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D1-7-formal: Collapse算子的形式化定义

机器验证元数据​

type: definition
verification: machine_ready
dependencies: ["D1-5-formal.md", "D1-6-formal.md"]
verification_points:
  - entropy_increase
  - irreversibility
  - self_reference
  - observer_dependence

核心定义​

定义 D1-7（Collapse算子）​

CollapseOperator(S : SelfReferentialComplete) : Prop ≡
  ∃Ĉ : Function[P(S) × O → S × R] .
    EntropyIncrease(Ĉ) ∧
    Irreversible(Ĉ) ∧
    SelfReferential(Ĉ) ∧
    ObserverDependent(Ĉ)

四个核心条件​

条件1：熵增性​

EntropyIncrease(Ĉ) : Prop ≡
  ∀𝒮 ∈ P(S), o ∈ O . 
    let (s_collapsed, r) = Ĉ(𝒮, o) in
    H({s_collapsed} ∪ {r}) > H(𝒮)

条件2：不可逆性​

Irreversible(Ĉ) : Prop ≡
  ¬∃Ĉ⁻¹ : S × R → P(S) . 
    ∀𝒮, o . Ĉ⁻¹(Ĉ(𝒮, o)) = 𝒮

条件3：自指性​

SelfReferential(Ĉ) : Prop ≡
  Ĉ ∈ S → 
    ∀𝒮 ∈ P(S), o ∈ O . Ĉ ∈ 𝒮 → 
      Ĉ(𝒮, o) is well-defined

条件4：观察者依赖性​

ObserverDependent(Ĉ) : Prop ≡
  ∃𝒮 ∈ P(S), o₁, o₂ ∈ O . 
    o₁ ≠ o₂ → Ĉ(𝒮, o₁) ≠ Ĉ(𝒮, o₂)

数学表述​

标准形式​

Ĉ(𝒮, o) = (s_collapsed, r_measurement)

where
  s_collapsed := select(𝒮, measure(o))
  r_measurement := record(𝒮, s_collapsed, o)

概率形式​

P(s_collapsed = sᵢ | 𝒮, o) = wᵢ(o) / Σⱼ wⱼ(o)

where
  wᵢ(o) : Weight function of observer o for state sᵢ

Collapse过程阶段​

阶段定义​

CollapseStages := Enum {
  PreCollapse,      // 𝒮_pre = {s₁, s₂, ..., sₙ}
  ObserverIntervention,  // measurement(o) : 𝒮_pre → I_o
  StateSelection,   // s_selected = selection_rule(𝒮_pre, result)
  RecordGeneration  // 𝒮_post = {s_selected} ∪ {record} ∪ {Desc(record)}
}

Collapse算子性质​

性质1：非线性性​

NonLinear(Ĉ) : Prop ≡
  ∃α, β ∈ ℝ, 𝒮₁, 𝒮₂ ∈ P(S), o ∈ O .
    Ĉ(α𝒮₁ + β𝒮₂, o) ≠ αĈ(𝒮₁, o) + βĈ(𝒮₂, o)

性质2：观察者特异性​

ObserverSpecific(Ĉ) : Prop ≡
  ∃𝒮 ∈ P(S), o₁, o₂ ∈ O .
    o₁ ≠ o₂ → Ĉ(𝒮, o₁) ≠ Ĉ(𝒮, o₂)

性质3：递归适用性​

RecursivelyApplicable(Ĉ) : Prop ≡
  ∀𝒮 ∈ P(S), o₁, o₂ ∈ O .
    let (s₁, r₁) = Ĉ(𝒮, o₁) in
    Ĉ({s₁}, o₂) is well-defined

特殊Collapse类型​

CollapseType := Enum {
  Complete,   // Ĉ_complete : P(S) × O → {single state} × R
  Partial,    // Ĉ_partial : P(S) × O → P'(S) × R, P' ⊂ P
  Soft,       // Ĉ_soft : P(S) × O → ProbDist(S) × R
  Delayed     // Ĉ_delayed : P(S) × O × Time → S × R
}

反作用效应​

观察者反作用​

ObserverBackaction(o_pre, collapse_result) : Observer ≡
  o_post = o_pre ⊕ experience(collapse_result)

系统反作用​

SystemBackaction(S_pre, collapse_result) : System ≡
  S_post = S_pre ∪ ΔS_collapse

信息理论解释​

信息获得与成本​

InformationGain(𝒮_pre, 𝒮_post) : Real⁺ ≡
  H(𝒮_pre) - H(𝒮_post)

TotalEntropyIncrease(S_pre, S_post) : Real⁺ ≡
  H_total(S_post) - H_total(S_pre) > 0

类型定义​

Type P(S) := PowerSet[SystemState]
Type O := Set[Observer]
Type R := Set[MeasurementResult]
Type Weight := Observer × State → Real⁺

机器验证检查点​

检查点1：熵增验证​

def verify_entropy_increase(collapse_op, state_set, observer):
    pre_entropy = compute_entropy(state_set)
    collapsed_state, record = collapse_op(state_set, observer)
    post_entropy = compute_entropy({collapsed_state, record})
    return post_entropy > pre_entropy

检查点2：不可逆性验证​

def verify_irreversibility(collapse_op, state_set, observer):
    original = state_set.copy()
    result = collapse_op(state_set, observer)
    # 验证无法从结果恢复原始状态集
    return cannot_reconstruct(result, original)

检查点3：自指性验证​

def verify_self_reference(collapse_op, system):
    if collapse_op in system:
        state_set_with_op = {state for state in system} | {collapse_op}
        result = collapse_op(state_set_with_op, observer)
        return result is not None  # Well-defined

检查点4：观察者依赖性验证​

def verify_observer_dependence(collapse_op, state_set):
    observer1 = create_observer("O1")
    observer2 = create_observer("O2")
    result1 = collapse_op(state_set, observer1)
    result2 = collapse_op(state_set, observer2)
    return result1 != result2  # Different observers, different results

形式化验证状态​

  • 定义语法正确
  • 核心条件完整
  • 过程阶段明确
  • 类型系统清晰
  • 最小完备