actual_endpoint_time_reversal_laws

Plain-English statement

- Exact source-sealed actual endpoint/time reversal law.

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_endpoint_time_reversal_laws
    {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_endpoint_time_reversal_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 Rec := (ℝ≥0 × (E × E)) ⊕ Unit
  let Hist : ℕ → Type _ := fun n => Fin (n + 1) → Rec
  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 Path : ℝ≥0 → Type _ := fun T => {s : ℝ≥0 // s ≤ T} → E × E
  let Ψ : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Hist n → Path T :=
    fun y xRef z₀ n T h s =>
      Fin.foldl (n + 1) (fun z j => (h j).elim
        (fun a => if a.1 ≤ s.val then Φ y xRef ((s.val - a.1 : ℝ≥0) : ℝ) a.2 else z)
        (fun _ => z)) z₀
  let Γ : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Prefix n T → Path T :=
    fun y xRef z₀ n T t => Ψ y xRef z₀ n T (K y xRef z₀ n T t)
  let R : (E × E) → (E × E) := fun z => (z.1, -z.2)
  let ell : Measure (E × E) := (volume : Measure E).prod (volume : Measure E)
  change
    ∀ n : ℕ, ∀ T : ℝ≥0, ∃ rho : Prefix n T → Prefix n T,
      (∀ t : Prefix n T, ∀ i : Fin n, (rho t).val i = (T : ℝ) - t.val i.rev) ∧
      Measurable rho ∧ Function.Involutive rho ∧ MeasurePreserving rho (m n T) (m n T) ∧
      let A : E → E → (E × E) → Prefix n T → (E × E) :=
        fun y r z t => Γ y r z n T t ⟨T, le_rfl⟩
      let Theta : E → E → ((E × E) × Prefix n T) → ((E × E) × Prefix n T) :=
        fun y r q => (R (A y r q.1 q.2), rho q.2)
      (∀ y r : E, ∀ z : E × E, ∀ t : Prefix n T,
        A y r z t = Φ y r ((T : ℝ) -
          (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) (Fin.last n))
          (zAt y r z n T t (Fin.last n))) ∧
      Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
        A p.1.1.1 p.1.1.2 p.1.2 p.2) ∧
      (∀ y r : E, ∀ t : Prefix n T,
        MeasurePreserving (fun z => A y r z t) ell ell) ∧
      (∀ y r : E, ∀ z : E × E, ∀ t : Prefix n T,
        A y r (R (A y r z t)) (rho t) = R z) ∧
      Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
        Theta p.1.1.1 p.1.1.2 (p.1.2, p.2)) ∧
      (∀ y r : E, Function.Involutive (Theta y r)) ∧
      (∀ y r : E, MeasurePreserving (Theta y r) (ell.prod (m n T)) (ell.prod (m n T)))

  have hed := endpoint_data hα hαβ hV hH hη hβη
  obtain ⟨hΦM, hSM, hRateM, hΦP, hSP, hRP, hΦI, hSI, hRI, hRΦ, hRS, hRate, hInt⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseReversalVolume.actual_phase_reversal_volume_laws
      hα hαβ hV hH hη hβη
  intro n T
  let rho₀ : Prefix n T → Prefix n T := rho n (T : ℝ)
  have hrM : Measurable rho₀ := rho_measurable n (T : ℝ)
  have hrI : Function.Involutive rho₀ := rho_involutive n (T : ℝ)
  have hrP : MeasurePreserving rho₀ (m n T) (m n T) := rho_volume n (T : ℝ) (hed n T).1
  refine ⟨rho₀, (fun t i => rfl), hrM, hrI, hrP, ?_⟩
  let A : E → E → (E × E) → Prefix n T → (E × E) :=
    fun y r z t => Γ y r z n T t ⟨T, le_rfl⟩
  let Theta : E → E → ((E × E) × Prefix n T) → ((E × E) × Prefix n T) :=
    fun y r q => (R (A y r q.1 q.2), rho₀ q.2)
  change
      (∀ y r : E, ∀ z : E × E, ∀ t : Prefix n T,
        A y r z t = Φ y r ((T : ℝ) -
          (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n + 1) → ℝ) (Fin.last n))
          (zAt y r z n T t (Fin.last n))) ∧
      Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
        A p.1.1.1 p.1.1.2 p.1.2 p.2) ∧
      (∀ y r : E, ∀ t : Prefix n T,
        MeasurePreserving (fun z => A y r z t) ell ell) ∧
      (∀ y r : E, ∀ z : E × E, ∀ t : Prefix n T,
        A y r (R (A y r z t)) (rho₀ t) = R z) ∧
      Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
        Theta p.1.1.1 p.1.1.2 (p.1.2, p.2)) ∧
      (∀ y r : E, Function.Involutive (Theta y r)) ∧
      (∀ y r : E, MeasurePreserving (Theta y r) (ell.prod (m n T)) (ell.prod (m n T)))

  have hAm : Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
      A p.1.1.1 p.1.1.2 p.1.2 p.2) := (hed n T).2.2.1
  have hAid (y r : E) (z : E × E) (t : Prefix n T) :
      A y r z t = endpoint (Φ y r) (S r) n (T : ℝ) t.val z := (hed n T).2.2.2 y r z t
  have hAP (y r : E) (t : Prefix n T) : MeasurePreserving (fun z => A y r z t) ell ell := by
    have he : (fun z => A y r z t) = endpoint (Φ y r) (S r) n (T : ℝ) t.val := funext (fun z => hAid y r z t)
    rw [he]
    exact endpoint_volume ell (Φ y r) (S r) (hSI r) (hSP r) (hΦP y r) n (T : ℝ) t.val
  have hAI (y r : E) (z : E × E) (t : Prefix n T) :
      A y r (R (A y r z t)) (rho₀ t) = R z := by
    rw [hAid, hAid]
    exact endpoint_reversed (Φ y r) (S r) R (hSI r) hRI (hRS r)
      (fun u z => (hΦI y r u z).2) (hRΦ y r) n (T : ℝ) t.val z
  have hThetaM : Measurable (fun p : ((E × E) × (E × E)) × Prefix n T =>
      Theta p.1.1.1 p.1.1.2 (p.1.2, p.2)) :=
    (hRP.measurable.comp hAm).prodMk (hrM.comp measurable_snd)
  refine ⟨(hed n T).2.2.2, hAm, hAP, hAI, hThetaM, ?_, ?_⟩
  · intro y r q
    apply Prod.ext
    · change R (A y r (R (A y r q.1 q.2)) (rho₀ q.2)) = q.1
      rw [hAI]
      exact hRI q.1
    · exact hrI q.2
  · intro y r
    have hs := (hed n T).1
    have hmσ : SigmaFinite ((m n T).map (fun t : Prefix n T => t.val)) := by
      change SigmaFinite ((Measure.comap (fun t : Prefix n T => t.val)
        (volume : Measure (Fin n → ℝ))).map (fun t : Prefix n T => t.val))
      rw [map_comap_subtype_coe hs]
      infer_instance
    letI : SigmaFinite (m n T) := SigmaFinite.of_map (m n T) measurable_subtype_coe.aemeasurable hmσ
    have hm : Measurable (fun q : (E × E) × Prefix n T => R (A y r q.1 q.2)) :=
      hRP.measurable.comp (hAm.comp
        (((measurable_const : Measurable (fun _ : (E × E) × Prefix n T => (y, r))).prodMk
          measurable_fst).prodMk measurable_snd))
    have hp := hrP.skew_product (g := fun t z => R (A y r z t)) (hm.comp measurable_swap)
      (Filter.Eventually.of_forall fun t => (hRP.comp (hAP y r t)).map_eq)
    have hout := (Measure.measurePreserving_swap (μ := m n T) (ν := ell)).comp
      (hp.comp (Measure.measurePreserving_swap (μ := ell) (ν := m n T)))
    exact hout

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualEndpointTimeReversal

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualEndpointTimeReversal.lean:399published source at ed709a7f320d

Proof architecture

Actual finite PBPS endpoint/time change of variables; 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.
Totalized `fderiv` values must not be read as a differentiability theorem.
Almost-everywhere hypotheses depend on the stated measure and representative.
A Gibbs expression is not automatically a probability law or an invariant law.