Actual PBPS scalar defect: unique positive real root

Let E be a finite-dimensional real Hilbert Borel space with canonical volume, including the explicit rank-zero extension. Let V:E→ℝ be C², let 0<α≤β, and assume α‖v‖²≤D²V(x)[v,v]≤β‖v‖² for every x,v. Assume η>0 and βη≤1. Define μ(dx) proportional to exp(−V(x))dx, J as the law of (X,X+√η Z) for independent X∼μ,Z∼N(0,I), ν=J.snd, and Λ as the law of (Y,2X−Y) under J. The laws μ,J,ν are probabilities. There exists a Markov kernel S whose every-state normalized density is S(y)=volume.tilted(x↦−V((y+x)/2)−‖y−x‖²/(8η)), with Λ.IsCondKernel S and Λ.fst=Λ.snd=ν. There exists a bounded selfadjoint T on the actual real L²(ν) such that for every u, ‖Tu‖≤‖u‖, Tu(y)=∫u(x)S(y,dx) ν-almost everywhere, and ∫Tu dν=∫u dν. Its full scalar defect D=I−T² is positive. There exists a bounded positive REAL Γ with Γ²=D and for EVERY scalar u, ‖Γu‖²=‖u‖²−‖Tu‖². Probability, kernel, T, D positivity and Γ are conclusions; no supplied CFC/root/commutation or finite-dimensional L² certificate is assumed. For EVERY bounded positive REAL alternative Γprime on this SAME marginal whose square is this SAME D, Γprime=Γ. No energy identity is assumed for an alternative. This uniqueness quantifier lies outside the earlier every-u energy assertion. Printed joint/macro ΓP and B.11 still require SAME onto-M transport and the typed B*B identity, which remain separate.

Braces mark parameters Lean can infer; square brackets request structures such as a measurable space or probability measure. Named hypotheses are mathematical premises, not facts established by this declaration. Section parameters are described in the mathematical hypotheses above; the module link retains their exact source context.

theorem actual_unique_positive_real_defect_root
    {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) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := (μ.prod (stdGaussian E)).map
      (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
    let ν := J.snd
    let Λ := J.map (fun p : E × E => (p.2,(2 : ℝ) • p.1-p.2))
    IsProbabilityMeasure μ ∧ IsProbabilityMeasure J ∧ IsProbabilityMeasure ν ∧
    ∃ S : Kernel E E, IsMarkovKernel S ∧
      (∀ y, S y = (volume : Measure E).tilted
        (fun x => -V ((1/2 : ℝ) • (y+x)) - ‖y-x‖^2/(8*η))) ∧
      Λ.IsCondKernel S ∧ Λ.fst = ν ∧ Λ.snd = ν ∧
      ∃ T : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν,
        IsSelfAdjoint T ∧
        (∀ u : Lp ℝ 2 ν, ‖T u‖ ≤ ‖u‖ ∧
          (T u : E → ℝ) =ᵐ[ν] (fun y => ∫ x, u x ∂S y) ∧
          (∫ y, T u y ∂ν) = ∫ y, u y ∂ν) ∧
        ((1 : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν)-T*T).IsPositive ∧
        ∃ Γ : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν,
          Γ.IsPositive ∧ Γ*Γ = ((1 : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν)-T*T) ∧
          (∀ u : Lp ℝ 2 ν, ‖Γ u‖^2 = ‖u‖^2-‖T u‖^2) ∧
            ∀ Γ' : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν,
              Γ'.IsPositive →
              Γ'*Γ' = ((1 : Lp ℝ 2 ν →L[ℝ] Lp ℝ 2 ν)-T*T) → Γ'=Γ

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