actual_sharp_corrector_bound

Plain-English statement

Independently exact-science68 VERIFIED 3ad3b127b5a645be9cf71b3d14520b2d8fea3122. Original finite real Hilbert/Borel/C2/two Hessians/positive capped eta inputs produce SAME actual root/inverse/coefficient and sharp |C(u,v)|<=(norm(u)²+norm(v)²)/(2gamma). Every globally centered actual f has SAME fP/fV, the actual norm budget and |C(fP,fV)|<=norm(f)²/(2gamma). Genuine original-input Test proves half/three-halves energy equivalence and exact perturbation bounds for every0<omegaWeight<=gamma. Rank0/alphaeta1 retained; literal private Props are representations only. B21 reflection rotation, H1/B2, B4 dynamics, main results, implementation errors, costs and composition remain open.

Lean statement

theorem actual_sharp_corrector_bound
    {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_sharp_corrector_bound_statement (E := E) (V := V) (α := α) (β := β) (η := η) hα hαβ hV hH hη hβη := by
  classical
  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 mY : MeasurableSpace (E × E) := MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)
  letI : MeasurableSpace (E × E) := Prod.instMeasurableSpace
  letI : Fact (mY ≤ (inferInstance : MeasurableSpace (E × E))) := ⟨measurable_snd.comap_le⟩
  let HP := lpMeas ℝ ℝ mY 2 J
  letI : NormedAddCommGroup HP := HP.normedAddCommGroup
  letI : InnerProductSpace ℝ HP := HP.innerProductSpace
  let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
    HP.subtypeL ∘L condExpL2 ℝ ℝ (μ := J) measurable_snd.comap_le
  have hBase := AutoSamplingTheory.ExampleCases.ProximalBPS.ActualRootCommutation.actual_same_root_inverse_commutation
    (E:=E) (V:=V) (α:=α) (β:=β) (η:=η) hα hαβ hV hH hη hβη
  have parentSlice0 := hBase
  have hμ := parentSlice0.1
  have parentSlice1 := parentSlice0.2
  have hJ := parentSlice1.1
  have parentSlice2 := parentSlice1.2
  have hν := parentSlice2.1
  have parentSlice3 := parentSlice2.2
  have hRange := parentSlice3.1
  have parentSlice4 := parentSlice3.2
  have hPf := parentSlice4.1
  have parentSlice5 := parentSlice4.2
  obtain ⟨S, parentSlice6⟩ := parentSlice5
  have hS := parentSlice6.1
  have parentSlice7 := parentSlice6.2
  have hSd := parentSlice7.1
  have parentSlice8 := parentSlice7.2
  have hSc := parentSlice8.1
  have parentSlice9 := parentSlice8.2
  have hSf := parentSlice9.1
  have parentSlice10 := parentSlice9.2
  have hSs := parentSlice10.1
  have parentSlice11 := parentSlice10.2
  obtain ⟨e, parentSlice12⟩ := parentSlice11
  have he := parentSlice12.1
  have parentSlice13 := parentSlice12.2
  obtain ⟨U, parentSlice14⟩ := parentSlice13
  have hU := parentSlice14.1
  have parentSlice15 := parentSlice14.2
  have hUi := parentSlice15.1
  have parentSlice16 := parentSlice15.2
  have hUs := parentSlice16.1
  have parentSlice17 := parentSlice16.2
  obtain ⟨T, parentSlice18⟩ := parentSlice17
  have hTs := parentSlice18.1
  have parentSlice19 := parentSlice18.2
  have hAll := parentSlice19.1
  have parentSlice20 := parentSlice19.2
  have hAeq := parentSlice20.1
  have parentSlice21 := parentSlice20.2
  have hAs := parentSlice21.1
  have parentSlice22 := parentSlice21.2
  have hAn := parentSlice22.1
  have parentSlice23 := parentSlice22.2
  have hPB := parentSlice23.1
  have parentSlice24 := parentSlice23.2
  obtain ⟨Γ, parentSlice25⟩ := parentSlice24
  have hΓ := parentSlice25.1
  have parentSlice26 := parentSlice25.2
  have hΓSq := parentSlice26.1
  have parentSlice27 := parentSlice26.2
  have hCommRoot := parentSlice27.1
  have parentSlice28 := parentSlice27.2
  have hΓP := parentSlice28.1
  have parentSlice29 := parentSlice28.2
  have hΓPSq := parentSlice29.1
  have parentSlice30 := parentSlice29.2
  have hCommP := parentSlice30.1
  have parentSlice31 := parentSlice30.2
  have hGram := parentSlice31.1
  have parentSlice32 := parentSlice31.2
  obtain ⟨q, parentSlice33⟩ := parentSlice32
  have hqa := parentSlice33.1
  have parentSlice34 := parentSlice33.2
  have hTq := parentSlice34.1
  have parentSlice35 := parentSlice34.2
  have hq := parentSlice35.1
  have parentSlice36 := parentSlice35.2
  have hCenter := parentSlice36.1
  have parentSlice37 := parentSlice36.2
  have hΓPq := parentSlice37.1
  have parentSlice38 := parentSlice37.2
  have hγ := parentSlice38.1
  have parentSlice39 := parentSlice38.2
  obtain ⟨ΓP0, parentSlice40⟩ := parentSlice39
  have hΓP0 := parentSlice40.1
  have parentSlice41 := parentSlice40.2
  have hPos0 := parentSlice41.1
  have parentSlice42 := parentSlice41.2
  have hOrder0 := parentSlice42.1
  have parentSlice43 := parentSlice42.2
  have hUnit := parentSlice43.1
  have parentSlice44 := parentSlice43.2
  obtain ⟨Inv, parentSlice45⟩ := parentSlice44
  have hLeft := parentSlice45.1
  have parentSlice46 := parentSlice45.2
  have hRight := parentSlice46.1
  have parentSlice47 := parentSlice46.2
  have hNormInv := parentSlice47.1
  have parentSlice48 := parentSlice47.2
  have hInvSelf := parentSlice48.1
  have parentSlice49 := parentSlice48.2
  obtain ⟨A0, parentSlice50⟩ := parentSlice49
  have hA0 := parentSlice50.1
  have parentSlice51 := parentSlice50.2
  have hA0self := parentSlice51.1
  have parentSlice52 := parentSlice51.2
  have hComm0 := parentSlice52.1
  have parentSlice53 := parentSlice52.2
  have hSquare0 := parentSlice53.1
  have parentSlice54 := parentSlice53.2
  have hCommInv := parentSlice54.1
  have parentSlice55 := parentSlice54.2
  obtain ⟨B0, parentSlice56⟩ := parentSlice55
  have hB0 := parentSlice56.1
  have parentSlice57 := parentSlice56.2
  obtain ⟨V0, parentSlice58⟩ := parentSlice57
  have hVdef := parentSlice58.1
  have parentSlice59 := parentSlice58.2
  have hFactor := parentSlice59.1
  have parentSlice60 := parentSlice59.2
  have hVadj := parentSlice60.1
  have parentSlice61 := parentSlice60.2
  have hVnorm := parentSlice61.1
  have parentSlice62 := parentSlice61.2
  obtain ⟨R, parentSlice63⟩ := parentSlice62
  have hR := parentSlice63.1
  have parentSlice64 := parentSlice63.2
  have hAmbient := parentSlice64.1
  have parentSlice65 := parentSlice64.2
  have hGlobal := parentSlice65
  let A : HP →L[ℝ] HP := HP.orthogonalProjectionOnto ∘L U.toContinuousLinearMap ∘L HP.subtypeL
  let B : HP →L[ℝ] Lp ℝ 2 J :=
    (((1 : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J)-P)*U.toContinuousLinearMap*P) ∘L HP.subtypeL
  let ΓP : HP →L[ℝ] HP := e.conjStarAlgEquiv Γ
  let qP : HP := e q
  let HP0 := (innerSL ℝ qP).ker
  letI : NormedAddCommGroup HP0 := HP0.normedAddCommGroup
  letI : InnerProductSpace ℝ HP0 := HP0.innerProductSpace
  letI : CompleteSpace HP0 := (innerSL ℝ qP).isClosed_ker.completeSpace_coe
  let Hperp := P.ker
  letI : NormedAddCommGroup Hperp := Hperp.normedAddCommGroup
  letI : InnerProductSpace ℝ Hperp := Hperp.innerProductSpace
  letI : CompleteSpace Hperp := P.isClosed_ker.completeSpace_coe
  change HP0 →L[ℝ] Hperp at B0 V0
  change HP0 →L[ℝ] HP0 at ΓP0 Inv A0
  change Lp ℝ 2 J →L[ℝ] Hperp at R
  change (∀ g : Lp ℝ 2 J, (R g : Lp ℝ 2 J)=g-P g) at hR
  change (∀ g : Lp ℝ 2 J,
    B.adjoint g=HP0.subtypeL (B0.adjoint (R g))) at hAmbient
  change (∀ f : Lp ℝ 2 J, (∫ z, f z ∂J)=0 →
    ∃ fP : HP0,
      HP0.subtypeL fP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le f ∧
      let fperp : Hperp := R f
      let fV : HP0 := V0.adjoint fperp
      B.adjoint (fperp : Lp ℝ 2 J)=HP0.subtypeL (ΓP0 fV) ∧
      HP0.subtypeL (ΓP0 fV)=ΓP (HP0.subtypeL fV) ∧
      ‖f‖^2=‖fP‖^2+‖fperp‖^2 ∧ ‖fV‖≤‖fperp‖) at hGlobal
  have hKself : IsSelfAdjoint (A0*Inv) := (hA0self.commute_iff hInvSelf).mp hCommInv
  have hCoefficient : (1 : HP0 →L[ℝ] HP0)+(A0*Inv)*(A0*Inv)=Inv*Inv := by
    have hFirst : (ΓP0*ΓP0)*(Inv*Inv)=(1 : HP0 →L[ℝ] HP0) := by
      calc
        (ΓP0*ΓP0)*(Inv*Inv)=ΓP0*((ΓP0*Inv)*Inv) := by noncomm_ring
        _ = (1 : HP0 →L[ℝ] HP0) := by rw [hRight,one_mul,hRight]
    have hSecond : (A0*Inv)*(A0*Inv)=(A0*A0)*(Inv*Inv) := by
      calc
        (A0*Inv)*(A0*Inv)=A0*((Inv*A0)*Inv) := by noncomm_ring
        _ = A0*((A0*Inv)*Inv) := by rw [hCommInv.eq.symm]
        _ = (A0*A0)*(Inv*Inv) := by noncomm_ring
    calc
      (1 : HP0 →L[ℝ] HP0)+(A0*Inv)*(A0*Inv)=
          (ΓP0*ΓP0)*(Inv*Inv)+(A0*A0)*(Inv*Inv) := by rw [hFirst,hSecond]
      _ = (ΓP0*ΓP0+A0*A0)*(Inv*Inv) := by rw [add_mul]
      _ = Inv*Inv := by rw [add_comm,hSquare0,one_mul]
  let γ : ℝ := 2*Real.sqrt ((α : ℝ)*η)/(1+(α : ℝ)*η)
  change 0 < γ at hγ
  change ‖Inv‖ ≤ 1/γ at hNormInv
  let C : HP0 → HP0 → ℝ := fun u v =>
    (1/2 : ℝ)*(‖u‖^2-‖v‖^2)-inner ℝ ((A0*Inv) u) v
  have hPair (u v : HP0) : |C u v| ≤ (‖u‖^2+‖v‖^2)/(2*γ) := by
    have h := AutoSamplingTheory.TechnicalLemmas.Analysis.HilbertCorrectorBound.quadratic_corrector_bound_of_square_identity
      (A0*Inv) Inv hKself hInvSelf hCoefficient (1/γ) (by positivity) hNormInv u v
    change |C u v| ≤ ((1/γ)/2)*(‖u‖^2+‖v‖^2) at h
    calc
      |C u v| ≤ ((1/γ)/2)*(‖u‖^2+‖v‖^2) := h
      _ = (‖u‖^2+‖v‖^2)/(2*γ) := by
        field_simp
  have hFinal : ∀ f : Lp ℝ 2 J, (∫ z, f z ∂J)=0 →
      ∃ fP : HP0,
        HP0.subtypeL fP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le f ∧
        let fperp : Hperp := R f
        let fV : HP0 := V0.adjoint fperp
        B.adjoint (fperp : Lp ℝ 2 J)=HP0.subtypeL (ΓP0 fV) ∧
        HP0.subtypeL (ΓP0 fV)=ΓP (HP0.subtypeL fV) ∧
        ‖f‖^2=‖fP‖^2+‖fperp‖^2 ∧ ‖fV‖≤‖fperp‖ ∧
        ‖fP‖^2+‖fV‖^2≤‖f‖^2 ∧ |C fP fV|≤‖f‖^2/(2*γ) := by
    intro f hf
    obtain ⟨fP,hGlobalFields⟩ := hGlobal f hf
    have hfP := hGlobalFields.1
    have hBf := hGlobalFields.2.1
    have hΓf := hGlobalFields.2.2.1
    have hPyth := hGlobalFields.2.2.2.1
    have hNormV := hGlobalFields.2.2.2.2
    let fperp : Hperp := R f
    let fV : HP0 := V0.adjoint fperp
    have hBudget : ‖fP‖^2+‖fV‖^2 ≤ ‖f‖^2 := by
      have hs := (sq_le_sq₀ (norm_nonneg fV) (norm_nonneg fperp)).2 hNormV
      nlinarith only [hPyth, hs]
    refine ⟨fP,hfP,hBf,hΓf,hPyth,hNormV,hBudget,?_⟩
    exact (hPair fP fV).trans (div_le_div_of_nonneg_right hBudget (by positivity))
  unfold actual_sharp_corrector_bound_statement
  refine ⟨hμ, ?_⟩
  refine ⟨hJ, ?_⟩
  refine ⟨hν, ?_⟩
  refine ⟨hRange, ?_⟩
  refine ⟨hPf, ?_⟩
  refine ⟨S, ?_⟩
  refine ⟨hS, ?_⟩
  refine ⟨hSd, ?_⟩
  refine ⟨hSc, ?_⟩
  refine ⟨hSf, ?_⟩
  refine ⟨hSs, ?_⟩
  refine ⟨e, ?_⟩
  refine ⟨he, ?_⟩
  refine ⟨U, ?_⟩
  refine ⟨hU, ?_⟩
  refine ⟨hUi, ?_⟩
  refine ⟨hUs, ?_⟩
  refine ⟨T, ?_⟩
  refine ⟨hTs, ?_⟩
  refine ⟨hAll, ?_⟩
  refine ⟨hAeq, ?_⟩
  refine ⟨hAs, ?_⟩
  refine ⟨hAn, ?_⟩
  refine ⟨hPB, ?_⟩
  refine ⟨Γ, ?_⟩
  refine ⟨hΓ, ?_⟩
  refine ⟨hΓSq, ?_⟩
  refine ⟨hCommRoot, ?_⟩
  refine ⟨hΓP, ?_⟩
  refine ⟨hΓPSq, ?_⟩
  refine ⟨hCommP, ?_⟩
  refine ⟨hGram, ?_⟩
  refine ⟨q, ?_⟩
  refine ⟨hqa, ?_⟩
  refine ⟨hTq, ?_⟩
  refine ⟨hq, ?_⟩
  refine ⟨hCenter, ?_⟩
  refine ⟨hΓPq, ?_⟩
  refine ⟨hγ, ?_⟩
  refine ⟨ΓP0, ?_⟩
  refine ⟨hΓP0, ?_⟩
  refine ⟨hPos0, ?_⟩
  refine ⟨hOrder0, ?_⟩
  refine ⟨hUnit, ?_⟩
  refine ⟨Inv, ?_⟩
  refine ⟨hLeft, ?_⟩
  refine ⟨hRight, ?_⟩
  refine ⟨hNormInv, ?_⟩
  refine ⟨hInvSelf,A0,hA0,hA0self,hComm0,hSquare0,hCommInv,hKself,hCoefficient,hPair,?_⟩
  refine ⟨B0, ?_⟩
  refine ⟨hB0, ?_⟩
  refine ⟨V0, ?_⟩
  refine ⟨hVdef, ?_⟩
  refine ⟨hFactor, ?_⟩
  refine ⟨hVadj, ?_⟩
  refine ⟨hVnorm, ?_⟩
  refine ⟨R, ?_⟩
  refine ⟨hR, ?_⟩
  refine ⟨hAmbient, ?_⟩
  exact hFinal


end
end AutoSamplingTheory.ExampleCases.ProximalBPS.SharpCorrectorEnergy

#print axioms AutoSamplingTheory.ExampleCases.ProximalBPS.SharpCorrectorEnergy.actual_sharp_corrector_bound

Open AutoSamplingTheory/ExampleCases/ProximalBPS/SharpCorrectorEnergy.lean:129published source at ed709a7f320d

Proof architecture

PBPS actual consumer; no SALD admission

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.
• `refine` instantiates a reusable theorem while leaving explicit subgoals.
• `exact` closes the current goal with an already typed term.

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.
• Totalized `fderiv` values must not be read as a differentiability theorem.
• Almost-everywhere hypotheses depend on the stated measure and representative.