finiteCoord_named_weightedDivergence_langevinGenerator_algebra

Plain-English statement

- Named finite-coordinate wrapper for `finiteCoord_weightedDivergence_langevinGenerator_algebra`. The hypotheses `hlap` and `hinner` are supplied identifications of the coordinate sums with a named Laplacian scalar and a named gradient inner-product scalar. The theorem does not prove those identifications.

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 finiteCoord_named_weightedDivergence_langevinGenerator_algebra
    {ι : Type*} [Fintype ι]
    {rho lapF divWeighted innerGradVGradF : ℝ}
    {divCoord hessDiag gradRho gradV gradF : ι → ℝ}
    (hdivWeighted : divWeighted = ∑ i, divCoord i)
    (hdiv : ∀ i, divCoord i = rho * hessDiag i + gradRho i * gradF i)
    (hgrad : ∀ i, gradRho i = -rho * gradV i)
    (hlap : lapF = ∑ i, hessDiag i)
    (hinner : innerGradVGradF = ∑ i, gradV i * gradF i) :
    divWeighted = rho * (lapF - innerGradVGradF) := by
  calc
    divWeighted = ∑ i, divCoord i := hdivWeighted
    _ = rho * ((∑ i, hessDiag i) - ∑ i, gradV i * gradF i) :=
      finiteCoord_weightedDivergence_langevinGenerator_algebra hdiv hgrad
    _ = rho * (lapF - innerGradVGradF) := by rw [hlap, hinner]

/-- Finite-coordinate Langevin divergence-form handoff using the Mathlib
`EuclideanSpace` inner-product notation for the coordinate gradients.

The coordinate product rule, Gibbs-weight chain rule, and Laplacian-coordinate
identification are still supplied as hypotheses.  This theorem only combines
the finite-coordinate algebra with the reusable Euclidean coordinate
inner-product bridge. -/

Open AutoSamplingTheory/TechnicalLemmas/StochasticProcesses/Langevin.lean:147published source at 7bcd37294df1

Proof architecture

Chewi SDE/DENS root: source-facing finite-coordinate handoff to named `divWeighted = rho * (lapF - innerGradVGradF)`

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Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.

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Hidden assumptions and non-claims

A Gibbs expression is not automatically a probability law or an invariant law.
A formal generator display does not establish a closed operator domain.