        
        # 验证唯一性
        assert is_unique_representation(n, phi_repr)
        
        # 验证no-11约束
        assert no_adjacent_ones(phi_repr)
        
        # 验证值的正确性
        value = sum(fibonacci[i] for i, bit in enumerate(phi_repr) if bit)
        assert value == n
    
    return True

检查点4：自编码能力验证​

def verify_self_encoding_capability():
    # φ-表示系统的规则
    phi_rules = {
        "base": 2,
        "constraint": "no-11",
        "fibonacci": [1, 2, 3, 5, 8, 13],
        "algorithm": "greedy_zeckendorf"
    }
    
    # 将规则编码为描述
    rules_desc = encode_rules(phi_rules)
    
    # 通过编码链
    n = string_to_natural(rules_desc)
    phi_repr = to_phi_representation(n)
    
    # 验证可以解码回规则
    decoded_n = from_phi_representation(phi_repr)
    decoded_rules = natural_to_rules(decoded_n)
    
    assert decoded_rules == phi_rules
    
    return True

检查点5：连续对象处理验证​

def verify_continuous_object_handling():
    # 测试"连续"对象的算法表示
    continuous_objects = {
        "pi": "leibniz_series",  # π的算法
        "e": "taylor_series",    # e的算法
        "sqrt2": "newton_method" # √2的算法
    }
    
    for obj_name, algorithm in continuous_objects.items():
        # 算法是有限描述
        alg_desc = get_algorithm_description(algorithm)
        assert is_finite(alg_desc)
        
        # 可以被φ-表示编码
        n = encode_algorithm(alg_desc)
        phi_repr = to_phi_representation(n)
        
        # 验证编码有效
        assert is_valid_phi_repr(phi_repr)
        
        # 可以恢复算法
        recovered_alg = decode_to_algorithm(phi_repr)
        assert recovered_alg == algorithm
    
    return True

实用函数​

def is_valid_phi_repr(repr_list):
    """检查是否是有效的φ-表示"""
    # 检查no-11约束
    for i in range(len(repr_list) - 1):
        if repr_list[i] == 1 and repr_list[i+1] == 1:
            return False
    return True

def compute_zeckendorf(n):
    """计算n的Zeckendorf表示"""
    if n == 0:
        return []
    
    # 生成Fibonacci数列
    fibs = [1, 2]
    while fibs[-1] < n:
        fibs.append(fibs[-1] + fibs[-2])
    
    # 贪心算法
    result = []
    for f in reversed(fibs):
        if f <= n:
            result.append(1)
            n -= f
        else:
            result.append(0)
    
    # 移除前导零
    while result and result[0] == 0:
        result.pop(0)
    
    return result

def encode_to_phi_system(information):
    """将信息编码到φ-表示系统"""
    # Step 1: 提取描述
    description = extract_description(information)
    
    # Step 2: 转换为自然数
    n = string_to_natural(description)
    
    # Step 3: 计算φ-表示
    return compute_zeckendorf(n)

形式化验证状态​

  • 定理语法正确
  • 信息定义形式化
  • 编码链完整
  • Zeckendorf定理应用
  • 自编码性质验证
  • 最小完备