coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt

Plain-English statement

- A.e. bridge from ASTIS wrapped coordinate divergence to Mathlib's Pi-space trace summand, assuming the Pi-space derivative exists a.e. on the restricted box. This theorem intentionally does not derive the a.e. differentiability assumption from an open-box/off-countable hypothesis. Boundary-null and countable-null transfers are separate analytic leaves.

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 coordinateDivergence_wrapped_toPi_trace_ae_of_ae_hasFDerivAt
    {n : ℕ}
    (a b : Fin (n + 1) → ℝ)
    (F : (Fin (n + 1) → ℝ) → Fin (n + 1) → ℝ)
    (F' : (Fin (n + 1) → ℝ) →
      (Fin (n + 1) → ℝ) →L[ℝ] (Fin (n + 1) → ℝ))
    (hF_ae : ∀ᵐ x ∂volume.restrict (Set.Icc a b), HasFDerivAt F (F' x) x) :
      (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 := by
  filter_upwards [hF_ae] with x hx
  exact coordinateDivergence_wrapped_toPi_trace_of_hasFDerivAt (ι := Fin (n + 1)) hx

/-- A.e. bridge from an open-box/off-countable `HasFDerivAt` hypothesis to the
`hdiv_ae` shape required by the finite-box face-term wrapper.

This discharges only the a.e. equality assumption.  It does not prove
integrability of the divergence integrand, weighted IBP, no-boundary limits, or
invariant-law consequences. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Divergence.lean:499published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 calculus root: discharge the finite-box face-term wrapper's `hdiv_ae` equality from an explicit a.e. Pi-space differentiability hypothesis

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