support_smul_subset_univ_pi_Ioo_of_scalar_tsupport_subset_univ_pi_Ioo

Plain-English statement

- If the topological support of a scalar cutoff is contained in the open Pi-box, then multiplying any vector field by that cutoff is plain-supported in the same open box. This is the direct consumer-facing bridge from Mathlib's `tsupport` cutoff output to the cutoff-smul support hypothesis used by the finite-box zero-face route. It does not prove cutoff construction, regularity of the smul field, tail decay, or whole-space integration by parts.

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 support_smul_subset_univ_pi_Ioo_of_scalar_tsupport_subset_univ_pi_Ioo
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ)
    (χ : (Fin (n + 1) → ℝ) → ℝ)
    (G : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
    (hχtsupp : tsupport χ ⊆
      Set.univ.pi (fun i => Set.Ioo (a i) (b i))) :
    Function.support (fun x => χ x • G x) ⊆
      (Set.univ.pi fun i => Set.Ioo (a i) (b i)) :=
  support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo a b χ G
    (support_subset_univ_pi_Ioo_of_tsupport_subset_univ_pi_Ioo hχtsupp)

/-- Closed-box continuity for a scalar cutoff times a Pi-space vector field.

This packages Mathlib's `ContinuousOn.smul` in the exact finite-box shape used
by the cutoff-smul divergence-theorem route.  It does not prove smooth cutoff
construction, differentiability, trace integrability, or any boundary result. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1041published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff route: a scalar cutoff with topological support inside the open Pi-box forces the cutoff-smul vector field to be plain-supported in that open box

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Hidden assumptions and non-claims

Integrability is an input or proved output; a displayed integral alone does not supply it.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
Pointwise support, topological support, and compact support retain distinct meanings.
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.