reflected_conditional_gradient_variance

Plain-English statement

Actual reflected conditional kernel and every-y density; true differentiability and exact pointwise gradient variance bound for every smooth compact observable. Identify both parent S internally from density, produce centered L2 domains, evaluate derivative at actual gradient and cancel zero-aware. Positive alpha<=beta, eta>0, beta*eta<=1; finite Hilbert/rank0 extension explicit. Actual Gaussian precision/rank0 Tests. No supplied law/coherence/domain/desired inequality certificate. Outer C.2/B.13/full L2-H1 and mains/error/cost separate. Independently verified 8d950c37e41bd5d816c4132c6a5cdde0e30dcbee.

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 reflected_conditional_gradient_variance
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : ℝ≥0} {η : ℝ}
    (hα : 0 < (α : ℝ)) (hαβ : α ≤ β) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x a : E,
      (α : ℝ) * ‖a‖^2 ≤ (fderiv ℝ (fderiv ℝ V) x a) a ∧
      (fderiv ℝ (fderiv ℝ V) x a) a ≤ (β : ℝ) * ‖a‖^2)
    (hη : 0 < η) (hβη : (β : ℝ) * η ≤ 1) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2 : ℝ) • x - y)) ∧
      (∀ y, S y = (volume : Measure E).tilted
        (fun u => -V ((1/2 : ℝ) • (y+u)) - ‖y-u‖^2/(8*η))) ∧
      ∀ (f : E → ℝ), ContDiff ℝ ∞ f → HasCompactSupport f →
        ∀ y, DifferentiableAt ℝ (fun z => ∫ u, f u ∂S z) y ∧
          ‖gradient (fun z => ∫ u, f u ∂S z) y‖^2 ≤
            (1/η-(α : ℝ))^2/(4*((α : ℝ)+1/η)) *
              AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance
                (S y) f := by
  dsimp only
  obtain ⟨R, S, hR, hS, hcond, hSR, hSd, hder⟩ :=
    ConditionalScore.reflected_conditional_covariance hα hV hH hη
  obtain ⟨R', S', _, _, _, _, hvar⟩ :=
    ConditionalScoreVariance.conditional_centered_domain_and_score_variance
      hα hαβ hV hH hη hβη
  have hSS : S' = S := by
    ext y : 1
    rw [(hvar y).1, hSd y]
    congr 1
    funext u
    simp only [norm_sub_rev u y, neg_add_rev]
    ring
  subst S'
  let : IsMarkovKernel S := hS
  refine ⟨R, S, hR, hS, hcond, hSR, hSd, ?_⟩
  intro f hf hfc y
  obtain ⟨hsI, hfsI, hfd⟩ := hder f hf hfc y
  refine ⟨hfd.differentiableAt, ?_⟩
  let s := fun u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
    (1/(4*η)) • innerSL ℝ (y-u)
  let G := gradient (fun z => ∫ u, f u ∂S z) y
  let Y := fun u => s u G
  obtain ⟨_, _, _, _, D, _, _, _, _, _, hscore⟩ := hvar y
  obtain ⟨hYad, hYvar⟩ := hscore G
  obtain ⟨M, hM⟩ := hf.continuous.norm.bddAbove_range_of_hasCompactSupport hfc.norm
  have hf2 : MemLp f 2 (S y) := MemLp.of_bound
    hf.continuous.aestronglyMeasurable M (Filter.Eventually.of_forall
      (fun u => hM (Set.mem_range_self u)))
  let Xc := fun u => f u - ∫ v, f v ∂S y
  let Yc := fun u => Y u - ∫ v, Y v ∂S y
  have hXc2 : MemLp Xc 2 (S y) := hf2.sub (memLp_const _)
  have hYc2 : MemLp Yc 2 (S y) :=
    (memLp_two_iff_integrable_sq
      (hYad.1.aestronglyMeasurable.sub aestronglyMeasurable_const)).mpr hYad.2.1
  have hY2 : MemLp Y 2 (S y) := by
    convert hYc2.add (memLp_const (∫ v, Y v ∂S y)) using 1
    funext u
    simp [Yc]
  have hpair (a b : E → ℝ) (ha : MemLp a 2 (S y)) (hb : MemLp b 2 (S y)) :
      ⟪ha.toLp a, hb.toLp b⟫_ℝ = ∫ u, a u * b u ∂S y := by
    rw [L2.inner_def]
    apply integral_congr_ae
    filter_upwards [ha.coeFn_toLp, hb.coeFn_toLp] with u hua hub
    simp [hua, hub, mul_comm]
  have hcs := real_inner_mul_inner_self_le (hXc2.toLp Xc) (hYc2.toLp Yc)
  rw [hpair Xc Yc hXc2 hYc2, hpair Xc Xc hXc2 hXc2,
    hpair Yc Yc hYc2 hYc2] at hcs
  have hcov : ProbabilityTheory.covariance f Y (S y) = ‖G‖^2 := by
    rw [covariance_eq_sub hf2 hY2]
    have hg := congrArg (fun L : E →L[ℝ] ℝ => L G) hfd.fderiv
    have hgG : fderiv ℝ (fun z => ∫ u, f u ∂S z) y G = ‖G‖^2 := by
      rw [← inner_gradient_left, real_inner_self_eq_norm_sq]
    rw [hgG] at hg
    simp only [sub_apply, smul_apply,
      smul_eq_mul, ContinuousLinearMap.integral_apply hfsI G,
      ContinuousLinearMap.integral_apply hsI G] at hg
    exact hg.symm
  have hnorm : (‖G‖^2)^2 ≤
      AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance (S y) f *
      AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance (S y) Y := by
    have he : (∫ u, Xc u * Yc u ∂S y) = ‖G‖^2 := hcov
    rw [he] at hcs
    simpa only [← pow_two, Xc, Yc,
      AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance] using hcs
  have hbound := hnorm.trans (mul_le_mul_of_nonneg_left hYvar
    (AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance_nonneg))
  change ‖G‖^2 ≤ _
  by_cases hz : ‖G‖^2 = 0
  · rw [hz]
    exact mul_nonneg (by positivity)
      AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance_nonneg
  · have hp : 0 < ‖G‖^2 := lt_of_le_of_ne (sq_nonneg _) (Ne.symm hz)
    apply (mul_le_mul_iff_right₀ hp).mp
    simpa only [pow_two, mul_assoc, mul_comm, mul_left_comm] using hbound

end AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradientVariance

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ConditionalGradientVariance.lean:16published source at ed709a7f320d

Proof architecture

PBPS actual reflected conditional mean; no SALD admission

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
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`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
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`refine` instantiates a reusable theorem while leaving explicit subgoals.
`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

Measurability is represented explicitly or must be supplied by a dependency.
Integrability is an input or proved output; a displayed integral alone does not supply it.
Totalized `fderiv` values must not be read as a differentiability theorem.
Genuine differentiability is localized to the hypotheses shown in the Lean statement.
Pointwise support, topological support, and compact support retain distinct meanings.
Almost-everywhere hypotheses depend on the stated measure and representative.
Density statements retain normalization and absolute-continuity prerequisites.