gradient_expNegPotential_eq_of_differentiableAt

Plain-English statement

- Pointwise Mathlib-gradient form of the Gibbs weight chain rule. If `V` is differentiable at `x`, then Mathlib's total `gradient` of `fun y => exp (-V y)` agrees with the expected vector `-exp (-V x) • gradient V x`. This is still only a pointwise calculus leaf; it does not prove weighted divergence, product-rule, integration-by-parts, stationarity, or invariant-law statements.

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 gradient_expNegPotential_eq_of_differentiableAt
    {F : Type*} [NormedAddCommGroup F] [InnerProductSpace ℝ F] [CompleteSpace F]
    {V : F → ℝ} {x : F}
    (hV : DifferentiableAt ℝ V x) :
    gradient (fun y : F => Real.exp (-V y)) x =
      -(Real.exp (-V x)) • gradient V x := by
  exact gradient_expNegPotential_eq_of_hasGradientAt hV.hasGradientAt

/-- Coordinate form of
`gradient_expNegPotential_eq_of_differentiableAt` on finite Euclidean space.

This is the narrow reusable leaf that supplies the Gibbs-weight chain-rule
coordinate equality used by the Langevin algebra handoffs. -/

Open AutoSamplingTheory/TechnicalLemmas/Analysis/Calculus/Gradient.lean:84published source at 7bcd37294df1

Proof architecture

Chewi SDE/DENS root: turn `DifferentiableAt ℝ V x` into the Mathlib total-gradient identity `gradient (exp(-V)) x = -exp(-V x) • gradient V x` before weighted-divergence handoffs

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`exact` closes the current goal with an already typed term.

Why the statement has this shape

The declaration is kept at the reusable level recorded by its Registry tags and direct consumers. Explicit measures, spaces, wrappers, and regularity hypotheses expose interfaces that paper notation often infers. A theorem card explains those interfaces but never widens the compiled statement.

Hidden assumptions and non-claims

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.