actual_macroscopic_centered_range

Plain-English statement

Original C2, both Hessian bounds, positive capped eta. Actual Gibbs/Gaussian J and nu; canonical snd pullback range equals true conditional projection range; mean transport and full centered image. Finite real Hilbert/Borel/rank0 extension explicit. Real Test joins same55 reflection/mean with57 contraction to give allmacro sharp contraction and squared defect gap. No Gamma/root/inverse/weakH1/dynamics/cost/composition closure. Independently verified 8c8847715c1d4c3033224b069d8dd694f2a4bd30.

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 actual_macroscopic_centered_range
    {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
    let hp : MeasurePreserving (Prod.snd : E × E → E) J ν :=
      ⟨measurable_snd,rfl⟩
    let M : Lp ℝ 2 ν →ₗᵢ[ℝ] Lp ℝ 2 J :=
      Lp.compMeasurePreservingₗᵢ ℝ Prod.snd hp
    let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
      (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E))
        2 J).subtypeL ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
    IsProbabilityMeasure J ∧ IsProbabilityMeasure ν ∧
      M.toLinearMap.range = P.toLinearMap.range ∧
      (∀ u : Lp ℝ 2 ν, (∫ p, (M u) p ∂J) = ∫ y, u y ∂ν) ∧
      M '' {u : Lp ℝ 2 ν | (∫ y, u y ∂ν)=0} =
        {f : Lp ℝ 2 J | f ∈ P.toLinearMap.range ∧ (∫ p, f p ∂J)=0} := by
  classical
  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 hp : MeasurePreserving (Prod.snd : E × E → E) J ν := ⟨measurable_snd,rfl⟩
  let M : Lp ℝ 2 ν →ₗᵢ[ℝ] Lp ℝ 2 J := Lp.compMeasurePreservingₗᵢ ℝ Prod.snd hp
  let S := lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd
    (inferInstance : MeasurableSpace E)) 2 J
  let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
    S.subtypeL ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
  obtain ⟨_,hJ,_⟩ := GibbsAugmentation.normalized_augmentation_density
    hα hV (fun x v => (hH x v).1) hη
  letI : IsProbabilityMeasure J := hJ
  have hν : IsProbabilityMeasure ν := by infer_instance
  letI : IsProbabilityMeasure ν := hν
  letI : Fact ((MeasurableSpace.comap Prod.snd
    (inferInstance : MeasurableSpace E)) ≤ (inferInstance : MeasurableSpace (E × E))) :=
    ⟨measurable_snd.comap_le⟩
  have hMrange : M.toLinearMap.range = S :=
    AutoSamplingTheory.TechnicalLemmas.Measure.L2PullbackRange.l2_pullback_range_eq_lpMeas hp
  have hPdef : P = S.starProjection := rfl
  have hPrange : P.toLinearMap.range = S := by
    change P.range = S
    rw [hPdef]
    exact S.range_starProjection
  have hRange : M.toLinearMap.range = P.toLinearMap.range := hMrange.trans hPrange.symm
  have hMean (u : Lp ℝ 2 ν) : (∫ p, (M u) p ∂J) = ∫ y, u y ∂ν := by
    calc
      (∫ p, (M u) p ∂J) = ∫ p : E × E, u p.2 ∂J :=
        integral_congr_ae (Lp.coeFn_compMeasurePreserving u hp)
      _ = ∫ y, u y ∂ν :=
        (integral_map measurable_snd.aemeasurable (Lp.aestronglyMeasurable u)).symm
  refine ⟨hJ,hν,hRange,hMean,?_⟩
  ext f
  constructor
  · rintro ⟨u,hu,rfl⟩
    refine ⟨?_,?_⟩
    · rw [← hRange]
      exact ⟨u,rfl⟩
    · rw [hMean]
      exact hu
  · rintro ⟨hf,hfmean⟩
    have hfM : f ∈ M.toLinearMap.range := by rwa [hRange]
    obtain ⟨u,hu⟩ := hfM
    refine ⟨u,?_,hu⟩
    change M u=f at hu
    change (∫ y, u y ∂ν)=0
    rw [← hMean u,hu]
    exact hfmean

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicRange

Open AutoSamplingTheory/ExampleCases/ProximalBPS/MacroscopicRange.lean:17published source at ed709a7f320d

Proof architecture

Actual 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.
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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.
Totalized `fderiv` values must not be read as a differentiability theorem.
Almost-everywhere hypotheses depend on the stated measure and representative.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.