exists_contDiff_cutoff_tsupport_subset_univ_pi_Ioo

Plain-English statement

- Smooth finite-dimensional cutoff localized inside a Pi-open box. For any point of `Set.univ.pi (fun i => Set.Ioo (a i) (b i))`, Mathlib's finite-dimensional bump theorem supplies a smooth real-valued cutoff whose topological support is contained in the open box, has compact support, takes values in `[0, 1]`, and is equal to `1` at the chosen point. This is only the local smooth-cutoff existence leaf. It does not choose an exhausting family of boxes, prove derivative formulas for a specific cutoff, perform a tail limit, or prove weighted integration by parts/invariance.

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_tsupport_subset_univ_pi_Ioo
    {n : ℕ} {a b x : Fin (n + 1) → ℝ}
    (hx : x ∈ Set.univ.pi fun i => Set.Ioo (a i) (b i)) :
    ∃ χ : (Fin (n + 1) → ℝ) → ℝ,
      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 ∧
      χ x = 1 := by
  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 exists_contDiff_tsupport_subset (n := (⊤ : ℕ∞)) (hopen.mem_nhds hx)

/-- Topological-support containment implies plain function-support containment
inside a finite Pi-open box.

This is the bridge needed by the finite-box cutoff route: Mathlib's smooth
cutoff theorem naturally returns `tsupport`, while the already-compiled
zero-face handoffs are phrased using `Function.support`.  The lemma is only a
support-API conversion; it does not construct a cutoff or prove any derivative,
tail, or whole-space integration-by-parts statement. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:814published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff route: construct a local smooth real-valued cutoff whose topological support is contained in a finite Pi-open box and which equals one at a chosen interior point

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`have` creates a named intermediate mathematical fact.
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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.