signedFaceTermSum_eq_zero_of_support_subset_univ_pi_Ioo

Plain-English statement

- 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.

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_eq_zero_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))) :
    ∑ i : Fin (n + 1),
        ((∫ x in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove),
            F (i.insertNth (b i) x) i) -
          ∫ x in Set.Icc (a ∘ i.succAbove) (b ∘ i.succAbove),
            F (i.insertNth (a i) x) i) = 0 :=
  signedFaceTermSum_eq_zero_of_eq_zero_off_univ_pi_Ioo a b F
    (eq_zero_off_univ_pi_Ioo_of_support_subset_univ_pi_Ioo a b F hsupp)

/-- If a scalar cutoff vanishes outside 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 is a plain support-containment leaf for finite-box boundary staging.  It
does not construct a smooth cutoff, prove topological compact support, or
discharge any differentiability/integrability hypotheses for the cutoff-smul
field. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:977published source at 7bcd37294df1

Proof architecture

log-concave sampling Ch.1 boundary route: a support subset of the open Pi-box implies Mathlib's finite-box signed face-term sum is zero

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Hidden assumptions and non-claims

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