laplacian_eq_sum_stdOrthonormalBasis

Plain-English statement

- Mathlib's finite-dimensional standard-orthonormal-basis formula for the Laplacian, exposed as an ASTIS calculus leaf. For Ch.1 Langevin this is the coordinate bridge behind a supplied Laplacian identifier `lapF = ∑ᵢ ∂ᵢᵢ f`. It is not an integration-by-parts or invariant measure theorem.

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 laplacian_eq_sum_stdOrthonormalBasis
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] (f : E → ℝ) :
    Laplacian.laplacian f =
      fun x => ∑ i, iteratedFDeriv ℝ 2 f x
        ![(stdOrthonormalBasis ℝ E) i, (stdOrthonormalBasis ℝ E) i] :=
  InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis f

/-- Handoff form of `laplacian_eq_sum_stdOrthonormalBasis` for source-defined
Laplacian functionals.

This is useful when a paper defines a weak-generator or test-function action by
the coordinate second-derivative sum and the local Lean proof needs to rewrite
that action as Mathlib's `Laplacian.laplacian`. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Laplacian.lean:29published source at 7bcd37294df1

Proof architecture

Chewi SDE/ANALYSIS root: identify Mathlib's finite-dimensional Laplacian with the standard orthonormal-basis second-derivative sum used in Langevin generator displays

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Hidden assumptions and non-claims

Totalized `fderiv` values must not be read as a differentiability theorem.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.