exists_contDiff_cutoff_eq_one_on_Icc_tsupport_subset_outer_univ_pi_Ioo

Plain-English statement

- Smooth plateau for a finite closed Pi-box inside a strictly larger open Pi-box. The extra hypothesis `a ≤ b` records that the inner box is nonempty in the intended exhaustion use. The construction comes from the generic compact-in-open plateau theorem in `Analysis.Calculus.Cutoff`; it gives both plain- and topological-support containment, compact support, smoothness, `[0, 1]` range, and equality to one on the whole inner box. This is one chosen cutoff, not yet an exhausting family with derivative bounds. Tail passage, whole-space weighted integration by parts, generator domains, invariant Gibbs law, reversibility, and KL/FI dissipation remain separate obligations.

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 exists_contDiff_cutoff_eq_one_on_Icc_tsupport_subset_outer_univ_pi_Ioo
    {n : ℕ} {a b A B : Fin (n + 1) → ℝ}
    (hab : a ≤ b)
    (hA : ∀ i, A i < a i)
    (hB : ∀ i, b i < B i) :
    ∃ χ : (Fin (n + 1) → ℝ) → ℝ,
      Function.support χ ⊆ Set.univ.pi (fun i => Set.Ioo (A i) (B i)) ∧
      tsupport χ ⊆ Set.univ.pi (fun i => Set.Ioo (A i) (B i)) ∧
      HasCompactSupport χ ∧
      ContDiff ℝ (⊤ : ℕ∞) χ ∧
      Set.range χ ⊆ Set.Icc 0 1 ∧
      Set.EqOn χ 1 (Set.Icc a b) := by
  have _hinner : (Set.Icc a b).Nonempty := ⟨a, le_rfl, hab⟩
  have hopen : IsOpen (Set.univ.pi fun i => Set.Ioo (A i) (B i)) := by
    exact isOpen_set_pi Set.finite_univ fun _ _ => isOpen_Ioo
  exact Cutoff.exists_contDiff_eq_one_tsupport_subset
    isCompact_Icc hopen (Icc_subset_univ_pi_Ioo_of_strict_bounds hA hB)

/-- Local smooth cutoff for a point in an inner closed Pi-box, supported in a
strictly larger open Pi-box.

This packages the closed-box-to-open-box inclusion with
`exists_contDiff_cutoff_support_subset_univ_pi_Ioo`.  It is a local cutoff at a
single point of the inner box; it does not construct one cutoff equal to `1` on
the whole inner box, choose an exhausting family, prove derivative bounds, or
pass to whole-space limits. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:930published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 finite-box route: one smooth compactly supported cutoff equals one on the entire inner closed Pi-box and is topologically supported in the outer open Pi-box

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Why the statement has this shape

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Hidden assumptions and non-claims

Pointwise support, topological support, and compact support retain distinct meanings.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.