gaussian_marginal_gradient_closable

Plain-English statement

Any probability mu on a finite real Hilbert/Borel space, eta>0, including rank0. For literal J=law(X,X+sqrt(eta)G), nu=J.snd, produce one dense genuine smooth-compact scalar/vector L2 gradient graph, closability and closed closure. Positive C2 Gaussian density, density L1, real normalizer1 and exact normalized tilted-law equality are internally derived. No caller moment/density/potential/integrability/normalization/closed-gradient certificate. Actual PBPS consumer retains original curvature and step assumptions; noncentered rank0 constant has mean1 and zero gradient. Authored background elaboration of C.1 density/closedness prerequisite. Literal Tf closure membership, full rough B13, Gamma/half-turn/main/errors/expected costs/composition remain separate. Independently verified 19f7e6bed7975fac9b7e1ea0b0b95d0c084083f6.

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 gaussian_marginal_gradient_closable
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (μ : Measure E) [IsProbabilityMeasure μ] {η : ℝ} (hη : 0 < η) :
    let J := (μ.prod (stdGaussian E)).map
      (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
    let ν := J.snd
    ∃ D : Lp ℝ 2 ν →ₗ.[ℝ] Lp E 2 ν,
      Dense (D.domain : Set (Lp ℝ 2 ν)) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
      ∀ (u : Lp ℝ 2 ν) (v : Lp E 2 ν), (u,v) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[ν] f ∧ v =ᵐ[ν] gradient f := by
  let J := (μ.prod (stdGaussian E)).map
    (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
  let ν := J.snd
  change ∃ D : Lp ℝ 2 ν →ₗ.[ℝ] Lp E 2 ν,
    Dense (D.domain : Set (Lp ℝ 2 ν)) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
      ∀ (u : Lp ℝ 2 ν) (v : Lp E 2 ν), (u,v) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[ν] f ∧ v =ᵐ[ν] gradient f
  let C := ((Real.sqrt (2*Real.pi*η))⁻¹)^Module.finrank ℝ E
  let Z := fun y : E => ∫ x, Real.exp (-‖y-x‖^2/(2*η)) ∂μ
  let ρ := fun y : E => C*Z y
  let W := fun y : E => -Real.log (ρ y)
  have hpair : Measurable (fun p : E × E =>
      (p.1, p.1 + Real.sqrt η • p.2)) := by fun_prop
  let : IsProbabilityMeasure J := Measure.isProbabilityMeasure_map hpair.aemeasurable
  let : IsProbabilityMeasure ν := Measure.isProbabilityMeasure_map measurable_snd.aemeasurable
  have hmap : ν = (μ.prod (stdGaussian E)).map
      (fun p : E × E => p.1 + Real.sqrt η • p.2) :=
    Measure.map_map measurable_snd hpair
  obtain ⟨hdensity,hpos,hZ,hW⟩ :=
    AutoSamplingTheory.TechnicalLemmas.Measure.GaussianConvolutionRegularity.gaussian_convolution_potential_c2 μ hη
  have hνdensity : ν = (volume : Measure E).withDensity
      (fun y => ENNReal.ofReal (ρ y)) := hmap.trans hdensity
  have hρ : Continuous ρ := continuous_const.mul hZ.continuous
  have hnn : 0 ≤ᵐ[(volume : Measure E)] ρ :=
    Filter.Eventually.of_forall fun y => (hpos y).le
  have hmass : (∫⁻ y, ENNReal.ofReal (ρ y) ∂(volume : Measure E)) = 1 := by
    have hm := congrArg (fun m : Measure E => m Set.univ) hνdensity
    rw [withDensity_apply _ MeasurableSet.univ] at hm
    simpa only [measure_univ,Measure.restrict_univ] using hm.symm
  have hI : Integrable ρ (volume : Measure E) :=
    (lintegral_ofReal_ne_top_iff_integrable hρ.aestronglyMeasurable hnn).mp
      (by rw [hmass]; exact ENNReal.one_ne_top)
  have hnorm : (∫ y, ρ y ∂(volume : Measure E)) = 1 :=
    ENNReal.ofReal_eq_one.mp ((ofReal_integral_eq_lintegral_ofReal hI hnn).trans hmass)
  have hexp (y : E) : Real.exp (-W y) = ρ y := by
    dsimp only [W]
    rw [neg_neg,Real.exp_log (hpos y)]
  have hWI : Integrable (fun y => Real.exp (-W y)) (volume : Measure E) := by
    simpa only [hexp] using hI
  have htilt : (volume : Measure E).tilted (fun y => -W y) = ν := by
    rw [Measure.tilted]
    simp_rw [hexp]
    rw [hnorm]
    simpa only [div_one] using hνdensity.symm
  rw [← htilt]
  exact WeightedGradient.compact_gradient_closable W
    (hW.of_le (by norm_num)) hWI

end
end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianMarginalGradient

Open AutoSamplingTheory/TechnicalLemmas/FunctionalInequalities/GaussianMarginalGradient.lean:27published source at ed709a7f320d

Proof architecture

Shared Gaussian outer-marginal gradient; no SALD admission

Lean proof walkthrough

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`simpa` closes the goal after a controlled simplification of a typed result.

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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.
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.