actual_gaussian_marginal_centered_poincare

Plain-English statement

Actual normalized Gibbs/Gaussian marginal nu, one genuine dense closable compact-gradient G and closed closure BEFORE every centered domain z; exact alpha/(1+alpha eta) PI. Original C2/two Hessian/positive capped eta retained; finite Hilbert/Borel/rank0 extension explicit. Internal normalization, posteriorL2 and exact alpha/(1+alphaeta),beta/(1+betaeta) Hessian bounds use six real admitted parents; no caller law/moment/curvature/PI/domain certificate or private provider. SAME56 closed pair and actual55 stationary mean consumer derive centering, C3 and sharp rho=(1-alphaeta)/(1+alphaeta) contraction including endpoint. WeakH1/macro-range/fullB13/Gamma/dynamics/main/errors/cost/composition OPEN. Independently verified e8a9044ba5a945eaa4b4aecd110b63494fe6c68e.

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Lean statement

theorem actual_gaussian_marginal_centered_poincare
    {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 v : E,
      (α : ℝ)*‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
      (fderiv ℝ (fderiv ℝ V) x v) v ≤ (β : ℝ)*‖v‖^2)
    (hη : 0 < η) (hβη : (β : ℝ)*η ≤ 1) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := (μ.prod (stdGaussian E)).map
      (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
    let ν := J.snd
    IsProbabilityMeasure ν ∧
      ∃ G : Lp ℝ 2 ν →ₗ.[ℝ] Lp E 2 ν,
        Dense (G.domain : Set (Lp ℝ 2 ν)) ∧ G.IsClosable ∧ G.closure.IsClosed ∧
        (∀ (u : Lp ℝ 2 ν) (v : Lp E 2 ν), (u,v) ∈ G.graph ↔
          ∃ φ : E → ℝ, ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧
            u =ᵐ[ν] φ ∧ v =ᵐ[ν] gradient φ) ∧
        ∀ z : G.closure.domain,
          (∫ x, (z : Lp ℝ 2 ν) x ∂ν) = 0 →
          ((α : ℝ)/(1+(α : ℝ)*η))*‖(z : Lp ℝ 2 ν)‖^2 ≤ ‖G.closure z‖^2 := by
  classical
  letI : CompleteSpace E := FiniteDimensional.complete ℝ E
  dsimp only
  let μ := (volume : Measure E).tilted (fun x => -V x)
  let J := (μ.prod (stdGaussian E)).map
    (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
  let ν := J.snd
  let C := ((Real.sqrt (2*Real.pi*η))⁻¹)^Module.finrank ℝ E
  let A := fun y : E => C*∫ x, Real.exp (-V x-‖y-x‖^2/(2*η)) ∂(volume : Measure E)
  let Vη := fun y : E => -Real.log (A y)
  let R := fun y : E => μ.tilted (fun x => -‖x-y‖^2/(2*η))
  let U := fun y : E => -Real.log (C*∫ x, Real.exp (-‖y-x‖^2/(2*η)) ∂μ)
  let ZV := ∫ x, Real.exp (-V x) ∂(volume : Measure E)
  obtain ⟨hZ,hμ,_hlaw,hIA,hIntA,hTilt,hA,hC2,hshift⟩ :=
    AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothedGibbsPotential.smoothed_gibbs_potential
      hα hV (fun x a => (hH x a).1) hη
  letI : IsProbabilityMeasure μ := hμ
  have hpair : Measurable (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) := by fun_prop
  letI : IsProbabilityMeasure J := Measure.isProbabilityMeasure_map hpair.aemeasurable
  letI : IsProbabilityMeasure ν := Measure.isProbabilityMeasure_map measurable_snd.aemeasurable
  have htilt : ν = (volume : Measure E).tilted (fun y => -Vη y) :=
    (Measure.map_map measurable_snd hpair).trans hTilt
  have hC : 0<C := by dsimp only [C]; positivity
  obtain ⟨hRm,_hD,hDD⟩ :=
    AutoSamplingTheory.TechnicalLemmas.Measure.GaussianConvolutionRegularity.gaussian_convolution_derivatives μ hη hC
  have hi := AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGibbsIntegrability.integrable_exp_neg_of_strongConvexOn
    hα (hV.differentiable (by norm_num))
    (AutoSamplingTheory.TechnicalLemmas.Analysis.HessianStrongConvexity.strongConvexOn_univ_of_fderiv2_lower hV (fun x a => (hH x a).1))
  have hcov (y : E) : ∀ a : E, covarianceBilin (R y) a a ≤ ‖a‖^2/((α : ℝ)+η⁻¹) := by
    let W := fun x => V x+η⁻¹/2*‖x-y‖^2
    have hW : ContDiff ℝ 2 W := hV.add (contDiff_const.mul ((contDiff_id.sub contDiff_const).norm_sq (𝕜:=ℝ)))
    have hVd : Differentiable ℝ V := hV.differentiable (by norm_num)
    have hVdd : Differentiable ℝ (fderiv ℝ V) := (hV.fderiv_right (m:=1) (by norm_num)).differentiable_one
    have hq (x : E) : HasFDerivAt (fun z => η⁻¹/2*‖z-y‖^2)
        (η⁻¹ • innerSL ℝ (x-y)) x := by
      convert (((hasFDerivAt_id x).sub_const y).norm_sq).const_mul (η⁻¹/2)
        using 1 <;> first | rfl | (ext a; simp;ring)
    have hWfd (x : E) : fderiv ℝ W x = fderiv ℝ V x+η⁻¹ • innerSL ℝ (x-y) :=
      ((hVd x).hasFDerivAt.add (hq x)).fderiv
    let J : E →L[ℝ] (E →L[ℝ] ℝ) :=
      {toFun := fun a => innerSL ℝ a
       map_add' := by intros;ext;simp
       map_smul' := by intros;ext;simp
       cont := (innerSL ℝ (E:=E)).continuous}
    have hWdd (x : E) : HasFDerivAt (fderiv ℝ W)
        (fderiv ℝ (fderiv ℝ V) x+η⁻¹ • J) x := by
      rw [show fderiv ℝ W=(fun z => fderiv ℝ V z+η⁻¹ • innerSL ℝ (z-y)) from funext hWfd]
      convert (hVdd x).hasFDerivAt.add ((J.hasFDerivAt.comp x ((hasFDerivAt_id x).sub_const y)).const_smul η⁻¹) using 1 <;> rfl
    have hb (x a : E) : ((α : ℝ)+η⁻¹)*‖a‖^2 ≤ fderiv ℝ (fderiv ℝ W) x a a ∧
        fderiv ℝ (fderiv ℝ W) x a a ≤ ((β : ℝ)+η⁻¹)*‖a‖^2 := by
      rw [(hWdd x).fderiv]
      change ((α : ℝ)+η⁻¹)*‖a‖^2 ≤ fderiv ℝ (fderiv ℝ V) x a a+η⁻¹*inner ℝ a a ∧
        fderiv ℝ (fderiv ℝ V) x a a+η⁻¹*inner ℝ a a ≤ ((β : ℝ)+η⁻¹)*‖a‖^2
      rw [real_inner_self_eq_norm_sq]
      constructor <;> nlinarith [(hH x a).1,(hH x a).2]
    have hm : 0<(α : ℝ)+η⁻¹ := add_pos hα (inv_pos.mpr hη)
    have hmM : (α : ℝ)+η⁻¹≤(β : ℝ)+η⁻¹ := by
      simpa only [add_comm] using (add_le_add_right (show (α : ℝ) ≤ (β : ℝ) from hαβ) η⁻¹)
    have hiW := AutoSamplingTheory.TechnicalLemmas.Analysis.StrongConvexGibbsIntegrability.integrable_exp_neg_of_strongConvexOn
      hm (hW.differentiable (by norm_num))
      (AutoSamplingTheory.TechnicalLemmas.Analysis.HessianStrongConvexity.strongConvexOn_univ_of_fderiv2_lower hW (fun x a => (hb x a).1))
    have hRW : R y=(volume : Measure E).tilted (fun x => -W x) := by
      dsimp only [R,μ]
      rw [tilted_tilted hi]
      congr 1
      funext x
      simp only [Pi.add_apply,W]
      field_simp [hη.ne']
      ring
    have hX : MemLp id 2 ((volume : Measure E).tilted (fun x => -W x)) := hRW ▸ (hRm y).2
    rw [hRW]
    exact AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GibbsLinearCovarianceUpper.gibbs_linear_covariance_upper
      W hW hiW (integral_exp_pos hiW) ((α : ℝ)+η⁻¹) ((β : ℝ)+η⁻¹) hm hmM
      (fun x a => (hb x a).1) (fun x a => (hb x a).2) hX
  have he : U=(fun y => Vη y+Real.log ZV) := funext hshift
  have hd : fderiv ℝ U=fderiv ℝ Vη := by
    rw [he]
    funext y
    exact fderiv_add_const _
  change ∀ y a b, fderiv ℝ (fderiv ℝ U) y a b = _ at hDD
  rw [hd] at hDD
  let m := (α : ℝ)/(1+(α : ℝ)*η)
  let M := (β : ℝ)/(1+(β : ℝ)*η)
  have hden : 0<1+(α : ℝ)*η := by positivity
  have hbden : 0<1+(β : ℝ)*η := by positivity
  have hm : 0<m := div_pos hα hden
  have hmM : m≤M := by
    dsimp only [m,M]
    apply (div_le_div_iff₀ hden hbden).2
    nlinarith [show (α : ℝ) ≤ (β : ℝ) from hαβ]
  have hlower (y a : E) : m*‖a‖^2 ≤ fderiv ℝ (fderiv ℝ Vη) y a a := by
    rw [hDD y a a,real_inner_self_eq_norm_sq]
    have hp : 0<(α : ℝ)+η⁻¹ := add_pos hα (inv_pos.mpr hη)
    calc
      m*‖a‖^2 = ‖a‖^2/η-(‖a‖^2/((α : ℝ)+η⁻¹))/η^2 := by
        dsimp only [m]
        field_simp [hη.ne',hp.ne',hden.ne']
        ring
      _ ≤ ‖a‖^2/η-covarianceBilin (R y) a a/η^2 :=
        sub_le_sub_left (div_le_div_of_nonneg_right (hcov y a) (sq_nonneg η)) _
  have hupper : ∀ y a : E, fderiv ℝ (fderiv ℝ Vη) y a a ≤ M*‖a‖^2 :=
    (AutoSamplingTheory.ExampleCases.SmoothedPicardHMC.SmoothedHessianUpper.smoothed_hessian_upper
      (show 0<α from hα) hαβ hV hH hη).2
  have hI : Integrable (fun y => Real.exp (-Vη y)) (volume : Measure E) := by
    have hexp (y : E) : Real.exp (-Vη y)=A y := by
      dsimp only [Vη];rw [neg_neg,Real.exp_log (hA y)]
    simpa only [hexp] using hIA
  have hZη : 0<∫ y, Real.exp (-Vη y) ∂(volume : Measure E) := integral_exp_pos hI
  obtain ⟨G,hDense,hClose,hClosed,hGraph⟩ :=
    GaussianMarginalGradient.gaussian_marginal_gradient_closable μ hη
  refine ⟨inferInstance,G,hDense,hClose,hClosed,hGraph,?_⟩
  have hPI : ∀ (D : Lp ℝ 2 ν →ₗ.[ℝ] Lp E 2 ν), D.IsClosable →
      (∀ (a : Lp ℝ 2 ν) (F : Lp E 2 ν), (a,F) ∈ D.graph ↔
        ∃ φ : E → ℝ, ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧
          a =ᵐ[ν] φ ∧ F =ᵐ[ν] gradient φ) →
      ∀ z : D.closure.domain, (∫ x, (z : Lp ℝ 2 ν) x ∂ν)=0 →
        m*‖(z : Lp ℝ 2 ν)‖^2 ≤ ‖D.closure z‖^2 := by
    rw [htilt]
    exact CenteredDomainPoincare.gibbs_centered_domain_poincare
      Vη hC2 hI hZη m M hm hmM hlower hupper
  exact hPI G hClose hGraph

end
end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianMarginalPoincare

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/GaussianMarginalPoincare.lean:28published source at ed709a7f320d

Proof architecture

Reusable Gaussian smoothing PI with PBPS consumer; 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.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`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.
A Gibbs expression is not automatically a probability law or an invariant law.