actual_position_l2_averaging

Plain-English statement

Independent science 86e2da02b6b1a855916adcd46363e34d8a6ecb5b. Under the original six assumptions, retain the same108 global actual process Z, phase family A and canonical flip J. Construct the position lift B and Gaussian adjoint M=B*. For every arbitrary MemLp phase representative, derive Gaussian section integrability, output L2 membership and the literal Gaussian mean class. Prove JB=B and MJ=M, then K_T=M A_T B is selfadjoint, norm at most1 and K_0=id. Identify its literal Gaussian-outer clock-inner position expectation, with both nested AE integrability statements. Rank0 included. No new source premise. Markov positivity/constants, strong continuity, restart/semigroup, separate kernel witnesses, independent-reference averaging, hypocoercivity, errors, actual expected costs, composition and all 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_position_l2_averaging
    {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_position_l2_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_position_l2_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, h108⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2FlipDuality.actual_l2_flip_duality
      hα hαβ hV hH hη hβη
  refine ⟨Z, hZm, hZa, hZf, hZ0, h106, hBorel, ?_⟩
  intro y xRef
  obtain ⟨L, hL0, hL, hR, hJrep, hJJ, hLadj⟩ := h108 y xRef
  refine ⟨L, hL0, hL, hR, ?_⟩
  intro J
  refine ⟨hJrep, hJJ, hLadj, ?_⟩
  intro mu
  have hfst : MeasurePreserving Prod.fst (nu y) mu := measurePreserving_fst
  refine ⟨hfst, ?_⟩
  intro B M
  have hBrep (f : Lp ℝ 2 mu) :
      (B f : (E × E) → ℝ) =ᵐ[nu y] fun z => f z.1 :=
    Lp.coeFn_compMeasurePreserving f hfst
  have hBn : ‖B.toContinuousLinearMap‖ ≤ 1 := B.norm_toContinuousLinearMap_le
  have hMn : ‖M‖ ≤ 1 := by
    change ‖B.toContinuousLinearMap.adjoint‖ ≤ 1
    rw [ContinuousLinearMap.adjoint.norm_map]
    exact hBn
  have hMB : M.comp B.toContinuousLinearMap = ContinuousLinearMap.id ℝ (Lp ℝ 2 mu) :=
    B.adjoint_comp_self
  have havg (G : (E × E) → ℝ) (hG : MemLp G 2 (nu y)) :
      (∀ᵐ x ∂mu, Integrable (fun p => G (x, p)) (stdGaussian E)) ∧
      ∃ hbar : MemLp (fun x => ∫ p, G (x, p) ∂stdGaussian E) 2 mu,
        M (hG.toLp G) = hbar.toLp (fun x => ∫ p, G (x, p) ∂stdGaussian E) := by
    have hprod : Integrable G (mu.prod (stdGaussian E)) := hG.integrable (by norm_num)
    have hbar1 : Integrable (fun x => ∫ p, G (x, p) ∂stdGaussian E) mu :=
      hprod.integral_prod_left
    have hMg : Integrable (M (hG.toLp G) : E → ℝ) mu :=
      (Lp.memLp (M (hG.toLp G))).integrable (by norm_num)
    have heq : (M (hG.toLp G) : E → ℝ) =ᵐ[mu]
        (fun x => ∫ p, G (x, p) ∂stdGaussian E) := by
      apply Integrable.ae_eq_of_forall_setIntegral_eq _ _ hMg hbar1
      intro s hs hms
      let i : Lp ℝ 2 mu := indicatorConstLp 2 hs hms.ne (1 : ℝ)
      have hi : (i : E → ℝ) =ᵐ[mu] s.indicator (fun _ => (1 : ℝ)) :=
        indicatorConstLp_coeFn
      have hi' := hfst.quasiMeasurePreserving.ae_eq hi
      let H : E × E → ℝ := fun z => s.indicator (fun _ => (1 : ℝ)) z.1 * G z
      have hH : Integrable H (mu.prod (stdGaussian E)) := by
        have hfun : H = (Prod.fst ⁻¹' s).indicator G := by
          funext z
          by_cases hz : z.1 ∈ s <;> simp [H, Set.indicator, hz]
        rw [hfun]
        exact hprod.indicator (hs.preimage measurable_fst)
      calc
        (∫ x in s, M (hG.toLp G) x ∂mu) = ⟪i, M (hG.toLp G)⟫_ℝ :=
          (L2.inner_indicatorConstLp_one hs hms.ne _).symm
        _ = ⟪B i, hG.toLp G⟫_ℝ := ContinuousLinearMap.adjoint_inner_right _ _ _
        _ = ∫ z, H z ∂nu y := by
          rw [L2.inner_def]
          apply integral_congr_ae
          filter_upwards [hBrep i, hi', hG.coeFn_toLp] with z hb hi hg
          change i z.1 = s.indicator (fun _ => (1 : ℝ)) z.1 at hi
          simp only [hb, hi, hg, H, RCLike.inner_apply, conj_trivial]
          exact mul_comm _ _
        _ = ∫ x, ∫ p, H (x, p) ∂stdGaussian E ∂mu := integral_prod H hH
        _ = ∫ x, s.indicator (fun _ => (1 : ℝ)) x *
            (∫ p, G (x, p) ∂stdGaussian E) ∂mu := by
          simp only [H, integral_const_mul]
        _ = ∫ x in s, (∫ p, G (x, p) ∂stdGaussian E) ∂mu := by
          rw [← integral_indicator hs]
          congr 1
          funext x
          by_cases hx : x ∈ s <;> simp [Set.indicator, hx]
    have hbar : MemLp (fun x => ∫ p, G (x, p) ∂stdGaussian E) 2 mu :=
      MemLp.ae_eq heq (Lp.memLp (M (hG.toLp G)))
    refine ⟨hprod.prod_right_ae, hbar, ?_⟩
    apply Lp.ext
    exact heq.trans hbar.coeFn_toLp.symm
  have hJB (f : Lp ℝ 2 mu) : J (B f) = B f := by
    apply Lp.ext
    filter_upwards [hJrep (B f), hR.quasiMeasurePreserving.ae_eq_comp (hBrep f),
      hBrep f] with z hj hb hb'
    change (J (B f)) z = (B f) z
    rw [hj, hb']
    simpa only [Function.comp_def, R] using hb
  have hJadj : J.toContinuousLinearMap.adjoint = J.toContinuousLinearMap := by
    apply ContinuousLinearMap.ext
    intro f
    apply ext_inner_right ℝ
    intro g
    rw [ContinuousLinearMap.adjoint_inner_left]
    change ⟪f, J g⟫_ℝ = ⟪J f, g⟫_ℝ
    calc
      ⟪f, J g⟫_ℝ = ⟪J f, J (J g)⟫_ℝ := (J.inner_map_map f (J g)).symm
      _ = ⟪J f, g⟫_ℝ := by rw [hJJ]
  have hJBmap : J.toContinuousLinearMap.comp B.toContinuousLinearMap = B.toContinuousLinearMap :=
    ContinuousLinearMap.ext hJB
  have hMJ : M.comp J.toContinuousLinearMap = M := by
    have hh := congrArg (fun U => ContinuousLinearMap.adjoint U) hJBmap
    simpa only [ContinuousLinearMap.adjoint_comp, hJadj] using hh
  refine ⟨hBrep, hBn, hMn, hMB, havg, hJB, hMJ, ?_⟩
  intro Kpos
  refine ⟨?_, ?_⟩
  · change M.comp ((L 0).comp B.toContinuousLinearMap) = _
    rw [hL0, ContinuousLinearMap.id_comp, hMB]
  · intro T
    refine ⟨?_, ?_, ?_⟩
    · change (M.comp ((L T).comp B.toContinuousLinearMap)).adjoint = _
      rw [ContinuousLinearMap.adjoint_comp, ContinuousLinearMap.adjoint_comp]
      change (M.comp (L T).adjoint).comp B.toContinuousLinearMap.adjoint.adjoint = _
      rw [ContinuousLinearMap.adjoint_adjoint, hLadj T]
      apply ContinuousLinearMap.ext
      intro f
      change M (J (L T (J (B f)))) = M (L T (B f))
      rw [hJB]
      exact congrArg (fun U : Lp ℝ 2 (nu y) →L[ℝ] Lp ℝ 2 mu => U (L T (B f))) hMJ
    · change ‖M.comp ((L T).comp B.toContinuousLinearMap)‖ ≤ 1
      have hn : ‖(L T).comp B.toContinuousLinearMap‖ ≤ 1 := by
        calc
          _ ≤ ‖L T‖ * ‖B.toContinuousLinearMap‖ := ContinuousLinearMap.opNorm_comp_le _ _
          _ ≤ 1 * 1 := mul_le_mul (hL T).1 hBn (norm_nonneg _) zero_le_one
          _ = 1 := one_mul 1
      calc
        _ ≤ ‖M‖ * ‖(L T).comp B.toContinuousLinearMap‖ := ContinuousLinearMap.opNorm_comp_le _ _
        _ ≤ 1 * 1 := mul_le_mul hMn hn (norm_nonneg _) zero_le_one
        _ = 1 := one_mul 1
    · intro f hf kTf
      let F : E × E → ℝ := fun z => f z.1
      have hF : MemLp F 2 (nu y) := hf.comp_measurePreserving hfst
      have hBF : B (hf.toLp f) = hF.toLp F := by
        apply Lp.ext
        exact (hBrep (hf.toLp f)).trans
          ((hfst.quasiMeasurePreserving.ae_eq_comp hf.coeFn_toLp).trans hF.coeFn_toLp.symm)
      obtain ⟨hsec, ⟨hop, heop⟩, _⟩ := (hL T).2 F hF
      obtain ⟨houter, hk, he⟩ := havg (fun z => ∫ w, F (Z y xRef z T w) ∂P) hop
      refine ⟨Measure.ae_ae_of_ae_prod hsec, houter, hk, ?_⟩
      change M (L T (B (hf.toLp f))) = _
      rw [hBF, heop]
      exact he

#print axioms actual_position_l2_averaging
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPositionL2Averaging

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualPositionL2Averaging.lean:201published source at ed709a7f320d

Proof architecture

Actual PBPS position averaging; 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.