reflected_conditional_gradient_energy

Plain-English statement

Actual same reflected R/S with genuine probability laws and every-y density. For every signed smooth compact f, produce true differentiability, f/Tf/gradient L2, conditional variance L1 and exact variance defect. Genuine disintegration, reflection invariance and finite-domain Fubini give integrated variance=integral f squared minus integral Tf squared. Integrate true pointwise gradient domination to exact eta coefficient (1-alpha eta)^2/[4(1+alpha eta)]. Positive alpha<=beta, global C2 Hessian bounds, eta>0 and beta eta<=1; finite Hilbert/rank0 extension explicit. No supplied law, domain or desired-bound certificate. Full rough L2-to-H1/closed-gradient and literal block/Gamma adapters, mains/errors/cost/composition remain separate. Independently verified 5ba91a1c9a553b13a38d48f02f0725a69af146a4.

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_energy
    {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))
    let ν := J.snd
    IsProbabilityMeasure μ ∧ IsProbabilityMeasure J ∧ IsProbabilityMeasure ν ∧
    ∃ 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 →
        let Tf := fun y => ∫ u, f u ∂S y
        Differentiable ℝ Tf ∧ MemLp f 2 ν ∧ MemLp Tf 2 ν ∧
        MemLp (gradient Tf) 2 ν ∧
        Integrable (fun y =>
          AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance
            (S y) f) ν ∧
        (∫ y, AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance
          (S y) f ∂ν) = (∫ y, (f y)^2 ∂ν) - (∫ y, (Tf y)^2 ∂ν) ∧
        η * (∫ y, ‖gradient Tf y‖^2 ∂ν) ≤
          (1-(α : ℝ)*η)^2/(4*(1+(α : ℝ)*η)) *
            ((∫ y, (f y)^2 ∂ν) - (∫ y, (Tf y)^2 ∂ν)) := by
  dsimp only
  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))
  let ν := J.snd
  obtain ⟨hZ,hJ,_⟩ := GibbsAugmentation.normalized_augmentation_density
    hα hV (fun x a => (hH x a).1) hη
  have hexp : Integrable (fun x => Real.exp (-V x)) (volume : Measure E) := by
    by_contra hn
    exact (ne_of_gt hZ) (integral_undef hn)
  have hμ : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hexp
  letI := hμ
  letI : IsProbabilityMeasure J := hJ
  have hν : IsProbabilityMeasure ν := inferInstance
  letI := hν
  obtain ⟨R,S,hR,hS,hcond,hSR,hSd,hpoint⟩ :=
    ConditionalGradientVariance.reflected_conditional_gradient_variance hα hαβ hV hH hη hβη
  letI := hR
  letI := hS
  refine ⟨hμ,hJ,hν,R,S,hR,hS,hcond,hSR,hSd,?_⟩
  intro f hf hfc
  let Tf := fun y => ∫ u, f u ∂S y
  obtain ⟨M,hMb⟩ := hf.continuous.norm.bddAbove_range_of_hasCompactSupport hfc.norm
  have hM (u : E) : ‖f u‖ ≤ M := hMb (Set.mem_range_self u)
  have hM0 : 0 ≤ M := (norm_nonneg (f 0)).trans (hM 0)
  have hf2 : MemLp f 2 ν := MemLp.of_bound hf.continuous.aestronglyMeasurable M
    (Filter.Eventually.of_forall hM)
  have hfy2 (y : E) : MemLp f 2 (S y) :=
    MemLp.of_bound hf.continuous.aestronglyMeasurable M (Filter.Eventually.of_forall hM)
  have hTfsm : StronglyMeasurable Tf := hf.continuous.stronglyMeasurable.integral_kernel
  have hTfbound (y : E) : ‖Tf y‖ ≤ M := by
    simpa using norm_integral_le_of_norm_le_const (μ := S y) (Filter.Eventually.of_forall hM)
  have hTf2 : MemLp Tf 2 ν := MemLp.of_bound hTfsm.aestronglyMeasurable M
    (Filter.Eventually.of_forall hTfbound)
  let vf := fun y =>
    AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance (S y) f
  have hv (y : E) : vf y = (∫ u, (f u)^2 ∂S y)-(Tf y)^2 := by
    dsimp only [vf, AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance]
    rw [← variance_eq_integral (hfy2 y).aemeasurable, variance_eq_sub (hfy2 y)]
    rfl
  have hfsqsm : StronglyMeasurable (fun y => ∫ u, (f u)^2 ∂S y) :=
    (hf.continuous.pow 2).stronglyMeasurable.integral_kernel
  have hfsqbound (u : E) : ‖(f u)^2‖ ≤ M^2 := by
    rw [norm_pow]
    exact (sq_le_sq₀ (norm_nonneg _) hM0).mpr (hM u)
  have hfsq1 : MemLp (fun y => ∫ u, (f u)^2 ∂S y) 1 ν :=
    MemLp.of_bound hfsqsm.aestronglyMeasurable (M^2) (Filter.Eventually.of_forall
      fun y => by simpa using (norm_integral_le_of_norm_le_const (μ := S y)
        (Filter.Eventually.of_forall hfsqbound)))
  have hfsqI := hfsq1.integrable le_rfl
  have hvI : Integrable vf ν := (hfsqI.sub hTf2.integrable_sq).congr
    (Filter.Eventually.of_forall fun y => (hv y).symm)
  have hvdef : (∫ y, vf y ∂ν) = (∫ y, (f y)^2 ∂ν)-(∫ y, (Tf y)^2 ∂ν) := by
    rw [integral_congr_ae (Filter.Eventually.of_forall hv), integral_sub hfsqI hTf2.integrable_sq]
    rw [reflected_second_moment μ η hη R S f hf.continuous M hM hcond hSR]
  have hgradsm : StronglyMeasurable (gradient Tf) := by
    exact ((toDual ℝ E).symm.continuous.measurable.comp
      (measurable_fderiv (𝕜 := ℝ) (f := Tf))).stronglyMeasurable
  let C := (1/η-(α : ℝ))^2/(4*((α : ℝ)+1/η))
  have hgradI : Integrable (fun y => ‖gradient Tf y‖^2) ν :=
    (hvI.const_mul C).mono_nonneg (hgradsm.norm.pow 2).aestronglyMeasurable
      (Filter.Eventually.of_forall fun _ => sq_nonneg _)
      (Filter.Eventually.of_forall fun y => (hpoint f hf hfc y).2)
  have hgrad2 : MemLp (gradient Tf) 2 ν :=
    (memLp_two_iff_integrable_sq_norm hgradsm.aestronglyMeasurable).mpr hgradI
  refine ⟨fun y => (hpoint f hf hfc y).1,hf2,hTf2,hgrad2,hvI,hvdef,?_⟩
  have hi : (∫ y, ‖gradient Tf y‖^2 ∂ν) ≤ C*(∫ y, vf y ∂ν) := by
    calc
      (∫ y, ‖gradient Tf y‖^2 ∂ν) ≤ ∫ y, C*vf y ∂ν :=
        integral_mono hgradI (hvI.const_mul C) (fun y => (hpoint f hf hfc y).2)
      _ = C*(∫ y, vf y ∂ν) := integral_const_mul _ _
  have hden : 0 < (α : ℝ)+1/η := by positivity
  have hnewden : 0 < 1+(α : ℝ)*η := by positivity
  have hcoeff : η*C = (1-(α : ℝ)*η)^2/(4*(1+(α : ℝ)*η)) := by
    dsimp only [C]
    field_simp [ne_of_gt hη,ne_of_gt hden,ne_of_gt hnewden]
    <;> ring
  calc
    η*(∫ y, ‖gradient Tf y‖^2 ∂ν) ≤ η*(C*(∫ y, vf y ∂ν)) :=
      mul_le_mul_of_nonneg_left hi hη.le
    _ = (η*C)*(∫ y, vf y ∂ν) := by ring
    _ = (1-(α : ℝ)*η)^2/(4*(1+(α : ℝ)*η))*
        ((∫ y, (f y)^2 ∂ν)-(∫ y, (Tf y)^2 ∂ν)) := by rw [hcoeff,hvdef]

end AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradientEnergy

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ConditionalGradientEnergy.lean:67published source at ed709a7f320d

Proof architecture

PBPS actual outer gradient energy; 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.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.
`refine` instantiates a reusable theorem while leaving explicit subgoals.
`exact` closes the current goal with an already typed term.
`simpa` closes the goal after a controlled simplification of a typed result.

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.