exists_contDiff_cutoff_support_subset_outer_univ_pi_Ioo_of_mem_Icc

Plain-English statement

- 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.

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_support_subset_outer_univ_pi_Ioo_of_mem_Icc
    {n : ℕ} {a b A B x : Fin (n + 1) → ℝ}
    (hA : ∀ i, A i < a i)
    (hB : ∀ i, b i < B i)
    (hx : x ∈ Set.Icc a b) :
    ∃ 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 ∧
      χ x = 1 :=
  exists_contDiff_cutoff_support_subset_univ_pi_Ioo
    (Icc_subset_univ_pi_Ioo_of_strict_bounds hA hB hx)

/-- Support contained in the open box implies Mathlib's finite-box signed
face-term sum is zero.

This is still only a finite-box support-to-face producer.  It does not prove
that a concrete Langevin/cutoff vector field has this support, and it does not
prove whole-space integration by parts or stationarity. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:956published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff/exhaustion route: every point of an inner closed Pi-box has a local smooth cutoff whose support and topological support lie in a strictly larger open Pi-box

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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.