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Actual PBPS corrector perturbation with the same witnesses

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang, arXiv:2609.06905v1, B20 and Lemma B4 perturbation expansion Ex28–Ex34. ASTIS original-input integration of the same perturbation algebra; all actual71 clauses retained.

arXiv:2609.06905v1; Lean4.33.0; fixed Mathlib db584cd6d46c92f209a44c0f1c829460d327499d · PBPS B20 and Appendix B4 Ex28--Ex34 perturbation algebra; future actual B27/B28 adapter · faithful paraphrase

Complete source statement (Samplinglib restatement)

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, 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 and Z~N(0,I), ν=J.snd, and Λ as the law of (Y,2X−Y) under J. These are probability laws. Put mY=comap(snd), H_P=L² classes measurable with respect to mY, P=i∘E[·|Y] and the canonical M:u↦u∘snd. Then H_P=ran(P), P is identity on H_P, and there exists an onto linear isometry e:L²(ν;ℝ)≃H_P with i e=M. There is a Markov kernel S whose every-state normalized law is S(y)=volume.tilted(x↦−V((y+x)/2)−‖y−x‖²/(8η)); Λ.IsCondKernel S and both Λ marginals equal ν. The same joint reflection F(x,y)=(x,2x−y) gives the involutive selfadjoint isometry U, with Uf=f∘F J-almost everywhere for every f. There is a bounded selfadjoint T on the full real L²(ν), with ‖Tu‖≤‖u‖, Tu(y)=∫u(x)S(y,dx) ν-almost everywhere for each u, and preserved integral. On H_P put A=PUP restricted via i and orthogonal projection, and B=(I_J−P)UPi:H_P→L²(J). Then A=eTe⁻¹ is selfadjoint and contractive, PB=0. There is a bounded positive operator Γ on the real scalar-valued L² space with Γ²=I−T². For the SAME transported Γ_P=eΓe⁻¹, positivity holds and Γ_P²=I_H_P−A²=B*B. There is q∈L²(ν;ℝ) equal to one ν-almost everywhere, with Tq=q and ⟨q,u⟩=∫u dν for every u. Set q_P=e q and H_P^0=ker⟨q_P,·⟩, with its inherited Hilbert structure. Membership is exactly zero ν-mean after e⁻¹. The SAME Γ_P annihilates q_P. Put γ=2√(αη)/(1+αη)>0. There exists Γ_0:H_P^0→H_P^0 equal under inclusion to the restriction of Γ_P, positive with Γ_0−γ I positive. Γ_0 is a unit in the algebra of bounded real operators. There exists a bounded real Inv on this SAME centered space with Inv Γ_0=Γ_0 Inv=I and ‖Inv‖≤1/γ. All probability, normalization, range, root, centering, spectral-gap, unit and inverse facts are internally derived from the original inputs; none is an additional public premise. αη=1 and rank-zero E remain legal. No inverse on full H_P is asserted. Define the actual closed conditional-mean-zero space H_perp=ker P in L²(J;ℝ), with its inherited complete Hilbert structure; this is not merely the globally zero-mean joint space. There exists B_0:H_P^0→H_perp whose inclusion equals B i_0. There exists V_0:H_P^0→H_perp with V_0=B_0 Inv, B_0=V_0 Γ_0, V_0* V_0=I and ‖V_0 f‖=‖f‖ for every f∈H_P^0. All these objects and identities are conclusions from the same original inputs. There exists the canonical bounded residual map R:L²(J;ℝ)→H_perp with i_perp Rg=g−Pg for every joint g. The SAME ambient adjoint satisfies B* g=i_0 B_0* Rg for every joint g; the printed intrinsic block adjoint is recovered on H_perp. For every actual joint f∈L²(J;ℝ) with ∫f dJ=0, there exists f_P∈H_P^0 whose inclusion is E[f|Y]. Put f_perp=Rf and f_V=V_0* f_perp∈the SAME H_P^0. Then B* i_perp f_perp=i_0 Γ_0 f_V=Γ_P i_0 f_V, ‖f‖²=‖f_P‖²+‖f_perp‖² and ‖f_V‖≤‖f_perp‖. The global zero-mean condition is a quantified input restriction to this concluded implication, not an extra assumption on the potential or an operator certificate. The SAME root commutes with T and its transport commutes with A. The SAME Inv on H_P^0 is selfadjoint. There exists an internally constructed A_0:H_P^0→H_P^0 whose inclusion equals A after inclusion; it is selfadjoint,commutes with Γ_0,satisfies A_0²+Γ_0²=I,and commutes with the SAME Inv. For the same U,R and inclusion of H_perp=ker P, define D=R U inclusion. Its full-domain identity V0*D=-A0 V0* is retained. For EVERY globally mean-zero f in L2(J), let fP be its actual centered conditional-expectation component, fperp=R f and fV=V0* fperp. Define the actual output g=U(P f-(f-P f)), not an abstract pair chosen to satisfy a matrix identity. The theorem proves its global mean is zero internally from invariance of the actual reflected augmentation law and preservation of the integral by conditional expectation. Let gP be the actual centered conditional-expectation component of this SAME g, gperp=R g and gV=V0* gperp. Then gP=A0 fP-Gamma0 fV, gV=Gamma0 fP+A0 fV, and ||gP||^2+||gV||^2=||fP||^2+||fV||^2. Both output components are produced by these actual conditional/polar semantics before proving the formulas; neither is defined as its right-hand side. All previous law/kernel/root/inverse/polar/full-micro and centered-observable conclusions persist with the SAME twelve witnesses and the original six analytic caller conditions. No surjectivity, sharp-energy68 bound, extra regularity or zero-output-mean premise is introduced. One private literal Prop stores the complete sealed mathematical conclusion; it supplies no proof provider or additional hypothesis. For the SAME centered A_0 and inverse Inv, define K=A_0 Inv and C(u,v)=(||u||²−||v||²)/2−⟨K u,v⟩ for u,v in H_P^0. For every globally centered actual f and the SAME actual g=U(Pf−(f−Pf)), the already produced conditional/polar components satisfy C(g_P,g_V)−C(f_P,f_V)=−||f_P||²+||f_V||². The half, signs, coefficient one and the same inverse are exact. This identity is proved internally from A_0²+Γ_0²=I, selfadjointness, commutation, the two-sided inverse and the actual projected rotation. No corrector-change certificate, sharp corrector bound, extra domain/smoothness, onto-V_0 or strict endpoint is an additional premise. Every previous clause and all twelve common witnesses persist. For the SAME C and the SAME produced A_0,Γ_0,Inv on H_P^0, for every u,v,r∈H_P^0 one further has C(u+Γ_0 r,v−A_0 r)−C(u,v)=⟨u,Inv r⟩+||r||²/2. All structural conditions of the auxiliary Hilbert theorem are internally supplied by the same parent witnesses, including selfadjoint Γ_0 from its positivity. The universal identity lies syntactically inside each centered f/output-component branch; C has no f dependence. Every earlier clause, the original six analytic caller conditions and twelve common witnesses remain. No new analytic premise or arbitrary kernel is introduced. This closes the exact perturbation algebra needed after a future actual B27 adapter. It does not construct the actual conditional half-turn H, K/B7 or the source residual r_rho, and does not assert the remaining B4/B28 dynamics, nonexplosion, full paper results, errors, expected costs or composition.

  • Finite-dimensional real Hilbert Borel base E with canonical volume, including the disclosed rank-zero extension; V is C²; 0<α≤β; both global Hessian bounds; η>0 and βη≤1.
  • Canonical normalized laws/kernel, onto pullback e, SAME U/T, full and centered roots, exact centering, gap, unit and bounded inverse are conclusions. Conditional actions are per-observable AE. L² is not assumed finite-dimensional.
  • γ=2√(αη)/(1+αη); the inverse acts only on H_P^0=ker⟨e1,·⟩ with inherited Hilbert structure; the legal endpoint αη=1 is retained.
  • H_perp is exactly ker P (conditional expectation given Y), while H_P^0 is the zero ν-mean macroscopic space. Their complete Hilbert structures, B_0 and V_0 are internally constructed; no onto-kerP or coisometry assumption is introduced.
  • The concluded global-input implication quantifies every f∈L²(J) with integral zero. R,conditional centering,constant annihilation,adjoint range and norm decomposition are derived internally. Private statement definitions expand literally to the original sealed propositions and add no premise.
  • The global output mean, actual projected component formulas and pair-energy identity are internally proved conclusions. There is no additional zero-output-mean, onto-V0, sharp-energy or higher-smoothness hypothesis.
  • C is the exact concluded functional on the same H_P^0, with the exact half and A_0 Inv cross term. Its actual B21 change is proved; no new corrector-bound, onto, mean-preservation, strict-endpoint or smoothness caller is added.
  • The new perturbation identity is a conclusion on the same H_P^0 and same A_0,Γ_0,Inv; all generic hypotheses are internally produced. Arbitrary r is not the source r_rho.

Actual PBPS corrector perturbation with the same witnesses

\[C(u+\Gamma_0r,v-A_0r)-C(u,v)=\langle u,\operatorname{Inv}r\rangle+\frac{\|r\|^2}{2},\qquad u,v,r\in H_P^0.\]

Read the formalized proofs

Each statement and proof below has its own closed Lean disclosure. Samplinglib parents, Mathlib calls and external mathematical sources are distinguished in each proof.

Samplinglib mathematical lesson

Actual PBPS corrector perturbation with the same witnesses

AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCorrectorPerturbation.actual_corrector_perturbation · theorem · Teaching coverage

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Mathematical statement

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, 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 and Z~N(0,I), ν=J.snd, and Λ as the law of (Y,2X−Y) under J. These are probability laws. Put mY=comap(snd), H_P=L² classes measurable with respect to mY, P=i∘E[·|Y] and the canonical M:u↦u∘snd. Then H_P=ran(P), P is identity on H_P, and there exists an onto linear isometry e:L²(ν;ℝ)≃H_P with i e=M. There is a Markov kernel S whose every-state normalized law is S(y)=volume.tilted(x↦−V((y+x)/2)−‖y−x‖²/(8η)); Λ.IsCondKernel S and both Λ marginals equal ν. The same joint reflection F(x,y)=(x,2x−y) gives the involutive selfadjoint isometry U, with Uf=f∘F J-almost everywhere for every f. There is a bounded selfadjoint T on the full real L²(ν), with ‖Tu‖≤‖u‖, Tu(y)=∫u(x)S(y,dx) ν-almost everywhere for each u, and preserved integral. On H_P put A=PUP restricted via i and orthogonal projection, and B=(I_J−P)UPi:H_P→L²(J). Then A=eTe⁻¹ is selfadjoint and contractive, PB=0. There is a bounded positive operator Γ on the real scalar-valued L² space with Γ²=I−T². For the SAME transported Γ_P=eΓe⁻¹, positivity holds and Γ_P²=I_H_P−A²=B*B. There is q∈L²(ν;ℝ) equal to one ν-almost everywhere, with Tq=q and ⟨q,u⟩=∫u dν for every u. Set q_P=e q and H_P^0=ker⟨q_P,·⟩, with its inherited Hilbert structure. Membership is exactly zero ν-mean after e⁻¹. The SAME Γ_P annihilates q_P. Put γ=2√(αη)/(1+αη)>0. There exists Γ_0:H_P^0→H_P^0 equal under inclusion to the restriction of Γ_P, positive with Γ_0−γ I positive. Γ_0 is a unit in the algebra of bounded real operators. There exists a bounded real Inv on this SAME centered space with Inv Γ_0=Γ_0 Inv=I and ‖Inv‖≤1/γ. All probability, normalization, range, root, centering, spectral-gap, unit and inverse facts are internally derived from the original inputs; none is an additional public premise. αη=1 and rank-zero E remain legal. No inverse on full H_P is asserted. Define the actual closed conditional-mean-zero space H_perp=ker P in L²(J;ℝ), with its inherited complete Hilbert structure; this is not merely the globally zero-mean joint space. There exists B_0:H_P^0→H_perp whose inclusion equals B i_0. There exists V_0:H_P^0→H_perp with V_0=B_0 Inv, B_0=V_0 Γ_0, V_0* V_0=I and ‖V_0 f‖=‖f‖ for every f∈H_P^0. All these objects and identities are conclusions from the same original inputs. There exists the canonical bounded residual map R:L²(J;ℝ)→H_perp with i_perp Rg=g−Pg for every joint g. The SAME ambient adjoint satisfies B* g=i_0 B_0* Rg for every joint g; the printed intrinsic block adjoint is recovered on H_perp. For every actual joint f∈L²(J;ℝ) with ∫f dJ=0, there exists f_P∈H_P^0 whose inclusion is E[f|Y]. Put f_perp=Rf and f_V=V_0* f_perp∈the SAME H_P^0. Then B* i_perp f_perp=i_0 Γ_0 f_V=Γ_P i_0 f_V, ‖f‖²=‖f_P‖²+‖f_perp‖² and ‖f_V‖≤‖f_perp‖. The global zero-mean condition is a quantified input restriction to this concluded implication, not an extra assumption on the potential or an operator certificate. The SAME root commutes with T and its transport commutes with A. The SAME Inv on H_P^0 is selfadjoint. There exists an internally constructed A_0:H_P^0→H_P^0 whose inclusion equals A after inclusion; it is selfadjoint,commutes with Γ_0,satisfies A_0²+Γ_0²=I,and commutes with the SAME Inv. For the same U,R and inclusion of H_perp=ker P, define D=R U inclusion. Its full-domain identity V0*D=-A0 V0* is retained. For EVERY globally mean-zero f in L2(J), let fP be its actual centered conditional-expectation component, fperp=R f and fV=V0* fperp. Define the actual output g=U(P f-(f-P f)), not an abstract pair chosen to satisfy a matrix identity. The theorem proves its global mean is zero internally from invariance of the actual reflected augmentation law and preservation of the integral by conditional expectation. Let gP be the actual centered conditional-expectation component of this SAME g, gperp=R g and gV=V0* gperp. Then gP=A0 fP-Gamma0 fV, gV=Gamma0 fP+A0 fV, and ||gP||^2+||gV||^2=||fP||^2+||fV||^2. Both output components are produced by these actual conditional/polar semantics before proving the formulas; neither is defined as its right-hand side. All previous law/kernel/root/inverse/polar/full-micro and centered-observable conclusions persist with the SAME twelve witnesses and the original six analytic caller conditions. No surjectivity, sharp-energy68 bound, extra regularity or zero-output-mean premise is introduced. One private literal Prop stores the complete sealed mathematical conclusion; it supplies no proof provider or additional hypothesis. For the SAME centered A_0 and inverse Inv, define K=A_0 Inv and C(u,v)=(||u||²−||v||²)/2−⟨K u,v⟩ for u,v in H_P^0. For every globally centered actual f and the SAME actual g=U(Pf−(f−Pf)), the already produced conditional/polar components satisfy C(g_P,g_V)−C(f_P,f_V)=−||f_P||²+||f_V||². The half, signs, coefficient one and the same inverse are exact. This identity is proved internally from A_0²+Γ_0²=I, selfadjointness, commutation, the two-sided inverse and the actual projected rotation. No corrector-change certificate, sharp corrector bound, extra domain/smoothness, onto-V_0 or strict endpoint is an additional premise. Every previous clause and all twelve common witnesses persist. For the SAME C and the SAME produced A_0,Γ_0,Inv on H_P^0, for every u,v,r∈H_P^0 one further has C(u+Γ_0 r,v−A_0 r)−C(u,v)=⟨u,Inv r⟩+||r||²/2. All structural conditions of the auxiliary Hilbert theorem are internally supplied by the same parent witnesses, including selfadjoint Γ_0 from its positivity. The universal identity lies syntactically inside each centered f/output-component branch; C has no f dependence. Every earlier clause, the original six analytic caller conditions and twelve common witnesses remain. No new analytic premise or arbitrary kernel is introduced. This closes the exact perturbation algebra needed after a future actual B27 adapter. It does not construct the actual conditional half-turn H, K/B7 or the source residual r_rho, and does not assert the remaining B4/B28 dynamics, nonexplosion, full paper results, errors, expected costs or composition.

\[C(u+\Gamma_0r,v-A_0r)-C(u,v)=\langle u,\operatorname{Inv}r\rangle+\frac{\|r\|^2}{2},\qquad u,v,r\in H_P^0.\]
Lean statement · actual_corrector_perturbation

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, 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 and Z~N(0,I), ν=J.snd, and Λ as the law of (Y,2X−Y) under J. These are probability laws. Put mY=comap(snd), H_P=L² classes measurable with respect to mY, P=i∘E[·|Y] and the canonical M:u↦u∘snd. Then H_P=ran(P), P is identity on H_P, and there exists an onto linear isometry e:L²(ν;ℝ)≃H_P with i e=M. There is a Markov kernel S whose every-state normalized law is S(y)=volume.tilted(x↦−V((y+x)/2)−‖y−x‖²/(8η)); Λ.IsCondKernel S and both Λ marginals equal ν. The same joint reflection F(x,y)=(x,2x−y) gives the involutive selfadjoint isometry U, with Uf=f∘F J-almost everywhere for every f. There is a bounded selfadjoint T on the full real L²(ν), with ‖Tu‖≤‖u‖, Tu(y)=∫u(x)S(y,dx) ν-almost everywhere for each u, and preserved integral. On H_P put A=PUP restricted via i and orthogonal projection, and B=(I_J−P)UPi:H_P→L²(J). Then A=eTe⁻¹ is selfadjoint and contractive, PB=0. There is a bounded positive operator Γ on the real scalar-valued L² space with Γ²=I−T². For the SAME transported Γ_P=eΓe⁻¹, positivity holds and Γ_P²=I_H_P−A²=B*B. There is q∈L²(ν;ℝ) equal to one ν-almost everywhere, with Tq=q and ⟨q,u⟩=∫u dν for every u. Set q_P=e q and H_P^0=ker⟨q_P,·⟩, with its inherited Hilbert structure. Membership is exactly zero ν-mean after e⁻¹. The SAME Γ_P annihilates q_P. Put γ=2√(αη)/(1+αη)>0. There exists Γ_0:H_P^0→H_P^0 equal under inclusion to the restriction of Γ_P, positive with Γ_0−γ I positive. Γ_0 is a unit in the algebra of bounded real operators. There exists a bounded real Inv on this SAME centered space with Inv Γ_0=Γ_0 Inv=I and ‖Inv‖≤1/γ. All probability, normalization, range, root, centering, spectral-gap, unit and inverse facts are internally derived from the original inputs; none is an additional public premise. αη=1 and rank-zero E remain legal. No inverse on full H_P is asserted. Define the actual closed conditional-mean-zero space H_perp=ker P in L²(J;ℝ), with its inherited complete Hilbert structure; this is not merely the globally zero-mean joint space. There exists B_0:H_P^0→H_perp whose inclusion equals B i_0. There exists V_0:H_P^0→H_perp with V_0=B_0 Inv, B_0=V_0 Γ_0, V_0* V_0=I and ‖V_0 f‖=‖f‖ for every f∈H_P^0. All these objects and identities are conclusions from the same original inputs. There exists the canonical bounded residual map R:L²(J;ℝ)→H_perp with i_perp Rg=g−Pg for every joint g. The SAME ambient adjoint satisfies B* g=i_0 B_0* Rg for every joint g; the printed intrinsic block adjoint is recovered on H_perp. For every actual joint f∈L²(J;ℝ) with ∫f dJ=0, there exists f_P∈H_P^0 whose inclusion is E[f|Y]. Put f_perp=Rf and f_V=V_0* f_perp∈the SAME H_P^0. Then B* i_perp f_perp=i_0 Γ_0 f_V=Γ_P i_0 f_V, ‖f‖²=‖f_P‖²+‖f_perp‖² and ‖f_V‖≤‖f_perp‖. The global zero-mean condition is a quantified input restriction to this concluded implication, not an extra assumption on the potential or an operator certificate. The SAME root commutes with T and its transport commutes with A. The SAME Inv on H_P^0 is selfadjoint. There exists an internally constructed A_0:H_P^0→H_P^0 whose inclusion equals A after inclusion; it is selfadjoint,commutes with Γ_0,satisfies A_0²+Γ_0²=I,and commutes with the SAME Inv. For the same U,R and inclusion of H_perp=ker P, define D=R U inclusion. Its full-domain identity V0*D=-A0 V0* is retained. For EVERY globally mean-zero f in L2(J), let fP be its actual centered conditional-expectation component, fperp=R f and fV=V0* fperp. Define the actual output g=U(P f-(f-P f)), not an abstract pair chosen to satisfy a matrix identity. The theorem proves its global mean is zero internally from invariance of the actual reflected augmentation law and preservation of the integral by conditional expectation. Let gP be the actual centered conditional-expectation component of this SAME g, gperp=R g and gV=V0* gperp. Then gP=A0 fP-Gamma0 fV, gV=Gamma0 fP+A0 fV, and ||gP||^2+||gV||^2=||fP||^2+||fV||^2. Both output components are produced by these actual conditional/polar semantics before proving the formulas; neither is defined as its right-hand side. All previous law/kernel/root/inverse/polar/full-micro and centered-observable conclusions persist with the SAME twelve witnesses and the original six analytic caller conditions. No surjectivity, sharp-energy68 bound, extra regularity or zero-output-mean premise is introduced. One private literal Prop stores the complete sealed mathematical conclusion; it supplies no proof provider or additional hypothesis. For the SAME centered A_0 and inverse Inv, define K=A_0 Inv and C(u,v)=(||u||²−||v||²)/2−⟨K u,v⟩ for u,v in H_P^0. For every globally centered actual f and the SAME actual g=U(Pf−(f−Pf)), the already produced conditional/polar components satisfy C(g_P,g_V)−C(f_P,f_V)=−||f_P||²+||f_V||². The half, signs, coefficient one and the same inverse are exact. This identity is proved internally from A_0²+Γ_0²=I, selfadjointness, commutation, the two-sided inverse and the actual projected rotation. No corrector-change certificate, sharp corrector bound, extra domain/smoothness, onto-V_0 or strict endpoint is an additional premise. Every previous clause and all twelve common witnesses persist. For the SAME C and the SAME produced A_0,Γ_0,Inv on H_P^0, for every u,v,r∈H_P^0 one further has C(u+Γ_0 r,v−A_0 r)−C(u,v)=⟨u,Inv r⟩+||r||²/2. All structural conditions of the auxiliary Hilbert theorem are internally supplied by the same parent witnesses, including selfadjoint Γ_0 from its positivity. The universal identity lies syntactically inside each centered f/output-component branch; C has no f dependence. Every earlier clause, the original six analytic caller conditions and twelve common witnesses remain. No new analytic premise or arbitrary kernel is introduced. This closes the exact perturbation algebra needed after a future actual B27 adapter. It does not construct the actual conditional half-turn H, K/B7 or the source residual r_rho, and does not assert the remaining B4/B28 dynamics, nonexplosion, full paper results, errors, expected costs or composition.

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.

Download module (.lean)
theorem actual_corrector_perturbation
    {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_corrector_perturbation_statement (E:=E) (V:=V) (α:=α) (β:=β) (η:=η) hα hαβ hV hH hη hβη

Exact module and namespace context

Full Lean proposition (definition) · actual_corrector_perturbation_statement

This literal definition stores the complete proposition asserted above. It supplies no mathematical proof or additional hypothesis; the theorem proof must establish its value.

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.

Download module (.lean)
private def actual_corrector_perturbation_statement
    {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) : Prop :=
    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))
    let F := fun p : E × E => (p.1,(2 : ℝ) • p.1-p.2)
    let mY := 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
    let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
      HP.subtypeL ∘L condExpL2 ℝ ℝ (μ := J) measurable_snd.comap_le
    let hp : MeasurePreserving (Prod.snd : E × E → E) J ν := ⟨measurable_snd,rfl⟩
    let M : Lp ℝ 2 ν →ₗᵢ[ℝ] Lp ℝ 2 J := Lp.compMeasurePreservingₗᵢ ℝ Prod.snd hp
    IsProbabilityMeasure μ ∧ IsProbabilityMeasure J ∧ IsProbabilityMeasure ν ∧
    HP = P.toLinearMap.range ∧
    (∀ f : HP, P (f : Lp ℝ 2 J) = f) ∧
    ∃ 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 ∧
      ∃ e : Lp ℝ 2 ν ≃ₗᵢ[ℝ] HP,
        (∀ u : Lp ℝ 2 ν, (e u : Lp ℝ 2 J) = M u) ∧
        ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
          (∀ g : Lp ℝ 2 J, (U g : E × E → ℝ) =ᵐ[J] g ∘ F) ∧
          Function.Involutive U ∧ IsSelfAdjoint U.toContinuousLinearMap ∧
          ∃ 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) ∧
            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
            A = e.conjStarAlgEquiv T ∧ IsSelfAdjoint A ∧
            (∀ f : HP, ‖A f‖ ≤ ‖f‖) ∧
            (∀ f : HP, P (B f) = 0) ∧
            ∃ 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 ∧
              Commute Γ T ∧
              let ΓP : HP →L[ℝ] HP := e.conjStarAlgEquiv Γ
              ΓP.IsPositive ∧ ΓP*ΓP=(1 : HP →L[ℝ] HP)-A*A ∧
              Commute ΓP A ∧
              B.adjoint ∘L B = ΓP*ΓP ∧
              ∃ q : Lp ℝ 2 ν,
                (q : E → ℝ) =ᵐ[ν] (fun _ => (1 : ℝ)) ∧ T q = q ∧
                (∀ u : Lp ℝ 2 ν, inner ℝ q u = ∫ y, u y ∂ν) ∧
                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 γ : ℝ := 2*Real.sqrt ((α : ℝ)*η)/(1+(α : ℝ)*η)
                (∀ f : HP, f ∈ HP0 ↔ (∫ y, (e.symm f) y ∂ν) = 0) ∧
                ΓP qP = 0 ∧ 0 < γ ∧
                ∃ ΓP0 : HP0 →L[ℝ] HP0,
                  (∀ f : HP0, (ΓP0 f : HP) = ΓP (f : HP)) ∧
                  ΓP0.IsPositive ∧ (ΓP0-γ • (1 : HP0 →L[ℝ] HP0)).IsPositive ∧
                  IsUnit ΓP0 ∧
                  ∃ Inv : HP0 →L[ℝ] HP0,
                    Inv*ΓP0=(1 : HP0 →L[ℝ] HP0) ∧
                    ΓP0*Inv=(1 : HP0 →L[ℝ] HP0) ∧ ‖Inv‖ ≤ 1/γ ∧
                    IsSelfAdjoint Inv ∧
                    ∃ A0 : HP0 →L[ℝ] HP0,
                      (∀ u : HP0, (A0 u : HP)=A (u : HP)) ∧
                      IsSelfAdjoint A0 ∧ Commute A0 ΓP0 ∧
                      A0*A0+ΓP0*ΓP0=(1 : HP0 →L[ℝ] HP0) ∧
                      Commute A0 Inv ∧
                    let Hperp := P.ker
                    letI : NormedAddCommGroup Hperp := Hperp.normedAddCommGroup
                    letI : InnerProductSpace ℝ Hperp := Hperp.innerProductSpace
                    letI : CompleteSpace Hperp := P.isClosed_ker.completeSpace_coe
                    ∃ B0 : HP0 →L[ℝ] Hperp,
                      (∀ f : HP0, (B0 f : Lp ℝ 2 J) = B (f : HP)) ∧
                      ∃ V0 : HP0 →L[ℝ] Hperp,
                        V0 = B0 ∘L Inv ∧ B0 = V0 ∘L ΓP0 ∧
                        V0.adjoint ∘L V0 = (1 : HP0 →L[ℝ] HP0) ∧
                        (∀ f : HP0, ‖V0 f‖ = ‖f‖) ∧
                        ∃ R : Lp ℝ 2 J →L[ℝ] Hperp,
                          (∀ g : Lp ℝ 2 J, (R g : Lp ℝ 2 J)=g-P g) ∧
                          (∀ g : Lp ℝ 2 J,
                            B.adjoint g=HP0.subtypeL (B0.adjoint (R g))) ∧
                          let D : Hperp →L[ℝ] Hperp :=
                            R ∘L U.toContinuousLinearMap ∘L Hperp.subtypeL
                          (∀ h : Hperp, (D h : Lp ℝ 2 J)=
                            U (h : Lp ℝ 2 J)-P (U (h : Lp ℝ 2 J))) ∧
                          V0.adjoint ∘L D = -(A0 ∘L V0.adjoint) ∧
                          (∀ 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‖ ∧
                              let g : Lp ℝ 2 J := U (P f-(f-P f))
                              (∫ z, g z ∂J)=0 ∧
                              ∃ gP : HP0,
                                HP0.subtypeL gP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le g ∧
                                let gperp : Hperp := R g
                                let gV : HP0 := V0.adjoint gperp
                                gP=A0 fP-ΓP0 fV ∧
                                gV=ΓP0 fP+A0 fV ∧
                                ‖gP‖^2+‖gV‖^2=‖fP‖^2+‖fV‖^2 ∧
                                let C : HP0 → HP0 → ℝ := fun u v =>
                                  (‖u‖^2-‖v‖^2)/2-inner ℝ (A0 (Inv u)) v
                                C gP gV-C fP fV= -‖fP‖^2+‖fV‖^2 ∧
                                (∀ u v r : HP0,
                                  C (u+ΓP0 r) (v-A0 r)-C u v=
                                    inner ℝ u (Inv r)+‖r‖^2/2))

Exact module and namespace context

Proof roadmap

  1. 1Retain every actual law, conditional operator and parent witness

    Apply the accepted actual corrector-change theorem with the original six analytic callers. Extract the same twelve witnesses and every earlier clause. Expose the inherited complete Hilbert spaces H_P^0 and H_perp=ker P, and retain D and its full-domain intertwining identity. None of these structural facts becomes a new caller premise.

  2. 2Instantiate the Hilbert algebra on the same centered space

    Positivity of Γ_0 supplies its selfadjointness. Apply the generic compiled identity with H=H_P^0, A=A_0, G=Γ_0 and the same Inv. The other selfadjointness, commutation, inverse and square-sum facts are parent conclusions. This is a production use of the auxiliary theorem.

  3. 3Adjoin the universal identity to actual conditional components

    For each globally centered f, retain its actual f_P, f_V, g, g_P and g_V, output mean, projected formulas, pair energy and B21 corrector change. Add only the identity for arbitrary u,v,r in the same H_P^0 and same C scope. No identification of r with the still-unconstructed source residual is made.

  4. 4Reassemble the original-input theorem without an extra premise

    Keep the all-ker-P intertwining and every previous law, kernel, root, inverse and polar conclusion with the same witness tuple. The final theorem now contains the perturbation identity. Producing actual H/K/r_rho/B27 and using it in B28 remains a separate source dependency.

Full assumptions and notation

All objects and hypotheses

  • Finite-dimensional real Hilbert Borel base E with canonical volume, including the disclosed rank-zero extension; V is C²; 0<α≤β; both global Hessian bounds; η>0 and βη≤1.
  • Canonical normalized laws/kernel, onto pullback e, SAME U/T, full and centered roots, exact centering, gap, unit and bounded inverse are conclusions. Conditional actions are per-observable AE. L² is not assumed finite-dimensional.
  • γ=2√(αη)/(1+αη); the inverse acts only on H_P^0=ker⟨e1,·⟩ with inherited Hilbert structure; the legal endpoint αη=1 is retained.
  • H_perp is exactly ker P (conditional expectation given Y), while H_P^0 is the zero ν-mean macroscopic space. Their complete Hilbert structures, B_0 and V_0 are internally constructed; no onto-kerP or coisometry assumption is introduced.
  • The concluded global-input implication quantifies every f∈L²(J) with integral zero. R,conditional centering,constant annihilation,adjoint range and norm decomposition are derived internally. Private statement definitions expand literally to the original sealed propositions and add no premise.
  • The global output mean, actual projected component formulas and pair-energy identity are internally proved conclusions. There is no additional zero-output-mean, onto-V0, sharp-energy or higher-smoothness hypothesis.
  • C is the exact concluded functional on the same H_P^0, with the exact half and A_0 Inv cross term. Its actual B21 change is proved; no new corrector-bound, onto, mean-preservation, strict-endpoint or smoothness caller is added.
  • The new perturbation identity is a conclusion on the same H_P^0 and same A_0,Γ_0,Inv; all generic hypotheses are internally produced. Arbitrary r is not the source r_rho.

Mathematical proof

1. Retain every actual law, conditional operator and parent witness

Apply the accepted actual corrector-change theorem with the original six analytic callers. Extract the same twelve witnesses and every earlier clause. Expose the inherited complete Hilbert spaces H_P^0 and H_perp=ker P, and retain D and its full-domain intertwining identity. None of these structural facts becomes a new caller premise.

\[\Gamma_0^2+A_0^2=I,\quad\operatorname{Inv}\Gamma_0=\Gamma_0\operatorname{Inv}=I,\quad g=U(Pf-(f-Pf)).\]
Corresponding Lean step
  unfold actual_corrector_perturbation_statement
  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.ActualCorrectorChange.actual_corrector_change
    (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 hDparent := parentSlice65.1
  have parentSlice66 := parentSlice65.2
  have hIntertwineParent := parentSlice66.1
  have hGlobal := parentSlice66.2
  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
  let D : Hperp →L[ℝ] Hperp := R ∘L U.toContinuousLinearMap ∘L Hperp.subtypeL
  have hD (h : Hperp) : (D h : Lp ℝ 2 J)=
      U (h : Lp ℝ 2 J)-P (U (h : Lp ℝ 2 J)) := hDparent h
  have hIntertwine : V0.adjoint ∘L D= -(A0 ∘L V0.adjoint) := hIntertwineParent

Exact compiled contiguous BODY span; preceding local definitions and inherited Hilbert structures remain in scope.

2. Instantiate the Hilbert algebra on the same centered space

Positivity of Γ_0 supplies its selfadjointness. Apply the generic compiled identity with H=H_P^0, A=A_0, G=Γ_0 and the same Inv. The other selfadjointness, commutation, inverse and square-sum facts are parent conclusions. This is a production use of the auxiliary theorem.

\[C(u+\Gamma_0r,v-A_0r)-C(u,v)=\langle u,\operatorname{Inv}r\rangle+\frac{\|r\|^2}{2},\qquad u,v,r\in H_P^0.\]
Corresponding Lean step
  have hΓself : IsSelfAdjoint ΓP0 := (show ΓP0.IsPositive from hPos0).isSelfAdjoint
  have hPerturb (u v r : HP0) :
      let C : HP0 → HP0 → ℝ := fun u v =>
        (‖u‖^2-‖v‖^2)/2-inner ℝ (A0 (Inv u)) v
      C (u+ΓP0 r) (v-A0 r)-C u v=inner ℝ u (Inv r)+‖r‖^2/2 := by
    exact AutoSamplingTheory.TechnicalLemmas.Analysis.HilbertCorrectorPerturbation.quadratic_corrector_perturbation
      A0 ΓP0 Inv hA0self hΓself hInvSelf hCommInv hLeft hRight hSquare0 u v r

Exact compiled contiguous BODY span; preceding local definitions and inherited Hilbert structures remain in scope.

3. Adjoin the universal identity to actual conditional components

For each globally centered f, retain its actual f_P, f_V, g, g_P and g_V, output mean, projected formulas, pair energy and B21 corrector change. Add only the identity for arbitrary u,v,r in the same H_P^0 and same C scope. No identification of r with the still-unconstructed source residual is made.

\[C(g_P,g_V)-C(f_P,f_V)=-\|f_P\|^2+\|f_V\|^2,\qquad\forall u,v,r\in H_P^0:\ C(u+\Gamma_0r,v-A_0r)-C(u,v)=\langle u,\operatorname{Inv}r\rangle+\tfrac12\|r\|^2.\]
Corresponding Lean step
  have hGlobal72 :
      (∀ 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‖ ∧
                              let g : Lp ℝ 2 J := U (P f-(f-P f))
                              (∫ z, g z ∂J)=0 ∧
                              ∃ gP : HP0,
                                HP0.subtypeL gP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le g ∧
                                let gperp : Hperp := R g
                                let gV : HP0 := V0.adjoint gperp
                                gP=A0 fP-ΓP0 fV ∧
                                gV=ΓP0 fP+A0 fV ∧
                                ‖gP‖^2+‖gV‖^2=‖fP‖^2+‖fV‖^2 ∧
                                let C : HP0 → HP0 → ℝ := fun u v =>
                                  (‖u‖^2-‖v‖^2)/2-inner ℝ (A0 (Inv u)) v
                                C gP gV-C fP fV= -‖fP‖^2+‖fV‖^2 ∧
                                (∀ u v r : HP0,
                                  C (u+ΓP0 r) (v-A0 r)-C u v=
                                    inner ℝ u (Inv r)+‖r‖^2/2)) := by
    intro f hf
    obtain ⟨fP,hpf,hB,hEmbed,hEnergy,hVbound,hgMean,gP,hgpf,hgp,hgv,hPair,hCorrector⟩ :=
      hGlobal f hf
    refine ⟨fP,hpf,hB,hEmbed,hEnergy,hVbound,hgMean,gP,hgpf,hgp,hgv,hPair,hCorrector,?_⟩
    exact hPerturb

Exact compiled contiguous BODY span; preceding local definitions and inherited Hilbert structures remain in scope.

4. Reassemble the original-input theorem without an extra premise

Keep the all-ker-P intertwining and every previous law, kernel, root, inverse and polar conclusion with the same witness tuple. The final theorem now contains the perturbation identity. Producing actual H/K/r_rho/B27 and using it in B28 remains a separate source dependency.

\[C(u+\Gamma_0r,v-A_0r)-C(u,v)=\langle u,\operatorname{Inv}r\rangle+\frac{\|r\|^2}{2},\qquad u,v,r\in H_P^0.\]
Corresponding Lean step
  have hFinal :
      (∀ h : Hperp, (D h : Lp ℝ 2 J)=U (h : Lp ℝ 2 J)-P (U (h : Lp ℝ 2 J))) ∧
      V0.adjoint ∘L D= -(A0 ∘L V0.adjoint) ∧
      (∀ 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‖ ∧
                              let g : Lp ℝ 2 J := U (P f-(f-P f))
                              (∫ z, g z ∂J)=0 ∧
                              ∃ gP : HP0,
                                HP0.subtypeL gP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le g ∧
                                let gperp : Hperp := R g
                                let gV : HP0 := V0.adjoint gperp
                                gP=A0 fP-ΓP0 fV ∧
                                gV=ΓP0 fP+A0 fV ∧
                                ‖gP‖^2+‖gV‖^2=‖fP‖^2+‖fV‖^2 ∧
                                let C : HP0 → HP0 → ℝ := fun u v =>
                                  (‖u‖^2-‖v‖^2)/2-inner ℝ (A0 (Inv u)) v
                                C gP gV-C fP fV= -‖fP‖^2+‖fV‖^2 ∧
                                (∀ u v r : HP0,
                                  C (u+ΓP0 r) (v-A0 r)-C u v=
                                    inner ℝ u (Inv r)+‖r‖^2/2)) := ⟨hD,hIntertwine,hGlobal72⟩
  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,?_⟩
  refine ⟨B0, ?_⟩
  refine ⟨hB0, ?_⟩
  refine ⟨V0, ?_⟩
  refine ⟨hVdef, ?_⟩
  refine ⟨hFactor, ?_⟩
  refine ⟨hVadj, ?_⟩
  refine ⟨hVnorm, ?_⟩
  refine ⟨R, ?_⟩
  refine ⟨hR, ?_⟩
  refine ⟨hAmbient, ?_⟩
  exact hFinal

Exact compiled contiguous BODY span; preceding local definitions and inherited Hilbert structures remain in scope.

Lean verification

The exact proof begins folded immediately after the mathematical argument; the statement is folded beside the claim above.

Lean proof · actual_corrector_perturbation

Retain every actual law, conditional operator and parent witness; Instantiate the Hilbert algebra on the same centered space; Adjoin the universal identity to actual conditional components; Reassemble the original-input theorem without an extra premise

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.

Download module (.lean)
theorem actual_corrector_perturbation
    {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_corrector_perturbation_statement (E:=E) (V:=V) (α:=α) (β:=β) (η:=η) hα hαβ hV hH hη hβη := by
  unfold actual_corrector_perturbation_statement
  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.ActualCorrectorChange.actual_corrector_change
    (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 hDparent := parentSlice65.1
  have parentSlice66 := parentSlice65.2
  have hIntertwineParent := parentSlice66.1
  have hGlobal := parentSlice66.2
  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
  let D : Hperp →L[ℝ] Hperp := R ∘L U.toContinuousLinearMap ∘L Hperp.subtypeL
  have hD (h : Hperp) : (D h : Lp ℝ 2 J)=
      U (h : Lp ℝ 2 J)-P (U (h : Lp ℝ 2 J)) := hDparent h
  have hIntertwine : V0.adjoint ∘L D= -(A0 ∘L V0.adjoint) := hIntertwineParent
  have hΓself : IsSelfAdjoint ΓP0 := (show ΓP0.IsPositive from hPos0).isSelfAdjoint
  have hPerturb (u v r : HP0) :
      let C : HP0 → HP0 → ℝ := fun u v =>
        (‖u‖^2-‖v‖^2)/2-inner ℝ (A0 (Inv u)) v
      C (u+ΓP0 r) (v-A0 r)-C u v=inner ℝ u (Inv r)+‖r‖^2/2 := by
    exact AutoSamplingTheory.TechnicalLemmas.Analysis.HilbertCorrectorPerturbation.quadratic_corrector_perturbation
      A0 ΓP0 Inv hA0self hΓself hInvSelf hCommInv hLeft hRight hSquare0 u v r
  have hGlobal72 :
      (∀ 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‖ ∧
                              let g : Lp ℝ 2 J := U (P f-(f-P f))
                              (∫ z, g z ∂J)=0 ∧
                              ∃ gP : HP0,
                                HP0.subtypeL gP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le g ∧
                                let gperp : Hperp := R g
                                let gV : HP0 := V0.adjoint gperp
                                gP=A0 fP-ΓP0 fV ∧
                                gV=ΓP0 fP+A0 fV ∧
                                ‖gP‖^2+‖gV‖^2=‖fP‖^2+‖fV‖^2 ∧
                                let C : HP0 → HP0 → ℝ := fun u v =>
                                  (‖u‖^2-‖v‖^2)/2-inner ℝ (A0 (Inv u)) v
                                C gP gV-C fP fV= -‖fP‖^2+‖fV‖^2 ∧
                                (∀ u v r : HP0,
                                  C (u+ΓP0 r) (v-A0 r)-C u v=
                                    inner ℝ u (Inv r)+‖r‖^2/2)) := by
    intro f hf
    obtain ⟨fP,hpf,hB,hEmbed,hEnergy,hVbound,hgMean,gP,hgpf,hgp,hgv,hPair,hCorrector⟩ :=
      hGlobal f hf
    refine ⟨fP,hpf,hB,hEmbed,hEnergy,hVbound,hgMean,gP,hgpf,hgp,hgv,hPair,hCorrector,?_⟩
    exact hPerturb
  have hFinal :
      (∀ h : Hperp, (D h : Lp ℝ 2 J)=U (h : Lp ℝ 2 J)-P (U (h : Lp ℝ 2 J))) ∧
      V0.adjoint ∘L D= -(A0 ∘L V0.adjoint) ∧
      (∀ 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‖ ∧
                              let g : Lp ℝ 2 J := U (P f-(f-P f))
                              (∫ z, g z ∂J)=0 ∧
                              ∃ gP : HP0,
                                HP0.subtypeL gP=condExpL2 ℝ ℝ (μ:=J) measurable_snd.comap_le g ∧
                                let gperp : Hperp := R g
                                let gV : HP0 := V0.adjoint gperp
                                gP=A0 fP-ΓP0 fV ∧
                                gV=ΓP0 fP+A0 fV ∧
                                ‖gP‖^2+‖gV‖^2=‖fP‖^2+‖fV‖^2 ∧
                                let C : HP0 → HP0 → ℝ := fun u v =>
                                  (‖u‖^2-‖v‖^2)/2-inner ℝ (A0 (Inv u)) v
                                C gP gV-C fP fV= -‖fP‖^2+‖fV‖^2 ∧
                                (∀ u v r : HP0,
                                  C (u+ΓP0 r) (v-A0 r)-C u v=
                                    inner ℝ u (Inv r)+‖r‖^2/2)) := ⟨hD,hIntertwine,hGlobal72⟩
  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,?_⟩
  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.ActualCorrectorPerturbation

#print axioms AutoSamplingTheory.ExampleCases.ProximalBPS.ActualCorrectorPerturbation.actual_corrector_perturbation

Exact module and namespace context

Source assumptions versus formal assumptions

SourceActual LeanDifference kindWhy it matters
Finite-dimensional real Hilbert Borel base E with canonical volume, including the disclosed rank-zero extension; V is C²; 0<α≤β; both global Hessian bounds; η>0 and βη≤1.Finite-dimensional real Hilbert Borel base E with canonical volume, including the disclosed rank-zero extension; V is C²; 0<α≤β; both global Hessian bounds; η>0 and βη≤1.sameExact original six analytic callers and all twelve witnesses are retained.
Actual H/K/r_rho/B27/B28 and full paper conclusions.The bounded corrector perturbation identity is focused-compiled. Arbitrary r is not the actual source residual r_rho. Actual conditional half-turn H, K/B7, r/r_rho/B27, B4/B28 dynamics, H1/B2, invariance/nonexplosion, full PBPS/SPHMC results, implementation errors/caps, expected-query costs and actual-input composition remain open. The perturbation identity alone establishes no Exposition Seal, PURIFIED, main/live, whole-paper or Goal completion; independent review and serialized integration are separate admissions.unresolvedThe perturbation algebra does not produce the algorithm-specific residual or dynamics/cost results.

The bounded corrector perturbation identity is focused-compiled. Arbitrary r is not the actual source residual r_rho. Actual conditional half-turn H, K/B7, r/r_rho/B27, B4/B28 dynamics, H1/B2, invariance/nonexplosion, full PBPS/SPHMC results, implementation errors/caps, expected-query costs and actual-input composition remain open. The perturbation identity alone establishes no Exposition Seal, PURIFIED, main/live, whole-paper or Goal completion; independent review and serialized integration are separate admissions.

Encoder–denoiser: source-reviewed · equivalent-after-elaboration

Detected semantic differences

  • objects: The literal private definition stores the complete result and supplies no provider or additional hypothesis. Its exact full helper is adjacent and initially folded. — actual20–141,143–152; StageB.unit.1.bounded-adjacent-fold.html
  • domains: AE L2/closed conditional subspaces and normalized tilted measures elaborate source notation; no extra input normalizer or integrability premise. — actual29–43,45–52; Tilted.lean36–44; inherited probability conclusions
  • assumptions: GammaP0 positivity supplies selfadjointness internally on complete HP0; decoder selfadjoint/nonnegative positivity wording is equivalent here. — actual82–83,90,314; Positive.lean266–281
  • quantifiers: Twelve global witnesses and two branch-local components are distinct scopes, exactly retained; early observer counting all fourteen together was corrected without candidate change. — StageB.parent-retention.exact-check.json; failed PID33468 EXIT1, corrected PID8608 EXIT0
  • domains: V0 maps into all kerP as an isometric embedding; no onto-V0 or range-only D identity. e is onto HP by equivalence and is a separate object. — actual51–52,103–118; decoder full-domain paragraph
  • conclusion: Actual g and real condExpL2 gP / V0*Rg gV are defined before their formulas; mean(g)=0 is an inherited internal conclusion. — actual119–138,321–349; exact parent retention
  • scopes: Decoder K=kerP and source-contract K=A0 Inv are independent local aliases, neither the still-open algorithm Markov K. Exact definitions resolve the name ambiguity. — decoder text Hperp paragraph; source contract cross operator; StageA S17; complete literal uses Hperp and A0(Inv u)
  • scopes: Final forall u v r is nested inside each actual f/gP/C branch; C is independent of f. This consumes actual witnesses without asserting r=r_rho or B27 outputs. — actual139–141,315–349; source Ex27/B27 outside result
  • scopes: Parent71/B21 retained for integration, not a pure-Hilbert algebra parent; no invented68 sharp-energy dependency. Two cells expose one SAU with two connected canonical declarations. — actual imports/parent call and generic Mathlib-only imports; source graph vs Lean graph distinction
  • constant_dependencies: Rank0 and alpha eta=1 remain legal; same centered inverse and exact half, signs/order survive; no rho, cost or higher-regularity assumption. — actual20–28,84–102,136–141; generic18–21
  • scopes: Reader status16-field v3 overlay changes only boundary/process strings; context hash changes are acknowledged and final packet3 is exact approved proposed bytes. — reader-status-overlay72.v3.independent-decision.json; StageB.final-binding-checks.json

A generalization is not a source correction. Proposed missing conditions require separate independent repair review. No proposed repair silently changes the original theorem.

Scope and omitted-condition boundaries
  • The bounded corrector perturbation identity is focused-compiled. Arbitrary r is not the actual source residual r_rho. Actual conditional half-turn H, K/B7, r/r_rho/B27, B4/B28 dynamics, H1/B2, invariance/nonexplosion, full PBPS/SPHMC results, implementation errors/caps, expected-query costs and actual-input composition remain open. The perturbation identity alone establishes no Exposition Seal, PURIFIED, main/live, whole-paper or Goal completion; independent review and serialized integration are separate admissions.
Sources, dependencies, and reuse

Source and reuse

Samplinglib parents called

Mathlib API called (external library)

  • IsSelfAdjoint.isSymmetric
  • ContinuousLinearMap.mul_apply
  • ContinuousLinearMap.one_apply
  • real_inner_self_eq_norm_sq
  • norm_add_sq_real
  • norm_sub_sq_real
  • real_inner_comm
  • congrArg
  • ContinuousLinearMap.IsPositive.isSelfAdjoint

Mathematical sources

  • PBPS B20 and Appendix B4 Ex28--Ex34 perturbation algebra; future actual B27/B28 adapter — The bounded corrector perturbation identity is focused-compiled. Arbitrary r is not the actual source residual r_rho. Actual conditional half-turn H, K/B7, r/r_rho/B27, B4/B28 dynamics, H1/B2, invariance/nonexplosion, full PBPS/SPHMC results, implementation errors/caps, expected-query costs and actual-input composition remain open. The perturbation identity alone establishes no Exposition Seal, PURIFIED, main/live, whole-paper or Goal completion; independent review and serialized integration are separate admissions.

Samplinglib prose is not a quotation or a source-equivalence certificate. Definitions and aliases are explained as constructions, not counted as new mathematical proofs.

Which proof obligations and prerequisites are covered?

  • Formalized proof edge; source adapter/review separate Exact signed perturbation algebra
  • TODO — not closed by these contributions Actual H/K/r_rho/B27/B28 and full paper/cost/composition