Plain-English statement
- Finite-box coordinate-divergence integral vanishes for a cutoff-smul vector field when the scalar cutoff vanishes outside the open Pi-box, deriving the regularity hypotheses from separate cutoff/vector-field assumptions and deriving the product-rule trace integrability from a closed-box trace-continuity hypothesis. This closes only the compact-box trace-integrability side condition for the cutoff-smul route. It still does not construct a smooth cutoff, prove tail limits, whole-space weighted IBP, generator domains, invariant laws, reversibility, or KL/FI.
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Proof architecture
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Lean statement
theorem integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_trace_continuous
{n : ℕ}
(a b : Fin (n + 1) → ℝ) (hle : a ≤ b)
(χ : (Fin (n + 1) → ℝ) → ℝ)
(χ' : (Fin (n + 1) → ℝ) → (Fin (n + 1) → ℝ) →L[ℝ] ℝ)
(G : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
(G' : (Fin (n + 1) → ℝ) →
(Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
(s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable)
(hχc : ContinuousOn χ (Set.Icc a b))
(hGc : ContinuousOn G (Set.Icc a b))
(hχd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
HasFDerivAt χ (χ' x) x)
(hGd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
HasFDerivAt G (G' x) x)
(htrace : ContinuousOn
(fun x => ∑ i, ((χ x • G' x + (χ' x).smulRight (G x))
(Pi.single i (1 : ℝ))) i)
(Set.Icc a b))
(hχzero : ∀ x ∉ (Set.univ.pi fun i => Set.Ioo (a i) (b i)), χ x = 0) :
∫ x in Set.Icc a b, coordinateDivergence
(fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
(WithLp.toLp 2 ((χ (WithLp.ofLp y)) • G (WithLp.ofLp y)) :
EuclideanSpace ℝ (Fin (n + 1))))
(WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) = 0 := by
exact integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_regularity
a b hle χ χ' G G' s hs hχc hGc hχd hGd
(integrableOn_smul_vectorField_trace_of_continuousOn a b χ χ' G G' htrace)
hχzero
/-- Component-continuity version of
`integral_coordinateDivergence_toPi_box_eq_zero_of_cutoff_eq_zero_off_univ_pi_Ioo_of_trace_continuous`.
It derives closed-box trace continuity from separate continuity assumptions on
`χ`, `χ'`, `G`, and `G'`, then discharges compact-box trace integrability. It
still assumes the derivative hypotheses and cutoff vanishing needed by the
finite-box zero-face handoff, and remains below smooth cutoff construction,
tail limits, weighted IBP, generator domains, invariant laws, and reversibility. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1564published source at 7bcd37294df1
Proof architecture
log-concave sampling Ch.1 finite-box cutoff route: derive cutoff-smul regularity and compact-box trace integrability before applying the zero-face coordinate-divergence handoff
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
- `exact` closes the current goal with an already typed term.
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
- Integrability is an input or proved output; a displayed integral alone does not supply it.
- Totalized `fderiv` values must not be read as a differentiability theorem.
- 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.