actual_gaussian_polynomial_ridge_L2_derivative_bounds

Plain-English statement

Same genuine Gaussian derivative with positive uniform operator and operator-Lipschitz constants on the closed unit ball. Constants may depend on the polynomial and ambient space. Fresh independent source and compiler audits accepted. Final root/reader gates and postmerge purification remain separate; no whole-model or convergence credit.

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_gaussian_polynomial_ridge_L2_derivative_bounds
    {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (p : Polynomial ℝ) :
    let γ := ProbabilityTheory.stdGaussian E
    let f : E → E → ℝ := fun w x => p.eval ⟪w,x⟫_ℝ
    let df : E → E → E → ℝ := fun w v x => p.derivative.eval ⟪w,x⟫_ℝ * ⟪v,x⟫_ℝ
    ∃ hf : ∀ w, MemLp (f w) 2 γ,
    ∃ hdf : ∀ w v, MemLp (df w v) 2 γ,
    ∃ D : E → E →L[ℝ] Lp ℝ 2 γ,
      (∀ w v, D w v = (hdf w v).toLp (df w v)) ∧
      (∀ w, HasFDerivAt (fun u => (hf u).toLp (f u)) (D w) w) ∧
      ∃ M₁ M₂ : ℝ, 0 < M₁ ∧ 0 < M₂ ∧
        (∀ w : E, ‖w‖ ≤ 1 → ‖D w‖ ≤ M₁) ∧
        ∀ w u : E, ‖w‖ ≤ 1 → ‖u‖ ≤ 1 → ‖D u - D w‖ ≤ M₂ * ‖u-w‖ := by
  dsimp only
  obtain ⟨hf,hdf,D,hD,hDiff⟩ := actual_gaussian_polynomial_ridge_L2_frechet_derivative (E := E) p
  exact ⟨hf,hdf,D,hD,hDiff,produced_operator_bounds p hdf D hD⟩


end AutoSamplingTheory.TechnicalLemmas.Probability.GaussianPolynomialDifferentiation

Open AutoSamplingTheory/TechnicalLemmas/Probability/GaussianPolynomialDifferentiation.lean:374published source at ed709a7f320d

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Measurability is represented explicitly or must be supplied by a dependency.
Totalized `fderiv` values must not be read as a differentiability theorem.