signedFaceTermSum_smul_eq_zero_of_scalar_support_subset_univ_pi_Ioo

Plain-English statement

- Scalar cutoff support contained in the open Pi-box implies Mathlib's finite-box signed face-term sum is zero for the cutoff-smul vector field. This is still a finite-box support-to-face producer, not a smooth-cutoff construction or whole-space no-boundary theorem.

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 signedFaceTermSum_smul_eq_zero_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))) :
    ∑ i : Fin (n + 1),
        ((∫ x in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove),
            (χ (i.insertNth (b i) x) • G (i.insertNth (b i) x)) i) -
          ∫ x in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove),
            (χ (i.insertNth (a i) x) • G (i.insertNth (a i) x)) i) = 0 :=
  signedFaceTermSum_eq_zero_of_support_subset_univ_pi_Ioo a b (fun x => χ x • G x)
    (support_smul_subset_univ_pi_Ioo_of_scalar_support_subset_univ_pi_Ioo a b χ G hχsupp)

/-- Scalar cutoff topological support contained in the open Pi-box implies
Mathlib's finite-box signed face-term sum is zero for the cutoff-smul vector
field.

This is a direct `tsupport`-API handoff for the local smooth-cutoff route.  It
does not construct the cutoff, prove regularity of the cutoff-smul field, pass
to a whole-space limit, or prove weighted integration by parts. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1231published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 cutoff route: scalar cutoff support inside the open Pi-box implies the cutoff-smul finite-box signed face-term sum is zero

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

Integrability is an input or proved output; a displayed integral alone does not supply it.
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.