fderiv_apply_eq_inner_of_hasGradientAt

Plain-English statement

- Apply the Frechet derivative to a vector when a gradient representative is supplied. This is the basic bridge from Mathlib's `fderiv` to the inner-product gradient convention. It is pointwise only: it does not choose coordinates, define divergence, prove integration by parts, or assert any global regularity.

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 fderiv_apply_eq_inner_of_hasGradientAt
    {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F]
    {f : F → ℝ} {x grad v : F}
    (hf : HasGradientAt f grad x) :
    fderiv ℝ f x v = inner ℝ grad v := by
  have hfd : fderiv ℝ f x = InnerProductSpace.toDual ℝ F grad :=
    hf.differentiableAt.hasFDerivAt.unique hf.hasFDerivAt
  calc
    fderiv ℝ f x v = InnerProductSpace.toDual ℝ F grad v := by rw [hfd]
    _ = inner ℝ grad v := by rw [InnerProductSpace.toDual_apply_apply]

/-- Apply the Frechet derivative to a vector and rewrite the result using
Mathlib's total `gradient`.

This is the pointwise `fderiv`/`gradient` bridge used before finite-coordinate
Langevin displays. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:128published source at 7bcd37294df1

Proof architecture

Chewi Ch.1 calculus root: rewrite a pointwise Frechet derivative application through a supplied Mathlib gradient representative before coordinate Langevin displays

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`have` creates a named intermediate mathematical fact.
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Hidden assumptions and non-claims

Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.