actual_macroscopic_l2_mean

Plain-English statement

Actual SAME Gibbs/Gaussian mu/J/nu and every-y literal reflected source S internally produce canonical snd isometry M and bounded T BEFORE all rough L2 inputs. PM=M, MT=AM, source conditional mean and fiber L1/squareL1 only AE; actual integrated variance is the joint reflection block/norm defect. Original C2/two Hessian/capped eta retained; finite Hilbert/rank0 extension disclosed. Actual rough same-T difference and noncentered rank0 class1 fixed/variance0 tested. Five real existing parents, no new private/background copy/caller law/domain/cost premise. Rough gradient/weakH1/B13/Gamma/dynamics/main/errors/cost/composition remain open. Independently verified 4f71a36d56500fda7f86e8080f695a514913950a.

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_l2_mean
    {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 Λ := J.map (fun p : E × E => (p.2,(2 : ℝ) • p.1-p.2))
    let F := fun p : E × E => (p.1,(2 : ℝ) • p.1-p.2)
    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 μ ∧ IsProbabilityMeasure J ∧ IsProbabilityMeasure ν ∧
    ∃ S : Kernel E E, IsMarkovKernel S ∧
      (∀ y, S y = (volume : Measure E).tilted
        (fun x => -V ((1/2 : ℝ) • (y+x)) - ‖y-x‖^2/(8*η))) ∧
      Λ.IsCondKernel S ∧ Λ.fst = ν ∧ Λ.snd = ν ∧
      ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
        (∀ g : Lp ℝ 2 J, (U g : E × E → ℝ) =ᵐ[J] g ∘ F) ∧
        Function.Involutive U ∧ IsSelfAdjoint U.toContinuousLinearMap ∧
        ∃ M : Lp ℝ 2 ν →ₗᵢ[ℝ] Lp ℝ 2 J,
          (∀ u : Lp ℝ 2 ν, (M u : E × E → ℝ) =ᵐ[J] u ∘ Prod.snd) ∧
          ∃ T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν,
            let A := P * U.toContinuousLinearMap * P
            let B := (1-P) * U.toContinuousLinearMap * P
            ∀ u : Lp ℝ 2 ν,
              P (M u) = M u ∧ M (T u) = A (M u) ∧ ‖T u‖ ≤ ‖u‖ ∧
              (T u : E → ℝ) =ᵐ[ν] (fun y => ∫ x, u x ∂S y) ∧
              (∀ᵐ y ∂ν, Integrable (u : E → ℝ) (S y) ∧
                Integrable (fun x => (u x)^2) (S y)) ∧
              Integrable (fun y =>
                AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance
                  (S y) (u : E → ℝ)) ν ∧
              (∫ y, AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance
                (S y) (u : E → ℝ) ∂ν) = ‖B (M u)‖^2 ∧
              ‖B (M u)‖^2 = ‖u‖^2-‖T u‖^2 := by
  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 Λ := J.map (fun p : E × E => (p.2,(2 : ℝ) • p.1-p.2))
  let F := fun p : E × E => (p.1,(2 : ℝ) • p.1-p.2)
  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
  obtain ⟨hμ,hJ,hν,R,S,hR,hS,hRc,hSr,hSd,hSc,hSf,U,hU,hUi,hUs,hops⟩ :=
    MacroscopicEnergy.actual_macroscopic_gradient_energy_blocks hα hαβ hV hH hη hβη
  let : IsProbabilityMeasure μ := hμ
  let : IsProbabilityMeasure J := hJ
  let : IsProbabilityMeasure ν := hν
  obtain ⟨hBA,hBD,hdef,hcompact⟩ := hops
  let A := P * U.toContinuousLinearMap * P
  have 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
  have hM (u : Lp ℝ 2 ν) : (M u : E × E → ℝ) =ᵐ[J] u ∘ Prod.snd :=
    Lp.coeFn_compMeasurePreserving u hp
  obtain ⟨D,hDense,hClose,hClosed,hGraph⟩ :=
    AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.GaussianMarginalGradient.gaussian_marginal_gradient_closable μ hη
  have hCore (u : Lp ℝ 2 ν) (hu : u ∈ D.domain) :
      P (M u) = M u ∧ A (M u) ∈ M.toLinearMap.range := by
    obtain ⟨f,hf,hfc,huf,hdf⟩ := (hGraph u (D ⟨u,hu⟩)).mp (D.mem_graph ⟨u,hu⟩)
    obtain ⟨hTd,hfLp,hTfLp,hgrad,hvar,g,hg,hPg,hAg,hn,hAn,hvb,hEnergy⟩ := hcompact f hf hfc
    have hufJ : (fun p : E × E => u p.2) =ᵐ[J] (fun p => f p.2) :=
      ae_of_ae_map measurable_snd.aemeasurable huf
    have hMg : M u = g := Lp.ext ((hM u).trans (hufJ.trans hg.symm))
    refine ⟨by rw [hMg]; exact hPg, ?_⟩
    let v := hTfLp.toLp (fun y => ∫ x, f x ∂S y)
    have hvJ : (fun p : E × E => v p.2) =ᵐ[J]
        (fun p => ∫ x, f x ∂S p.2) :=
      ae_of_ae_map measurable_snd.aemeasurable hTfLp.coeFn_toLp
    have hAv : A (M u) = M v := by
      rw [hMg]
      exact Lp.ext (hAg.trans ((hM v).trans hvJ).symm)
    exact ⟨v,hAv.symm⟩
  have hPM (u : Lp ℝ 2 ν) : P (M u) = M u := by
    exact hDense.induction (P := fun v => P (M v) = M v)
      (fun v hv => (hCore v hv).1)
      (isClosed_eq (P.continuous.comp M.continuous) M.continuous) u
  have hAM (u : Lp ℝ 2 ν) : A (M u) ∈ M.toLinearMap.range := by
    exact hDense.induction (P := fun v => A (M v) ∈ M.toLinearMap.range)
      (fun v hv => (hCore v hv).2)
      (M.isometry.isClosedEmbedding.isClosed_range.preimage (A.continuous.comp M.continuous)) u
  let Ar := (A.comp M.toContinuousLinearMap).codRestrict M.toLinearMap.range hAM
  let T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν :=
    M.equivRange.symm.toContinuousLinearEquiv.toContinuousLinearMap.comp Ar
  have hMT (u : Lp ℝ 2 ν) : M (T u) = A (M u) := by
    have he := M.equivRange.apply_symm_apply (Ar u)
    exact congrArg Subtype.val he
  have hPn (v : Lp ℝ 2 J) : ‖P v‖ ≤ ‖v‖ :=
    norm_condExpL2_coe_le measurable_snd.comap_le v
  have hTn (u : Lp ℝ 2 ν) : ‖T u‖ ≤ ‖u‖ := by
    calc
      ‖T u‖ = ‖M (T u)‖ := (M.norm_map _).symm
      _ = ‖A (M u)‖ := by rw [hMT]
      _ = ‖P (U (M u))‖ := by
        change ‖P (U (P (M u)))‖ = ‖P (U (M u))‖
        rw [hPM]
      _ ≤ ‖U (M u)‖ := hPn _
      _ = ‖u‖ := by rw [U.norm_map,M.norm_map]

  let : IsMarkovKernel R := hR
  let : IsMarkovKernel S := hS
  let : (J.map Prod.swap).IsCondKernel R := hRc
  let : Λ.IsCondKernel S := hSc
  obtain ⟨hFi,hFmap⟩ := GaussianReflection.reflection_preserves_augmentation μ η hη
  have hFm : Measurable F := by fun_prop
  have hFp : MeasurePreserving F J J := ⟨hFm,hFmap⟩
  have hΛsnd : Λ.snd = ν := by
    calc
      Λ.snd = (J.map F).snd := by
        dsimp only [Λ,Measure.snd]
        rw [Measure.map_map measurable_snd (by fun_prop),Measure.map_map measurable_snd hFm]
        rfl
      _ = ν := by rw [hFmap]
  have hcompS : ν ⊗ₘ S = Λ := by
    calc
      ν ⊗ₘ S = Λ.fst ⊗ₘ S := congrArg (fun ρ => ρ ⊗ₘ S) hSf.symm
      _ = Λ := Measure.disintegrate Λ S
  have hstation : S ∘ₘ ν = ν := by
    rw [← Measure.snd_compProd,hcompS]
    exact hΛsnd
  obtain ⟨R₀,hR₀,hR₀d,hR₀c⟩ :=
    AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel.exists_tilted_isCondKernel μ hη
  let : IsMarkovKernel R₀ := hR₀
  let : (J.map Prod.swap).IsCondKernel R₀ := hR₀c
  obtain ⟨R₁,hR₁,hR₁d,U₁,hU₁,hUi₁,hUs₁,hPk,hAlgebra₁⟩ :=
    ReflectionL2.actual_reflection_block_identities μ hη
  have hR₁₀ : R₁ = R₀ := by
    ext y s hs
    rw [hR₁d y,hR₀d y]
  have hR₀R : R₀ =ᵐ[ν] R := by
    have h₀ : R₀ =ᵐ[(J.map Prod.swap).fst] (J.map Prod.swap).condKernel := eq_condKernel_of_measure_eq_compProd R₀ (Measure.disintegrate (J.map Prod.swap) R₀).symm
    have hr : R =ᵐ[(J.map Prod.swap).fst] (J.map Prod.swap).condKernel := eq_condKernel_of_measure_eq_compProd R (Measure.disintegrate (J.map Prod.swap) R).symm
    simpa only [Measure.fst_map_swap] using (Filter.EventuallyEq.trans h₀ (Filter.EventuallyEq.symm hr))
  have hR₁R : R₁ =ᵐ[ν] R := by rw [hR₁₀]; exact hR₀R
  have hSource (u : Lp ℝ 2 ν) :
      (T u : E → ℝ) =ᵐ[ν] (fun y => ∫ x, u x ∂S y) := by
    have huS : Integrable (u : E → ℝ) (S ∘ₘ ν) := by
      rw [hstation]
      exact (Lp.memLp u).integrable (by norm_num)
    have hfiberS := Measure.ae_integrable_of_integrable_comp huS
    have hUM : (U (M u) : E × E → ℝ) =ᵐ[J]
        (fun p => u ((2 : ℝ) • p.1-p.2)) :=
      (hU (M u)).trans (hFp.quasiMeasurePreserving.ae_eq (hM u))
    have hswap : (fun p : E × E => (U (M u)) (p.2,p.1)) =ᵐ[J.map Prod.swap]
        (fun p => u ((2 : ℝ) • p.2-p.1)) := by
      change ∀ᵐ p ∂J.map (MeasurableEquiv.prodComm : E × E ≃ᵐ E × E),
        (U (M u)) (p.2,p.1) = u ((2 : ℝ) • p.2-p.1)
      rw [← (MeasurableEquiv.prodComm : E × E ≃ᵐ E × E).map_ae]
      change ∀ᵐ p ∂J, (U (M u)) p = u ((2 : ℝ) • p.1-p.2)
      exact hUM
    have hswap' : (fun p : E × E => (U (M u)) (p.2,p.1)) =ᵐ[ν ⊗ₘ R]
        (fun p => u ((2 : ℝ) • p.2-p.1)) := by
      have hcR : ν ⊗ₘ R = J.map Prod.swap := by
        simpa only [Measure.fst_map_swap] using Measure.disintegrate (J.map Prod.swap) R
      rw [hcR]
      exact hswap
    have hfiberR := Measure.ae_ae_of_ae_compProd hswap'
    have hSourceJ : (fun p : E × E => (T u) p.2) =ᵐ[J]
        (fun p => ∫ x, u x ∂S p.2) := by
      have hMTae : (M (T u) : E × E → ℝ) =ᵐ[J]
          (fun p => ∫ x, (U (M u)) (x,p.2) ∂R₁ p.2) := by
        rw [hMT]
        change (P (U (P (M u))) : E × E → ℝ) =ᵐ[J] _
        rw [hPM]
        exact hPk (U (M u))
      have hkJ := ae_of_ae_map measurable_snd.aemeasurable hR₁R
      have hfJ := ae_of_ae_map measurable_snd.aemeasurable hfiberR
      have hfiJ := ae_of_ae_map measurable_snd.aemeasurable hfiberS
      filter_upwards [(hM (T u)).symm.trans hMTae,hkJ,hfJ,hfiJ] with p hmean hk hf hfi
      change (T u) p.2 = ∫ x, (U (M u)) (x,p.2) ∂R₁ p.2 at hmean
      rw [hmean,hk,integral_congr_ae hf,hSr p.2]
      exact (integral_map (by fun_prop) (by simpa only [hSr p.2] using hfi.aestronglyMeasurable)).symm
    exact (ae_map_iff measurable_snd.aemeasurable
      (measurableSet_eq_fun (Lp.stronglyMeasurable (T u)).measurable
        ((Lp.stronglyMeasurable u).integral_kernel.measurable))).mpr hSourceJ
  let B := (1-P) * U.toContinuousLinearMap * P
  refine ⟨hμ,hJ,hν,S,hS,hSd,hSc,hSf,hΛsnd,U,hU,hUi,hUs,M,hM,T,?_⟩
  intro u
  have huS : Integrable (u : E → ℝ) (S ∘ₘ ν) := by
    rw [hstation]
    exact (Lp.memLp u).integrable (by norm_num)
  have huSqS : Integrable (fun x => (u x)^2) (S ∘ₘ ν) := by
    rw [hstation]
    exact (Lp.memLp u).integrable_sq
  have hfS := Measure.ae_integrable_of_integrable_comp huS
  have hfSqS := Measure.ae_integrable_of_integrable_comp huSqS
  have hfs : ∀ᵐ y ∂ν, Integrable (u : E → ℝ) (S y) ∧
      Integrable (fun x => (u x)^2) (S y) := hfS.and hfSqS
  have hmean := hSource u
  let m := fun y => ∫ x, u x ∂S y
  have hmLp : MemLp m 2 ν := MemLp.ae_eq hmean (Lp.memLp (T u))
  have hsqI : Integrable (fun y => ∫ x, (u x)^2 ∂S y) ν := by
    have hi := Measure.integrable_integral_norm_of_integrable_comp huSqS
    simpa only [Real.norm_eq_abs,abs_pow,sq_abs] using hi
  have hvarEq : (fun y => AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance
      (S y) (u : E → ℝ)) =ᵐ[ν] (fun y => (∫ x, (u x)^2 ∂S y)-(m y)^2) := by
    filter_upwards [hfSqS] with y hy
    have hLpY := (memLp_two_iff_integrable_sq (Lp.stronglyMeasurable u).aestronglyMeasurable).mpr hy
    dsimp only [AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance,m]
    rw [← variance_eq_integral hLpY.aemeasurable,variance_eq_sub hLpY]
    rfl
  have hvarI : Integrable (fun y =>
      AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance (S y) (u : E → ℝ)) ν :=
    (hsqI.sub hmLp.integrable_sq).congr hvarEq.symm
  have hsecond : (∫ y, ∫ x, (u x)^2 ∂S y ∂ν) = ∫ x, (u x)^2 ∂ν := by
    have hj : Integrable (fun p : E × E => (u p.2)^2) (ν ⊗ₘ S) := by
      rw [hcompS]
      apply ((Lp.memLp u).comp_measurePreserving
        (show MeasurePreserving (Prod.snd : E × E → E) Λ ν from ⟨measurable_snd,hΛsnd⟩)).integrable_sq
    rw [← Measure.integral_compProd hj,hcompS]
    calc
      (∫ p : E × E, (u p.2)^2 ∂Λ) = ∫ x, (u x)^2 ∂Λ.snd :=
        (integral_map (μ := Λ) measurable_snd.aemeasurable
          (f := fun x => (u x)^2) ((Lp.stronglyMeasurable u).pow 2).aestronglyMeasurable).symm
      _ = ∫ x, (u x)^2 ∂ν := by rw [hΛsnd]
  have hn (v : Lp ℝ 2 ν) : ‖v‖^2 = ∫ y, (v y)^2 ∂ν := by
    calc
      ‖v‖^2 = inner ℝ v v := (real_inner_self_eq_norm_sq v).symm
      _ = ∫ y, (v y)^2 ∂ν := by rw [L2.inner_def]; simp only [real_inner_self_eq_norm_sq,Real.norm_eq_abs,sq_abs]
  have hDef : ‖B (M u)‖^2 = ‖u‖^2-‖T u‖^2 := by
    calc
      ‖B (M u)‖^2 = ‖M u‖^2-‖A (M u)‖^2 := hdef _ (hPM u)
      _ = ‖u‖^2-‖T u‖^2 := by rw [M.norm_map,← hMT,M.norm_map]
  have hvarDef : (∫ y,
      AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.variance (S y) (u : E → ℝ) ∂ν) =
      ‖B (M u)‖^2 := by
    rw [integral_congr_ae hvarEq,integral_sub hsqI hmLp.integrable_sq,hsecond,hDef,hn u,hn (T u)]
    congr 1
    exact (integral_congr_ae (Filter.EventuallyEq.fun_comp hmean (fun x : ℝ => x^2))).symm
  exact ⟨hPM u,hMT u,hTn u,hmean,hfs,hvarI,hvarDef,hDef⟩

end AutoSamplingTheory.ExampleCases.ProximalBPS.L2MacroscopicMean

Open AutoSamplingTheory/ExampleCases/ProximalBPS/L2MacroscopicMean.lean:9published source at ed709a7f320d

Proof architecture

Source-specific PBPS integration; 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.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.

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.
Almost-everywhere hypotheses depend on the stated measure and representative.
A Gibbs expression is not automatically a probability law or an invariant law.