actual_bounded_gibbs_duality

Plain-English statement

- One actual global PBPS process preserves the Gibbs phase law at each fixed time and satisfies bounded signed momentum-flipped duality.

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_bounded_gibbs_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_bounded_gibbs_duality_statement hα hαβ hV hH hη hβη := by
  classical
  unfold actual_bounded_gibbs_duality_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, h105⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCountableEndpointLaw.actual_countable_endpoint_law
      hα hαβ hV hH hη hβη
  refine ⟨Z, hZm, hZa, hZf, hZ0, ?_⟩
  intro y r T op endpoint
  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 hemb : Measurable (fun a : (E × E) × (ℕ → ℝ) => (((y, r), a.1), T, a.2)) :=
    (measurable_const.prodMk measurable_fst).prodMk (measurable_const.prodMk measurable_snd)
  have he : Measurable endpoint := hZm.comp hemb
  let M : (E × E) × (ℕ → ℝ) → (E × E) × (E × E) := fun a => (a.1, endpoint a)
  let Q : Measure ((E × E) × (E × E)) := Measure.map M ((nu y).prod P)
  let J : (E × E) × (E × E) → (E × E) × (E × E) := fun a => (R a.2, R a.1)
  have hM : Measurable M := measurable_fst.prodMk he
  have hJ : MeasurePreserving J Q Q := (h105 y r T).2.2
  letI : IsProbabilityMeasure Q := (h105 y r T).2.1
  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]
  have hMP : MeasurePreserving M ((nu y).prod P) Q := ⟨hM, rfl⟩
  have hfst : MeasurePreserving Prod.fst Q (nu y) := by
    refine ⟨measurable_fst, ?_⟩
    change Measure.map Prod.fst (Measure.map M ((nu y).prod P)) = nu y
    rw [Measure.map_map measurable_fst hM]
    change Measure.map Prod.fst ((nu y).prod P) = nu y
    simp only [Measure.map_fst_prod, measure_univ, one_smul]
  have hinvariant : MeasurePreserving endpoint ((nu y).prod P) (nu y) := by
    have h := hR.comp ((hfst.comp hJ).comp hMP)
    simpa only [Function.comp_def, M, J, R, neg_neg] using h
  refine ⟨hinvariant, ?_⟩
  have htest (F G : (E × E) → ℝ) (CF CG : ℝ)
      (hF : Measurable F) (hG : Measurable G) (hCF : 0 ≤ CF) (hCG : 0 ≤ CG)
      (hFb : ∀ z, |F z| ≤ CF) (hGb : ∀ z, |G z| ≤ CG) :
      Integrable (fun z => F z * op G z) (nu y) ∧
      (∫ z, F z * op G z ∂nu y) = ∫ a, F a.1 * G a.2 ∂Q := by
    let H : (E × E) × (ℕ → ℝ) → ℝ := fun a => F a.1 * G (endpoint a)
    have hHm : Measurable H := (hF.comp measurable_fst).mul (hG.comp he)
    have hHi : Integrable H ((nu y).prod P) :=
      Integrable.of_bound hHm.aestronglyMeasurable (CF * CG)
        (Filter.Eventually.of_forall fun a => by
          change |F a.1 * G (endpoint a)| ≤ CF * CG
          rw [abs_mul]
          exact mul_le_mul (hFb a.1) (hGb (endpoint a)) (abs_nonneg _) hCF)
    have houter : Integrable (fun z => F z * op G z) (nu y) := by
      simpa only [H, endpoint, integral_const_mul] using hHi.integral_prod_left
    refine ⟨houter, ?_⟩
    have hpair : Measurable (fun a : (E × E) × (E × E) => F a.1 * G a.2) :=
      (hF.comp measurable_fst).mul (hG.comp measurable_snd)
    calc
      (∫ z, F z * op G z ∂nu y) = ∫ a, H a ∂((nu y).prod P) := by
        rw [integral_prod H hHi]
        simp only [H, endpoint, op, integral_const_mul]
      _ = ∫ a, F a.1 * G a.2 ∂Q :=
        (integral_map hM.aemeasurable hpair.aestronglyMeasurable).symm
  intro F G CF CG hF hG hCF hCG hFb hGb
  have hleft := htest F G CF CG hF hG hCF hCG hFb hGb
  have hright := htest (fun z => G (R z)) (fun z => F (R z)) CG CF
    (hG.comp hRm) (hF.comp hRm) hCG hCF (fun z => hGb (R z)) (fun z => hFb (R z))
  refine ⟨hleft.1, hright.1, ?_⟩
  rw [hleft.2, hright.2]
  have hpair : Measurable (fun a : (E × E) × (E × E) => F a.1 * G a.2) :=
    (hF.comp measurable_fst).mul (hG.comp measurable_snd)
  calc
    (∫ a, F a.1 * G a.2 ∂Q) = ∫ a, F (J a).1 * G (J a).2 ∂Q := by
      conv_lhs => rw [← hJ.map_eq]
      exact integral_map hJ.measurable.aemeasurable hpair.aestronglyMeasurable
    _ = ∫ a, G (R a.1) * F (R a.2) ∂Q := by
      apply integral_congr_ae
      exact Filter.Eventually.of_forall fun a => mul_comm _ _

#print axioms actual_bounded_gibbs_duality
end
end AutoSamplingTheory.ExampleCases.ProximalBPS.ActualBoundedGibbsDuality

Open AutoSamplingTheory/ExampleCases/ProximalBPS/ActualBoundedGibbsDuality.lean:152published source at ed709a7f320d

Proof architecture

Actual bounded duality and fixed-time invariance; 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.