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Registry leaf card · pbps.actualL2FlipDuality

actual_l2_flip_duality

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Independent science 25a3e47e3e171f2d5584a6568ce685c6f546e529. Under the original six assumptions, the same global actual PBPS process and the same real L2 expectation family from107 retain all path, Borel, Gibbs, bounded-duality, integrability, contraction and initialization conclusions. The canonical momentum-flip pullback J is an involutive L2 isometry and A_T.adjoint = J A_T J for every nonnegative T, on all real L2 classes. Gaussian reflection and product maps prove phase preservation. Bounded simple representatives give the signed pairing; continuity and two-variable density extend it to all L2, then adjoint uniqueness closes Eq A.3. No new source premise. Restart/Markov/semigroup, strong continuity, separate kernel witnesses, actual position averaging, hypocoercivity, errors, actual expected costs, composition and four complete papers remain open.

Plain-English statement

Independent science 25a3e47e3e171f2d5584a6568ce685c6f546e529. Under the original six assumptions, the same global actual PBPS process and the same real L2 expectation family from107 retain all path, Borel, Gibbs, bounded-duality, integrability, contraction and initialization conclusions. The canonical momentum-flip pullback J is an involutive L2 isometry and A_T.adjoint = J A_T J for every nonnegative T, on all real L2 classes. Gaussian reflection and product maps prove phase preservation. Bounded simple representatives give the signed pairing; continuity and two-variable density extend it to all L2, then adjoint uniqueness closes Eq A.3. No new source premise. Restart/Markov/semigroup, strong continuity, separate kernel witnesses, actual position averaging, hypocoercivity, errors, actual expected costs, composition and four complete papers remain open.

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_l2_flip_duality
    {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_l2_flip_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_l2_flip_statement
  intro c Φ S rate Λ τ next record P ε history Rec Hist readTime Δ Prefix m zAt K qReal q A eventTime I N liveTime d survival weight C Path Ψ Γ ρ R Vy nu
  obtain ⟨Z, hZm, hZa, hZf, hZ0, h106, hBorel, h107⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2Expectation.actual_l2_expectation_contraction
      hα hαβ hV hH hη hβη
  refine ⟨Z, hZm, hZa, hZf, hZ0, h106, hBorel, ?_⟩
  intro y xRef
  obtain ⟨L, hL0, hL⟩ := h107 y xRef
  refine ⟨L, hL0, hL, ?_⟩
  have hRm : Measurable R := by dsimp [R]; fun_prop
  have hneg : Measure.map (fun p : E => -p) (stdGaussian E) = stdGaussian E := by
    simpa using (stdGaussian_map (LinearIsometryEquiv.neg ℝ (E := E)))
  have hR : MeasurePreserving R (nu y) (nu y) := by
    refine ⟨hRm, ?_⟩
    change Measure.map (Prod.map id (fun p : E => -p))
      (((volume : Measure E).tilted (fun x => -Vy y x)).prod (stdGaussian E)) = _
    rw [← Measure.map_prod_map _ _ measurable_id (by fun_prop), hneg, Measure.map_id]
  refine ⟨hR, ?_⟩
  intro J
  have hJrep (f : Lp ℝ 2 (nu y)) :
      (J f : (E × E) → ℝ) =ᵐ[nu y] fun z => f (R z) :=
    Lp.coeFn_compMeasurePreserving f hR
  have hJJ (f : Lp ℝ 2 (nu y)) : J (J f) = f := by
    apply Lp.ext
    filter_upwards [hJrep (J f), hR.quasiMeasurePreserving.ae_eq_comp (hJrep f)] with z hz hz'
    simpa only [Function.comp_def, R, neg_neg] using hz.trans hz'
  refine ⟨hJrep, hJJ, ?_⟩
  intro T
  have hrep (G : (E × E) → ℝ) (hG : MemLp G 2 (nu y)) :
      (L T (hG.toLp G) : (E × E) → ℝ) =ᵐ[nu y]
        fun z => ∫ w, G (Z y xRef z T w) ∂P := by
    obtain ⟨_, ⟨hOp, he⟩, _⟩ := (hL T).2 G hG
    rw [he]
    exact hOp.coeFn_toLp
  have hsimple (f g : Lp.simpleFunc ℝ 2 (nu y)) :
      ⟪(f : Lp ℝ 2 (nu y)), L T g⟫_ℝ = ⟪J g, L T (J f)⟫_ℝ := by
    let F := Lp.simpleFunc.toSimpleFunc f
    let G := Lp.simpleFunc.toSimpleFunc g
    have hF : MemLp F 2 (nu y) := Lp.simpleFunc.memLp f
    have hG : MemLp G 2 (nu y) := Lp.simpleFunc.memLp g
    have hFrep : (f : (E × E) → ℝ) =ᵐ[nu y] F :=
      (Lp.simpleFunc.toSimpleFunc_eq_toFun f).symm
    have hGrep : (g : (E × E) → ℝ) =ᵐ[nu y] G :=
      (Lp.simpleFunc.toSimpleFunc_eq_toFun g).symm
    have hFto : hF.toLp F = (f : Lp ℝ 2 (nu y)) := by
      apply Lp.ext
      exact hF.coeFn_toLp.trans hFrep.symm
    have hGto : hG.toLp G = (g : Lp ℝ 2 (nu y)) := by
      apply Lp.ext
      exact hG.coeFn_toLp.trans hGrep.symm
    have hFR : MemLp (fun z => F (R z)) 2 (nu y) := hF.comp_measurePreserving hR
    have hJfto : J f = hFR.toLp (fun z => F (R z)) := by
      apply Lp.ext
      exact ((hJrep f).trans (hR.quasiMeasurePreserving.ae_eq_comp hFrep)).trans
        hFR.coeFn_toLp.symm
    have hJg : (J g : (E × E) → ℝ) =ᵐ[nu y] fun z => G (R z) :=
      (hJrep g).trans (hR.quasiMeasurePreserving.ae_eq_comp hGrep)
    have hLg := hrep G hG
    rw [hGto] at hLg
    have hLJf := hrep (fun z => F (R z)) hFR
    rw [← hJfto] at hLJf
    obtain ⟨CF, hCF⟩ := F.exists_forall_norm_le
    obtain ⟨CG, hCG⟩ := G.exists_forall_norm_le
    have hBF : ∀ z, |F z| ≤ max CF 0 := by
      intro z
      exact le_trans (by simpa only [Real.norm_eq_abs] using hCF z) (le_max_left _ _)
    have hBG : ∀ z, |G z| ≤ max CG 0 := by
      intro z
      exact le_trans (by simpa only [Real.norm_eq_abs] using hCG z) (le_max_left _ _)
    have hd := ((h106 y xRef T).2 F G (max CF 0) (max CG 0)
      F.measurable G.measurable (le_max_right _ _) (le_max_right _ _) hBF hBG).2.2
    calc
      ⟪(f : Lp ℝ 2 (nu y)), L T g⟫_ℝ =
          ∫ z, F z * (∫ w, G (Z y xRef z T w) ∂P) ∂nu y := by
        rw [L2.inner_def]
        apply integral_congr_ae
        filter_upwards [hFrep, hLg] with z hz hz'
        simp only [RCLike.inner_apply, conj_trivial, hz, hz', mul_comm]
      _ = ∫ z, G (R z) * (∫ w, F (R (Z y xRef z T w)) ∂P) ∂nu y := hd
      _ = ⟪J g, L T (J f)⟫_ℝ := by
        rw [L2.inner_def]
        apply integral_congr_ae
        filter_upwards [hJg, hLJf] with z hz hz'
        simp only [RCLike.inner_apply, conj_trivial, hz, hz', mul_comm]
  have hpair (f g : Lp ℝ 2 (nu y)) :
      ⟪f, L T g⟫_ℝ = ⟪J g, L T (J f)⟫_ℝ := by
    apply (Lp.simpleFunc.denseRange (E := ℝ) (μ := nu y) (p := 2) (by norm_num)).induction_on₂
      (p := fun f g => ⟪f, L T g⟫_ℝ = ⟪J g, L T (J f)⟫_ℝ) _ hsimple f g
    exact isClosed_eq (continuous_fst.inner ((L T).continuous.comp continuous_snd))
      ((J.continuous.comp continuous_snd).inner
        ((L T).continuous.comp (J.continuous.comp continuous_fst)))
  apply ContinuousLinearMap.ext
  intro f
  apply ext_inner_right ℝ
  intro g
  calc
    ⟪(L T).adjoint f, g⟫_ℝ = ⟪f, L T g⟫_ℝ := (L T).adjoint_inner_left g f
    _ = ⟪J g, L T (J f)⟫_ℝ := hpair f g
    _ = ⟪g, J (L T (J f))⟫_ℝ := by
      rw [← J.inner_map_map g (J (L T (J f))), hJJ]
    _ = ⟪J (L T (J f)), g⟫_ℝ := real_inner_comm _ _

#print axioms actual_l2_flip_duality
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2FlipDuality

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualL2FlipDuality.lean:175published source at ed709a7f320d

Proof architecture

Actual same-process full real-L2 flip duality; no SALD claim

Lean proof walkthrough

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  • `rw` rewrites by an established identity.
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Why the statement has this shape

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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.
  • 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.
Common pitfall. A wrapper or display identity is not a new analytic theorem merely because it has its own Lean name. Check the statement, hypotheses, and downstream consumers before interpreting its mathematical contribution.