gradient_quadratic_representative

Plain-English statement

- The smooth representative of the log-density ratio between two restricted Gaussian fibers has the constant score `(y-y') / eta`. This is the differential identity in Lee--Shen--Tian equation (11). The normalizing constants are retained in `representative`; they disappear only because they are constant in the spatial variable.

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_quadratic_representative
    (mu : Measure E) {eta : ℝ} (heta : eta ≠ 0) (y y' x : E) :
    gradient
        (fun t =>
          -(‖t - y‖ ^ 2 / (2 * eta)) -
            Real.log (∫ z, Real.exp (-(‖z - y‖ ^ 2 / (2 * eta))) ∂mu) -
          (-(‖t - y'‖ ^ 2 / (2 * eta))) +
            Real.log (∫ z, Real.exp (-(‖z - y'‖ ^ 2 / (2 * eta))) ∂mu)) x =
      eta⁻¹ • (y - y') := by
  have hy := hasGradientAt_quadraticExponent heta y x
  have hy' := hasGradientAt_quadraticExponent heta y' x
  apply HasGradientAt.gradient
  rw [hasGradientAt_iff_hasFDerivAt]
  change HasFDerivAt
    (fun t => quadraticExponent eta y t -
      Real.log (∫ z, Real.exp (quadraticExponent eta y z) ∂mu) -
      quadraticExponent eta y' t +
      Real.log (∫ z, Real.exp (quadraticExponent eta y' z) ∂mu))
    (InnerProductSpace.toDual ℝ E (eta⁻¹ • (y - y'))) x
  let c := Real.log (∫ z, Real.exp (quadraticExponent eta y z) ∂mu)
  let c' := Real.log (∫ z, Real.exp (quadraticExponent eta y' z) ∂mu)
  have hderiv : HasFDerivAt
      (fun t => quadraticExponent eta y t - c - quadraticExponent eta y' t + c')
      ((InnerProductSpace.toDual ℝ E (-eta⁻¹ • (x - y))) -
        InnerProductSpace.toDual ℝ E (-eta⁻¹ • (x - y'))) x :=
    ((hy.hasFDerivAt.sub_const c).sub hy'.hasFDerivAt).add_const c'
  have hvec :
      (-eta⁻¹ • (x - y)) - (-eta⁻¹ • (x - y')) =
        eta⁻¹ • (y - y') := by
    module
  rw [← hvec, map_sub]
  simpa only [c, c'] using hderiv

end Quadratic

end

end TiltedLogRatio
end InformationTheory
end TechnicalLemmas
end AutoSamplingTheory

Open AutoSamplingTheory/TechnicalLemmas/InformationTheory/TiltedLogRatio.lean:113published source at ed709a7f320d

Proof architecture

SPHMC Lemma 4.17: compute the constant relative score (y-y')/eta for restricted-Gaussian fibers

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`have` creates a named intermediate mathematical fact.
`rw` rewrites by an established identity.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`simpa` closes the goal after a controlled simplification of a typed result.

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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

Measurability is represented explicitly or must be supplied by a dependency.
Totalized `fderiv` values must not be read as a differentiability theorem.