support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo

Plain-English statement

- If a scalar cutoff is supported in the open Pi-box, then multiplying any Pi-space vector field by this cutoff gives a vector field supported in the open Pi-box. This only uses `Function.support`; it is not a `HasCompactSupport` theorem and does not build 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 support_smul_subset_univ_pi_Ioo_of_scalar_support_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)
    (hχsupp : Function.support 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)) := by
  exact support_smul_subset_univ_pi_Ioo_of_eq_zero_off_univ_pi_Ioo a b χ G
    (by
      intro x hxbox
      by_contra hχne
      exact hxbox (hχsupp hχne))

/-- 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. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1018published source at 7bcd37294df1

Proof architecture

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

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Hidden assumptions and non-claims

Pointwise support, topological support, and compact support retain distinct meanings.
A cutoff lemma does not by itself prove a whole-space integration-by-parts identity.