integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_countable

Plain-English statement

- Box-level signed-face divergence theorem wrapper for ASTIS coordinate divergence. Mathlib's Bochner divergence theorem is stated on `Fin (n + 1) → ℝ`; ASTIS finite Euclidean pointwise calculations use `EuclideanSpace ℝ (Fin (n + 1))`. This theorem only bridges those interfaces under an explicit a.e. equality `hdiv_ae` between the ASTIS coordinate-divergence integrand and Mathlib's trace summand. The conclusion is exactly Mathlib's signed face-term formula. It does not derive `hdiv_ae`, prove box integrability, take a whole-space limit, prove boundary cancellation, perform weighted integration by parts, establish generator domains, or prove invariant/reversible Gibbs laws.

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 integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_countable
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ) (hle : a ≤ b)
    (F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
    (F' : (Fin (n + 1) → ℝ) →
      (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
    (s : Set (Fin (n + 1) → ℝ)) (hs : s.Countable)
    (Hc : ContinuousOn F (Set.Icc a b))
    (Hd : ∀ x ∈ (Set.univ.pi fun i => Set.Ioo (a i) (b i)) \ s,
      HasFDerivAt F (F' x) x)
    (hdiv_ae :
      (fun x => coordinateDivergence
        (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
          (WithLp.toLp 2 (F (WithLp.ofLp y)) :
            EuclideanSpace ℝ (Fin (n + 1))))
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
      =ᵐ[volume.restrict (Set.Icc a b)]
      fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) i)
    (Hi : IntegrableOn
      (fun x => coordinateDivergence
        (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
          (WithLp.toLp 2 (F (WithLp.ofLp y)) :
            EuclideanSpace ℝ (Fin (n + 1))))
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))))
      (Set.Icc a b) volume) :
    ∫ x in Set.Icc a b, coordinateDivergence
        (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
          (WithLp.toLp 2 (F (WithLp.ofLp y)) :
            EuclideanSpace ℝ (Fin (n + 1))))
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1))) =
      ∑ 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) := by
  have Hi_trace : IntegrableOn (fun x => ∑ i, F' x (Pi.single i (1 : ℝ)) i)
      (Set.Icc a b) volume :=
    Hi.congr_fun_ae hdiv_ae
  calc
    (∫ x in Set.Icc a b, coordinateDivergence
        (fun y : EuclideanSpace ℝ (Fin (n + 1)) =>
          (WithLp.toLp 2 (F (WithLp.ofLp y)) :
            EuclideanSpace ℝ (Fin (n + 1))))
        (WithLp.toLp 2 x : EuclideanSpace ℝ (Fin (n + 1)))) =
        ∫ x in Set.Icc a b, ∑ i, F' x (Pi.single i (1 : ℝ)) i := by
          exact MeasureTheory.integral_congr_ae hdiv_ae
    _ = ∑ 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) := by
          exact MeasureTheory.integral_divergence_of_hasFDerivAt_off_countable
            (a := a) (b := b) hle F F' s hs Hc Hd Hi_trace

/-- Box-level signed-face divergence theorem wrapper for ASTIS coordinate
divergence, using Mathlib's trace-integrability hypothesis directly.

Compared with `integral_coordinateDivergence_toPi_box_of_hasFDerivAt_off_countable`,
this version no longer asks callers to provide the `hdiv_ae` representation
bridge or coordinate-divergence integrability.  Both are derived from the
open-box/off-countable derivative hypothesis and the explicit trace-integrability
assumption.

It still proves only the finite-box signed face-term formula.  It does not
prove trace integrability for a concrete vector field, whole-space/no-boundary
limits, weighted IBP, generator domains, invariant Gibbs law, reversibility,
stationarity, or KL/FI dissipation. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:584published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 calculus root: integrate ASTIS coordinateDivergence over a finite Mathlib box and obtain only the signed face-term formula, assuming the a.e. bridge to Mathlib's trace integrand and box integrability explicitly

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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.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
Almost-everywhere hypotheses depend on the stated measure and representative.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.