actual_prescribed_path_density_rewrite

Plain-English statement

Independent science ad3b80a339bf1cf0dd88685f202632a92eeb0797. Under exactly the six original analytic assumptions and the literal actual93 prescribed interpolation, every finite strict event tuple has integrable path rate, an exact n+1-arc hazard integral sum, incoming phase and rate left limits, and rho=ofReal(exp(-whole-path hazard integral) times product of incoming jump rates). Terminal survival, n=0, T=0, rank0 and zero rates are retained. No additional path/integrability/limit provider. This deterministic ingredient does not yet compose the actual finite-count measure with the printed density or sum full A7. Restart/Markovness, invariance, hypocoercivity, implementation errors, expected cost, paper mains, actual-input composition and four-paper completion 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_prescribed_path_density_rewrite
    {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) :
    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 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 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 liveTime : Rec → ℝ≥0 := fun r => r.elim Prod.fst (fun _ => 0)
    let d : (n : ℕ) → ℝ≥0 → Hist n → ℝ≥0 :=
      fun n T h => T - liveTime (h ⟨n, Nat.lt_succ_self n⟩)
    let weight : E → E → Rec → ℝ≥0 → ℝ≥0∞ :=
      fun y xRef r u => r.elim
        (fun a => ENNReal.ofReal (Real.exp (-Λ y xRef a.2 u))) (fun _ => 0)
    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 ρ : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Prefix n T → ℝ≥0∞ :=
      fun y xRef z₀ n T t => q y xRef z₀ n T t *
        weight y xRef ((K y xRef z₀ n T t) ⟨n, Nat.lt_succ_self n⟩)
          (d n T (K y xRef z₀ n T t))
    ∀ y xRef : E, ∀ z₀ : E × E, ∀ n : ℕ, ∀ T : ℝ≥0, ∀ t : Prefix n T,
      let grid : Fin (n + 2) → ℝ :=
        Fin.snoc (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ)) (T : ℝ)
      let γ : ℝ → E × E := fun s =>
        Γ y xRef z₀ n T t ⟨min (Real.toNNReal s) T, min_le_right _ _⟩
      IntervalIntegrable (fun s => rate xRef (γ s)) volume 0 (T : ℝ) ∧
      (∫ s in (0 : ℝ)..(T : ℝ), rate xRef (γ s)) =
        ∑ j : Fin (n + 1), Λ y xRef (zAt y xRef z₀ n T t j)
          (Real.toNNReal (grid j.succ - grid j.castSucc)) ∧
      (∀ i : Fin n,
        Tendsto γ (𝓝[<] (t.val i))
          (𝓝 (Φ y xRef (t.val i - grid i.castSucc.castSucc)
            (zAt y xRef z₀ n T t i.castSucc))) ∧
        Tendsto (fun s => rate xRef (γ s)) (𝓝[<] (t.val i))
          (𝓝 (rate xRef (Φ y xRef (t.val i - grid i.castSucc.castSucc)
            (zAt y xRef z₀ n T t i.castSucc))))) ∧
      ρ y xRef z₀ n T t = ENNReal.ofReal
        (Real.exp (-(∫ s in (0 : ℝ)..(T : ℝ), rate xRef (γ s))) *
          ∏ i : Fin n, rate xRef (Φ y xRef (t.val i - grid i.castSucc.castSucc)
            (zAt y xRef z₀ n T t i.castSucc))) := 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 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 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 liveTime : Rec → ℝ≥0 := fun r => r.elim Prod.fst (fun _ => 0)
  let d : (n : ℕ) → ℝ≥0 → Hist n → ℝ≥0 :=
    fun n T h => T - liveTime (h ⟨n, Nat.lt_succ_self n⟩)
  let weight : E → E → Rec → ℝ≥0 → ℝ≥0∞ :=
    fun y xRef r u => r.elim
      (fun a => ENNReal.ofReal (Real.exp (-Λ y xRef a.2 u))) (fun _ => 0)
  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 ρ : E → E → (E × E) → (n : ℕ) → (T : ℝ≥0) → Prefix n T → ℝ≥0∞ :=
    fun y xRef z₀ n T t => q y xRef z₀ n T t *
      weight y xRef ((K y xRef z₀ n T t) ⟨n, Nat.lt_succ_self n⟩)
        (d n T (K y xRef z₀ n T t))
  change
    ∀ y xRef : E, ∀ z₀ : E × E, ∀ n : ℕ, ∀ T : ℝ≥0, ∀ t : Prefix n T,
      let grid : Fin (n + 2) → ℝ :=
        Fin.snoc (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ)) (T : ℝ)
      let γ : ℝ → E × E := fun s =>
        Γ y xRef z₀ n T t ⟨min (Real.toNNReal s) T, min_le_right _ _⟩
      IntervalIntegrable (fun s => rate xRef (γ s)) volume 0 (T : ℝ) ∧
      (∫ s in (0 : ℝ)..(T : ℝ), rate xRef (γ s)) =
        ∑ j : Fin (n + 1), Λ y xRef (zAt y xRef z₀ n T t j)
          (Real.toNNReal (grid j.succ - grid j.castSucc)) ∧
      (∀ i : Fin n,
        Tendsto γ (𝓝[<] (t.val i))
          (𝓝 (Φ y xRef (t.val i - grid i.castSucc.castSucc)
            (zAt y xRef z₀ n T t i.castSucc))) ∧
        Tendsto (fun s => rate xRef (γ s)) (𝓝[<] (t.val i))
          (𝓝 (rate xRef (Φ y xRef (t.val i - grid i.castSucc.castSucc)
            (zAt y xRef z₀ n T t i.castSucc))))) ∧
      ρ y xRef z₀ n T t = ENNReal.ofReal
        (Real.exp (-(∫ s in (0 : ℝ)..(T : ℝ), rate xRef (γ s))) *
          ∏ i : Fin n, rate xRef (Φ y xRef (t.val i - grid i.castSucc.castSucc)
            (zAt y xRef z₀ n T t i.castSucc)))
  intro y xRef z₀ n T t
  let grid : Fin (n+2) → ℝ := Fin.snoc (Fin.cons (0 : ℝ) t.val) (T : ℝ)
  let γ : ℝ → E × E := fun s =>
    Γ y xRef z₀ n T t ⟨min (Real.toNNReal s) T, min_le_right _ _⟩
  obtain ⟨hint, hsum, hleft⟩ :=
    actual_analysis hα hαβ hV hH hη hβη y xRef z₀ n T t
  refine ⟨hint, hsum, hleft, ?_⟩
  change ρ y xRef z₀ n T t = ENNReal.ofReal
    (Real.exp (-(∫ s in (0 : ℝ)..(T : ℝ), rate xRef (γ s))) *
      ∏ i : Fin n, rate xRef (Φ y xRef (t.val i-grid i.castSucc.castSucc)
        (zAt y xRef z₀ n T t i.castSucc)))
  let r : Fin n → ℝ := fun i =>
    rate xRef (Φ y xRef (t.val i-grid i.castSucc.castSucc) (zAt y xRef z₀ n T t i.castSucc))
  let h : Fin (n+1) → ℝ := fun j =>
    Λ y xRef (zAt y xRef z₀ n T t j) (Real.toNNReal (grid j.succ-grid j.castSucc))
  have hsum' : (∫ s in (0 : ℝ)..(T : ℝ), rate xRef (γ s)) = ∑ j, h j := hsum
  have hstep (i : Fin n) : grid i.castSucc.succ = t.val i := by
    simp only [grid, Fin.succ_castSucc, Fin.snoc_castSucc, Fin.cons_succ]
  have hq : qReal y xRef z₀ n T t = ∏ i : Fin n, r i * Real.exp (-h i.castSucc) := by
    apply Finset.prod_congr rfl
    intro i _
    simp only [r, h, hstep, grid, Fin.snoc_castSucc]
  have hlast0 : 0 ≤ (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n+1) → ℝ) (Fin.last n) := by
    have hh := t.property.1.monotone (Fin.zero_le (Fin.last n))
    simpa only [Fin.cons_zero] using hh
  have hlastT : (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n+1) → ℝ) (Fin.last n) ≤ (T : ℝ) := by
    cases n with
    | zero => simpa using T.coe_nonneg
    | succ n => simpa using (t.property.2 (Fin.last n)).le
  have htail : T - Real.toNNReal ((Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n+1) → ℝ) (Fin.last n)) =
      Real.toNNReal ((T : ℝ) - (Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n+1) → ℝ) (Fin.last n)) := by
    apply NNReal.eq
    rw [NNReal.coe_sub ((Real.toNNReal_le_iff_le_coe).2 hlastT),
      Real.coe_toNNReal _ hlast0, Real.coe_toNNReal _ (sub_nonneg.mpr hlastT)]
  have hweight : weight y xRef ((K y xRef z₀ n T t) ⟨n, Nat.lt_succ_self n⟩)
      (d n T (K y xRef z₀ n T t)) = ENNReal.ofReal (Real.exp (-h (Fin.last n))) := by
    change ENNReal.ofReal (Real.exp (-Λ y xRef (zAt y xRef z₀ n T t (Fin.last n))
      (T - Real.toNNReal ((Fin.cons (0 : ℝ) (t.val : Fin n → ℝ) : Fin (n+1) → ℝ) (Fin.last n))))) = _
    rw [htail]
    simp only [h, grid, Fin.succ_last, Fin.snoc_last, Fin.snoc_castSucc]
  change ENNReal.ofReal (qReal y xRef z₀ n T t) * _ = _
  rw [hweight, ← ENNReal.ofReal_mul' (Real.exp_pos _).le, hq,
    product_survival, hsum']


#print axioms actual_prescribed_path_density_rewrite
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPrescribedPathDensity

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPrescribedPathDensity.lean:470published source at ed709a7f320d

Proof architecture

Finite PBPS prescribed-path density consumer; 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.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`simpa` closes the goal after a controlled simplification of a typed result.

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.