eq_zero_off_univ_pi_Ioo_of_support_subset_univ_pi_Ioo

Plain-English statement

- If the support of a Pi-space vector field is contained in the open box, then the field vanishes outside that open box. This is a support-to-boundary staging leaf. It uses plain `Function.support`; it does not assert compactness of the support and does not construct a cutoff.

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 eq_zero_off_univ_pi_Ioo_of_support_subset_univ_pi_Ioo
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ)
    (F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
    (hsupp : Function.support F ⊆ (Set.univ.pi fun i => Set.Ioo (a i) (b i))) :
    ∀ x ∉ (Set.univ.pi fun i => Set.Ioo (a i) (b i)), F x = 0 := by
  intro x hx
  by_contra hne
  exact hx (hsupp hne)

/-- 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. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:794published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 boundary route: a support subset of the open Pi-box implies the vector field is zero outside that open Pi-box

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
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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

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.