continuous_gaussian_polynomial_ridge_L2

Plain-English statement

Actual Gaussian polynomial L2 membership and continuity, with radial domination derived internally. 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 continuous_gaussian_polynomial_ridge_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)) := 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 ⟨hb, hbpos, hf, hnorm⟩ :=
    AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialRidge.standard_gaussian_polynomial_ridge_uniform_L2 (E := E) p
  refine ⟨hf, ?_⟩
  let F := fun w : {w : E // ‖w‖ = 1} => (hf w).toLp (fun x => p.eval ⟪w.1,x⟫_ℝ)
  change Continuous F
  apply continuous_iff_continuousAt.2
  intro w₀
  have hBsq : Integrable (fun x => 4 * (B x)^2) μ :=
    ((memLp_two_iff_integrable_sq hB.1).1 hB).const_mul 4
  have hn (w) : ‖F w - F w₀‖^2 =
      ∫ x, (p.eval ⟪w.1,x⟫_ℝ - p.eval ⟪w₀.1,x⟫_ℝ)^2 ∂μ := by
    rw [← real_inner_self_eq_norm_sq, L2.inner_def]
    apply integral_congr_ae
    filter_upwards [Lp.coeFn_sub (F w) (F w₀), (hf w).coeFn_toLp, (hf w₀).coeFn_toLp] with x hx hw hw₀
    simp only [hx, Pi.sub_apply, F] at *
    rw [hw,hw₀]
    simp [real_inner_self_eq_norm_sq, Real.norm_eq_abs, sq_abs]
  have hc : ContinuousAt (fun w : {w : E // ‖w‖ = 1} =>
      ∫ x, (p.eval ⟪w.1,x⟫_ℝ - p.eval ⟪w₀.1,x⟫_ℝ)^2 ∂μ) w₀ := by
    apply continuousAt_of_dominated (bound := fun x => 4 * (B x)^2)
    · exact Eventually.of_forall fun w => by fun_prop
    · exact Eventually.of_forall fun w => ae_of_all _ fun x => by
        rw [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg _)]
        have h := (norm_sub_le (p.eval ⟪w.1,x⟫_ℝ) (p.eval ⟪w₀.1,x⟫_ℝ)).trans
          (add_le_add (hBbound w x) (hBbound w₀ x))
        have hs := (sq_le_sq₀ (norm_nonneg _) (add_nonneg (hBpos x) (hBpos x))).2 h
        simpa only [Real.norm_eq_abs, sq_abs] using hs.trans_eq (by ring)
    · exact hBsq
    · exact ae_of_all _ fun x => by fun_prop
  have hsq : ContinuousAt (fun w => ‖F w - F w₀‖^2) w₀ := by
    simpa only [← hn] using hc
  rw [ContinuousAt, tendsto_iff_dist_tendsto_zero]
  simpa only [Function.comp_def, Real.sqrt_sq (norm_nonneg _), sub_self, norm_zero, zero_pow (by decide : 2 ≠ 0),
    Real.sqrt_zero, dist_eq_norm] using
    (Real.continuous_sqrt.continuousAt.comp hsq).tendsto

Open AutoSamplingTheory/TechnicalLemmas/Probability/GaussianPolynomialAveraging.lean:49published source at ed709a7f320d

Proof architecture

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

Lean proof walkthrough

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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.