continuous_laplacian_of_contDiff_two

Plain-English statement

- A globally `C²` real-valued function has a continuous Mathlib Laplacian. This packages the standard finite-dimensional route: expand the Laplacian in a standard orthonormal basis, use continuity of the second iterated Frechet derivative from `ContDiff`, and apply the resulting continuous multilinear map to fixed basis directions. It does not prove closed-box `ContDiffOn` regularity, divergence theorem, weighted integration by parts, boundary cancellation, generator domains, invariant laws, reversibility, or KL/FI dissipation.

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 continuous_laplacian_of_contDiff_two
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E]
    {f : E → ℝ} (hf : ContDiff ℝ 2 f) :
    Continuous (fun x : E => Laplacian.laplacian f x) := by
  rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis f]
  refine continuous_finsetSum _ ?_
  intro i _
  let v : Fin 2 → E := ![(stdOrthonormalBasis ℝ E) i, (stdOrthonormalBasis ℝ E) i]
  have h2 : Continuous (fun x : E => iteratedFDeriv ℝ 2 f x) := by
    exact (hf.iteratedFDeriv_right (m := 0) (i := 2) (by norm_num)).continuous
  exact (ContinuousMultilinearMap.apply ℝ (fun _ : Fin 2 => E) ℝ v).continuous.comp h2

/-- The Laplacian is bounded by dimension times the operator norm of the
second iterated Fréchet derivative.

The dimension factor comes only from summing the diagonal evaluations in a
standard orthonormal basis.  This is a pointwise finite-dimensional trace
bound; it assumes no compact support or integrability. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Laplacian.lean:64published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 Langevin root: derive continuity of Mathlib's finite-dimensional total Laplacian from global `C²` regularity before closed-box integrability handoffs

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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.
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