Plain-English statement
- Component-continuity scalar-support version of the cutoff-smul finite-box trace handoff. It derives the trace-continuity input from separate continuity assumptions on `χ`, `χ'`, `G`, and `G'`, then applies the scalar-support zero-face handoff. It is still finite-box only.
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Proof architecture
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Lean statement
theorem integral_coordinateDivergence_toPi_box_eq_zero_of_scalar_support_subset_univ_pi_Ioo_of_component_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))
(hχ'c : ∀ i, ContinuousOn
(fun x => χ' x (Pi.single i (1 : ℝ))) (Set.Icc a b))
(hG'c : ∀ i, ContinuousOn
(fun x => (G' x (Pi.single i (1 : ℝ))) i) (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)
(hχsupp : Function.support χ ⊆
(Set.univ.pi fun i => Set.Ioo (a i) (b i))) :
∫ 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_scalar_support_subset_univ_pi_Ioo_of_trace_continuous
a b hle χ χ' G G' s hs hχc hGc hχd hGd
(continuousOn_smul_vectorField_trace_of_component_continuousOn a b χ χ' G G' hχc
(fun i => (continuous_apply i).comp_continuousOn hGc) hχ'c hG'c)
hχsupp
/-- The whole-space coordinate-divergence integral of a compactly supported
`C¹` vector field is zero.
The field is represented in raw finite-Pi coordinates, while
`coordinateDivergence` is evaluated after the canonical `PiLp` transport to
Euclidean space. The proof encloses `tsupport F` in a strict finite box,
derives trace integrability from `C¹` regularity, invokes Mathlib's finite-box
divergence theorem through the ASTIS zero-face wrapper, and then removes the
box because the derivative vanishes off `tsupport F`.
This is a reusable whole-space no-boundary leaf. It contains no Gibbs,
Langevin, generator-domain, semigroup, or invariant-measure semantics. -/Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:1750published source at 7bcd37294df1
Proof architecture
log-concave sampling Ch.1 finite-box cutoff route: component trace continuity, regularity hypotheses, and scalar support containment imply the finite-box cutoff-smul coordinate-divergence integral is zero
Lean proof walkthrough
- Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
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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
- Measurability is represented explicitly or must be supplied by a dependency.
- 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.
- Pointwise support, topological support, and compact support retain distinct meanings.
- A Gibbs expression is not automatically a probability law or an invariant law.
- 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.