actual_kernel_l2_realization

Plain-English statement

- The actual fixed-reference phase kernel preserves its Gibbs law and realizes the same real-L2 family, with AE positivity and every constant class fixed. This does not prove process Markovness, restart or Chapman--Kolmogorov.

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_kernel_l2_realization
    {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_kernel_l2_realization_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_kernel_l2_realization_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, h110⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualL2StrongContinuity.actual_l2_strong_continuity
      hα hαβ hV hH hη hβη
  obtain ⟨Zk, _hZkm, hZka, hZkf, _hZkgood,
      Kphase, hK, hfiber, _hset, _hinit, _hbounded⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualPhaseTransitionKernel.actual_phase_transition_kernel
      hα hαβ hV hH hη hβη
  -- Same literal recursion and raw fallback identify the two path witnesses.
  have hZeq : Zk = Z := by
    funext y xRef z t w
    by_cases hx : ∃ n, w ∈ I y xRef z t n
    · obtain ⟨n, hn⟩ := hx
      obtain ⟨a, ha⟩ : ∃ a, record y xRef z (ε w) n = Sum.inl a := by
        cases hr : record y xRef z (ε w) n with
        | inl a => exact ⟨a, rfl⟩
        | inr u =>
          have hh := hn.1
          change (record y xRef z (ε w) n).elim
            (fun a => (a.1 : WithTop ℝ≥0)) (fun _ => ⊤) ≤ (t : WithTop ℝ≥0) at hh
          rw [hr] at hh
          exact False.elim ((WithTop.not_top_le_coe t) hh)
      exact (hZka y xRef z w t n a hn.1 hn.2 ha).trans
        (hZa y xRef z w t n a hn ha).symm
    · exact (hZkf y xRef z w t hx).trans (hZf y xRef z w t hx).symm
  rw [hZeq] at hfiber
  letI : IsProbabilityMeasure P :=
    AutoSamplingTheory.TechnicalLemmas.Probability.UnitExponentialProduct.unit_exponential_product_laws.1
  letI : IsMarkovKernel Kphase := hK
  have hzM (y xRef : E) (z : E × E) (T : ℝ≥0) :
      Measurable (Z y xRef z T) := hZm.comp
    (show Measurable (fun w : ℕ → ℝ => (((y, xRef), z), T, w)) from
      measurable_const.prodMk (measurable_const.prodMk measurable_id))
  -- A fixed-time endpoint pushforward, not a restart/Markov argument.
  have hInv (y xRef : E) (T : ℝ≥0) :
      (Kphase.comap (fun z : E × E => (((y, xRef), z), T))
        (by fun_prop)).Invariant (nu y) := by
    let κ : Kernel (E × E) (E × E) :=
      Kphase.comap (fun z : E × E => (((y, xRef), z), T)) (by fun_prop)
    have he := (h106 y xRef T).1
    change (nu y).bind κ = nu y
    calc
      (nu y).bind κ =
          Measure.map (fun a : (E × E) × (ℕ → ℝ) => Z y xRef a.1 T a.2)
            ((nu y).prod P) := by
        ext s hs
        rw [Measure.bind_apply hs κ.aemeasurable, Measure.map_apply he.measurable hs,
          Measure.prod_apply (he.measurable hs)]
        apply lintegral_congr_ae
        filter_upwards with z
        change Kphase (((y, xRef), z), T) s =
          P ((Z y xRef z T) ⁻¹' s)
        rw [hfiber, Measure.map_apply (hzM y xRef z T) hs]
      _ = nu y := he.map_eq
  refine ⟨Z, hZm, hZa, hZf, hZ0, h106, hBorel, Kphase, hK, hfiber, hInv, ?_⟩
  intro y xRef
  obtain ⟨L, hL0, hSC, hL, hRest⟩ := h110 y xRef
  have hnorm := ActualPhaseNormalization.actual_phase_gibbs_normalization
    hα hαβ hV hH hη hβη y
  letI : IsProbabilityMeasure (nu y) := by
    have hp := hnorm.2.2.2.2.2.1
    rw [hnorm.2.2.2.2.2.2] at hp
    exact hp
  have hreal (T : ℝ≥0) (f : Lp ℝ 2 (nu y)) :
      (∀ᵐ z ∂nu y, Integrable (f : (E × E) → ℝ)
        (Kphase (((y, xRef), z), T))) ∧
      (L T f : (E × E) → ℝ) =ᵐ[nu y]
        fun z => ∫ z', f z' ∂Kphase (((y, xRef), z), T) := by
    let opG : (E × E) → ℝ := fun z => ∫ w, f (Z y xRef z T w) ∂P
    have hclock := (hL T).2 (f : (E × E) → ℝ) (Lp.memLp f)
    obtain ⟨hOp, heq⟩ := hclock.2.1
    have heq' : L T f = hOp.toLp opG := by
      simpa only [Lp.toLp_coeFn] using heq
    have hae : (L T f : (E × E) → ℝ) =ᵐ[nu y] opG := by
      rw [heq']
      exact hOp.coeFn_toLp
    have hinter (z : E × E) :
        (∫ z', f z' ∂Kphase (((y, xRef), z), T)) = opG z := by
      rw [hfiber]
      exact integral_map (hzM y xRef z T).aemeasurable
        (Lp.stronglyMeasurable f).aestronglyMeasurable
    refine ⟨?_, ?_⟩
    · filter_upwards [hclock.1] with z hz
      rw [hfiber]
      exact (integrable_map_measure
        (Lp.stronglyMeasurable f).aestronglyMeasurable (hzM y xRef z T).aemeasurable).2 hz
    · filter_upwards [hae] with z hz
      exact hz.trans (hinter z).symm
  refine ⟨L, hL0, hSC, hreal, ?_, ?_, hL, hRest⟩
  · intro T f hf
    let κ : Kernel (E × E) (E × E) :=
      Kphase.comap (fun z : E × E => (((y, xRef), z), T)) (by fun_prop)
    have hcomp : ∀ᵐ z ∂((nu y).bind κ), 0 ≤ f z := by
      rw [(hInv y xRef T).def]
      exact hf
    have hfib := Measure.ae_ae_of_ae_comp (κ := κ) hcomp
    filter_upwards [(hreal T f).2, hfib] with z hz hzz
    rw [hz]
    exact integral_nonneg_of_ae hzz
  · intro T r
    have hr : MemLp (fun _ : E × E => r) 2 (nu y) := memLp_const r
    let f : Lp ℝ 2 (nu y) := hr.toLp (fun _ : E × E => r)
    let κ : Kernel (E × E) (E × E) :=
      Kphase.comap (fun z : E × E => (((y, xRef), z), T)) (by fun_prop)
    have hcae : (f : (E × E) → ℝ) =ᵐ[nu y] fun _ => r := hr.coeFn_toLp
    have hcomp : ∀ᵐ z ∂((nu y).bind κ), f z = r := by
      rw [(hInv y xRef T).def]
      exact hcae
    have hfib := Measure.ae_ae_of_ae_comp (κ := κ) hcomp
    refine ⟨hr, ?_⟩
    change L T f = f
    apply Lp.ext
    filter_upwards [(hreal T f).2, hcae, hfib] with z hz hzr hfz
    calc
      L T f z = ∫ z', f z' ∂Kphase (((y, xRef), z), T) := hz
      _ = ∫ _ : E × E, r ∂Kphase (((y, xRef), z), T) := integral_congr_ae hfz
      _ = r := by simp
      _ = f z := hzr.symm

end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualKernelL2Realization

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualKernelL2Realization.lean:232published source at ed709a7f320d

Proof architecture

Same-process PBPS transition/operator bridge; 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.
Density statements retain normalization and absolute-continuity prerequisites.
A Gibbs expression is not automatically a probability law or an invariant law.