actual_l2_expectation_contraction

Plain-English statement

Independent science 2b657840a2f21c5d8001c83e4625a1ab13ef70ba. Under the original six assumptions, the same global jointly measurable actual PBPS Z retains live arcs, fallback, parameterwise initialization, Gibbs preservation and bounded flip duality. Every Borel test has a joint Borel total clock expectation; a uniform finite bound proves every section L1. For each y,r an internally constructed real L2 contraction family A_T realizes the literal clock expectation of every MemLp representative, with almost-everywhere section L1, output L2 membership, exact quotient identity, square-integral contraction and A_0=id. Actual106, normalization103 and UnitExponentialProduct are the three producers. Adjoint pullbacks, product L1, signed Fubini and indicator uniqueness are internal dependencies. Representative independence is a consequence of universal quotient identification; the planned direct-AE route remains an unselected alternative. Full L2 flip identity A.3, restart/Markov/semigroup, separate kernel witness identification, strong continuity, hypocoercivity, errors, actual expected costs, composition and all four 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_expectation_contraction
    {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_expectation_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_l2_expectation_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⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedGibbsDuality.actual_bounded_gibbs_duality
      hα hαβ hV hH hη hβη
  have hpair_core
      (μ : Measure (E × E)) (ν : Measure (ℕ → ℝ)) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
      (Z : (E × E) × (ℕ → ℝ) → (E × E)) (hZ : MeasurePreserving Z (μ.prod ν) μ) :
      ∃ A : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ, ‖A‖ ≤ 1 ∧
        ∀ f g : Lp ℝ 2 μ,
          ⟪f, A g⟫_ℝ = ∫ p : (E × E) × (ℕ → ℝ), f p.1 * g (Z p) ∂μ.prod ν := by
    have hfst : MeasurePreserving Prod.fst (μ.prod ν) μ := by
      refine ⟨measurable_fst, ?_⟩
      simp only [Measure.map_fst_prod, measure_univ, one_smul]
    let B : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 (μ.prod ν) :=
      (Lp.compMeasurePreservingₗᵢ ℝ Prod.fst hfst).toContinuousLinearMap
    let C : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 (μ.prod ν) :=
      (Lp.compMeasurePreservingₗᵢ ℝ Z hZ).toContinuousLinearMap
    let A : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ := B.adjoint.comp C
    have hB : ‖B‖ ≤ 1 := LinearIsometry.norm_toContinuousLinearMap_le _
    have hC : ‖C‖ ≤ 1 := LinearIsometry.norm_toContinuousLinearMap_le _
    refine ⟨A, ?_, ?_⟩
    · calc
        ‖A‖ ≤ ‖B.adjoint‖ * ‖C‖ := ContinuousLinearMap.opNorm_comp_le _ _
        _ = ‖B‖ * ‖C‖ := by rw [ContinuousLinearMap.adjoint.norm_map]
        _ ≤ 1 * 1 := mul_le_mul hB hC (norm_nonneg _) zero_le_one
        _ = 1 := one_mul 1
    · intro f g
      change ⟪f, B.adjoint (C g)⟫_ℝ = _
      rw [ContinuousLinearMap.adjoint_inner_right, L2.inner_def]
      have hb : (B f : (E × E) × (ℕ → ℝ) → ℝ) =ᵐ[μ.prod ν] (fun p => f p.1) :=
        Lp.coeFn_compMeasurePreserving f hfst
      have hc : (C g : (E × E) × (ℕ → ℝ) → ℝ) =ᵐ[μ.prod ν] (fun p => g (Z p)) :=
        Lp.coeFn_compMeasurePreserving g hZ
      apply integral_congr_ae
      filter_upwards [hb, hc] with p hp hq
      rw [hp, hq]
      simp [RCLike.inner_apply, mul_comm]
  have haverage_core
      (μ : Measure (E × E)) (ν : Measure (ℕ → ℝ)) [IsProbabilityMeasure μ] [IsProbabilityMeasure ν]
      (Z : (E × E) × (ℕ → ℝ) → (E × E)) (hZ : MeasurePreserving Z (μ.prod ν) μ) :
      ∃ A : Lp ℝ 2 μ →L[ℝ] Lp ℝ 2 μ, ‖A‖ ≤ 1 ∧
        ∀ G : (E × E) → ℝ, ∀ hG : MemLp G 2 μ,
          (∀ᵐ x ∂μ, Integrable (fun w => G (Z (x, w))) ν) ∧
          ∃ hOp : MemLp (fun x => ∫ w, G (Z (x, w)) ∂ν) 2 μ,
            A (hG.toLp G) = hOp.toLp (fun x => ∫ w, G (Z (x, w)) ∂ν) := by
    classical
    obtain ⟨A, hA, hpair⟩ := hpair_core μ ν Z hZ
    refine ⟨A, hA, ?_⟩
    intro G hG
    have hfst : MeasurePreserving Prod.fst (μ.prod ν) μ := by
      refine ⟨measurable_fst, ?_⟩
      simp only [Measure.map_fst_prod, measure_univ, one_smul]
    have hprod : Integrable (fun p : (E × E) × (ℕ → ℝ) => G (Z p)) (μ.prod ν) :=
      (hG.comp_measurePreserving hZ).integrable (by norm_num)
    have hop : Integrable (fun x => ∫ w, G (Z (x, w)) ∂ν) μ := hprod.integral_prod_left
    have hAg : Integrable (A (hG.toLp G) : (E × E) → ℝ) μ :=
      (Lp.memLp (A (hG.toLp G))).integrable (by norm_num)
    have heq : (A (hG.toLp G) : (E × E) → ℝ) =ᵐ[μ] (fun x => ∫ w, G (Z (x, w)) ∂ν) := by
      apply Integrable.ae_eq_of_forall_setIntegral_eq _ _ hAg hop
      intro s hs hms
      let i : Lp ℝ 2 μ := indicatorConstLp 2 hs hms.ne (1 : ℝ)
      have hi : (i : (E × E) → ℝ) =ᵐ[μ] s.indicator (fun _ => (1 : ℝ)) := indicatorConstLp_coeFn
      have hi' := hfst.quasiMeasurePreserving.ae_eq hi
      have hg' := hZ.quasiMeasurePreserving.ae_eq hG.coeFn_toLp
      let H : (E × E) × (ℕ → ℝ) → ℝ := fun p => (s.indicator (fun _ => (1 : ℝ)) p.1) * G (Z p)
      have hH : Integrable H (μ.prod ν) := by
        have hfun : H = (Prod.fst ⁻¹' s).indicator (fun p : (E × E) × (ℕ → ℝ) => G (Z p)) := by
          funext p
          by_cases hp : p.1 ∈ s <;> simp [H, Set.indicator, hp]
        rw [hfun]
        exact hprod.indicator (hs.preimage measurable_fst)
      calc
        (∫ x in s, A (hG.toLp G) x ∂μ) = ⟪i, A (hG.toLp G)⟫_ℝ :=
          (L2.inner_indicatorConstLp_one hs hms.ne _).symm
        _ = ∫ p : (E × E) × (ℕ → ℝ), i p.1 * (hG.toLp G) (Z p) ∂μ.prod ν := hpair i (hG.toLp G)
        _ = ∫ p : (E × E) × (ℕ → ℝ), H p ∂μ.prod ν := by
          apply integral_congr_ae
          filter_upwards [hi', hg'] with p hp hq
          change i p.1 = s.indicator (fun _ => (1 : ℝ)) p.1 at hp
          change (hG.toLp G) (Z p) = G (Z p) at hq
          dsimp only [H]
          rw [hp, hq]
        _ = ∫ x, ∫ w, H (x, w) ∂ν ∂μ := integral_prod H hH
        _ = ∫ x, s.indicator (fun _ => (1 : ℝ)) x * (∫ w, G (Z (x, w)) ∂ν) ∂μ := by
          simp only [H, integral_const_mul]
        _ = ∫ x in s, (∫ w, G (Z (x, w)) ∂ν) ∂μ := by
          rw [← integral_indicator hs]
          congr 1
          funext x
          by_cases hx : x ∈ s <;> simp [Set.indicator, hx]
    have hOp : MemLp (fun x => ∫ w, G (Z (x, w)) ∂ν) 2 μ := MemLp.ae_eq heq (Lp.memLp (A (hG.toLp G)))
    refine ⟨hprod.prod_right_ae, hOp, ?_⟩
    apply Lp.ext
    exact heq.trans hOp.coeFn_toLp.symm
  have hsquare_core
      (μ : Measure (E × E)) (G : (E × E) → ℝ) (hG : MemLp G 2 μ) :
      (∫ x, |G x| ^ 2 ∂μ) = ‖hG.toLp G‖ ^ 2 := by
    calc
      (∫ x, |G x| ^ 2 ∂μ) = ∫ x, ⟪(hG.toLp G) x, (hG.toLp G) x⟫_ℝ ∂μ := by
        apply integral_congr_ae
        filter_upwards [hG.coeFn_toLp] with x hx
        rw [hx]
        simp [RCLike.inner_apply, sq_abs, pow_two]
      _ = ⟪hG.toLp G, hG.toLp G⟫_ℝ := rfl
      _ = ‖hG.toLp G‖ ^ 2 := real_inner_self_eq_norm_sq _
  refine ⟨Z, hZm, hZa, hZf, hZ0, h106, ?_, ?_⟩
  · letI : IsProbabilityMeasure P :=
      AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws.1
    intro G hG
    constructor
    · have hmap : Measurable (fun a : (((E × E) × (E × E)) × ℝ≥0) × (ℕ → ℝ) =>
          G (Z a.1.1.1.1 a.1.1.1.2 a.1.1.2 a.1.2 a.2)) :=
        hG.comp (hZm.comp (measurable_fst.fst.prodMk
          (measurable_fst.snd.prodMk measurable_snd)))
      exact hmap.stronglyMeasurable.integral_prod_right.measurable
    · intro C hC hbound y xRef z T
      have hemb : Measurable (fun w : ℕ → ℝ => (((y, xRef), z), T, w)) :=
        measurable_const.prodMk (measurable_const.prodMk measurable_id)
      have hm : Measurable (fun w => G (Z y xRef z T w)) := hG.comp (hZm.comp hemb)
      have hb : ∀ᵐ w ∂P, ‖G (Z y xRef z T w)‖ ≤ C :=
        Filter.Eventually.of_forall fun w => by simpa only [Real.norm_eq_abs] using hbound (Z y xRef z T w)
      refine ⟨Integrable.of_bound hm.aestronglyMeasurable C hb, ?_⟩
      simpa using norm_integral_le_of_norm_le_const hb
  · intro y xRef
    have h103 := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseNormalization.actual_phase_gibbs_normalization
      hα hαβ hV hH hη hβη y
    letI : IsProbabilityMeasure (nu y) := by
      have hp := h103.2.2.2.2.2.1
      rw [h103.2.2.2.2.2.2] at hp
      exact hp
    letI : IsProbabilityMeasure P :=
      AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws.1
    have hfor : ∀ T : ℝ≥0,
        ∃ L : Lp ℝ 2 (nu y) →L[ℝ] Lp ℝ 2 (nu y), ‖L‖ ≤ 1 ∧
          ∀ G : (E × E) → ℝ, ∀ hG : MemLp G 2 (nu y),
            (∀ᵐ z ∂nu y, Integrable (fun w => G (Z y xRef z T w)) P) ∧
            ∃ hOp : MemLp (fun z => ∫ w, G (Z y xRef z T w) ∂P) 2 (nu y),
              L (hG.toLp G) = hOp.toLp (fun z => ∫ w, G (Z y xRef z T w) ∂P) := by
      intro T
      exact haverage_core (nu y) P (fun a => Z y xRef a.1 T a.2) (h106 y xRef T).1
    choose L hL hrep using hfor
    refine ⟨L, ?_, ?_⟩
    · apply ContinuousLinearMap.ext
      intro f
      obtain ⟨_, hm, he⟩ := hrep 0 (f : (E × E) → ℝ) (Lp.memLp f)
      rw [Lp.toLp_coeFn] at he
      apply Lp.ext
      change (L 0 f : (E × E) → ℝ) =ᵐ[nu y] (f : (E × E) → ℝ)
      have hpoint : ∀ z, (∫ w, f (Z y xRef z 0 w) ∂P) = f z := by
        intro z
        exact integral_eq_const ((hZ0 y xRef z).mono (fun w hw => congrArg f hw))
      rw [he]
      exact hm.coeFn_toLp.trans (Filter.Eventually.of_forall hpoint)
    · intro T
      refine ⟨hL T, ?_⟩
      intro G hG opG
      obtain ⟨hsec, hOp, he⟩ := hrep T G hG
      refine ⟨hsec, ⟨hOp, he⟩, ?_⟩
      rw [hsquare_core (nu y) opG hOp, hsquare_core (nu y) G hG, ← he]
      have hn : ‖L T (hG.toLp G)‖ ≤ ‖hG.toLp G‖ :=
        (ContinuousLinearMap.le_opNorm (L T) (hG.toLp G)).trans
          (mul_le_of_le_one_left (norm_nonneg _) (hL T))
      nlinarith [norm_nonneg (L T (hG.toLp G)), norm_nonneg (hG.toLp G)]

#print axioms actual_l2_expectation_contraction
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2Expectation

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualL2Expectation.lean:168published source at ed709a7f320d

Proof architecture

Actual same-process L2 expectation contraction; no SALD claim

Lean proof walkthrough

Read the quantified variables and typeclass brackets as part of the mathematical statement; inferred arguments are not missing assumptions.
`intro` introduces quantified hypotheses into the local proof context.
`have` creates a named intermediate mathematical fact.
`calc` records an equality or inequality chain matching a paper calculation.
`rw` rewrites by an established identity.
`simp` normalizes through registered definitional and theorem rewrites.
`apply` reduces the goal to the hypotheses of a reusable theorem.
`refine` instantiates a reusable theorem while leaving explicit subgoals.

Why the statement has this shape

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.
A Gibbs expression is not automatically a probability law or an invariant law.