gaussian_polynomial_parameter_average_L2

Plain-English statement

Actual scalar parameter mean membership and its L2 class identified with the genuine Bochner feature mean. Independent fresh compilation and whole-module current-context semantic review accepted. Final root/reader gates and postmerge purification separate. No spatial gradient regularity, SDE or stochastic convergence claim.

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_polynomial_parameter_average_L2
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (p : Polynomial ℝ) :
    ∃ hfeature : ∀ w : {w : E // ‖w‖ = 1},
      MemLp (fun x : E => p.eval ⟪w.1, x⟫_ℝ) 2 (ProbabilityTheory.stdGaussian E),
      Continuous (fun w : {w : E // ‖w‖ = 1} =>
        (hfeature w).toLp (fun x : E => p.eval ⟪w.1, x⟫_ℝ)) ∧
      ∀ rho : Measure {w : E // ‖w‖ = 1}, IsProbabilityMeasure rho →
        (∀ x : E, Integrable (fun w : {w : E // ‖w‖ = 1} => p.eval ⟪w.1, x⟫_ℝ) rho) ∧
        Integrable (fun w : {w : E // ‖w‖ = 1} =>
          (hfeature w).toLp (fun x : E => p.eval ⟪w.1, x⟫_ℝ)) rho ∧
        ∃ hmean : MemLp (fun x : E => ∫ w : {w : E // ‖w‖ = 1},
            p.eval ⟪w.1, x⟫_ℝ ∂rho) 2 (ProbabilityTheory.stdGaussian E),
          hmean.toLp (fun x : E => ∫ w : {w : E // ‖w‖ = 1}, p.eval ⟪w.1, x⟫_ℝ ∂rho) =
            ∫ w : {w : E // ‖w‖ = 1},
              (hfeature w).toLp (fun x : E => p.eval ⟪w.1, x⟫_ℝ) ∂rho := by
  classical
  letI : CompleteSpace E := FiniteDimensional.complete ℝ E
  letI : SecondCountableTopology E := inferInstance
  let μ := stdGaussian E
  obtain ⟨B,hB,hBpos,hBbound⟩ := polynomial_norm_envelope (E := E) p
  obtain ⟨hf,hcont⟩ := AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialAveraging.continuous_gaussian_polynomial_ridge_L2 (E := E) p
  obtain ⟨hb, hbpos, hf8, hnorm⟩ :=
    AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialRidge.standard_gaussian_polynomial_ridge_uniform_L2 (E := E) p
  let U := {w : E // ‖w‖ = 1}
  let K : U → E → ℝ := fun w x => p.eval ⟪w.1,x⟫_ℝ
  let F := fun w : U => (hf w).toLp (K w)
  have hFcont : Continuous F := hcont
  have hFbound (w : U) : ‖F w‖ ≤ ‖hb.toLp (fun z : ℝ => p.eval z)‖ + 1 :=
    (hnorm w).2
  have hKcont : Continuous (fun z : U × E => K z.1 z.2) := by
    change Continuous (fun z : {w : E // ‖w‖ = 1} × E => p.eval ⟪z.1.1,z.2⟫_ℝ)
    fun_prop
  refine ⟨hf, hcont, ?_⟩
  intro rho hrho
  letI : IsProbabilityMeasure rho := hrho
  have hscalar (x : E) : Integrable (fun w : U => K w x) rho := by
    apply (integrable_const (B x)).mono' (by fun_prop)
    exact ae_of_all _ fun w => hBbound w x
  have hFint : Integrable F rho := by
    apply (integrable_const (‖hb.toLp (fun z : ℝ => p.eval z)‖ + 1)).mono'
      hFcont.aestronglyMeasurable
    exact ae_of_all _ hFbound
  let mean : E → ℝ := fun x => ∫ w, K w x ∂rho
  have hmeanmeas : StronglyMeasurable mean :=
    hKcont.stronglyMeasurable.integral_prod_left'
  have hmeanbound (x : E) : ‖mean x‖ ≤ B x := by
    have h := norm_integral_le_of_norm_le_const (μ := rho) (f := fun w : U => K w x)
      (ae_of_all _ fun w => hBbound w x)
    simpa only [probReal_univ, mul_one] using h
  have hmean : MemLp mean 2 μ := hB.mono' hmeanmeas.aestronglyMeasurable
    (ae_of_all _ hmeanbound)
  refine ⟨hscalar,hFint,hmean,?_⟩
  apply ext_inner_left ℝ
  intro g
  have hC : Integrable (fun x : E => ‖g x‖ * B x) μ :=
    (Lp.memLp g).norm.integrable_mul hB
  have hHmeas : AEStronglyMeasurable (fun z : U × E => g z.2 * K z.1 z.2) (rho.prod μ) :=
    (Lp.aestronglyMeasurable g).comp_snd.mul hKcont.aestronglyMeasurable
  have hHint : Integrable (fun z : U × E => g z.2 * K z.1 z.2) (rho.prod μ) := by
    apply (hC.comp_snd rho).mono' hHmeas
    exact ae_of_all _ fun z => by
      rw [norm_mul]
      exact mul_le_mul_of_nonneg_left (hBbound z.1 z.2) (norm_nonneg _)
  have hin (w : U) : ⟪g,F w⟫_ℝ = ∫ x, g x * K w x ∂μ := by
    rw [L2.inner_def]
    apply integral_congr_ae
    filter_upwards [(hf w).coeFn_toLp] with x hx
    change inner ℝ (g x) ((hf w).toLp (K w) x) = _
    rw [hx]
    simp only [RCLike.inner_apply, conj_trivial, K, mul_comm]
  calc
    ⟪g,hmean.toLp mean⟫_ℝ = ∫ x, g x * mean x ∂μ := by
      rw [L2.inner_def]
      apply integral_congr_ae
      filter_upwards [hmean.coeFn_toLp] with x hx
      rw [hx]
      simp only [RCLike.inner_apply, conj_trivial, K, mul_comm]
    _ = ∫ x, ∫ w, g x * K w x ∂rho ∂μ := by
      apply integral_congr_ae
      exact ae_of_all _ fun x => (integral_const_mul (g x) (fun w : U => K w x)).symm
    _ = ∫ w, ∫ x, g x * K w x ∂μ ∂rho := (integral_integral_swap hHint).symm
    _ = ∫ w, ⟪g,F w⟫_ℝ ∂rho := integral_congr_ae (ae_of_all _ fun w => (hin w).symm)
    _ = ⟪g,∫ w, F w ∂rho⟫_ℝ := integral_inner hFint g

end AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialAveraging

Open AutoSamplingTheory/TechnicalLemmas/Probability/GaussianPolynomialAveraging.lean:98published source at ed709a7f320d

Proof architecture

Generic probability-law calculus and feature averaging; no model completion

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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.
Almost-everywhere hypotheses depend on the stated measure and representative.