actual_fixed_reference_physical_time_cover

Plain-English statement

Independently verified science 56e4b7e101a1016e03df2971018b1f018f03e6c1. With the original six analytic binders and canonical actual input, every finite physical time almost surely lies in a unique actual half-open event interval and selects a unique actual live record with stored time<=t and finite elapsed offset<actual next wait. No positive-wait/index/live-record/clock/nonaccumulation provider premise. Last live arc remains available when next wait is infinity. No global interpolation/measurability/origin/Markov/invariance/kernel/hypocoercivity/main/error/expected-cost/composition or whole-paper/Goal completion. Aggregate/visual/main/PURIFIED/live remain separate.

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_fixed_reference_physical_time_cover
    {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_fixed_reference_physical_time_cover_statement hα hαβ hV hH hη hβη := by
  classical
  let P : Measure (ℕ → ℝ) := Measure.infinitePi (fun _ : ℕ => expMeasure (1 : ℝ))
  let ε : (ℕ → ℝ) → ℕ → ℝ≥0 := fun sample k => Real.toNNReal (sample k)
  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 eventTime : E → E → (E × E) → (ℕ → ℝ≥0) → ℕ → WithTop ℝ≥0 :=
    fun y xRef z₀ e n => (record y xRef z₀ e n).elim
      (fun a => (a.1 : WithTop ℝ≥0)) (fun _ => ⊤)
  change ∀ y xRef : E, ∀ z₀ : E × E, ∀ᵐ sample ∂P,
    (∀ t : ℝ≥0, ∃! n : ℕ,
      eventTime y xRef z₀ (ε sample) n ≤ (t : WithTop ℝ≥0) ∧
      (t : WithTop ℝ≥0) < eventTime y xRef z₀ (ε sample) (n + 1)) ∧
    (∀ t : ℝ≥0, ∀ n : ℕ,
      eventTime y xRef z₀ (ε sample) n ≤ (t : WithTop ℝ≥0) →
      (t : WithTop ℝ≥0) < eventTime y xRef z₀ (ε sample) (n + 1) →
      ∃! a : ℝ≥0 × (E × E),
        record y xRef z₀ (ε sample) n = Sum.inl a ∧ a.1 ≤ t ∧
        ((t - a.1 : ℝ≥0) : WithTop ℝ≥0) < τ y xRef a.2 (ε sample n))
  -- Exact deterministic initialization/monotonicity and guarded live-record updates.
  rcases ActualFiniteJumpRecursion.actual_fixed_reference_finite_jump_recursion
      hα hαβ hV hH hη hβη with
    ⟨_, hstep, _, _, _, htime, _, _, _, _⟩
  have hinput := ActualNonaccumulation.actual_fixed_reference_event_time_nonaccumulation
    hα hαβ hV hH hη hβη
  intro y xRef z₀
  filter_upwards [hinput y xRef z₀] with sample hgood
  have hzero : eventTime y xRef z₀ (ε sample) 0 = 0 :=
    (htime y xRef z₀ (ε sample)).1
  have hmono : Monotone (eventTime y xRef z₀ (ε sample)) :=
    (htime y xRef z₀ (ε sample)).2
  constructor
  · -- The first clock strictly above t has an initialized finite predecessor.
    intro t
    obtain ⟨N, hN⟩ := eventually_atTop.mp (hgood.1 t)
    have hex : ∃ k : ℕ, (t : WithTop ℝ≥0) < eventTime y xRef z₀ (ε sample) k :=
      ⟨N, hN N le_rfl⟩
    let k : ℕ := Nat.find hex
    have hk : (t : WithTop ℝ≥0) < eventTime y xRef z₀ (ε sample) k :=
      Nat.find_spec hex
    have hkpos : 0 < k := by
      by_contra h
      have hk0 : k = 0 := Nat.eq_zero_of_not_pos h
      rw [hk0, hzero] at hk
      exact (not_lt_of_ge (show (0 : WithTop ℝ≥0) ≤ t from bot_le)) hk
    let n : ℕ := k - 1
    have hsucc : n + 1 = k := by dsimp [n]; omega
    have hlow : eventTime y xRef z₀ (ε sample) n ≤ (t : WithTop ℝ≥0) := by
      apply le_of_not_gt
      exact Nat.find_min hex (by change k - 1 < k; omega)
    have hupp : (t : WithTop ℝ≥0) < eventTime y xRef z₀ (ε sample) (n + 1) := by
      simpa only [hsucc] using hk
    refine ⟨n, ⟨hlow, hupp⟩, ?_⟩
    -- Monotonicity makes distinct half-open event intervals disjoint.
    intro m hm
    apply le_antisymm
    · by_contra h
      have hnm : n + 1 ≤ m := by omega
      exact (not_lt_of_ge ((hmono hnm).trans hm.1)) hupp
    · by_contra h
      have hmn : m + 1 ≤ n := by omega
      exact (not_lt_of_ge ((hmono hmn).trans hlow)) hm.2
  · -- A clock below finite t cannot be a stopped record.
    intro t n hlow hupp
    cases hr : record y xRef z₀ (ε sample) n with
    | inr u =>
      have hevent : eventTime y xRef z₀ (ε sample) n = ⊤ := by
        change (record y xRef z₀ (ε sample) n).elim _ _ = ⊤
        rw [hr]
        rfl
      rw [hevent] at hlow
      exact False.elim ((not_le_of_gt (WithTop.coe_lt_top t)) hlow)
    | inl a =>
      have haClock : eventTime y xRef z₀ (ε sample) n = (a.1 : WithTop ℝ≥0) := by
        change (record y xRef z₀ (ε sample) n).elim _ _ = _
        rw [hr]
        rfl
      have hat : a.1 ≤ t := WithTop.coe_le_coe.mp (by
        rw [← haClock]
        exact hlow)
      refine ⟨a, ⟨rfl, hat, ?_⟩, ?_⟩
      · -- Infinite wait keeps the last live arc available at every finite offset.
        by_cases hwtop : τ y xRef a.2 (ε sample n) = ⊤
        · simp only [hwtop, WithTop.coe_lt_top]
        -- Finite wait: the actual next clock is a.time plus that same wait.
        · let q : ℝ≥0 := (τ y xRef a.2 (ε sample n)).untop hwtop
          have hq : (q : WithTop ℝ≥0) = τ y xRef a.2 (ε sample n) :=
            WithTop.coe_untop _ _
          have hn : record y xRef z₀ (ε sample) (n + 1) =
              Sum.inl (a.1 + (τ y xRef a.2 (ε sample n)).untopD 0,
                S xRef (Φ y xRef (((τ y xRef a.2 (ε sample n)).untopD 0 : ℝ≥0) : ℝ) a.2)) :=
            ((hstep y xRef z₀ (ε sample) n).2 a hr).2 hwtop
          rw [← hq] at hn ⊢
          have hnTime : eventTime y xRef z₀ (ε sample) (n + 1) =
              ((a.1 + q : ℝ≥0) : WithTop ℝ≥0) := by
            change (record y xRef z₀ (ε sample) (n + 1)).elim _ _ = _
            rw [hn]
            rfl
          have hsmall : t < a.1 + q := WithTop.coe_lt_coe.mp (by
            simpa only [hnTime] using hupp)
          exact WithTop.coe_lt_coe.mpr ((tsub_lt_iff_right hat).mpr (by
            simpa only [add_comm] using hsmall))
      · -- The stored record is a function; injectivity gives uniqueness.
        intro b hb
        exact Sum.inl.inj hb.1.symm

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhysicalTimeCover

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPhysicalTimeCover.lean:71published source at ed709a7f320d

Proof architecture

PBPS measurable physical-time interpolation prerequisite; 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.
`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.