type: definition
verification: machine_ready
dependencies: ["D1-5-formal.md", "D1-6-formal.md"]
verification_points:
- entropy_increase
- irreversibility
- self_reference
- observer_dependence
CollapseOperator(S : SelfReferentialComplete) : Prop ≡
∃Ĉ : Function[P(S) × O → S × R] .
EntropyIncrease(Ĉ) ∧
Irreversible(Ĉ) ∧
SelfReferential(Ĉ) ∧
ObserverDependent(Ĉ)
EntropyIncrease(Ĉ) : Prop ≡
∀𝒮 ∈ P(S), o ∈ O .
let (s_collapsed, r) = Ĉ(𝒮, o) in
H({s_collapsed} ∪ {r}) > H(𝒮)
Irreversible(Ĉ) : Prop ≡
¬∃Ĉ⁻¹ : S × R → P(S) .
∀𝒮, o . Ĉ⁻¹(Ĉ(𝒮, o)) = 𝒮
SelfReferential(Ĉ) : Prop ≡
Ĉ ∈ S →
∀𝒮 ∈ P(S), o ∈ O . Ĉ ∈ 𝒮 →
Ĉ(𝒮, o) is well-defined
ObserverDependent(Ĉ) : Prop ≡
∃𝒮 ∈ P(S), o₁, o₂ ∈ O .
o₁ ≠ o₂ → Ĉ(𝒮, o₁) ≠ Ĉ(𝒮, o₂)
Ĉ(𝒮, o) = (s_collapsed, r_measurement)
where
s_collapsed := select(𝒮, measure(o))
r_measurement := record(𝒮, s_collapsed, o)
P(s_collapsed = sᵢ | 𝒮, o) = wᵢ(o) / Σⱼ wⱼ(o)
where
wᵢ(o) : Weight function of observer o for state sᵢ
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)}
}
NonLinear(Ĉ) : Prop ≡
∃α, β ∈ ℝ, 𝒮₁, 𝒮₂ ∈ P(S), o ∈ O .
Ĉ(α𝒮₁ + β𝒮₂, o) ≠ αĈ(𝒮₁, o) + βĈ(𝒮₂, o)
ObserverSpecific(Ĉ) : Prop ≡
∃𝒮 ∈ P(S), o₁, o₂ ∈ O .
o₁ ≠ o₂ → Ĉ(𝒮, o₁) ≠ Ĉ(𝒮, o₂)
RecursivelyApplicable(Ĉ) : Prop ≡
∀𝒮 ∈ P(S), o₁, o₂ ∈ O .
let (s₁, r₁) = Ĉ(𝒮, o₁) in
Ĉ({s₁}, o₂) is well-defined
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
}
ObserverBackaction(o_pre, collapse_result) : Observer ≡
o_post = o_pre ⊕ experience(collapse_result)
SystemBackaction(S_pre, collapse_result) : System ≡
S_post = S_pre ∪ ΔS_collapse
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
Type P(S) := PowerSet[SystemState]
Type O := Set[Observer]
Type R := Set[MeasurementResult]
Type Weight := Observer × State → Real⁺
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
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)
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
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