actual_ordered_prefix_law

Plain-English statement

Independent science a4599f28131fcf91d6b0754003facbc46e23ad2e. With the six original analytic hypotheses, canonical iid Exp(1) product P and literal stopped recursion, every finite n and T>=0 has a strict ordered-prefix complete-history law. Delta_n(T) is the open ordered time simplex with actual subtype comap of product Lebesgue measure. K gives all n+1 live time/state records; q is the product of n pre-bounce rates and elapsed segment survivals. Derive joint Borel K/q, actual history/observer/guard measurability, nonnegative real product, pointwise full-history reconstruction, H_*(P restricted to B)=K_*(q m), all nonnegative extended Borel test identities and integral q=P(B)<=1. Covers n0/T0/rank0/stopped/top/zero-input boundaries without new density or integrability providers. No terminal survival: not an exactly-n count law or full A7. Terminal survival/path/count assembly, restart/Markovness, invariance/hypocoercivity/errors/unbounded costs/main/composition/four-paper Goal remain OPEN.

Scope guard. This card records a compiled local declaration. Its mathematical scope is exactly the Lean statement below; the Registry note and source correspondence may describe motivation but do not strengthen it.

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Lean statement

theorem actual_ordered_prefix_law
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E,
      (α : ℝ) * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ) * ‖v‖ ^ 2)
    (hη : 0 < η) (hβη : (β : ℝ) * η ≤ 1) :
    actual_ordered_prefix_statement hα hαβ hV hH hη hβη := by
  classical
  let c : E → E → E := fun y xRef => y - η • gradient V xRef
  let Φ : E → E → ℝ → (E × E) → (E × E) := fun y xRef t z =>
    (c y xRef + Real.cos t • (z.1 - c y xRef) +
        (Real.sqrt η * Real.sin t) • z.2,
     (-Real.sin t / Real.sqrt η) • (z.1 - c y xRef) + Real.cos t • z.2)
  let S : E → (E × E) → (E × E) := fun xRef z =>
    (z.1, z.2 - (2 * inner ℝ z.2 (gradient V z.1 - gradient V xRef) /
      ‖gradient V z.1 - gradient V xRef‖ ^ 2) • (gradient V z.1 - gradient V xRef))
  let rate : E → (E × E) → ℝ := fun xRef z =>
    Real.sqrt η * max 0 (inner ℝ z.2 (gradient V z.1 - gradient V xRef))
  let Λ : E → E → (E × E) → ℝ≥0 → ℝ := fun y xRef z t =>
    ∫ s in (0 : ℝ)..(t : ℝ), rate xRef (Φ y xRef s z)
  let τ : E → E → (E × E) → ℝ≥0 → WithTop ℝ≥0 := fun y xRef z e =>
    MeasureTheory.hittingAfter
      (fun t : ℝ≥0 => fun a : (E × E) × (E × E) × ℝ≥0 =>
        Λ a.1.1 a.1.2 a.2.1 t - (a.2.2 : ℝ))
      (Set.Ici (0 : ℝ)) 0 ((y, xRef), z, e)
  let next : E → E → ℝ≥0 → ((ℝ≥0 × (E × E)) ⊕ Unit) →
      ((ℝ≥0 × (E × E)) ⊕ Unit) := fun y xRef e r =>
    match r with
    | Sum.inr _ => Sum.inr ()
    | Sum.inl a =>
      if τ y xRef a.2 e = ⊤ then Sum.inr () else
        Sum.inl (a.1 + (τ y xRef a.2 e).untopD 0,
          S xRef (Φ y xRef (((τ y xRef a.2 e).untopD 0 : ℝ≥0) : ℝ) a.2))
  let record : E → E → (E × E) → (ℕ → ℝ≥0) → ℕ →
      ((ℝ≥0 × (E × E)) ⊕ Unit) := fun y xRef z₀ e n =>
    Nat.rec (Sum.inl (0, z₀)) (fun k r => next y xRef (e k) r) n
  let P : Measure (ℕ → ℝ) := Measure.infinitePi (fun _ : ℕ => expMeasure (1 : ℝ))
  let ε : (ℕ → ℝ) → ℕ → ℝ≥0 := fun w k => Real.toNNReal (w k)
  let history : E → E → (E × E) → (n : ℕ) → (ℕ → ℝ) →
      (Fin (n + 1) → ((ℝ≥0 × (E × E)) ⊕ Unit)) :=
    fun y xRef z₀ n w j => record y xRef z₀ (ε w) j.val
  let Rec := (ℝ≥0 × (E × E)) ⊕ Unit
  let Hist : ℕ → Type _ := fun n => Fin (n + 1) → Rec
  let readTime : Rec → ℝ := fun r => match r with
    | Sum.inl q => (q.1 : ℝ)
    | Sum.inr _ => 0
  let Δ : (n : ℕ) → ℝ≥0 → Set (Fin n → ℝ) := fun n T =>
    {t | StrictMono (Fin.cons (0 : ℝ) t) ∧ ∀ i, t i < (T : ℝ)}
  let Prefix : (n : ℕ) → ℝ≥0 → Type := fun n T => {t : Fin n → ℝ // t ∈ Δ n T}
  let m : (n : ℕ) → (T : ℝ≥0) → Measure (Prefix n T) := fun n T =>
    Measure.comap (fun t : Prefix n T => t.val) (volume : Measure (Fin n → ℝ))
  let zAt : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → Fin (n + 1) → (E × E) := fun y xRef z₀ n T t =>
    Fin.induction z₀ (fun i z =>
      S xRef (Φ y xRef (t.val i - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) i.castSucc) z))
  let K : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → Hist n := fun y xRef z₀ n T t j =>
    Sum.inl (Real.toNNReal ((Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) j), zAt y xRef z₀ n T t j)
  let qReal : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → ℝ := fun y xRef z₀ n T t =>
    ∏ i : Fin n,
      rate xRef (Φ y xRef (t.val i - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) i.castSucc)
        (zAt y xRef z₀ n T t i.castSucc)) *
      Real.exp (-Λ y xRef (zAt y xRef z₀ n T t i.castSucc)
        (Real.toNNReal (t.val i - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) i.castSucc)))
  let q : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) →
      Prefix n T → ℝ≥0∞ := fun y xRef z₀ n T t =>
    ENNReal.ofReal (qReal y xRef z₀ n T t)
  let A : (n : ℕ) → ℝ≥0 → Set (Hist n) := fun n T =>
    {h | (∀ j, ∃ a : ℝ≥0 × (E × E), h j = Sum.inl a) ∧
      StrictMono (fun j => readTime (h j)) ∧
      ∀ i : Fin n, readTime (h i.succ) < (T : ℝ)}
  change ∀ n : ℕ, ∀ T : ℝ≥0,
      MeasurableSet (Δ n T) ∧ MeasurableSet (A n T) ∧
      Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
        K p.1.1.1 p.1.1.2 p.1.2 n T p.2) ∧
      Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
        q p.1.1.1 p.1.1.2 p.1.2 n T p.2) ∧
      (∀ y xRef : E, ∀ z₀ : E × E,
        let H : (ℕ → ℝ) → Hist n := history y xRef z₀ n
        let B : Set (ℕ → ℝ) := H ⁻¹' A n T
        let obs : (ℕ → ℝ) → Fin n → ℝ := fun w i => readTime (H w i.succ)
        Measurable H ∧ MeasurableSet B ∧ Measurable obs ∧
        (∀ t : Prefix n T, 0 ≤ qReal y xRef z₀ n T t) ∧
        (∀ w ∈ B, ∃ t : Prefix n T, t.val = obs w ∧ H w = K y xRef z₀ n T t) ∧
        Measure.map H (P.restrict B) =
          Measure.map (K y xRef z₀ n T) ((m n T).withDensity (q y xRef z₀ n T)) ∧
        (∀ F : Hist n → ℝ≥0∞, Measurable F →
          (∫⁻ w in B, F (H w) ∂P) =
            ∫⁻ t : Prefix n T, F (K y xRef z₀ n T t) * q y xRef z₀ n T t ∂m n T) ∧
        (∫⁻ t : Prefix n T, q y xRef z₀ n T t ∂m n T) = P B ∧
        P B ≤ 1)
  intro n T
  have hg := ordered_prefix_borel_data hα hαβ hV hH hη hβη n T
  refine ⟨hg.1, hg.2.1, hg.2.2.1, hg.2.2.2.1, ?_⟩
  intro y xRef z₀
  dsimp only
  have h91 := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualAbsoluteNextEventTime.actual_absolute_next_event_time_law
    hα hαβ hV hH hη hβη y xRef z₀ n
  have hHist : Measurable (history y xRef z₀ n) := h91.1
  have hB : MeasurableSet (history y xRef z₀ n ⁻¹' A n T) := hg.2.1.preimage hHist
  have hread : readTime = @AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOrderedPrefix.readTime (E × E) := by
    funext r; cases r <;> rfl
  have hObs : Measurable (fun w : ℕ → ℝ => fun i : Fin n =>
      readTime (history y xRef z₀ n w i.succ)) := by
    rw [hread]
    apply measurable_pi_lambda
    intro i
    exact readTime_measurable.comp ((measurable_pi_apply i.succ).comp hHist)
  have hparams : Measurable (fun t : Prefix n T => (((y, xRef), z₀), t)) :=
    measurable_const.prodMk measurable_id
  have hKjoint : Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
      K p.1.1.1 p.1.1.2 p.1.2 n T p.2) := hg.2.2.1
  have hqjoint : Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
      q p.1.1.1 p.1.1.2 p.1.2 n T p.2) := hg.2.2.2.1
  have hK := hKjoint.comp hparams
  have hq := hqjoint.comp hparams
  dsimp only [Function.comp_def] at hK hq
  have htest := actual_prefix_all_integrals hα hαβ hV hH hη hβη n T y xRef z₀
  letI : IsProbabilityMeasure P :=
    AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws.1
  have hclosure := law_and_mass_of_all_tests P (m n T)
    (history y xRef z₀ n ⁻¹' A n T) (history y xRef z₀ n)
    (K y xRef z₀ n T) (q y xRef z₀ n T) hHist hK hq htest
  refine ⟨hHist, hB, hObs, hg.2.2.2.2 y xRef z₀, ?_, hclosure.1, htest,
    hclosure.2.1, hclosure.2.2⟩
  exact actual_prefix_identification hα hαβ hV hH hη hβη n T y xRef z₀

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOrderedPrefix

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualOrderedPrefix.lean:1554published source at ed709a7f320d

Proof architecture

Actual PBPS finite path-law substrate; no SALD claim

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`rw` rewrites by an established identity.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.
Density statements retain normalization and absolute-continuity prerequisites.