actual_outer_bounded_l2_continuity

Plain-English statement

- For the actual PBPS phase, bounded continuous tests have a measurable clock expectation whose squared difference from the initial test tends to zero under the exact conditional Gibbs-times-Gaussian initial law. The C_b class is an explicit ASTIS extension of the source C_c ingredient in arXiv2609.06905v1 Appendix A.1 Ex22; this is not the all-L2 operator result.

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_outer_bounded_l2_continuity
    {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) :
    actual_outer_bounded_l2_continuity_statement hα hαβ hV hH hη hβη := by
  classical
  let P : Measure (ℕ → ℝ) := Measure.infinitePi (fun _ : ℕ => expMeasure (1 : ℝ))
  let q : E → Measure E := fun y =>
    (volume : Measure E).tilted (fun x => -V x - ‖x - y‖ ^ 2 / (2 * η))
  let ν : E → Measure (E × E) := fun y => (q y).prod (stdGaussian E)
  obtain ⟨Z, hZM, hcovered, hfallback, hAE, hshort, htest⟩ :=
    ActualBoundedTestContinuity.actual_bounded_test_expectation_continuity
      hα hαβ hV hH hη hβη
  have hZV := (GibbsAugmentation.normalized_augmentation_density
    hα hV (fun x v => (hH x v).1) hη).1
  have hi : Integrable (fun x : E => Real.exp (-V x)) (volume : Measure E) :=
    Integrable.of_integral_ne_zero hZV.ne'
  let μ : Measure E := (volume : Measure E).tilted (fun x => -V x)
  letI : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hi
  obtain ⟨R, hR, hRy, _⟩ :=
    AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel.exists_tilted_isCondKernel μ hη
  letI : IsMarkovKernel R := hR
  have hRq (y : E) : R y = q y := by
    rw [hRy y, show μ = (volume : Measure E).tilted (fun x => -V x) from rfl,
      tilted_tilted hi]
    congr 1
    funext z
    dsimp only [Pi.add_apply]
    ring
  have hqprob (y : E) : IsProbabilityMeasure (q y) := by
    rw [← hRq y]
    infer_instance
  have hνprob (y : E) : IsProbabilityMeasure (ν y) := by
    letI := hqprob y
    dsimp only [ν]
    infer_instance
  letI : IsProbabilityMeasure P :=
    AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws.1
  refine ⟨Z, hZM, hcovered, hfallback, hAE, hshort, htest, ?_⟩
  refine ⟨fun y => ⟨hqprob y, hνprob y⟩, ?_⟩
  intro y xRef f hf M hM hbound
  letI := hνprob y
  let A : ℝ≥0 → (E × E) → ℝ := fun t z₀ => ∫ sample, f (Z y xRef z₀ t sample) ∂P
  have hAM (t : ℝ≥0) : Measurable (A t) := by
    have hargs : Measurable (fun a : (E × E) × (ℕ → ℝ) =>
        (((y, xRef), a.1), t, a.2)) :=
      (measurable_const.prodMk measurable_fst).prodMk (measurable_const.prodMk measurable_snd)
    exact ((hf.measurable.comp (hZM.comp hargs)).stronglyMeasurable.integral_prod_right').measurable
  have hAbound (t : ℝ≥0) (z₀ : E × E) : |A t z₀| ≤ M := by
    have hb : ∀ᵐ sample ∂P, ‖f (Z y xRef z₀ t sample)‖ ≤ M :=
      Filter.Eventually.of_forall fun sample => by
        simpa only [Real.norm_eq_abs] using hbound (Z y xRef z₀ t sample)
    simpa only [A, Real.norm_eq_abs, probReal_univ, mul_one] using
      norm_integral_le_of_norm_le_const hb
  have hsquare (t : ℝ≥0) (z₀ : E × E) : (A t z₀ - f z₀) ^ 2 ≤ 4 * M ^ 2 := by
    have ha := (abs_le.mp (hAbound t z₀))
    have hb := (abs_le.mp (hbound z₀))
    nlinarith [sq_nonneg (A t z₀ - f z₀ - 2 * M),
      mul_nonneg (show 0 ≤ 2 * M - (A t z₀ - f z₀) by linarith)
        (show 0 ≤ 2 * M + (A t z₀ - f z₀) by linarith)]
  have hsM (t : ℝ≥0) : Measurable (fun z₀ : E × E => (A t z₀ - f z₀) ^ 2) :=
    ((hAM t).sub hf.measurable).pow_const 2
  have hsi (t : ℝ≥0) : Integrable (fun z₀ : E × E => (A t z₀ - f z₀) ^ 2) (ν y) := by
    apply Integrable.of_bound (hsM t).aestronglyMeasurable (4 * M ^ 2)
    exact Filter.Eventually.of_forall fun z₀ => by
      simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (A t z₀ - f z₀))] using hsquare t z₀
  refine ⟨fun t => ⟨hAM t, hsi t, hsquare t⟩, ?_⟩
  have hlimit (z₀ : E × E) : Tendsto (fun t : ℝ≥0 => (A t z₀ - f z₀) ^ 2)
      (𝓝 0) (𝓝 0) := by
    have ht : Tendsto (fun t : ℝ≥0 => A t z₀) (𝓝 0) (𝓝 (f z₀)) :=
      (htest y xRef z₀ f hf M hM hbound).2
    have hc : Tendsto (fun _ : ℝ≥0 => f z₀) (𝓝 0) (𝓝 (f z₀)) := tendsto_const_nhds
    simpa only [sub_self, zero_pow (by decide : 2 ≠ 0)] using
      (ht.sub hc).pow 2
  have hDCT := tendsto_integral_filter_of_dominated_convergence (μ := ν y)
    (F := fun t : ℝ≥0 => fun z₀ : E × E => (A t z₀ - f z₀) ^ 2)
    (f := fun _ : E × E => (0 : ℝ)) (fun _ : E × E => 4 * M ^ 2)
    (Filter.Eventually.of_forall fun t => (hsM t).aestronglyMeasurable)
    (Filter.Eventually.of_forall fun t => Filter.Eventually.of_forall fun z₀ => by
      simpa only [Real.norm_eq_abs, abs_of_nonneg (sq_nonneg (A t z₀ - f z₀))] using hsquare t z₀)
    (integrable_const _) (Filter.Eventually.of_forall hlimit)
  simpa only [integral_zero] using hDCT

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualOuterBoundedL2Continuity

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualOuterBoundedL2Continuity.lean:132published source at ed709a7f320d

Proof architecture

Source Ex22 bounded-test outer square-integral ingredient; no SALD claim

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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.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.