Planned local formalization

Proximal BPS: high accuracy from a warm start

Consumer: a Renyi-warm input becomes a high-accuracy position law through an augmented-state discrete chain.

Common setting and conventions

\[d\ge1,\quad V\in C^2(\mathbb R^d),\quad 0<\alpha I\preceq\nabla^2V(x)\preceq\beta I,\quad \kappa=\beta/\alpha,\quad \pi(dx)=Z^{-1}e^{-V(x)}dx\]

The Hessian inequalities hold at every point; alpha and beta are positive real numbers. Separate Hessian-to-chord and Gibbs-integrability proofs support the compiled normalized PBPS augmentation certificate: positive Gibbs integral, probability and exact product-volume joint density. Consult linked cells for independent-review and admission states. Conditional kernels, the sampling process and its guarantees remain separate.
TV(P,Q) means sup over measurable events of |P(A)-Q(A)|. For probability laws with densities it is half their L1 distance.
R_2(P||Q)=log integral (dP/dQ)^2 dQ when P is absolutely continuous with respect to Q, and infinity otherwise. The PBPS paper writes D_2 and calls the target mu; here the common target is pi.
Expected oracle queries are not worst-case runtime. Reference-point preparation, proximal access and gradient-only reductions must be charged under the particular theorem's model.

RED · source theorem known · local proof open

Optimized expected gradient complexity

Source: 2609.06905v1. PBPS Corollary 4.4, (4.19)–(4.20); Theorem 4.3, (4.8)–(4.9)

Statement

With universal positive tuning constants, run the implementable algorithm with the source parameter choices. Its output is epsilon-close to the target in TV and has the following expected gradient-query bound. No fixed worst-case running-time claim is made.

The common setting; an input probability law mu_0 with R_2(mu_0||pi) <= Delta, where Delta >= 1.
0 < epsilon < 1/4; query access to grad V and a sample from mu_0.
Choose L and eta below and the iteration, resampling, error and solver budgets in source (4.8)–(4.9). The constants c_opt and K_opt are universal, not arbitrary user choices.

\[\begin{gathered}L=\Delta+\log\frac{K_{\rm opt}d\kappa}{\varepsilon},\quad\eta=\frac{c_{\rm opt}}{\beta(\sqrt{dL}+L)},\quad\operatorname{TV}(\mathcal L(\widehat X),\pi)\le\varepsilon,\\\mathbb E Q_\nabla\le K_{\rm opt}\sqrt\kappa(d+L)^{1/4}L^{3/4}\left(\Delta+\log\frac4\varepsilon\right)\log(\kappa L).\end{gathered}\]

Lean statement — not formalized yet

A future declaration must bind the probability laws, normalization, regularity, source algorithm and oracle model explicitly. The formula is a source theorem contract, not Lean code.

No corresponding ASTIS declaration is asserted. Search locations below are candidates, not established dependencies or copied library proofs.

Proof architecture and calculations

This is a source-linked proof route, not a complete reconstruction of all cited lemmas. The dependencies below remain separate formalization tasks.

Augment, then reflect the auxiliary variable
\[\pi_\eta(dx,dy)=\pi(dx)\mathcal N(x,\eta I)(dy),\qquad (x,y)\mapsto(x,2x-y)\]
Gaussian symmetry preserves the augmented law under this reflection. It differs from velocity reflection at a bounce. A conditional half-turn and occasional RGO resampling form the ideal transition; they are not independent ordinary Gibbs redraws at every step.
Source: 2609.06905v1. Proposition 2.1; (3.11); Proposition 3.4
Use discrete hypocoercivity, not a reversible gap shortcut
\[f=\mathbb E[f\mid Y]+(f-\mathbb E[f\mid Y]),\qquad\chi^2(\nu_0\mathcal K^{JN}\Vert\pi_\eta)\le e^{-J}\chi^2(\nu_0\Vert\pi_\eta)\]
The source chooses eta, rho and a block length N as in Theorem 3.5. Conditional resampling damps the fluctuation; reflection couples it to the conditional mean. A modified L2 energy and half-turn operator bounds are essential. The local unrefreshed process alone need not mix.
Source: 2609.06905v1. Theorem 3.5; Appendices B–C
Keep implementation mismatch separate from ideal mixing
\[\mathbb P(\widehat X\ne X_T^{\rm id})\le3Ta=\varepsilon/2,\qquad\operatorname{TV}(\mathcal L(\widehat X),\pi)\le\varepsilon/2+\varepsilon/2\]
Solver failure, approximate RGO draws and capped bounce rates each consume an error budget a. Lifted Renyi control bounds bad events along the ideal chain; a coupling union bound transfers its convergence to the implemented output.
Source: 2609.06905v1. Theorem 4.3, (4.14)–(4.18)
Balance localization against block length
\[(\alpha\eta)^{-1/2}=\sqrt{\kappa/c_{\rm opt}}(\sqrt{dL}+L)^{1/2}\]
Substitute the chosen eta into the iteration bound and charge proximal solves, RGO calls and candidate bounce queries. Retaining L prevents a hidden warmness or precision dependence from disappearing into the dimension exponent.
Source: 2609.06905v1. Corollary 4.4, (4.21)–(4.23)

Lean proof — not formalized yet

Formalization will first match the named technology interfaces, then assemble this source theorem. No placeholder proof or source-cited wrapper has been added.

No corresponding ASTIS declaration is asserted. Search locations below are candidates, not established dependencies or copied library proofs.

Strict boundary

This is the discrete augmented Proximal BPS algorithm with curved conditional flights. It is not vanilla continuous-time BPS, and its acceleration does not follow just from invariance or an algebraic reflection identity.

Expand reusable prerequisites and next packets →

The Gaussian posterior is an actual measurable conditional kernel

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang. ASTIS supplies the standard conditional-kernel derivation; no author endorsement or complete-paper certificate is implied.

arXiv:2609.06905v1 · Section 2.2, equations (2.7)-(2.8), backward conditional-law clause; Definition 2.2 and Section 3.2 consumers · faithful paraphrase

Complete source statement (ASTIS restatement)

PBPS v1 §2.2, (2.7)-(2.8), takes X with normalized Gibbs law μ(dx) proportional to exp(−V(x))dx, an independent standard Gaussian G, and Y=X+sqrt(η)G. The selected backward conditional-law clause gives X conditional on Y=y density proportional to exp(−V(x)−‖x−y‖²/(2η)). ASTIS formalizes its normalized conditional-kernel mechanism: for any probability μ on a finite-dimensional real inner-product Borel space E and any η>0, the everywhere-defined normalized tilt R(y,dx)=exp(−‖x−y‖²/(2η))μ(dx)/∫exp(−‖z−y‖²/(2η))μ(dz) is a measurable Markov kernel satisfying L(Y,X)=L(Y)⊗R. The source's Gibbs, C² curvature and η≤1/β restrictions are not needed for this identity; this generalization is explicit, and the source Gibbs probability certificate is a separate compiled parent. This is an ASTIS mathematical restatement and standard expansion, not an original quotation. Equality with other conditional versions is only marginal-almost-everywhere. No RGO implementation, PBPS process invariance or quantitative sampling bound is asserted.

E is a finite-dimensional real inner-product space with its Borel sigma algebra; μ is a probability measure, possibly singular and without finite moments.
The variance η is strictly positive. No curvature bound, density for μ, or restriction η≤1/β is needed for this law identity.
A kernel means a measure-valued function with measurable dependence on its input. Every returned fiber is a probability law; these properties are proved, not supplied.
The explicit formula selects a version at every y. Conditional laws chosen in other ways need agree with it only almost everywhere under the observation marginal.

source Gibbs specialization

\[\mathcal L(X\mid Y=y)(dx)\propto e^{-V(x)-\|x-y\|^2/(2\eta)}\,dx.\]

normalized generic conditional kernel

\[R(y,dx)=\frac{e^{-\|x-y\|^2/(2\eta)}\mu(dx)}{\int_E e^{-\|z-y\|^2/(2\eta)}\mu(dz)},\qquad\mathcal L(Y,X)=\mathcal L(Y)\otimes R.\]

Read the formalized proofs

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

ASTIS mathematical exposition

The Gaussian posterior is an actual measurable conditional kernel

AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel.exists_tilted_isCondKernel · theorem · Teaching coverage

Statement

Let μ be any probability measure on a finite-dimensional real inner-product Borel space E, and let η>0. For independent X∼μ and G∼N(0,I), set Y=X+√η G and let J be the law of (X,Y). There exists a measurable Markov kernel R from E to E such that for every y its law is the normalized quadratic tilt μ.tilted(x↦−‖x−y‖²/(2η)), and R disintegrates the law of (Y,X) with respect to the actual observation marginal.

\[R(y,dx)=\frac{e^{-\|x-y\|^2/(2\eta)}\mu(dx)}{\int_E e^{-\|z-y\|^2/(2\eta)}\mu(dz)},\qquad\mathcal L(Y,X)=\mathcal L(Y)\otimes R.\]

All objects and hypotheses

E is a finite-dimensional real inner-product space with its Borel sigma algebra; μ is a probability measure, possibly singular and without finite moments.
The variance η is strictly positive. No curvature bound, density for μ, or restriction η≤1/β is needed for this law identity.
A kernel means a measure-valued function with measurable dependence on its input. Every returned fiber is a probability law; these properties are proved, not supplied.
The explicit formula selects a version at every y. Conditional laws chosen in other ways need agree with it only almost everywhere under the observation marginal.

Mathematical proof

1. A bounded positive likelihood has a genuine normalizer

For each observation y put w(y,x)=exp(−‖x−y‖²/(2η)). Positivity of η gives 0<w≤1. Joint continuity makes w measurable; the finite input measure makes every fiber integrable. Since μ has mass one and w is strictly positive everywhere, its integral is strictly positive. Thus neither division nor the tilted-measure API can fall back to a zero law.

\[w(y,x)=e^{-\|x-y\|^2/(2\eta)},\qquad Z(y)=\int_Ew(y,x)\,\mu(dx),\qquad0<Z(y)\le1.\]

Corresponding Lean step

hw proves joint measurability. hI uses integrable_const and Integrable.mono' with the explicit bound by one; hZpos invokes integral_exp_pos. The upper bound is the domination used in hI, not a separate production declaration.

2. Make the normalized fibers into one measurable Markov kernel

Measurability of parameter integrals makes y↦Z(y) measurable. Consequently w(y,x)/Z(y) is jointly measurable. Weight the constant kernel μ by this density. Its fiber at every y is exactly the existing normalized tilted measure, whose probability theorem applies because hI has already been proved.

\[R(y,dx)=\frac{w(y,x)}{Z(y)}\mu(dx),\qquad R(y,E)=1.\]

Corresponding Lean step

hZ uses StronglyMeasurable.integral_prod_right. hd supplies the required joint measurability to Kernel.withDensity_apply; without it that totalized constructor would return zero. hRfiber identifies the exact Measure.tilted expression, and hR uses isProbabilityMeasure_tilted.

3. Express the actual joint law in observation-first order

Take independent X with law μ and a standard Gaussian G, and set Y=X+√η G. Reuse the already compiled Gaussian augmentation density, then swap its coordinates. The symmetry ‖y−x‖=‖x−y‖ gives the displayed density with respect to dy μ(dx). No density of μ relative to volume is needed.

\[J^\leftrightarrow(dy,dx)=C_\eta w(y,x)\,dy\,\mu(dx),\qquad C_\eta=(\sqrt{2\pi\eta})^{-\dim E}.\]

Corresponding Lean step

hJ calls the ASTIS-owned GaussianAugmentation.augmentation_eq_withDensity, transports the density through MeasurableEquiv.prodComm using the existing ASTIS RadonNikodym lemma, then uses Measure.prod_swap and norm_sub_rev.

4. Cancel only the proven positive normalizer

Introduce the auxiliary observation measure ν(dy)=Cη Z(y)dy. This measure is s-finite, so its composition-product with R is defined by the usual kernel integral. The density-product identities combine the factors; Z(y)>0 justifies cancellation at every point. This proves the full joint-measure identity, not merely proportionality of densities.

\[(\nu\otimes R)(dy,dx)=C_\eta Z(y)\frac{w(y,x)}{Z(y)}\,dy\,\mu(dx)=J^\leftrightarrow(dy,dx).\]

Corresponding Lean step

hcomp uses Measure.compProd_withDensity, Measure.compProd_const, prod_withDensity_left and withDensity_mul. ENNReal.ofReal_mul converts the nonnegative density product, and field_simp receives (hZpos y).ne'. No cancellation of a possibly zero integral is permitted.

5. Identify the observation marginal and certify disintegration

Take first marginals of the joint identity. Because each R(y,·) has total mass one, the first marginal of ν⊗R is ν. Therefore ν is the actual law of Y, and the joint identity is precisely the conditional-kernel certificate. Taking second marginals also recovers the original input law after applying R to the law of Y.

\[\nu=J^\leftrightarrow_1=\mathcal L(Y),\qquad\mathcal L(Y)\otimes R=\mathcal L(Y,X),\qquad\int R(y,\cdot)\,\mathcal L(Y)(dy)=\mu.\]

Corresponding Lean step

hf applies congrArg Measure.fst and Measure.fst_compProd. Measure.IsCondKernel packages exactly the resulting equality. The separate focused test takes second marginals with Measure.snd_compProd and checks the backward-law recovery consumer.

Lean statement · exists_tilted_isCondKernel

The let-bound J is the pushforward of μ.prod(stdGaussian E) by (x,g)↦(x,x+sqrt η•g), so independence and the actual observation law are encoded explicitly. IsMarkovKernel and the all-y fiber formula are returned together with (J.map Prod.swap).IsCondKernel R. Measure.tilted already expresses normalized exponential reweighting; no duplicate posterior definition is introduced.

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 exists_tilted_isCondKernel (μ : Measure E) [IsProbabilityMeasure μ]
    {η : ℝ} (hη : 0 < η) :
    let J := Measure.map
      (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    ∃ R : Kernel E E, IsMarkovKernel R ∧
      (∀ y, R y = μ.tilted (fun x => -‖x - y‖ ^ 2 / (2 * η))) ∧
      (J.map Prod.swap).IsCondKernel R

Exact module and namespace context

Lean proof · exists_tilted_isCondKernel

All weights, normalizers, densities and the constructed kernel are local proof terms. Their measurability and normalization witnesses are proved before density or composition APIs are used. The focused tests exercise a singular Dirac input and the general backward-law recovery identity, and print the theorem's axioms.

theorem exists_tilted_isCondKernel (μ : Measure E) [IsProbabilityMeasure μ]
    {η : ℝ} (hη : 0 < η) :
    let J := Measure.map
      (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    ∃ R : Kernel E E, IsMarkovKernel R ∧
      (∀ y, R y = μ.tilted (fun x => -‖x - y‖ ^ 2 / (2 * η))) ∧
      (J.map Prod.swap).IsCondKernel R := by
  classical
  let J := Measure.map
    (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
    (μ.prod (stdGaussian E))
  let w : E → E → ℝ := fun y x => Real.exp (-‖x - y‖ ^ 2 / (2 * η))
  let Z : E → ℝ := fun y => ∫ x, w y x ∂μ
  have hw : Measurable (Function.uncurry w) := by fun_prop
  have hI (y : E) : Integrable (w y) μ := by
    refine (integrable_const (1 : ℝ)).mono' (by fun_prop) ?_
    filter_upwards with x
    rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
    exact Real.exp_le_one_iff.mpr
      (div_nonpos_of_nonpos_of_nonneg (neg_nonpos.mpr (sq_nonneg _)) (by positivity))
  have hZpos (y : E) : 0 < Z y := integral_exp_pos (hI y)
  have hZ : Measurable Z := hw.stronglyMeasurable.integral_prod_right.measurable
  let d : E → E → ℝ≥0∞ := fun y x => ENNReal.ofReal (w y x / Z y)
  have hd : Measurable (Function.uncurry d) :=
    (hw.div (hZ.comp measurable_fst)).ennreal_ofReal
  let R : Kernel E E := (Kernel.const E μ).withDensity d
  have hRfiber (y : E) : R y = μ.tilted (fun x => -‖x - y‖ ^ 2 / (2 * η)) := by
    rw [show R = (Kernel.const E μ).withDensity d from rfl,
      Kernel.withDensity_apply _ hd]
    rfl
  have hR : IsMarkovKernel R := ⟨fun y => by
    rw [hRfiber]
    exact isProbabilityMeasure_tilted (hI y)⟩
  let C : ℝ := ((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E
  have hC : 0 ≤ C := by positivity
  let a : E → ℝ≥0∞ := fun y => ENNReal.ofReal (C * Z y)
  have ha : Measurable a := (measurable_const.mul hZ).ennreal_ofReal
  let ν : Measure E := volume.withDensity a
  have hJ : J.map Prod.swap = (volume.prod μ).withDensity
      (fun p : E × E => ENNReal.ofReal (C * w p.1 p.2)) := by
    dsimp only [J]
    rw [ExampleCases.ProximalBPS.GaussianAugmentation.augmentation_eq_withDensity μ η hη]
    change ((μ.prod volume).withDensity _).map
      (MeasurableEquiv.prodComm : E × E ≃ᵐ E × E) = _
    rw [Measure.RadonNikodym.measurableEquiv_map_withDensity
      (MeasurableEquiv.prodComm : E × E ≃ᵐ E × E) _ (by fun_prop)]
    change ((μ.prod volume).map Prod.swap).withDensity _ = _
    rw [Measure.prod_swap]
    congr 1
    funext p
    change ENNReal.ofReal (C * Real.exp (-‖p.1 - p.2‖ ^ 2 / (2 * η))) =
      ENNReal.ofReal (C * Real.exp (-‖p.2 - p.1‖ ^ 2 / (2 * η)))
    rw [norm_sub_rev p.1 p.2]
  have hcomp : ν ⊗ₘ R = J.map Prod.swap := by
    rw [hJ]
    change ν ⊗ₘ (Kernel.const E μ).withDensity d = _
    rw [Measure.compProd_withDensity hd, Measure.compProd_const]
    change ((volume.withDensity a).prod μ).withDensity _ = _
    rw [prod_withDensity_left ha]
    rw [← withDensity_mul _
      (show Measurable (fun p : E × E => a p.1) from ha.comp measurable_fst)
      (show Measurable (fun p : E × E => d p.1 p.2) from hd)]
    congr 1
    funext p
    change ENNReal.ofReal (C * Z p.1) * ENNReal.ofReal (w p.1 p.2 / Z p.1) = _
    rw [← ENNReal.ofReal_mul (mul_nonneg hC (hZpos p.1).le)]
    congr 1
    field_simp [(hZpos p.1).ne']
  have hf := congrArg Measure.fst hcomp
  rw [Measure.fst_compProd] at hf
  refine ⟨R, hR, hRfiber, ⟨?_⟩⟩
  rw [← hf]
  exact hcomp

end AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Normalized Gibbs input law in Euclidean space	Any probability μ on finite-dimensional real inner-product Borel E	generalization	The Gaussian likelihood argument permits singular input laws and zero dimension. Source Gibbs normalization is supplied separately, not silently assumed from a totalized integral.
η in (0,1/β] and source curvature assumptions	η>0 with no curvature or moment premise	generalization	These source restrictions support later curvature and complexity bounds; they are unnecessary for the selected exact conditional-law identity.
Conditional density proportional to the exponential	Everywhere positive normalizer, jointly measurable kernel, every fiber probability and actual disintegration	source-implicit	Bounded positive weights and parameter-integral measurability justify the omitted conditions without adding input hypotheses.
X conditional on Y=y	One explicit all-y kernel with IsCondKernel certificate	same	An explicit regular version is selected. The claim is not pointwise equality with all arbitrary conditional versions.

This closes the conditional-law construction behind PBPS (2.7)-(2.8), not a realizable RGO algorithm, PBPS trajectory, nonexplosion, invariant process law, mixing bound or expected query cost. Gibbs specialization uses the separately proved positive Gibbs normalizer and probability certificate. The generic theorem itself assumes an input probability law and does not prove those Gibbs facts again. The focused consumer test proves exact recovery of μ from the actual smoothed marginal via this kernel. This is one joint-law marginal identity, not stationarity of the PBPS process. Curvature, covariance, moment bounds and implementation error remain separate. No pointwise uniqueness among arbitrary conditional versions is claimed.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

domains: Arbitrary probability inputs replace the source Gibbs input. — This extension is explicitly disclosed and justified by the bounded, strictly positive likelihood. It does not claim that a singular input has a volume density.
assumptions: Curvature and η≤1/β are omitted for the conditional-law identity. — Neither is needed for normalization or disintegration; η>0 remains essential. This omission does not relax the conditions of the source algorithms or their quantitative estimates.
quantifiers: An explicit all-y version elaborates conditional-law notation. — The positive normalizer and measurable kernel construction justify this version. No all-y uniqueness claim is made.

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

This closes the conditional-law construction behind PBPS (2.7)-(2.8), not a realizable RGO algorithm, PBPS trajectory, nonexplosion, invariant process law, mixing bound or expected query cost.
Gibbs specialization uses the separately proved positive Gibbs normalizer and probability certificate. The generic theorem itself assumes an input probability law and does not prove those Gibbs facts again.
The focused consumer test proves exact recovery of μ from the actual smoothed marginal via this kernel. This is one joint-law marginal identity, not stationarity of the PBPS process.
Curvature, covariance, moment bounds and implementation error remain separate. No pointwise uniqueness among arbitrary conditional versions is claimed.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

integral_exp_pos
StronglyMeasurable.integral_prod_right
Kernel.withDensity_apply
isProbabilityMeasure_tilted
Measure.prod_swap
Measure.compProd_withDensity
Measure.compProd_const
prod_withDensity_left
withDensity_mul
Measure.fst_compProd

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS v1, §2.2, (2.7)-(2.8) — Conditional-law step, expanded by ASTIS with normalization, measurability and disintegration; no copied paper prose.
Chen, Chewi, Lu and Zhang, SPHMC v1, §3.4 — Intended consumer: the backward restricted Gaussian law. Algorithm implementation and quantitative guarantees are independent.
Rémy Degenne and Mathlib contributors, Apache-2.0 — Existing measurable density-kernel and composition APIs are reused, not counted as ASTIS-owned results.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Construct the actual normalized measurable Markov kernel and prove the joint disintegration
TODO — not closed by these contributions Construct an implementable restricted Gaussian sampler with its source error and cost bounds
TODO — not closed by these contributions Construct and analyze the PBPS process, nonexplosion and invariant-law/domain obligations

The actual normalized Gibbs augmentation law

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang. ASTIS mathematical restatement with explicit normalization details, not copied prose or author endorsement.

arXiv:2609.06905v1 · Section 2.2 equations (2.6)-(2.7); positive Hessian setting in introduction (1.1) · faithful paraphrase

Complete source statement (ASTIS restatement)

In the paper's Euclidean C² setting αI≤D²V≤βI with 0<α≤β and 0<η≤1/β, the target μ has normalized density Z_V^{-1}exp(-V). The joint density proportional to exp(-V(x)-||x-y||²/(2η)) is the law of X~μ and Y=X+sqrt(η)N for independent standard Gaussian N. Its exact normalizing factor is (2πη)^(-d/2)/Z_V. ASTIS makes finite positive Gibbs mass and probability of this law explicit.

The primary source assumes V∈C²(R^d), 0<α≤β and αI≤D²V(x)≤βI globally, in (1.1).
Section 2.2 fixes 0<η≤1/β. The noise N has covariance I and is independent of X.
The normalization convention uses Euclidean Lebesgue volume and Z_V=∫exp(-V(x))dx. The source writes the joint density up to proportionality; this packet supplies its exact factor and positive finite mass.

(2.6)-(2.7), explicit normalizer

\[Z_V=\int e^{-V(x)}dx>0,\quad \pi_\eta(dx,dy)=\frac{((\sqrt{2\pi\eta})^{-1})^d}{Z_V}e^{-V(x)-\|y-x\|^2/(2\eta)}dx\,dy=\mathcal L(X,X+\sqrt\eta N)(dx,dy).\]

Read the formalized proofs

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

ASTIS mathematical exposition

From a genuine Hessian bound to the normalized augmented Gibbs law

AutoSamplingTheory.ExampleCases.ProximalBPS.GibbsAugmentation.normalized_augmentation_density · theorem · Teaching coverage

Statement

Let E be a finite-dimensional real inner-product space with its Borel sigma algebra and canonical volume. Let V:E→R be twice continuously differentiable, with D²V(x)[v,v]≥α||v||² for every x,v and some α>0. For any η>0, the Gibbs integral Z_V is strictly positive; independently drawing X from volume tilted by -V and standard Gaussian N, then setting Y=X+sqrt(η)N, gives a probability joint law. This joint law has the following exact density with respect to product volume. No minimizer, integrability, normalizer, or probability-law identity is supplied as a hypothesis.

\[Z_V=\int_E e^{-V(x)}\,dx>0,\quad C_\eta=((\sqrt{2\pi\eta})^{-1})^d,\quad \mu(dx)=Z_V^{-1}e^{-V(x)}dx,\quad \mathcal L(X,X+\sqrt\eta N)(dx,dy)=\frac{C_\eta}{Z_V}\exp\!\left(-V(x)-\frac{\|y-x\|^2}{2\eta}\right)dx\,dy.\]

All objects and hypotheses

All quantifiers are global: V is C² on E and the genuine second Fréchet derivative lower bound holds for every position x and direction v. Positive α is essential to this normalization route. C² prevents unsupported uses of totalized fderiv.
E has finite real dimension d, including zero. Its measure is the canonical inner-product-space volume; an arbitrary Haar normalization is not silently substituted.
η>0. The source's upper Hessian bound βI and upper scale restriction η≤1/β are unnecessary for this law identity, though they remain assumptions of downstream sampler statements.
Independence is the product μ.prod(stdGaussian E). The normalized target μ is defined using Mathlib tilted; the theorem proves the positive integral and probability conclusions that exclude its zero totalized fallback.

Mathematical proof

1. Establish the Gibbs probability from curvature

The existing Hessian criterion converts the genuine C² lower bound into strong convexity with the same α. The existing minimizer-free Gibbs integrability theorem then proves exp(-V) is integrable. Its proof already supplies the quadratic tail bound by the first-order estimate at zero and Young's inequality. Since the exponential is strictly positive and volume is nonzero, Z_V>0. Mathlib can therefore normalize it into a probability measure.

\[D^2V\succeq\alpha I\ \Longrightarrow\ V\text{ strongly convex}\ \Longrightarrow\ e^{-V}\in L^1(dx),\qquad 0<Z_V<\infty,\quad \mu(E)=1.\]

Corresponding Lean step

HessianStrongConvexity.strongConvexOn_univ_of_fderiv2_lower and StrongConvexGibbsIntegrability.integrable_exp_neg_of_strongConvexOn are existing ASTIS proofs. ContDiff.differentiable supplies genuine differentiability. integral_exp_pos gives hZ; isProbabilityMeasure_tilted installs the actual probability instance. No parent proof is duplicated.

2. Use the generative Gaussian joint law

The augmentation map Φη(x,n)=(x,x+sqrt(η)n) is continuous and hence measurable. Its pushforward of the independent product of two probability laws is a probability. The already proved Gaussian augmentation theorem identifies its density qη(y-x), initially relative to μ(dx)dy; it includes the exact Gaussian normalization.

\[\pi_\eta=(\Phi_\eta)_\#(\mu\otimes\gamma_E),\quad \pi_\eta(E\times E)=1,\quad \pi_\eta(dx,dy)=\mu(dx)q_\eta(y-x)dy,\quad q_\eta(w)=C_\eta e^{-\|w\|^2/(2\eta)}.\]

Corresponding Lean step

The measurable-map witness hΦ is proved by fun_prop. Measure.isProbabilityMeasure_map proves the probability conclusion. GaussianAugmentation.augmentation_eq_withDensity is the third actual ASTIS parent and supplies the joint density, not an assumed law equality.

3. Change the reference measure to product volume

Insert μ(dx)=f(x)dx with f(x)=exp(-V(x))/Z_V. Continuity of V explicitly gives measurability of f; the Gaussian weight is also measurable. The product-density identity moves f to the first coordinate, then density multiplication combines the two weights. No new interchange assumption or conditional probability representative is supplied here: the invoked Mathlib product-density theorem handles its product integration contract, with canonical volume providing the required SFinite instance.

\[(\mu\otimes dx_y)q_\eta(y-x)=((dx_x\,dx_y)f(x))q_\eta(y-x)=(dx_x\,dx_y)[f(x)q_\eta(y-x)].\]

Corresponding Lean step

Unfolding only the existing Measure.tilted definition reveals withDensity(ofReal(exp(-V)/Z_V)). prod_withDensity_left uses hf; the reverse direction of withDensity_mul flattens the two weights, with hf.comp measurable_fst and hq as explicit measurable inputs. These are Mathlib calls, not new ASTIS wrappers.

4. Combine the exponents and retain nonvacuous normalization

The real factors are nonnegative, so converting their product to an ENNReal density preserves multiplication. The exponential addition rule gives the displayed source density. The theorem returns its measure equality together with Z_V>0 and probability of the joint law: equality alone would not exclude a zero totalized tilt. The focused test transfers both probability and the existing auxiliary reflection identity to this source-density measure.

\[\frac{e^{-V(x)}}{Z_V}C_\eta e^{-\|y-x\|^2/(2\eta)}=\frac{C_\eta}{Z_V}e^{-V(x)-\|y-x\|^2/(2\eta)}.\]

Corresponding Lean step

ENNReal.ofReal_mul uses exp positivity and hZ; Real.exp_add and ring perform the exact algebra. The conjunction returns hZ, the proved pushforward probability, and the exact density equality. Tests use all three conclusions and GaussianReflection.reflection_preserves_augmentation; this does not establish PBPS process invariance.

Lean statement · normalized_augmentation_density

The statement's let bindings merely name Z_V, μ and the generative joint measure; they add no hypotheses. IsProbabilityMeasure is a proposition-valued typeclass and appears explicitly in the conclusion. withDensity expects an ENNReal-valued density, so ENNReal.ofReal wraps the nonnegative real formula. Module.finrank is the natural dimension. The conclusion is equality of whole measures, not equality only of total masses or a chosen conditional representative.

theorem normalized_augmentation_density {V : E → ℝ} {α η : ℝ}
    (hα : 0 < α) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E, α * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v)
    (hη : 0 < η) :
    let ZV := ∫ x, Real.exp (-V x) ∂(volume : Measure E)
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let joint := Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    0 < ZV ∧ IsProbabilityMeasure joint ∧
      joint = ((volume : Measure E).prod volume).withDensity (fun p =>
        ENNReal.ofReal
          (((((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E) / ZV) *
            Real.exp (-V p.1 - ‖p.2 - p.1‖ ^ 2 / (2 * η))))

Exact module and namespace context

Lean proof · normalized_augmentation_density

One source-integration theorem joins three compiled ASTIS parents and existing Mathlib normalization/product-density APIs. Every auxiliary name f,q,Z_V,μ and every measurability/probability witness is local to the proof. The positive integral certifies meaningful normalization before any law identification. The formula uses ||y-x||, equal to the source's ||x-y||; the Gaussian variance is exactly η, not 2η.

theorem normalized_augmentation_density {V : E → ℝ} {α η : ℝ}
    (hα : 0 < α) (hV : ContDiff ℝ 2 V)
    (hH : ∀ x v : E, α * ‖v‖ ^ 2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v)
    (hη : 0 < η) :
    let ZV := ∫ x, Real.exp (-V x) ∂(volume : Measure E)
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let joint := Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    0 < ZV ∧ IsProbabilityMeasure joint ∧
      joint = ((volume : Measure E).prod volume).withDensity (fun p =>
        ENNReal.ofReal
          (((((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E) / ZV) *
            Real.exp (-V p.1 - ‖p.2 - p.1‖ ^ 2 / (2 * η)))) := by
  let ZV := ∫ x, Real.exp (-V x) ∂(volume : Measure E)
  let μ := (volume : Measure E).tilted (fun x => -V x)
  have hi : Integrable (fun x => Real.exp (-V x)) (volume : Measure E) :=
    TechnicalLemmas.Analysis.StrongConvexGibbsIntegrability.integrable_exp_neg_of_strongConvexOn
      hα (hV.differentiable (by norm_num))
      (TechnicalLemmas.Analysis.HessianStrongConvexity.strongConvexOn_univ_of_fderiv2_lower
        hV hH)
  have hZ : 0 < ZV := integral_exp_pos hi
  have : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hi
  have hΦ : Measurable (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2)) := by
    fun_prop
  refine ⟨hZ, Measure.isProbabilityMeasure_map hΦ.aemeasurable, ?_⟩
  change Measure.map _ (μ.prod (stdGaussian E)) = _
  rw [GaussianAugmentation.augmentation_eq_withDensity μ η hη]
  let f : E → ℝ≥0∞ := fun x => ENNReal.ofReal (Real.exp (-V x) / ZV)
  let q : E × E → ℝ≥0∞ := fun p => ENNReal.ofReal
    (((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E *
      Real.exp (-‖p.2 - p.1‖ ^ 2 / (2 * η)))
  have hf : Measurable f := by
    exact ((Real.continuous_exp.comp hV.continuous.neg).div_const ZV).measurable.ennreal_ofReal
  have hq : Measurable q := by fun_prop
  have hfp : Measurable (fun p : E × E => f p.1) := hf.comp measurable_fst
  change (((volume : Measure E).withDensity f).prod volume).withDensity q = _
  rw [prod_withDensity_left hf, ← withDensity_mul _ hfp hq]
  congr 1
  funext p
  dsimp only [Pi.mul_apply, f, q]
  rw [← ENNReal.ofReal_mul (div_nonneg (Real.exp_pos _).le hZ.le)]
  congr 1
  rw [show -V p.1 - ‖p.2 - p.1‖ ^ 2 / (2 * η) =
      -V p.1 + (-‖p.2 - p.1‖ ^ 2 / (2 * η)) by ring, Real.exp_add]
  ring

end AutoSamplingTheory.ExampleCases.ProximalBPS.GibbsAugmentation

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
C² Euclidean V and lower matrix bound αI≤D²V, α>0	ContDiff ℝ 2 V, hα:0<α, and ∀x v, α*\|\|v\|\|²≤(fderiv ℝ (fderiv ℝ V) x v) v	same	The quadratic-form version expresses the lower Hessian bound. C² guarantees genuine first and second derivatives; a zero totalized derivative without differentiability cannot replace this hypothesis.
Euclidean space and Lebesgue volume	Any finite-dimensional real inner-product Borel E, including dimension zero, with canonical volume	generalization	The shared analytic and Gaussian parents are coordinate-free and include zero dimension. The volume normalization is fixed by the inner product, not an arbitrary Haar measure.
Upper Hessian bound βI and 0<η≤1/β	No upper Hessian bound; η>0	generalization	Only lower curvature is used to normalize the Gibbs target. Gaussian augmentation and density multiplication hold at every positive scale. The omitted upper constraints are retained in separate sampler/curvature obligations.
Normalized Gibbs notation and joint-density proportionality	Explicit 0<Z_V and IsProbabilityMeasure joint as well as the exact density equality	source-implicit	Existing parent proofs establish exp(-V) integrability without a minimizer. Positive exponential mass excludes zero normalization; the explicit probability conclusion prevents the displayed equality from being satisfied only by a zero totalized tilted law.
Independent Gaussian construction (2.7)	Product measure and proved measurable pushforward, with measurable product-density multiplication	source-implicit	The Borel continuity witnesses and product measure contracts are supplied internally. No conditional representative, boundary integration or exchange of improper integrals is assumed.
Exponent -V(x)-\|\|x-y\|\|²/(2η), Gaussian covariance ηI	Exponent -V p.1-\|\|p.2-p.1\|\|²/(2η), prefactor ((sqrt(2πη))⁻¹)^finrank/Z_V	same	Norm symmetry changes x-y to y-x without changing the value; the natural-power square-root expression is exactly the same Gaussian normalization.

Exact normalized joint-law component (2.6)-(2.7), not conditional kernels, curvature conclusions, process invariance, mixing or query cost. Independent source-blind semantic audit is required for admission.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

domains: Finite-dimensional real inner-product Borel E, including dimension zero, is an explicit generalization of the paper's Euclidean presentation. — The source extraction, candidate assumptions, module documentation, lesson and anonymous reconstruction all disclose it. Canonical volume, rather than arbitrary Haar scaling, preserves the Gaussian constant. The full Fin 0 test passes.
assumptions: The formal identity omits the upper Hessian bound and upper scale restriction, so it applies to more inputs than the paper's standing regime. — The mathematical proof requires only C², the positive lower bound and eta>0. These broader assumptions are part of the selected source extraction and authored lesson; omitted upper controls are not silently removed from downstream sampler claims.
conclusion: Positive finite normalization, an explicit probability assertion and the exact density factor elaborate the paper's proportional joint-density notation. — The code derives genuine integrability and Z_V>0 before identifying the normalized density. Inserting f(x)=exp(-V(x))/Z_V into the proven Gaussian augmentation density changes the reference from mu×volume to volume×volume; this is the required source-law identification, not a remaining assumption.
scopes: The accepted result is the selected normalized-law proof edge only. Reflection is a tested consumer; conditionals, process invariance and sampler performance remain outside this theorem. — The production conclusion is exactly the triple, and the full lesson explicitly distinguishes its reflection test from PBPS process invariance. Adjacent source statements do not expand the declaration's formal credit.

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

This completes the normalized joint-law identification between the generative construction and the product-volume density, not the whole PBPS paper or Proposition 2.1.
The source upper Hessian bound and η≤1/β remain relevant elsewhere. Their absence here is a disclosed generalization, not a correction of the source sampler assumptions.
Conditional-law kernels and representatives, marginal curvature, process invariance, mixing, tail and moment estimates, and query cost remain separate obligations.
Compilation certifies this Lean proposition; independent source-blind reconstruction and source fidelity review remain distinct admission gates.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

MeasureTheory.integral_exp_pos
MeasureTheory.isProbabilityMeasure_tilted
MeasureTheory.Measure.isProbabilityMeasure_map
MeasureTheory.prod_withDensity_left
MeasureTheory.withDensity_mul
ENNReal.ofReal_mul
Real.exp_add
ContDiff.differentiable
ContDiff.continuous
Real.continuous_exp

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS v1 §2.2, equations (2.6)-(2.7) — Normalized joint-density and generative-law identification, with the positive Hessian setting in introduction (1.1). Original ASTIS exposition and explicit normalization argument, not copied paper prose.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Derive strictly positive finite Gibbs mass from genuine source curvature, without an assumed minimizer
Local proof component; source adapter/review separate Certify that the independent Gaussian augmentation is a probability measure
Local proof component; source adapter/review separate Identify that actual joint law with the exact source density relative to product volume

A Gibbs L2 representative is locally Lebesgue L2

Samplinglib-authored elementary prerequisite for the analytic route underlying Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; not original wording or a full paper theorem.

arXiv:2609.06905v1 · Appendix C.1 analytic prerequisite; background1310.2526v7 §2.5 · faithful paraphrase

Complete source statement (ASTIS restatement)

The squared norm of the selected representative a is locally integrable with respect to volume. For every compact K⊆E, that same representative belongs to L²(dx restricted to K), including its required almost-everywhere strong measurability.

E is a finite-dimensional real inner-product space, including dimension zero, with its Borel measurable structure and canonical volume dx. F is any normed additive commutative group; no scalar structure or completeness of F is assumed.
W:E→R is continuous and exp(-W) is volume-integrable. Let Z=∫exp(-W)dx and μ=Z⁻¹exp(-W)dx. The positive finite normalization is justified, not an unspecified density constant.
a is an arbitrary element of L²(μ;F). Its function coercion denotes the selected almost-everywhere representative. There is no continuity, differentiability or initially assumed volume measurability of a.

Local L2

\[a\in L^2(\mu;F)\quad\Longrightarrow\quad \|a\|^2\in L^1_{\mathrm{loc}}(dx),\qquad \forall K\Subset E,\ a\in L^2(K,dx;F).\]

Read the formalized proofs

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

ASTIS mathematical exposition

A Gibbs L2 representative is locally Lebesgue L2

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedLocalL2.lp_locallyMemLp_volume · theorem · Teaching coverage

Statement

\[a\in L^2(\mu;F)\quad\Longrightarrow\quad \|a\|^2\in L^1_{\mathrm{loc}}(dx),\qquad \forall K\Subset E,\ a\in L^2(K,dx;F).\]

All objects and hypotheses

E is a finite-dimensional real inner-product space, including dimension zero, with its Borel measurable structure and canonical volume dx. F is any normed additive commutative group; no scalar structure or completeness of F is assumed.
W:E→R is continuous and exp(-W) is volume-integrable. Let Z=∫exp(-W)dx and μ=Z⁻¹exp(-W)dx. The positive finite normalization is justified, not an unspecified density constant.
a is an arbitrary element of L²(μ;F). Its function coercion denotes the selected almost-everywhere representative. There is no continuity, differentiability or initially assumed volume measurability of a.

Mathematical proof

1. Use the squared-norm characterization of L2

Membership in Gibbs L2 supplies almost-everywhere strong measurability and integrability of the real-valued squared norm. Even if F has no real scalar action, this norm-squared function is a real function, so change of density can be applied to it.

\[\int \|a(x)\|^2\,d\mu(x)<\infty.\]

Corresponding Lean step

hs via memLp_two_iff_integrable_sq_norm and Lp.memLp

2. Remove only the Gibbs normalization

The tilted-measure integrability equivalence gives integrability of exp(-W) times the squared norm for volume. The assumption on exp(-W) makes the normalization finite; positivity prevents a degenerate normalized measure. No uniform bound on W is needed.

\[e^{-W}\|a\|^2\in L^1(dx).\]

Corresponding Lean step

hw via integrable_tilted_iff hI; smul_eq_mul

3. Invert the weight locally

The inverse weight exp(W) is continuous. A continuous multiplier preserves local integrability. Multiplying the preceding integrable function locally and cancelling the two exponentials proves the local squared-norm assertion. This step is local: the inverse weight may be unbounded on the whole space.

\[e^W(e^{-W}\|a\|^2)=\|a\|^2\in L^1_{\mathrm{loc}}(dx).\]

Corresponding Lean step

hw.locallyIntegrable.continuous_mul; Real.exp_add; hl

4. Transfer measurability to volume

Gibbs L2 initially gives strong measurability only almost everywhere for μ. The positive exponential density and finite normalization imply dx is absolutely continuous with respect to μ. Thus any exceptional μ-null set is also volume-null, and the same representative is strongly measurable almost everywhere for volume and its restrictions.

\[dx\ll\mu,\qquad a\text{ is a.e. strongly measurable for }dx|_K.\]

Corresponding Lean step

hm via AEStronglyMeasurable.mono_ac (absolutelyContinuous_tilted hI); hm.restrict

5. Restrict to an arbitrary compact set

Local integrability implies integrability on every compact K. Combine this squared-norm integral with the restricted strong measurability and apply the L2 characterization in reverse. The consumer test applies exactly this conclusion to an existing resolvent solution, its gradient, and its forcing, retaining the same weak PDE.

\[\forall K\Subset E,\quad\int_K\|a(x)\|^2\,dx<\infty\quad\Longrightarrow\quad a\in L^2(dx|_K;F).\]

Corresponding Lean step

hl.integrableOn_isCompact hK; memLp_two_iff_integrable_sq_norm hm.restrict

Lean statement · lp_locallyMemLp_volume

Exact spaces, representative coercion and compact-set quantifiers are shown in the closed disclosure.

theorem lp_locallyMemLp_volume {F : Type*} [NormedAddCommGroup F]
    (W : E → ℝ) (hW : Continuous W)
    (hI : Integrable (fun x => Real.exp (-W x)))
    (a : Lp F 2 ((volume : Measure E).tilted (fun x => -W x))) :
    LocallyIntegrable (fun x => ‖a x‖ ^ 2) (volume : Measure E) ∧
      ∀ K : Set E, IsCompact K → MemLp (fun x => a x) 2 (volume.restrict K)

Exact module and namespace context

Lean proof · lp_locallyMemLp_volume

Squared norm → remove weight → invert locally → transfer measurability → compact restriction.

theorem lp_locallyMemLp_volume {F : Type*} [NormedAddCommGroup F]
    (W : E → ℝ) (hW : Continuous W)
    (hI : Integrable (fun x => Real.exp (-W x)))
    (a : Lp F 2 ((volume : Measure E).tilted (fun x => -W x))) :
    LocallyIntegrable (fun x => ‖a x‖ ^ 2) (volume : Measure E) ∧
      ∀ K : Set E, IsCompact K → MemLp (fun x => a x) 2 (volume.restrict K) := by
  have hs := (memLp_two_iff_integrable_sq_norm (Lp.memLp a).aestronglyMeasurable).mp
    (Lp.memLp a)
  have hw : Integrable (fun x => Real.exp (-W x) * ‖a x‖ ^ 2) := by
    simpa only [smul_eq_mul] using (integrable_tilted_iff hI _).mp hs
  have hl : LocallyIntegrable (fun x => ‖a x‖ ^ 2) (volume : Measure E) := by
    have h := hw.locallyIntegrable.continuous_mul (Real.continuous_exp.comp hW)
    simpa only [Function.comp_apply, ← mul_assoc, ← Real.exp_add, add_neg_cancel,
      Real.exp_zero, one_mul]
      using h
  refine ⟨hl, ?_⟩
  intro K hK
  have hm : AEStronglyMeasurable (fun x => a x) (volume : Measure E) :=
    (Lp.memLp a).aestronglyMeasurable.mono_ac (absolutelyContinuous_tilted hI)
  exact (memLp_two_iff_integrable_sq_norm hm.restrict).mpr
    (hl.integrableOn_isCompact hK)

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedLocalL2

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
The paper's conditional Gibbs potentials have C2 regularity and curvature bounds.	Continuous W and integrable exp(-W) suffice for the elementary local L2 transfer.	generalization	No curvature or derivative estimate is asserted.
Weighted Sobolev arguments use integrability and representatives implicitly.	Positive finite Gibbs normalization, local squared norm integrability, explicit reverse absolute continuity, and every compact volume restriction.	source-implicit	Integrability does not by itself replace the almost-everywhere measurability requirement when changing measure.
The paper uses real functions and finite-dimensional vector gradients.	F is any normed additive commutative group, without scalar structure or completeness.	generalization	Only the real squared norm is multiplied by the weight; no additional paper conclusion follows.

Local square integrability only: no global unweighted L2 bound, quantitative embedding constant, derivative regularity, H2 theorem, Sobolev-domain converse, Poincare or paper completion.

Encoder–denoiser: accepted · equivalent-after-elaboration

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

Local square integrability only: no global unweighted L2 bound, quantitative embedding constant, derivative regularity, H2 theorem, Sobolev-domain converse, Poincare or paper completion.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

MeasureTheory.memLp_two_iff_integrable_sq_norm
MeasureTheory.Lp.memLp
MeasureTheory.integrable_tilted_iff
MeasureTheory.LocallyIntegrable.continuous_mul
MeasureTheory.absolutelyContinuous_tilted
MeasureTheory.AEStronglyMeasurable.mono_ac
MeasureTheory.LocallyIntegrable.integrableOn_isCompact

Mathematical sources

PBPS Appendix C.1: authored analytic prerequisite — Elementary localization supporting the closed-gradient/weighted-Poisson route, not a quoted paper theorem.
Kolesnikov–Milman §2.5: weighted Sobolev background — Background only; no compact-manifold regularity theorem is imported.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Squared norm local integrability and compact L2 with representative measurability

Gaussian-noise density prerequisite for the proximal augmentation

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang. ASTIS mathematical expansion of the standard Gaussian-noise prerequisite, not copied paper prose or a claim of complete joint-law formalization.

arXiv:2609.06905v1 · Section 2.2, equations (2.6)-(2.7): isotropic Gaussian-noise density implicit in the joint-law identification · faithful paraphrase

Complete source statement (ASTIS restatement)

For eta>0 and Z a standard Gaussian vector in R^d, sqrt(eta) Z has density (sqrt(2*pi*eta))^{-d} exp(-||z||^2/(2*eta)) with respect to Euclidean volume. This is the Gaussian-noise density used when passing from the independent construction (2.7) to the joint density (2.6).

The source uses d>=1, a real Euclidean state space, and independent standard Gaussian noise Z in the displayed augmentation.
The paper assumes 0<eta<=1/beta for the overall sampler. The Gaussian-noise density prerequisite only uses eta>0.
The target potential and its Hessian bounds are not hypotheses of this noise-only identity. They remain part of the paper's target and algorithmic setting.

isotropic-noise density prerequisite of (2.6)-(2.7)

\[(z\mapsto\sqrt\eta\,z)_\#\gamma_d=\mathrm{vol}_{\mathbb R^d}\,q_\eta,\qquad q_\eta(z)=((\sqrt{2\pi\eta})^{-1})^d\exp(-\|z\|^2/(2\eta)).\]

Read the formalized proofs

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

ASTIS mathematical exposition

The explicit density of isotropic Gaussian noise

AutoSamplingTheory.TechnicalLemmas.Measure.IsotropicGaussianDensity.map_sqrt_smul_stdGaussian_eq_withDensity · theorem · Teaching coverage

Statement

Let E be a finite-dimensional real inner-product space, equipped with its Borel sigma algebra and its canonical volume measure, and let eta be a positive real number. If Z has the standard Gaussian law on E, the entire law of sqrt(eta) Z equals canonical volume weighted by the density q_eta below. The dimension d is the real dimension of E and may be zero.

\[(z\mapsto\sqrt\eta\,z)_\#\gamma_E=\mathrm{vol}_E\,q_\eta,\qquad q_\eta(z)=\left((\sqrt{2\pi\eta})^{-1}\right)^d\exp\!\left(-\frac{\|z\|^2}{2\eta}\right),\quad d=\dim_{\mathbb R}E.\]

All objects and hypotheses

E is a normed additive commutative group carrying a real inner product, is finite-dimensional over the reals, and has the Borel measurable structure. The theorem uses Mathlib's canonical inner-product-space volume, not an arbitrary supplied Haar-measure normalization.
eta is a real number with eta>0. This makes the coordinate variance nonzero. In positive dimension eta=0 would give a singular Dirac law, not this Lebesgue density.
No Nontrivial E or positive-dimension assumption is imposed. In dimension zero, the empty product normalizer is one, every vector has norm zero, and canonical volume is the point mass on the unique vector.
No target potential, log-concavity, curvature bound, upper step-size restriction, input distribution, or additional integrability assumption is used. This is a reusable Gaussian-noise dependency, not the full PBPS augmentation identity.

Mathematical proof

1. Choose orthonormal coordinates

Choose an orthonormal basis indexed by Fin d. The measurable equivalence e sends a coordinate tuple to its orthonormal linear combination. The standard Gaussian on E is the pushforward under e of independent standard real Gaussians. Scalar multiplication commutes with e.

\[e(a)=\sum_{i=1}^d a_i b_i,\qquad\gamma_E=e_\#\bigotimes_{i=1}^d\mathcal N(0,1),\qquad\sqrt\eta\,e(a)=e(\sqrt\eta\,a).\]

Corresponding Lean step

The proof uses stdGaussian_eq_map_pi_orthonormalBasis, the measurable equivalence composed from MeasurableEquiv.toLp and the basis representation inverse, and the linear map's scalar-multiplication law. All finite-product and composition maps are shown measurable.

2. Scale each scalar Gaussian

The scalar Gaussian pushforward theorem multiplies variance by the square of the scale. Since eta is positive, (sqrt eta)^2=eta. The finite-product map theorem then identifies the scaled coordinate law with independent variance-eta real Gaussians.

\[(t\mapsto\sqrt\eta\,t)_\#\mathcal N(0,1)=\mathcal N(0,\eta),\qquad(\sqrt\eta\,\cdot)_\#\gamma_E=e_\#\bigotimes_{i=1}^d\mathcal N(0,\eta).\]

Corresponding Lean step

ProbabilityTheory.gaussianReal_map_const_mul and Real.sq_sqrt supply the scalar law. MeasureTheory.Measure.pi_map_pi, followed by Measure.map_map, proves the vector law. The nonnegative variance is explicitly eta with its proved nonnegativity witness.

3. Combine actual scalar densities

Every variance-eta scalar Gaussian is Lebesgue measure weighted by its known density. The existing ASTIS finite-product density theorem combines these into the product density. Its sigma-finiteness requirements hold because the individual density measures equal probability measures.

\[\bigotimes_{i=1}^d\mathcal N(0,\eta)=\mathrm{vol}_{\mathbb R^d}\,\prod_{i=1}^d\left[(\sqrt{2\pi\eta})^{-1}\exp\left(-\frac{a_i^2}{2\eta}\right)\right].\]

Corresponding Lean step

gaussianReal_of_var_ne_zero is applied after proving eta is nonzero. RadonNikodym.pi_withDensity_prod is used in reverse, with measurable Gaussian densities and probability instances obtained from the scalar Gaussian identity. No multidimensional density identity is assumed.

4. Transport volume and its density

The coordinate equivalence preserves canonical volume: passing between ordinary coordinate tuples and EuclideanSpace preserves volume, as does the orthonormal representation inverse. The existing ASTIS measurable-equivalence density theorem therefore transports the product density by composing it with e inverse.

\[e_\#\mathrm{vol}_{\mathbb R^d}=\mathrm{vol}_E,\qquad e_\#(\mathrm{vol}_{\mathbb R^d}f)=\mathrm{vol}_E(f\circ e^{-1}).\]

Corresponding Lean step

PiLp.volume_preserving_toLp and OrthonormalBasis.measurePreserving_repr_symm provide the volume identity. RadonNikodym.measurableEquiv_map_withDensity transports the actual product density. This avoids introducing a Jacobian theorem or an arbitrary volume-normalization assumption.

5. Collect the normalizer and quadratic exponent

There are exactly d equal scalar normalizing factors. Products of exponentials add the exponents, and orthonormal coordinates preserve the squared norm. These identities give the stated density pointwise, not only almost everywhere. Empty products and sums also prove the dimension-zero case.

\[\prod_i(\sqrt{2\pi\eta})^{-1}=((\sqrt{2\pi\eta})^{-1})^d,\qquad\sum_i(e^{-1}z)_i^2=\|z\|^2.\]

Corresponding Lean step

Finset.prod_mul_distrib, Finset.prod_const, Real.exp_sum, ENNReal.ofReal_prod_of_nonneg, EuclideanSpace.real_norm_sq_eq and preservation of the norm by the basis representation complete the equality. The real density is nonnegative before the finite-product ofReal conversion.

Lean statement · map_sqrt_smul_stdGaussian_eq_withDensity

Measure.map denotes pushforward, stdGaussian is Mathlib's basis-independent standard Gaussian measure, and withDensity takes an ENNReal-valued density. ENNReal.ofReal converts the displayed nonnegative real Gaussian density. Module.finrank is a natural number, so the constant uses a natural power of an inverse square root rather than a real-power notation. The canonical volume instance is inferred from the finite-dimensional inner-product Borel setting; no arbitrary MeasureSpace E is added as a parameter.

theorem map_sqrt_smul_stdGaussian_eq_withDensity (η : ℝ) (hη : 0 < η) :
    (stdGaussian E).map (fun z : E => Real.sqrt η • z) =
      (volume : Measure E).withDensity (fun z =>
        ENNReal.ofReal
          (((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E *
            Real.exp (-‖z‖ ^ 2 / (2 * η))))

Exact module and namespace context

Lean proof · map_sqrt_smul_stdGaussian_eq_withDensity

All auxiliary names are local to this one proof. The proof identifies distributions through scalar Gaussian scaling, finite-product densities, and a volume-preserving measurable equivalence; it does not infer equality merely from matching total mass. No density normalizing integral or change-of-variables determinant is reproved. Eta positivity is used to obtain a nonzero scalar variance and its square-root square identity.

theorem map_sqrt_smul_stdGaussian_eq_withDensity (η : ℝ) (hη : 0 < η) :
    (stdGaussian E).map (fun z : E => Real.sqrt η • z) =
      (volume : Measure E).withDensity (fun z =>
        ENNReal.ofReal
          (((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E *
            Real.exp (-‖z‖ ^ 2 / (2 * η)))) := by
  classical
  let ι := Fin (Module.finrank ℝ E)
  let b := stdOrthonormalBasis ℝ E
  let v : ℝ≥0 := ⟨η, hη.le⟩
  have hvcoe : (v : ℝ) = η := rfl
  have hv : v ≠ 0 := by
    intro h
    have : η = 0 := congrArg (fun x : ℝ≥0 => (x : ℝ)) h
    exact hη.ne' this
  let e : (ι → ℝ) ≃ᵐ E :=
    (MeasurableEquiv.toLp 2 (ι → ℝ)).trans b.measurableEquiv.symm
  let T : (ι → ℝ) → (ι → ℝ) := fun x i => Real.sqrt η * x i
  have he (x : ι → ℝ) : e x = ∑ i, x i • b i := by
    exact (b.sum_repr_symm (WithLp.toLp 2 x)).symm
  have hstd : stdGaussian E =
      (Measure.pi (fun _ : ι => gaussianReal 0 1)).map e := by
    rw [stdGaussian_eq_map_pi_orthonormalBasis b]
    congr 1
    funext x
    exact (he x).symm
  have hcomm : (fun z : E => Real.sqrt η • z) ∘ e = e ∘ T := by
    funext x
    change Real.sqrt η • b.repr.symm (WithLp.toLp 2 x) =
      b.repr.symm (WithLp.toLp 2 (T x))
    rw [← map_smul]
    rfl
  have hscalar : (gaussianReal 0 1).map (fun t : ℝ => Real.sqrt η * t) =
      gaussianReal 0 v := by
    rw [gaussianReal_map_const_mul]
    congr 1
    · simp
    · apply NNReal.eq
      change (Real.sqrt η) ^ 2 * 1 = η
      rw [mul_one, Real.sq_sqrt hη.le]
  have hpi : (Measure.pi (fun _ : ι => gaussianReal 0 1)).map T =
      Measure.pi (fun _ : ι => gaussianReal 0 v) := by
    rw [Measure.pi_map_pi (fun _ => by fun_prop)]
    simp_rw [hscalar]
  have hscaled : (stdGaussian E).map (fun z : E => Real.sqrt η • z) =
      (Measure.pi (fun _ : ι => gaussianReal 0 v)).map e := by
    rw [hstd, Measure.map_map (by fun_prop) e.measurable, hcomm,
      ← Measure.map_map e.measurable (by fun_prop : Measurable T), hpi]
  have : IsProbabilityMeasure ((volume : Measure ℝ).withDensity (gaussianPDF 0 v)) := by
    rw [← gaussianReal_of_var_ne_zero 0 hv]
    infer_instance
  have hproduct : Measure.pi (fun _ : ι => gaussianReal 0 v) =
      (volume : Measure (ι → ℝ)).withDensity (fun x => ∏ i, gaussianPDF 0 v (x i)) := by
    rw [volume_pi, RadonNikodym.pi_withDensity_prod (fun _ => measurable_gaussianPDF 0 v)]
    simp_rw [← gaussianReal_of_var_ne_zero 0 hv]
  have hvol : (volume : Measure (ι → ℝ)).map e = (volume : Measure E) :=
    (b.measurePreserving_repr_symm.comp (PiLp.volume_preserving_toLp ι)).map_eq
  rw [hscaled, hproduct,
    RadonNikodym.measurableEquiv_map_withDensity e _ (by fun_prop), hvol]
  congr 1
  funext z
  have hnorm : (∑ i, ((e.symm z) i) ^ 2) = ‖z‖ ^ 2 := by
    rw [← EuclideanSpace.real_norm_sq_eq (WithLp.toLp 2 (e.symm z))]
    change ‖b.repr z‖ ^ 2 = ‖z‖ ^ 2
    rw [b.repr.norm_map]
  simp only [gaussianPDF_def, gaussianPDFReal, sub_zero]
  rw [← ENNReal.ofReal_prod_of_nonneg (fun _ _ => by positivity)]
  congr 1
  rw [Finset.prod_mul_distrib, Finset.prod_const, Finset.card_univ, ← Real.exp_sum]
  simp only [ι, Fintype.card_fin, hvcoe]
  congr 1
  congr 1
  rw [← Finset.sum_div, Finset.sum_neg_distrib, hnorm]

end AutoSamplingTheory.TechnicalLemmas.Measure.IsotropicGaussianDensity

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
A standard Gaussian vector in R^d with d>=1	stdGaussian E for any finite-dimensional real inner-product Borel space E, including dimension zero	generalization	Orthonormal coordinate transport proves the same law with canonical volume. Empty products and sums make the zero-dimensional case valid; a focused test reduces its density to one.
The sampler takes 0<eta<=1/beta	eta is real and eta>0	generalization	The noise law only uses positivity to ensure nonzero variance and (sqrt eta)^2=eta. The upper bound remains an assumption of other source results.
Euclidean Lebesgue volume and the Gaussian density normalizer	Canonical volume supplied by the finite-dimensional real inner-product Borel instance; inverse square-root constant to the natural power Module.finrank R E	same	Orthonormal bases preserve this precise volume normalization. ((sqrt(2pieta))^{-1})^d equals the conventional (2pieta)^{-d/2} for eta>0.
Implicit measurable coordinate maps and independent Gaussian coordinates	Measurable equivalences and finite-product pushforwards, with measurability proved and Gaussian probability instances supplying sigma-finiteness	source-implicit	The change-of-law and product-density theorems require these measure-theoretic contracts; they are established from the displayed objects rather than assumed as the desired density identity.
The full joint law uses the target potential V and the Gibbs normalizer	No input law or potential occurs in the selected Gaussian-noise density statement	same	This binding targets only the noise prerequisite, not the full density-generative identification. The source's Gibbs assumptions and joint adapter are neither removed from their own statement nor claimed proved here.

Only the isotropic-noise density prerequisite. No PBPS joint/conditional law, target normalizer, process invariance, convergence or oracle cost is formalized by this binding.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

domains: General finite-dimensional E and dimension zero are explicit extensions beyond the source's coordinate presentation. — The extraction and lesson acknowledge the extension; orthonormal transport and empty finite products justify it mathematically. It is not a source repair.
assumptions: The isolated noise law removes the sampler upper bound on eta and target-potential assumptions. — These conditions are unused in scalar Gaussian scaling or the density transport. The complete sampler claim is not asserted under the reduced hypotheses.
constant_dependencies: The Gaussian constant is explicit and tied to canonical inner-product volume. — The proof combines known scalar constants; the volume-preserving coordinate equivalence rules out an unspecified normalization factor.
scopes: The fresh lesson uses exact ASTIS declaration IDs without appended explanatory prose. — Both IDs resolve to the existing RadonNikodym declarations and match visible proof calls. Their mathematical roles are still explained in lesson steps 3 and 4. This correction changes no mathematical boundary.

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

This proves only the explicit density of the scaled standard Gaussian law. The equality between the PBPS generative joint law (2.7) and joint density (2.6) still needs the shear/product-density adapter and the target's own density definition.
The paper assumes eta<=1/beta and d>=1. Those are not required for this Gaussian dependency, and the broader scope is explicit rather than a change to the full paper theorem.
No strong-convexity proof of target-normalizer finiteness, conditional-density representative, process invariance, PBPS convergence, or query complexity is asserted.
The PBPS and heat-smoothing consumers are planned dependencies, not claims that their downstream identities already compile. Independent source review and shared publication integration remain separate admission steps.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

ProbabilityTheory.stdGaussian_eq_map_pi_orthonormalBasis; gaussianReal_map_const_mul; gaussianReal_of_var_ne_zero; gaussianPDF_def; gaussianPDFReal.
MeasureTheory.Measure.pi_map_pi; Measure.map_map; MeasureTheory.volume_pi.
PiLp.volume_preserving_toLp; OrthonormalBasis.measurePreserving_repr_symm; OrthonormalBasis.sum_repr_symm.
EuclideanSpace.real_norm_sq_eq; LinearIsometryEquiv.norm_map; Real.sq_sqrt; Real.exp_sum; finite-product and finite-sum algebra.

Mathematical sources

Chen, Chewi, Lu and Zhang, Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler v1 — Gaussian-noise density dependency of equations (2.6)-(2.7), Section 2.2. Original ASTIS exposition; not a reproduction of paper prose or a claim to formalize the entire joint-law identity.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Identify the scaled standard Gaussian pushforward with its explicit canonical-volume density

Closed gradients satisfy weighted compact-test integration by parts

Samplinglib's explicit Euclidean analytic prerequisite to the proof route of Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; not a quotation or a claim to have proved Lemma B.1.

arXiv:2609.06905v1 · Appendix C.1 closed-gradient/density and conditional Poincare route; background Kolesnikov–Milman1310.2526v7§2.5 · faithful paraphrase

Complete source statement (ASTIS restatement)

Both φ⟨G,v⟩ and u(D_vφ−φD_vW) are μ-integrable, and their integrals have opposite signs. This holds for every graph-closure pair of the same operator D, not just for its original smooth core.

E is any finite-dimensional real inner-product space with its Borel sigma algebra, including dimension zero. W:E→R is C1 and exp(-W) is integrable for Euclidean volume; μ=volume.tilted(-W) is the actual normalized Gibbs law.
D is a given closable partial real-linear map from scalar L²(μ) to vector L²(μ). Its original graph is exactly the pairs represented μ-almost everywhere by (f,∇f) for smooth compactly supported f. This is the interface already proved by WeightedGradient, not an assumed integration-by-parts formula.
(u,G) belongs to the graph of D.closure. The test φ:E→R is C1 with compact support and v is any constant vector. No differentiability of the chosen measurable L² representatives u or G is required.

Weighted weak identity

\[\int \varphi\langle G,v\rangle\,d\mu=-\int u\,(D_v\varphi-\varphi D_vW)\,d\mu.\]

Read the formalized proofs

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

ASTIS mathematical exposition

Closed gradients satisfy weighted compact-test integration by parts

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientWeak.closed_gradient_weighted_ibp · theorem · Teaching coverage

Statement

\[\int \varphi\langle G,v\rangle\,d\mu=-\int u\,(D_v\varphi-\varphi D_vW)\,d\mu.\]

All objects and hypotheses

E is any finite-dimensional real inner-product space with its Borel sigma algebra, including dimension zero. W:E→R is C1 and exp(-W) is integrable for Euclidean volume; μ=volume.tilted(-W) is the actual normalized Gibbs law.
D is a given closable partial real-linear map from scalar L²(μ) to vector L²(μ). Its original graph is exactly the pairs represented μ-almost everywhere by (f,∇f) for smooth compactly supported f. This is the interface already proved by WeightedGradient, not an assumed integration-by-parts formula.
(u,G) belongs to the graph of D.closure. The test φ:E→R is C1 with compact support and v is any constant vector. No differentiability of the chosen measurable L² representatives u or G is required.

Mathematical proof

1. Localize the classical integration by parts

For a smooth core representative f, put F=e^(−W)φ. Both F and its directional derivative are continuous and compactly supported. Products with f or its continuous derivative are therefore integrable for volume. The classical full-space formula has no boundary term. No global boundedness of f or the score is needed.

\[D_vF=e^{-W}(D_v\varphi-\varphi D_vW),\qquad \int F D_vf\,dx=-\int f D_vF\,dx.\]

Corresponding Lean step

private compact_directional_ibp: hF,hFc,hDFc,h1,h2,h3,hi; genuine HasFDerivAt product rule

2. Normalize the actual Gibbs measure

The continuous strictly positive density exp(−W) is assumed integrable, so its normalizing integral is finite and positive and μ is a probability measure. Multiplying the volume identity by the reciprocal normalization gives the weighted identity. This uses the actual tilted measure, not an abstract measure with supplied IBP.

\[Z=\int e^{-W}\,dx,\qquad d\mu=Z^{-1}e^{-W}dx,\qquad \int\varphi D_vf\,d\mu=-\int f(D_v\varphi-\varphi D_vW)\,d\mu.\]

Corresponding Lean step

isProbabilityMeasure_tilted hI; ht uses integral_tilted and integral_const_mul

3. Represent the tests in L²

Set P=φv and q=D_vφ−φD_vW. The C1 assumptions make both continuous. The derivative of φ vanishes off its topological support, so both tests have compact support even if D_vW grows at infinity. Finite measure and compact continuity imply L² membership.

\[P(x)=\varphi(x)v,\qquad q(x)=D_v\varphi(x)-\varphi(x)D_vW(x),\qquad P\in L^2(\mu;E),\quad q\in L^2(\mu).\]

Corresponding Lean step

hP,hPc,hdφ,hdW,hdc,hq,hqc,hPL,hqL; pv and qv are toLp representatives

4. Establish genuine integrability and representative independence

Products of L² functions are integrable by Cauchy–Schwarz. The pointwise representatives supplied by toLp agree almost everywhere with the explicit tests; transfer both product integrabilities across these equalities. Changing u or G on a null set does not change the conclusion.

\[\int|\langle G,P\rangle|\,d\mu\le\|G\|_2\|P\|_2,\qquad\int|uq|\,d\mu\le\|u\|_2\|q\|_2.\]

Corresponding Lean step

hPe,hqe; L2.integrable_inner and Integrable.congr

5. Prove the pairing identity on the exact core

Take any pair in D.graph. Its exact graph characterization provides a smooth compact representative f and its genuine gradient. Transfer through the almost-everywhere equalities, use D_vf=⟨∇f,v⟩, and apply the normalized identity. This proves the equality on the whole original graph without assuming it.

\[(a,H)\in\operatorname{graph}D\ \Longrightarrow\ \langle H,P\rangle_{L^2}=-\langle a,q\rangle_{L^2}.\]

Corresponding Lean step

hsub uses hgraph, hleft, hright, ht and compact_directional_ibp

6. Pass to the graph closure by continuity

The two L² pairings are continuous functions on the product Hilbert space. Their equality set is closed and contains D.graph, hence contains its topological closure. Closability identifies that closure with D.closure.graph. This retains precisely the D already used in the weak resolvent.

\[\overline{\operatorname{graph}D}\subseteq\{(a,H):\langle H,P\rangle=-\langle a,q\rangle\},\qquad\overline{\operatorname{graph}D}=\operatorname{graph}(\overline D).\]

Corresponding Lean step

hclosed,closure_minimal,hmem; hD.graph_closure_eq_closure_graph

7. Return to the readable integral identity

Convert the two Hilbert pairings back to integrals using the same almost-everywhere test representatives. This establishes a weighted weak identity; it does not differentiate u pointwise. Passing to ordinary volume weak derivatives, or invoking elliptic regularity for the resolvent, still requires separate work.

\[\int \varphi\langle G,v\rangle\,d\mu=-\int u\,(D_v\varphi-\varphi D_vW)\,d\mu.\]

Corresponding Lean step

he; L2.inner_def,integral_congr_ae (hPe G),integral_congr_ae (hqe u)

Lean statement · closed_gradient_weighted_ibp

All spaces, the exact graph, closability, representatives, compact C1 test and integrabilities appear explicitly below.

theorem closed_gradient_weighted_ibp (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∀ (D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ), D.IsClosable →
      (∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[μ] f ∧ G =ᵐ[μ] gradient f) →
      ∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.closure.graph →
      ∀ (φ : E → ℝ), ContDiff ℝ 1 φ → HasCompactSupport φ → ∀ v : E,
        Integrable (fun x => φ x * inner ℝ (G x) v) μ ∧
        Integrable (fun x => u x * (fderiv ℝ φ x v - φ x * fderiv ℝ W x v)) μ ∧
        (∫ x, φ x * inner ℝ (G x) v ∂μ) =
          - ∫ x, u x * (fderiv ℝ φ x v - φ x * fderiv ℝ W x v) ∂μ

Exact module and namespace context

Lean proof · closed_gradient_weighted_ibp

Compact classical weighted IBP plus a closed equality of continuous L² pairings, with explicit representative and integrability transfers.

theorem closed_gradient_weighted_ibp (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∀ (D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ), D.IsClosable →
      (∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[μ] f ∧ G =ᵐ[μ] gradient f) →
      ∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.closure.graph →
      ∀ (φ : E → ℝ), ContDiff ℝ 1 φ → HasCompactSupport φ → ∀ v : E,
        Integrable (fun x => φ x * inner ℝ (G x) v) μ ∧
        Integrable (fun x => u x * (fderiv ℝ φ x v - φ x * fderiv ℝ W x v)) μ ∧
        (∫ x, φ x * inner ℝ (G x) v ∂μ) =
          - ∫ x, u x * (fderiv ℝ φ x v - φ x * fderiv ℝ W x v) ∂μ := by
  let μ := (volume : Measure E).tilted (fun x => -W x)
  let : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hI
  dsimp only
  intro D hD hgraph u G hu φ hφ hc v
  let P := fun x => φ x • v
  let q := fun x => fderiv ℝ φ x v - φ x * fderiv ℝ W x v
  have hP : Continuous P := hφ.continuous.smul continuous_const
  have hPc : HasCompactSupport P := hc.smul_right
  have hdφ : Continuous (fun x => fderiv ℝ φ x v) :=
    (hφ.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
  have hdW : Continuous (fun x => fderiv ℝ W x v) :=
    (hW.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
  have hdc : HasCompactSupport (fun x => fderiv ℝ φ x v) := by
    refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
    intro x hx
    by_contra hn
    exact hx (by simp [fderiv_of_notMem_tsupport ℝ hn])
  have hq : Continuous q := hdφ.sub (hφ.continuous.mul hdW)
  have hqc : HasCompactSupport q := hdc.sub hc.mul_right
  have hPL : MemLp P 2 μ := hP.memLp_of_hasCompactSupport hPc
  have hqL : MemLp q 2 μ := hq.memLp_of_hasCompactSupport hqc
  let pv : Lp E 2 μ := hPL.toLp P
  let qv : Lp ℝ 2 μ := hqL.toLp q
  have hPe (H : Lp E 2 μ) :
      (fun x => inner ℝ (H x) (pv x)) =ᵐ[μ] (fun x => φ x * inner ℝ (H x) v) := by
    filter_upwards [hPL.coeFn_toLp] with x hx
    rw [show pv x = P x from hx]
    simp [P, inner_smul_right]
  have hqe (a : Lp ℝ 2 μ) :
      (fun x => inner ℝ (a x) (qv x)) =ᵐ[μ] (fun x => a x * q x) := by
    filter_upwards [hqL.coeFn_toLp] with x hx
    rw [show qv x = q x from hx]
    simp [mul_comm]
  refine ⟨(L2.integrable_inner (𝕜 := ℝ) G pv).congr (hPe G),
    (L2.integrable_inner (𝕜 := ℝ) u qv).congr (hqe u), ?_⟩
  have ht (g : E → ℝ) : (∫ x, g x ∂μ) =
      (∫ x, Real.exp (-W x))⁻¹ * ∫ x, Real.exp (-W x) * g x := by
    rw [show μ = (volume : Measure E).tilted (fun x => -W x) from rfl, integral_tilted]
    rw [← integral_const_mul]
    apply integral_congr_ae
    filter_upwards [] with x
    change (Real.exp (-W x) / (∫ z, Real.exp (-W z))) • g x = _
    simp only [smul_eq_mul, div_eq_mul_inv]
    ring
  have hclosed : IsClosed {z : Lp ℝ 2 μ × Lp E 2 μ |
      inner ℝ z.2 pv = -inner ℝ z.1 qv} :=
    isClosed_eq (continuous_snd.inner continuous_const)
      (continuous_fst.inner continuous_const).neg
  have hsub : (D.graph : Set (Lp ℝ 2 μ × Lp E 2 μ)) ⊆
      {z | inner ℝ z.2 pv = -inner ℝ z.1 qv} := by
    rintro ⟨a,H⟩ hz
    obtain ⟨f,hf,hfc,ha,hH⟩ := (hgraph a H).mp hz
    change inner ℝ H pv = -inner ℝ a qv
    rw [L2.inner_def, L2.inner_def, integral_congr_ae (hPe H), integral_congr_ae (hqe a)]
    have hff := contDiff_infty.mp hf 1
    have hleft : (∫ x, φ x * inner ℝ (H x) v ∂μ) =
        ∫ x, φ x * fderiv ℝ f x v ∂μ := by
      apply integral_congr_ae
      filter_upwards [hH] with x hx
      rw [hx, AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt
        (hff.differentiable one_ne_zero x)]
    have hright : (∫ x, a x * q x ∂μ) = ∫ x, f x * q x ∂μ :=
      integral_congr_ae (ha.mul (Filter.EventuallyEq.refl _ _))
    rw [hleft, hright, ht, ht]
    have hi := compact_directional_ibp W f φ hW hff hφ hc v
    simp_rw [← mul_assoc] at ⊢
    rw [hi]
    simp only [mul_neg]
    congr 1
    congr 1
    apply integral_congr_ae
    filter_upwards [] with x
    dsimp [q]
    ring
  have hmem : (u,G) ∈ D.graph.topologicalClosure := by
    rwa [hD.graph_closure_eq_closure_graph]
  have he := (closure_minimal hsub hclosed) hmem
  change inner ℝ G pv = -inner ℝ u qv at he
  rw [L2.inner_def, L2.inner_def, integral_congr_ae (hPe G), integral_congr_ae (hqe u)] at he
  exact he

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientWeak

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Source uses a conditional Euclidean Sobolev gradient and its closedness in the Poincare/density route.	Any given closable exact smooth compact gradient graph for normalized exp(-W) volume.	source-implicit	The same operator is supplied by an existing construction; the weak test identity is proved, not assumed.
Actual conditional potential comes from a C2 strongly convex V with proved normalization.	C1 W and integrable exp(-W), no curvature bound.	generalization	Only the local product rule and compact tests enter; no curvature-dependent coercivity is claimed.
Kolesnikov–Milman background discusses compact weighted manifolds, boundary conditions and spectral properties.	Only a full-space finite-dimensional weighted test identity, including dimension zero.	API-limitation	Domain and conclusion boundary deliberately narrower than a full background regularity/spectral result; this is a prerequisite, not that theorem.
Representative choice, compact approximation and integrability are implicit in the analytic route.	L2 equivalence classes, exact graph AE witnesses, both integrabilities, genuinely C1 compact test.	source-implicit	No totalized derivative of an arbitrary L2 representative is used; compact support prevents any global score integrability premise.

Weighted compact-test identity for the same closed gradient only. Not an unweighted distributional derivative, H2 regularity, a D*D operator core, noncompact Bochner extension, Poincare, score variance, macroscopic coercivity or a complete paper.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

assumptions: The short route paraphrase is elaborated into C¹ regularity, integrable positive Gibbs density, a given closable exact smooth-compact gradient graph, and compact C¹ tests. These are explicit in the publication candidate and reconstruction; they are not silently supplied conclusions. — The declaration explicitly includes hW, hI, D.IsClosable, the universally quantified exact graph equivalence, closure-graph membership, and C¹ compact support of φ. The authored assumptions and blind reconstruction preserve each condition. The private helper establishes classical integration by parts under its stated C¹ hypotheses and compact support of the test.
conclusion: Both product-integrability claims and almost-everywhere representative transfers are explicit, making the source's meaningful integral identity precise. — The theorem concludes two Integrable propositions together with the signed identity. The proof obtains these using L2.integrable_inner and transfers through hPe and hqe; the final conversion uses L2.inner_def and integral_congr_ae. Lesson steps four and seven accurately describe these operations.
scopes: The general C¹ full-space theorem is an independently proved prerequisite applicable to the source route under specialization. Acceptance is not an equivalence claim between this theorem and either cited paper result. — The packet's source wording explicitly labels the result a route prerequisite. The module introduction and lesson source scopes exclude unweighted regularity, D*D operator-core claims, Poincaré, macroscopic coercivity and complete-paper claims. The supplied proof establishes only the weighted compact-test identity for D.closure.graph. The inspected primary PBPS Appendix C.1 and Kolesnikov–Milman §2.5 address broader analytic results that this module does not claim to prove.

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

Weighted compact-test identity for the same closed gradient only. Not an unweighted distributional derivative, H2 regularity, a D*D operator core, noncompact Bochner extension, Poincare, score variance, macroscopic coercivity or a complete paper.

Source and reuse

ASTIS parents called

AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt

Mathlib API called (external library)

integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
MeasureTheory.integral_tilted
MeasureTheory.L2.integrable_inner
MeasureTheory.L2.inner_def
LinearPMap.IsClosable.graph_closure_eq_closure_graph
closure_minimal

Mathematical sources

PBPS Appendix C.1: analytic prerequisite — Weighted compact-test identity for the same closed gradient only. Not an unweighted distributional derivative, H2 regularity, a D*D operator core, noncompact Bochner extension, Poincare, score variance, macroscopic coercivity or a complete paper.
Kolesnikov–Milman §2.5: weighted Sobolev background — Direct full-space proof, not the compact-manifold spectral theorem.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Identify the same closed gradient by compact tests, including integrability

Weighted Bochner identity from compact-test integration by parts

Kolesnikov and Milman background; Chen, Chewi, Lu and Zhang PBPS consumer; ASTIS expanded mathematical proof.

arXiv:1310.2526v7 · Theorem1.1 (1.3), Euclidean compact-test localized component · faithful paraphrase

Complete source statement (ASTIS restatement)

The weighted Reilly identity in Theorem 1.1 (1.3) is a compact weighted manifold formula with boundary terms. The selected ASTIS background component is its Euclidean smooth compact-test localization, proved directly: on R^d for d>=1, W C2 and f smooth compactly supported, Lf=Delta f-gradient W dot gradient f satisfies integral exp(-W)(Lf)^2 = integral exp(-W)sum_i norm(gradient(D_i f))^2 + integral exp(-W)D2W[gradient f,gradient f]. All tested weighted products are integrable by compact support; global weight integrability is not required. If D2W>=m I, dropping the nonnegative Hessian-square term gives m times weighted gradient energy at most weighted square-generator energy. This selected component does not claim the entire manifold theorem or a spectral gap.

Selected Euclidean localization: d>=1, W:R^d->R C2 and f smooth compactly supported. Haar volume and actual derivatives are used; no global probability normalization is needed for this compact-test formula.
For the energy consequence only, a real m satisfies D2W(x)[v,v]>=m norm(v)^2 for every x,v. The full background source has compact manifold and boundary hypotheses outside this selected localized component.

Bochner identity and curvature energy

\[\int e^{-W}(Lf)^2=\int e^{-W}H_f+\int e^{-W}D^2W[\nabla f,\nabla f],\qquad m\int e^{-W}\|\nabla f\|^2\le\int e^{-W}(Lf)^2.\]

Read the formalized proofs

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

ASTIS mathematical exposition

Weighted Bochner identity from compact-test integration by parts

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedBochner.integrated_bochner_identity · theorem · Teaching coverage

Statement

Let E be a finite-dimensional real inner-product space with its Borel structure, W a C2 real potential, and f a smooth compactly supported real function. Put Lf=Delta f-inner(gradient W,gradient f), H_f=sum_i norm(gradient(D_i f))^2 for a fixed orthonormal basis, and C_f=D2W[gradient f,gradient f]. All four weighted functions (Lf)^2, norm(gradient f)^2, H_f and C_f are integrable against exp(-W) times volume. Their actual integrals satisfy the Bochner identity. For every real m with genuine Hessian lower bound D2W(x)[v,v]>=m norm(v)^2, m times weighted gradient energy is at most the weighted square of Lf.

\[\int e^{-W}(Lf)^2=\int e^{-W}H_f+\int e^{-W}D^2W[\nabla f,\nabla f],\qquad m\int e^{-W}\|\nabla f\|^2\le\int e^{-W}(Lf)^2.\]

All objects and hypotheses

E is finite-dimensional real inner-product with Borel sigma algebra and canonical volume, including dimension zero.
W:E->R is C2; f:E->R is C-infinity and compactly supported. No C3 regularity of W is assumed.
All differential expressions use the actual Frechet derivative, Riesz gradient and canonical Laplacian. H_f is a coordinate Hessian-square expression; no separate abstract Hilbert-Schmidt norm theorem is claimed.
The identity needs no curvature or global integrability of exp(-W). All tested products are compactly supported. For the energy consequence, m is any real number and its pointwise genuine Hessian lower bound is an explicit premise.
This is the directly proved Euclidean compact-test localization of the background weighted Reilly formula. Positive-dimensional localization kills boundary terms on an enclosing ball; dimension zero is a disclosed extension, not a full manifold theorem.

Mathematical proof

1. Prove weighted integration by parts in one direction

For fixed v set F=exp(-W)g and H=D_v h, where W,g are C1 and h is C2 with compact support. H and D_v H are continuous and vanish outside the support of h. Thus F H, D_v F H and F D_v H are continuous compactly supported products and are genuinely integrable. Apply the Haar-volume integration-by-parts theorem, then expand D_v F=exp(-W)(D_v g-g D_v W). The API is applied to volume, not to a non-Haar Gibbs measure.

\[\int e^{-W}gD_{vv}h=-\int e^{-W}(D_vg-gD_vW)D_vh.\]

Corresponding Lean step

weighted_directional_ibp derives all three API integrability premises. All helper proofs are local steps inside the single public theorem and included in its full Lean proof.

2. Sum directions to recover the actual Langevin expression

Sum over the standard orthonormal basis. The diagonal genuine second derivatives sum to the canonical Laplacian, while the products of directional derivatives sum to the inner product of gradients. Finite sums and compact integrability justify splitting the integrals. This gives weighted bilinear IBP for g merely C1; h is C2 compact. No semigroup or generator-domain contract enters.

\[\int e^{-W}gLh=-\int e^{-W}\langle\nabla g,\nabla h\rangle.\]

Corresponding Lean step

directional_second and inner_gradient_eq_sum identify the actual derivatives. weighted_bilinear_ibp proves the integral identity rather than assuming it.

3. Differentiate Lf using only the second derivative of W

Smooth f permits the required directional derivative exchanges. The C2 Hessian symmetry theorem exchanges two genuine derivatives; the product rule gives the differentiated drift. Hence D_i Lf=L(D_i f)-sum_j D_ij W D_j f. W being C2 guarantees Lf is C1; no third derivative of W is used. Compact support is inherited by all directional derivatives and Lf.

\[D_iLf=L(D_if)-\sum_jD_{ij}W\,D_jf.\]

Corresponding Lean step

dir_comm, op_coords, op_contDiff, op_compact and dir_op are locally proved helpers. The displayed op is the actual Laplacian-minus-drift expression.

4. Apply integration by parts twice

First use g=Lf and h=f. Expand the gradient inner product in the orthonormal basis and substitute the commutator. For each basis direction use g=h=D_i f in the second IBP. Its negative sign cancels the first IBP sign and yields a positive squared gradient. Every cross term is integrable because a smooth compact derivative of f is one factor.

\[\int e^{-W}(Lf)^2=\sum_i\int e^{-W}\|\nabla D_if\|^2+\sum_{i,j}\int e^{-W}D_{ij}W D_ifD_jf.\]

Corresponding Lean step

bochner_coordinate_integrals derives both IBP applications and all finite-sum integral exchanges with explicit Integrable witnesses.

5. Recover the genuine Hessian quadratic form

Expand gradient f in the orthonormal basis using its actual derivative coordinates. Bilinearity and C2 symmetry identify the curvature sum with D2W[gradient f,gradient f]. The sum of squared gradients of D_i f is the coordinate expression for the Hessian Hilbert-Schmidt square. The formal result retains this explicit sum and does not assert a separate abstract norm identification.

\[\sum_{i,j}D_{ij}W D_ifD_jf=D^2W[\nabla f,\nabla f].\]

Corresponding Lean step

gradient_basis and curvature_sum provide the pointwise algebra; curvature_integrable_and_sum and integrated_identity preserve genuine integral equalities.

6. Derive the curvature-energy inequality

All four weighted integrability claims follow from the compact-test proof. If D2W>=m I, multiply the pointwise quadratic-form bound by the nonnegative weight and integrate. The Hessian-square term is nonnegative, so dropping it gives m times the gradient energy at most the square-generator energy. This is an estimate on the actual differential expression, not Poincare or closed-operator coercivity.

\[m\int e^{-W}\|\nabla f\|^2\le\int e^{-W}D^2W[\nabla f,\nabla f]\le\int e^{-W}(Lf)^2.\]

Corresponding Lean step

weighted_energy_lower_bound uses integral_mono with proved integrability; the public theorem packages that result with the full identity and input integrability.

Lean statement · integrated_bochner_identity

The public statement retains actual differential expressions and every integrability clause; the m-bound is universally quantified with an explicit genuine curvature premise.

theorem integrated_bochner_identity (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
    (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) :
    let L := fun x => Laplacian.laplacian f x - inner ℝ (gradient W x) (gradient f x)
    let H := fun x => ∑ i, ‖gradient (fun z => fderiv ℝ f z ((stdOrthonormalBasis ℝ E) i)) x‖^2
    let C := fun x => fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x)
    Integrable (fun x => Real.exp (-W x) * (L x)^2) ∧
    Integrable (fun x => Real.exp (-W x) * ‖gradient f x‖^2) ∧
    Integrable (fun x => Real.exp (-W x) * H x) ∧
    Integrable (fun x => Real.exp (-W x) * C x) ∧
    (∫ x, Real.exp (-W x) * (L x)^2) =
      (∫ x, Real.exp (-W x) * H x) + ∫ x, Real.exp (-W x) * C x ∧
    ∀ m : ℝ, (∀ x v, m * ‖v‖^2 ≤ fderiv ℝ (fderiv ℝ W) x v v) →
      m * (∫ x, Real.exp (-W x) * ‖gradient f x‖^2) ≤ ∫ x, Real.exp (-W x) * (L x)^2

Exact module and namespace context

Lean proof · integrated_bochner_identity

Two proved Haar-volume IBP applications, C2 Hessian symmetry and finite-basis algebra yield the identity; nonnegative integration gives the energy bound. All helper proofs are local haves/lets in this complete public proof.

theorem integrated_bochner_identity (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
    (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) :
    let L := fun x => Laplacian.laplacian f x - inner ℝ (gradient W x) (gradient f x)
    let H := fun x => ∑ i, ‖gradient (fun z => fderiv ℝ f z ((stdOrthonormalBasis ℝ E) i)) x‖^2
    let C := fun x => fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x)
    Integrable (fun x => Real.exp (-W x) * (L x)^2) ∧
    Integrable (fun x => Real.exp (-W x) * ‖gradient f x‖^2) ∧
    Integrable (fun x => Real.exp (-W x) * H x) ∧
    Integrable (fun x => Real.exp (-W x) * C x) ∧
    (∫ x, Real.exp (-W x) * (L x)^2) =
      (∫ x, Real.exp (-W x) * H x) + ∫ x, Real.exp (-W x) * C x ∧
    ∀ m : ℝ, (∀ x v, m * ‖v‖^2 ≤ fderiv ℝ (fderiv ℝ W) x v v) →
      m * (∫ x, Real.exp (-W x) * ‖gradient f x‖^2) ≤ ∫ x, Real.exp (-W x) * (L x)^2 := by
  have weighted_directional_ibp (W g h : E → ℝ) (hW : ContDiff ℝ 1 W) (hg : ContDiff ℝ 1 g)
      (hh : ContDiff ℝ 2 h) (hc : HasCompactSupport h) (v : E) :
      (∫ x, (Real.exp (-W x) * g x) *
        fderiv ℝ (fun z => fderiv ℝ h z v) x v) =
      - ∫ x, Real.exp (-W x) *
        (fderiv ℝ g x v - g x * fderiv ℝ W x v) * fderiv ℝ h x v := by
    let F := fun x => Real.exp (-W x) * g x
    let H := fun x => fderiv ℝ h x v
    have hF : ContDiff ℝ 1 F := hW.neg.exp.mul hg
    have hH : ContDiff ℝ 1 H :=
      (hh.fderiv_right (m := 1) (by norm_num)).clm_apply contDiff_const
    have hHc : HasCompactSupport H := by
      refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
      intro x hx
      by_contra hnot
      exact hx (by simp [H, fderiv_of_notMem_tsupport ℝ hnot])
    have hDHc : HasCompactSupport (fun x => fderiv ℝ H x v) := by
      refine HasCompactSupport.of_support_subset_isCompact hHc.isCompact ?_
      intro x hx
      by_contra hnot
      exact hx (by simp [fderiv_of_notMem_tsupport ℝ hnot])
    have hDF : Continuous (fun x => fderiv ℝ F x v) :=
      (hF.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
    have hDH : Continuous (fun x => fderiv ℝ H x v) :=
      (hH.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
    have h1 : Integrable (fun x => fderiv ℝ F x v * H x) :=
      (hDF.mul hH.continuous).integrable_of_hasCompactSupport hHc.mul_left
    have h2 : Integrable (fun x => F x * fderiv ℝ H x v) :=
      (hF.continuous.mul hDH).integrable_of_hasCompactSupport hDHc.mul_left
    have h3 : Integrable (fun x => F x * H x) :=
      (hF.continuous.mul hH.continuous).integrable_of_hasCompactSupport hHc.mul_left
    have hibp := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable h1 h2 h3
      (fun x _ => hF.differentiable one_ne_zero x)
      (fun x _ => hH.differentiable one_ne_zero x)
    change (∫ x, F x * fderiv ℝ H x v) = _
    rw [hibp]
    congr 1
    apply integral_congr_ae
    filter_upwards [] with x
    have hw := (hW.differentiable one_ne_zero x).hasFDerivAt.neg.exp
    have hder := hw.mul (hg.differentiable one_ne_zero x).hasFDerivAt
    have hd : fderiv ℝ F x v = Real.exp (-W x) *
        (fderiv ℝ g x v - g x * fderiv ℝ W x v) := by
      rw [show fderiv ℝ F x = _ from hder.fderiv]
      simp only [add_apply, smul_apply,
        neg_apply, smul_eq_mul, Pi.neg_apply]
      ring
    rw [hd]

  have directional_second (h : E → ℝ) (hh : ContDiff ℝ 2 h) (x v w : E) :
      fderiv ℝ (fun z => fderiv ℝ h z w) x v =
        fderiv ℝ (fderiv ℝ h) x v w := by
    have hd := ((hh.fderiv_right (m := 1) (by norm_num)).differentiable one_ne_zero x).hasFDerivAt
    have he := hd.clm_apply (hasFDerivAt_const w x)
    simpa using congrArg (fun T : E →L[ℝ] ℝ => T v) he.fderiv

  have inner_gradient_eq_sum (f g : E → ℝ)
      (hf : Differentiable ℝ f) (hg : Differentiable ℝ g) (x : E) :
      inner ℝ (gradient f x) (gradient g x) =
        ∑ i, fderiv ℝ f x ((stdOrthonormalBasis ℝ E) i) *
          fderiv ℝ g x ((stdOrthonormalBasis ℝ E) i) := by
    simp_rw [AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt (hf x),
      AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt (hg x)]
    rw [← (stdOrthonormalBasis ℝ E).sum_inner_mul_inner (gradient f x) (gradient g x)]
    apply Finset.sum_congr rfl
    intro i _
    rw [real_inner_comm ((stdOrthonormalBasis ℝ E) i) (gradient g x)]

  have weighted_bilinear_ibp (W g h : E → ℝ) (hW : ContDiff ℝ 1 W)
      (hg : ContDiff ℝ 1 g) (hh : ContDiff ℝ 2 h) (hc : HasCompactSupport h) :
      (∫ x, Real.exp (-W x) * g x *
        (Laplacian.laplacian h x - inner ℝ (gradient W x) (gradient h x))) =
      - ∫ x, Real.exp (-W x) * inner ℝ (gradient g x) (gradient h x) := by
    let b := stdOrthonormalBasis ℝ E
    let A := fun i x => (Real.exp (-W x) * g x) *
      fderiv ℝ (fun z => fderiv ℝ h z (b i)) x (b i)
    let B := fun i x => Real.exp (-W x) *
      (fderiv ℝ g x (b i) - g x * fderiv ℝ W x (b i)) * fderiv ℝ h x (b i)
    have hA (i) : Integrable (A i) := by
      have hD : ContDiff ℝ 1 (fun z => fderiv ℝ h z (b i)) :=
        (hh.fderiv_right (m := 1) (by norm_num)).clm_apply contDiff_const
      have hC : Continuous (fun x => fderiv ℝ (fun z => fderiv ℝ h z (b i)) x (b i)) :=
        (hD.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
      apply (((hW.neg.exp).continuous.mul hg.continuous).mul hC).integrable_of_hasCompactSupport
      refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
      intro x hx
      by_contra hn
      have hcD : tsupport (fun z => fderiv ℝ h z (b i)) ⊆ tsupport h := by
        apply closure_minimal _ isClosed_closure
        intro z hz
        by_contra hnz
        exact hz (by simp [fderiv_of_notMem_tsupport ℝ hnz])
      have hnD : x ∉ tsupport (fun z => fderiv ℝ h z (b i)) := fun hx => hn (hcD hx)
      exact hx (by simp [fderiv_of_notMem_tsupport ℝ hnD])
    have hB (i) : Integrable (B i) := by
      have hD (f : E → ℝ) (hf : ContDiff ℝ 1 f) : Continuous (fun x => fderiv ℝ f x (b i)) :=
        (hf.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
      have hDh : Continuous (fun x => fderiv ℝ h x (b i)) := hD h (hh.of_le (by norm_num))
      apply ((hW.neg.exp).continuous.mul ((hD g hg).sub (hg.continuous.mul (hD W hW))) |>.mul hDh).integrable_of_hasCompactSupport
      refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
      intro x hx
      by_contra hn
      exact hx (by simp [fderiv_of_notMem_tsupport ℝ hn])
    have heq : (∫ x, ∑ i, A i x) = - ∫ x, ∑ i, B i x := by
      rw [integral_finsetSum _ (fun i _ => hA i), integral_finsetSum _ (fun i _ => hB i)]
      simp_rw [show ∀ i, (∫ x, A i x) = -∫ x, B i x from fun i => weighted_directional_ibp W g h hW hg hh hc (b i)]
      exact Finset.sum_neg_distrib _
    have hLap (x : E) : Laplacian.laplacian h x =
        ∑ i, fderiv ℝ (fun z => fderiv ℝ h z (b i)) x (b i) := by
      rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis]
      apply Finset.sum_congr rfl
      intro i _
      rw [directional_second h hh, iteratedFDeriv_two_apply]
      rfl
    have hpoint (x : E) : (∑ i, A i x) + (∑ i, B i x) =
        Real.exp (-W x) * g x *
          (Laplacian.laplacian h x - inner ℝ (gradient W x) (gradient h x)) +
        Real.exp (-W x) * inner ℝ (gradient g x) (gradient h x) := by
      rw [hLap, inner_gradient_eq_sum W h (hW.differentiable one_ne_zero) (hh.differentiable (by norm_num)),
        inner_gradient_eq_sum g h (hg.differentiable one_ne_zero) (hh.differentiable (by norm_num))]
      simp only [A, B, mul_sub, sub_mul, mul_assoc, Finset.sum_sub_distrib, ← Finset.mul_sum, b]
      ring
    -- The finite sums establish compact integrability before splitting integrals.
    have hS1 : Integrable (fun x => ∑ i, A i x) := integrable_finsetSum _ (fun i _ => hA i)
    have hS2 : Integrable (fun x => ∑ i, B i x) := integrable_finsetSum _ (fun i _ => hB i)
    let G := fun x => Real.exp (-W x) * inner ℝ (gradient g x) (gradient h x)
    have hG : Integrable G := by
      have hgD : Continuous (fun x => fderiv ℝ g x) :=
        (hg.fderiv_right (m := 0) (by norm_num)).continuous
      have hhD : Continuous (fun x => fderiv ℝ h x) :=
        (hh.fderiv_right (m := 1) (by norm_num)).continuous
      have hrep : G = fun x => Real.exp (-W x) * ∑ i, fderiv ℝ g x (b i) * fderiv ℝ h x (b i) := by
        funext x
        exact congrArg (fun t => Real.exp (-W x) * t)
          (inner_gradient_eq_sum g h (hg.differentiable one_ne_zero) (hh.differentiable (by norm_num)) x)
      rw [hrep]
      apply ((hW.neg.exp).continuous.mul (continuous_finsetSum _ fun i _ =>
        (hgD.clm_apply continuous_const).mul (hhD.clm_apply continuous_const))).integrable_of_hasCompactSupport
      refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
      intro x hx
      by_contra hn
      exact hx (by simp [fderiv_of_notMem_tsupport ℝ hn])
    let L := fun x => Real.exp (-W x) * g x *
        (Laplacian.laplacian h x - inner ℝ (gradient W x) (gradient h x))
    have hLG : (fun x => (∑ i, A i x) + (∑ i, B i x)) = fun x => L x + G x := funext hpoint
    have hL : Integrable L := by
      have hp := hS1.add hS2
      change Integrable (fun x => (∑ i, A i x) + (∑ i, B i x)) at hp
      rw [hLG] at hp
      convert hp.sub hG using 1
      ext x
      simp only [Pi.sub_apply, add_sub_cancel_right]
    have hz : (∫ x, L x) + (∫ x, G x) = 0 := by
      rw [← integral_add hL hG, ← hLG, integral_add hS1 hS2, heq]
      ring
    exact eq_neg_of_add_eq_zero_left hz

  let dir (f : E → ℝ) (v : E) : E → ℝ := fun x => fderiv ℝ f x v

  have dir_contDiff {n : ℕ∞ω} (f : E → ℝ) (hf : ContDiff ℝ (n+1) f) (v : E) :
      ContDiff ℝ n (dir f v) := (hf.fderiv_right le_rfl).clm_apply contDiff_const

  have dir_compact (f : E → ℝ) (hc : HasCompactSupport f) (v : E) :
      HasCompactSupport (dir f v) := by
    refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
    intro x hx
    by_contra hn
    exact hx (by simp [dir, fderiv_of_notMem_tsupport ℝ hn])

  have dir_comm (f : E → ℝ) (hf : ContDiff ℝ 2 f) (v w : E) :
      dir (dir f v) w = dir (dir f w) v := by
    funext x
    change fderiv ℝ (fun z => fderiv ℝ f z v) x w =
      fderiv ℝ (fun z => fderiv ℝ f z w) x v
    rw [directional_second f hf, directional_second f hf]
    exact (hf.contDiffAt.isSymmSndFDerivAt (by norm_num)).eq w v

  have laplacian_dirs (f : E → ℝ) (hf : ContDiff ℝ 2 f) :
      Laplacian.laplacian f = fun x => ∑ i,
        dir (dir f ((stdOrthonormalBasis ℝ E) i)) ((stdOrthonormalBasis ℝ E) i) x := by
    funext x
    rw [InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis]
    apply Finset.sum_congr rfl
    intro i _
    change iteratedFDeriv ℝ 2 f x ![(stdOrthonormalBasis ℝ E) i, (stdOrthonormalBasis ℝ E) i] =
      fderiv ℝ (fun z => fderiv ℝ f z ((stdOrthonormalBasis ℝ E) i)) x ((stdOrthonormalBasis ℝ E) i)
    rw [directional_second f hf, iteratedFDeriv_two_apply]
    rfl

  have dir_sum {ι : Type} [Fintype ι] (f : ι → E → ℝ)
      (hf : ∀ i, Differentiable ℝ (f i)) (v : E) :
      dir (fun x => ∑ i, f i x) v = fun x => ∑ i, dir (f i) v x := by
    funext x
    dsimp [dir]
    rw [fderiv_fun_sum (fun i _ => hf i x)]
    simp

  have smooth_finite (f : E → ℝ) (hf : ContDiff ℝ ∞ f) (n : ℕ) :
      ContDiff ℝ n f := contDiff_infty.mp hf n

  have dir_mul (f g : E → ℝ) (hf : Differentiable ℝ f) (hg : Differentiable ℝ g)
      (v : E) : dir (fun x => f x * g x) v =
        fun x => dir f v x * g x + f x * dir g v x := by
    funext x
    dsimp [dir]
    change fderiv ℝ (f * g) x v = _
    rw [fderiv_mul (hf x) (hg x)]
    simp only [add_apply, smul_apply, smul_eq_mul]
    ring

  let op (W f : E → ℝ) : E → ℝ :=
    fun x => Laplacian.laplacian f x - inner ℝ (gradient W x) (gradient f x)

  have op_coords (W f : E → ℝ) (hW : ContDiff ℝ 1 W) (hf : ContDiff ℝ 2 f) :
      op W f = fun x => ∑ i,
        (dir (dir f ((stdOrthonormalBasis ℝ E) i)) ((stdOrthonormalBasis ℝ E) i) x -
          dir W ((stdOrthonormalBasis ℝ E) i) x * dir f ((stdOrthonormalBasis ℝ E) i) x) := by
    funext x
    dsimp [op]
    rw [laplacian_dirs f hf, inner_gradient_eq_sum W f (hW.differentiable one_ne_zero)
      (hf.differentiable (by norm_num))]
    exact (Finset.sum_sub_distrib _ _).symm

  have op_contDiff (W f : E → ℝ) (hW : ContDiff ℝ 2 W) (hf : ContDiff ℝ ∞ f) :
      ContDiff ℝ 1 (op W f) := by
    rw [op_coords W f (hW.of_le (by norm_num)) (smooth_finite f hf 2)]
    apply ContDiff.sum
    intro i _
    have hfd : ContDiff ℝ ∞ (dir f ((stdOrthonormalBasis ℝ E) i)) :=
      dir_contDiff f (by simpa using hf) _
    have hfdd : ContDiff ℝ ∞ (dir (dir f ((stdOrthonormalBasis ℝ E) i)) ((stdOrthonormalBasis ℝ E) i)) :=
      dir_contDiff _ (by simpa using hfd) _
    exact (smooth_finite _ hfdd 1).sub
      ((dir_contDiff W hW _).mul (smooth_finite _ hfd 1))

  have op_compact (W f : E → ℝ) (hW : ContDiff ℝ 1 W)
      (hf : ContDiff ℝ 2 f) (hc : HasCompactSupport f) : HasCompactSupport (op W f) := by
    rw [op_coords W f hW hf]
    refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
    intro x hx
    by_contra hn
    apply hx
    apply Finset.sum_eq_zero
    intro i _
    have hsub : tsupport (dir f ((stdOrthonormalBasis ℝ E) i)) ⊆ tsupport f := by
      apply closure_minimal _ isClosed_closure
      intro z hz
      by_contra hnz
      exact hz (by simp [dir, fderiv_of_notMem_tsupport ℝ hnz])
    have hnd : x ∉ tsupport (dir f ((stdOrthonormalBasis ℝ E) i)) := fun hx => hn (hsub hx)
    change (fderiv ℝ (dir f ((stdOrthonormalBasis ℝ E) i)) x) ((stdOrthonormalBasis ℝ E) i) -
      dir W ((stdOrthonormalBasis ℝ E) i) x * (fderiv ℝ f x) ((stdOrthonormalBasis ℝ E) i) = 0
    simp [fderiv_of_notMem_tsupport ℝ hn, fderiv_of_notMem_tsupport ℝ hnd]

  have dir_op (W f : E → ℝ) (hW : ContDiff ℝ 2 W) (hf : ContDiff ℝ ∞ f) (v : E) :
      dir (op W f) v = fun x => op W (dir f v) x -
        ∑ i, dir (dir W ((stdOrthonormalBasis ℝ E) i)) v x *
          dir f ((stdOrthonormalBasis ℝ E) i) x := by
    let b := stdOrthonormalBasis ℝ E
    have hfd (w : E) : ContDiff ℝ ∞ (dir f w) := dir_contDiff f (by simpa using hf) w
    have hfdd (w z : E) : ContDiff ℝ ∞ (dir (dir f w) z) :=
      dir_contDiff _ (by simpa using hfd w) z
    have hWd (w : E) : ContDiff ℝ 1 (dir W w) := dir_contDiff W hW w
    rw [op_coords W f (hW.of_le (by norm_num)) (smooth_finite f hf 2)]
    rw [dir_sum (fun i x => dir (dir f (b i)) (b i) x - dir W (b i) x * dir f (b i) x)
      (fun i => ((hfdd _ _).differentiable (by simp)).sub
      ((hWd _).differentiable one_ne_zero |>.mul ((hfd _).differentiable (by simp))))]
    funext x
    rw [op_coords W (dir f v) (hW.of_le (by norm_num)) (smooth_finite _ (hfd v) 2)]
    rw [← Finset.sum_sub_distrib]
    apply Finset.sum_congr rfl
    intro i _
    have hdsub : dir (fun x => dir (dir f (b i)) (b i) x - dir W (b i) x * dir f (b i) x) v x =
        dir (dir (dir f (b i)) (b i)) v x -
          dir (fun x => dir W (b i) x * dir f (b i) x) v x := by
      change fderiv ℝ (dir (dir f (b i)) (b i) - dir W (b i) * dir f (b i)) x v =
        fderiv ℝ (dir (dir f (b i)) (b i)) x v -
          fderiv ℝ (dir W (b i) * dir f (b i)) x v
      rw [fderiv_sub ((hfdd _ _).differentiable (by simp) x)
        (((hWd _).differentiable one_ne_zero).mul ((hfd _).differentiable (by simp)) x)]
      rfl
    rw [hdsub, dir_mul _ _ ((hWd _).differentiable one_ne_zero) ((hfd _).differentiable (by simp))]
    rw [dir_comm (dir f (b i)) (smooth_finite _ (hfd _) 2) (b i) v,
      dir_comm f (smooth_finite f hf 2) (b i) v]
    dsimp only [b]
    ring

  have weighted_mul_integrable (W f g : E → ℝ) (hW : Continuous W)
      (hf : Continuous f) (hg : Continuous g) (hc : HasCompactSupport g) :
      Integrable (fun x => Real.exp (-W x) * f x * g x) :=
    ((Real.continuous_exp.comp hW.neg).mul hf |>.mul hg).integrable_of_hasCompactSupport hc.mul_left

  have weighted_inner_integrable (W f g : E → ℝ) (hW : Continuous W)
      (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g) (hc : HasCompactSupport g) :
      Integrable (fun x => Real.exp (-W x) * inner ℝ (gradient f x) (gradient g x)) := by
    have hd (k : E → ℝ) (hk : ContDiff ℝ 1 k) (v : E) : Continuous (dir k v) :=
      (dir_contDiff (n := 0) k hk v).continuous
    have he : (fun x => Real.exp (-W x) * inner ℝ (gradient f x) (gradient g x)) =
        fun x => ∑ i, Real.exp (-W x) * dir f ((stdOrthonormalBasis ℝ E) i) x *
          dir g ((stdOrthonormalBasis ℝ E) i) x := by
      funext x
      rw [inner_gradient_eq_sum f g (hf.differentiable one_ne_zero) (hg.differentiable one_ne_zero)]
      simp only [Finset.mul_sum, dir, mul_assoc]
    rw [he]
    exact integrable_finsetSum _ (fun i _ => weighted_mul_integrable W _ _ hW
      (hd f hf _) (hd g hg _) (dir_compact g hc _))

  have bochner_coordinate_integrals (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
      (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) :
      (∫ x, Real.exp (-W x) * (op W f x)^2) =
        (∑ i, ∫ x, Real.exp (-W x) *
          ‖gradient (dir f ((stdOrthonormalBasis ℝ E) i)) x‖^2) +
        ∑ i, ∫ x, Real.exp (-W x) *
          (∑ j, dir (dir W ((stdOrthonormalBasis ℝ E) j)) ((stdOrthonormalBasis ℝ E) i) x *
            dir f ((stdOrthonormalBasis ℝ E) j) x) * dir f ((stdOrthonormalBasis ℝ E) i) x := by
    let b := stdOrthonormalBasis ℝ E
    have hfd (v : E) : ContDiff ℝ ∞ (dir f v) := dir_contDiff f (by simpa using hf) v
    have hop := op_contDiff W f hW hf
    have hW1 : ContDiff ℝ 1 W := hW.of_le (by norm_num)
    have hf1 : ContDiff ℝ 1 f := smooth_finite f hf 1
    have hfd1 (v : E) : ContDiff ℝ 1 (dir f v) := smooth_finite _ (hfd v) 1
    have hWdd (v w : E) : Continuous (dir (dir W w) v) :=
      (dir_contDiff (n := 0) _ (dir_contDiff (n := 1) W hW w) v).continuous
    let T := fun i x => Real.exp (-W x) * dir (op W f) (b i) x * dir f (b i) x
    let A := fun i x => Real.exp (-W x) * dir f (b i) x * op W (dir f (b i)) x
    let K := fun i x => Real.exp (-W x) *
      (∑ j, dir (dir W (b j)) (b i) x * dir f (b j) x) * dir f (b i) x
    have hTi (i) : Integrable (T i) := weighted_mul_integrable W _ _ hW.continuous
      (dir_contDiff (n := 0) _ hop _).continuous (hfd1 _).continuous (dir_compact f hc _)
    have hAi (i) : Integrable (A i) := weighted_mul_integrable W _ _ hW.continuous
      (hfd1 _).continuous (op_contDiff W _ hW (hfd _)).continuous
      (op_compact W _ hW1 (smooth_finite _ (hfd _) 2) (dir_compact f hc _))
    have hKi (i) : Integrable (K i) := weighted_mul_integrable W _ _ hW.continuous
      (continuous_finsetSum _ fun j _ => (hWdd _ _).mul (hfd1 _).continuous)
      (hfd1 _).continuous (dir_compact f hc _)
    have hpoint (i) : T i = fun x => A i x - K i x := by
      funext x
      dsimp only [T]
      rw [dir_op W f hW hf (b i)]
      dsimp only [A, K, b]
      ring
    have hfirst : (∫ x, Real.exp (-W x) * (op W f x)^2) = -∑ i, ∫ x, T i x := by
      have hibp := weighted_bilinear_ibp W (op W f) f hW1 hop (smooth_finite f hf 2) hc
      have hlhs : (fun x => Real.exp (-W x) * op W f x *
          (Laplacian.laplacian f x - inner ℝ (gradient W x) (gradient f x))) =
          fun x => Real.exp (-W x) * (op W f x)^2 := by
        funext x
        change Real.exp (-W x) * op W f x * op W f x = _
        ring
      rw [hlhs] at hibp
      rw [hibp]
      congr 1
      rw [← integral_finsetSum _ (fun i _ => hTi i)]
      apply integral_congr_ae
      filter_upwards [] with x
      rw [inner_gradient_eq_sum (op W f) f (hop.differentiable one_ne_zero) (hf1.differentiable one_ne_zero)]
      simp only [T, dir, b, Finset.mul_sum, mul_assoc]
    have hsecond (i) : (∫ x, A i x) = -∫ x, Real.exp (-W x) * ‖gradient (dir f (b i)) x‖^2 := by
      have hibp := weighted_bilinear_ibp W (dir f (b i)) (dir f (b i)) hW1
        (hfd1 _) (smooth_finite _ (hfd _) 2) (dir_compact f hc _)
      simpa only [A, op, real_inner_self_eq_norm_sq] using hibp
    rw [hfirst]
    have heach (i) : (∫ x, T i x) =
        -(∫ x, Real.exp (-W x) * ‖gradient (dir f (b i)) x‖^2) - ∫ x, K i x := by
      rw [hpoint, integral_sub (hAi i) (hKi i), hsecond]
    simp_rw [heach]
    rw [Finset.sum_sub_distrib, Finset.sum_neg_distrib]
    dsimp only [K, b]
    ring

  have gradient_basis (f : E → ℝ) (hf : Differentiable ℝ f) (x : E) :
      gradient f x = ∑ i, dir f ((stdOrthonormalBasis ℝ E) i) x • ((stdOrthonormalBasis ℝ E) i) := by
    rw [← (stdOrthonormalBasis ℝ E).sum_repr' (gradient f x)]
    apply Finset.sum_congr rfl
    intro i _
    congr 1
    rw [real_inner_comm]
    exact (AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt (hf x)).symm

  have curvature_sum (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
      (hf : Differentiable ℝ f) (x : E) :
      (∑ i, (∑ j, dir (dir W ((stdOrthonormalBasis ℝ E) j)) ((stdOrthonormalBasis ℝ E) i) x *
        dir f ((stdOrthonormalBasis ℝ E) j) x) * dir f ((stdOrthonormalBasis ℝ E) i) x) =
      fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x) := by
    have hdir (v w : E) : dir (dir W w) v x = fderiv ℝ (fderiv ℝ W) x v w :=
      directional_second W hW x v w
    simp_rw [hdir]
    rw [gradient_basis f hf x]
    simp only [map_sum, map_smul, sum_apply, smul_apply, smul_eq_mul]
    apply Finset.sum_congr rfl
    intro i _
    rw [Finset.sum_mul, Finset.mul_sum]
    apply Finset.sum_congr rfl
    intro j _
    rw [(hW.contDiffAt.isSymmSndFDerivAt (by norm_num)).eq ((stdOrthonormalBasis ℝ E) i) ((stdOrthonormalBasis ℝ E) j)]
    ring

  have curvature_component_integrable (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
      (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) (i : Fin (Module.finrank ℝ E)) :
      Integrable (fun x => Real.exp (-W x) *
        (∑ j, dir (dir W ((stdOrthonormalBasis ℝ E) j)) ((stdOrthonormalBasis ℝ E) i) x *
          dir f ((stdOrthonormalBasis ℝ E) j) x) * dir f ((stdOrthonormalBasis ℝ E) i) x) := by
    apply weighted_mul_integrable W _ _ hW.continuous
    · apply continuous_finsetSum
      intro j _
      exact ((dir_contDiff (n := 0) _ (dir_contDiff (n := 1) W hW _) _).continuous).mul
        (dir_contDiff (n := 0) f hf _).continuous
    · exact (dir_contDiff (n := 0) f hf _).continuous
    · exact dir_compact f hc _

  have curvature_integrable_and_sum (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
      (hf : ContDiff ℝ 1 f) (hc : HasCompactSupport f) :
      Integrable (fun x => Real.exp (-W x) *
        fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x)) ∧
      (∫ x, Real.exp (-W x) * fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x)) =
        ∑ i, ∫ x, Real.exp (-W x) *
          (∑ j, dir (dir W ((stdOrthonormalBasis ℝ E) j)) ((stdOrthonormalBasis ℝ E) i) x *
            dir f ((stdOrthonormalBasis ℝ E) j) x) * dir f ((stdOrthonormalBasis ℝ E) i) x := by
    let K := fun i x => Real.exp (-W x) *
      (∑ j, dir (dir W ((stdOrthonormalBasis ℝ E) j)) ((stdOrthonormalBasis ℝ E) i) x *
        dir f ((stdOrthonormalBasis ℝ E) j) x) * dir f ((stdOrthonormalBasis ℝ E) i) x
    have he : (fun x => Real.exp (-W x) * fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x)) =
        fun x => ∑ i, K i x := by
      funext x
      rw [← curvature_sum W f hW (hf.differentiable one_ne_zero)]
      rw [Finset.mul_sum]
      apply Finset.sum_congr rfl
      intro i _
      dsimp only [K]
      ring
    rw [he]
    exact ⟨integrable_finsetSum _ (fun i _ => curvature_component_integrable W f hW hf hc i),
      integral_finsetSum _ (fun i _ => curvature_component_integrable W f hW hf hc i)⟩

  have integrated_identity (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
      (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) :
      (∫ x, Real.exp (-W x) * (op W f x)^2) =
        (∫ x, Real.exp (-W x) * ∑ i, ‖gradient (dir f ((stdOrthonormalBasis ℝ E) i)) x‖^2) +
        ∫ x, Real.exp (-W x) * fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x) := by
    rw [(curvature_integrable_and_sum W f hW (smooth_finite f hf 1) hc).2,
      bochner_coordinate_integrals W f hW hf hc]
    congr 1
    have hfd (v : E) : ContDiff ℝ ∞ (dir f v) := dir_contDiff f (by simpa using hf) v
    have hI (i : Fin (Module.finrank ℝ E)) : Integrable (fun x =>
        Real.exp (-W x) * ‖gradient (dir f ((stdOrthonormalBasis ℝ E) i)) x‖^2) := by
      simpa only [real_inner_self_eq_norm_sq] using weighted_inner_integrable W
        (dir f ((stdOrthonormalBasis ℝ E) i)) (dir f ((stdOrthonormalBasis ℝ E) i)) hW.continuous
        (smooth_finite _ (hfd _) 1) (smooth_finite _ (hfd _) 1) (dir_compact f hc _)
    simp_rw [Finset.mul_sum]
    exact (integral_finsetSum _ (fun i _ => hI i)).symm

  have weighted_energy_lower_bound (W f : E → ℝ) (hW : ContDiff ℝ 2 W)
      (hf : ContDiff ℝ ∞ f) (hc : HasCompactSupport f) (m : ℝ)
      (hm : ∀ x v, m * ‖v‖^2 ≤ fderiv ℝ (fderiv ℝ W) x v v) :
      m * (∫ x, Real.exp (-W x) * ‖gradient f x‖^2) ≤
        ∫ x, Real.exp (-W x) * (op W f x)^2 := by
    have hI : Integrable (fun x => Real.exp (-W x) * ‖gradient f x‖^2) := by
      simpa only [real_inner_self_eq_norm_sq] using weighted_inner_integrable W f f hW.continuous
        (smooth_finite f hf 1) (smooth_finite f hf 1) hc
    have hC := (curvature_integrable_and_sum W f hW (smooth_finite f hf 1) hc).1
    have hbound : m * (∫ x, Real.exp (-W x) * ‖gradient f x‖^2) ≤
        ∫ x, Real.exp (-W x) * fderiv ℝ (fderiv ℝ W) x (gradient f x) (gradient f x) := by
      rw [← integral_const_mul]
      apply integral_mono (hI.const_mul m) hC
      intro x
      calc
        m * (Real.exp (-W x) * ‖gradient f x‖^2) =
          Real.exp (-W x) * (m * ‖gradient f x‖^2) := by ring
        _ ≤ _ := mul_le_mul_of_nonneg_left (hm x (gradient f x)) (Real.exp_nonneg _)
    rw [integrated_identity W f hW hf hc]
    exact hbound.trans (le_add_of_nonneg_left (integral_nonneg (fun x =>
      mul_nonneg (Real.exp_nonneg _) (Finset.sum_nonneg (fun i _ => sq_nonneg _)))))
  have hop := op_contDiff W f hW hf
  have hopc := op_compact W f (hW.of_le (by norm_num)) (smooth_finite f hf 2) hc
  have hL : Integrable (fun x => Real.exp (-W x) * (op W f x)^2) := by
    simpa only [pow_two, mul_assoc] using weighted_mul_integrable W (op W f) (op W f)
      hW.continuous hop.continuous hop.continuous hopc
  have hG : Integrable (fun x => Real.exp (-W x) * ‖gradient f x‖^2) := by
    simpa only [real_inner_self_eq_norm_sq] using weighted_inner_integrable W f f hW.continuous
      (smooth_finite f hf 1) (smooth_finite f hf 1) hc
  have hfd (v : E) : ContDiff ℝ ∞ (dir f v) := dir_contDiff f (by simpa using hf) v
  have hH : Integrable (fun x => Real.exp (-W x) *
      ∑ i, ‖gradient (dir f ((stdOrthonormalBasis ℝ E) i)) x‖^2) := by
    simp_rw [Finset.mul_sum]
    apply integrable_finsetSum
    intro i _
    simpa only [real_inner_self_eq_norm_sq] using weighted_inner_integrable W
      (dir f ((stdOrthonormalBasis ℝ E) i)) (dir f ((stdOrthonormalBasis ℝ E) i)) hW.continuous
      (smooth_finite _ (hfd _) 1) (smooth_finite _ (hfd _) 1) (dir_compact f hc _)
  exact ⟨hL,hG,hH,(curvature_integrable_and_sum W f hW (smooth_finite f hf 1) hc).1,
    integrated_identity W f hW hf hc,fun m hm => weighted_energy_lower_bound W f hW hf hc m hm⟩

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedBochner

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Selected Euclidean localization: d>=1, W:R^d->R C2 and f smooth compactly supported. Haar volume and actual derivatives are used; no global probability normalization is needed for this compact-test formula.	E is finite-dimensional real inner-product with Borel sigma algebra and canonical volume, including dimension zero.	generalization	Arbitrary finite-dimensional real inner-product spaces and zero dimension are allowed. The positive-dimensional Euclidean source is recovered by specialization; no full manifold result is claimed.
Full weighted manifold Reilly formula has boundary terms.	Direct whole-space Haar IBP for smooth compact tests, using W C2; all products are integrable.	API-limitation	Selected Euclidean compact-test localization kills boundary terms. This is not the full source theorem on a noncompact space.
Hessian Hilbert-Schmidt square term.	Explicit sum_i norm(gradient(D_i f))^2 using the standard orthonormal basis.	same	This is the coordinate expression in the identity; a separate abstract norm equivalence theorem is not claimed.
Compact-test integral manipulations.	Four explicit Integrable conclusions and a universally quantified real lower-curvature parameter.	source-implicit	All integrability and nonnegative-integral steps are proved. No global normalizer or spectral gap is assumed.

Compact-test Bochner identity and curvature-energy bound only. No curvature-to-Poincare theorem, closed weighted operator/core, dense range or resolvent construction, noncompact score extension, variance estimate, macro coercivity, process/mixing/error/cost or complete-paper result.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Disclosed abstract finite-dimensional and zero-dimensional extension. — Empty basis sums valid; extension explicitly disclosed.
scopes: Direct Euclidean localized component only, not the full manifold source theorem. — Enclosing-ball localization, zero-dimensional extension and absent abstract HS norm theorem disclosed.

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

Compact-test Bochner identity and curvature-energy bound only. No curvature-to-Poincare theorem, closed weighted operator/core, dense range or resolvent construction, noncompact score extension, variance estimate, macro coercivity, process/mixing/error/cost or complete-paper result.

Source and reuse

ASTIS parents called

AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt

Mathlib API called (external library)

integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
ContDiffAt.isSymmSndFDerivAt
InnerProductSpace.laplacian_eq_iteratedFDeriv_stdOrthonormalBasis
Continuous.integrable_of_hasCompactSupport
MeasureTheory.integral_finsetSum
MeasureTheory.integral_mono

Mathematical sources

Kolesnikov and Milman, weighted Reilly identity, Theorem 1.1 (1.3) — Euclidean compact-test localized version proved directly; not the full compact manifold/boundary theorem. Zero dimension is an additional extension.
Chen, Chewi, Lu and Zhang, PBPS v1 Appendix C.1 — Analytic prerequisite for the invoked conditional Poincare inequality. The Bochner edge does not discharge that inequality.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Genuine compact-test weighted Bochner identity and integrability
Local proof component; source adapter/review separate Actual curvature square-generator energy estimate

Bochner energy under the actual reflected conditional Gibbs kernel

Kolesnikov and Milman background; Chen, Chewi, Lu and Zhang PBPS consumer; ASTIS expanded mathematical proof.

arXiv:2609.06905v1 · Appendix C.1 conditional Poincare analytic prerequisite · faithful paraphrase

Complete source statement (ASTIS restatement)

PBPS Appendix C.1 applies a conditional Poincare inequality to the reflected conditional potential W_y(u)=V((y+u)/2)+norm(u-y)^2/(8 eta), whose lower curvature is (alpha+eta inverse)/4. The selected ASTIS prerequisite expands the missing analytic curvature route using the separately proved compact-test weighted Bochner identity: for the actual conditional law S_y, the square-generator integral equals the coordinate Hessian-square integral plus the D2W_y quadratic-form integral, and is at least (alpha+eta inverse)/4 times gradient energy. Global normalization and all tested integrability are established under the actual conditional law. This is a prerequisite expansion, not the Poincare or variance conclusion stated in C.1.

Paper setting: Euclidean d>=1, V C2, 0<alpha<=beta and actual Hessian bounds alpha I<=D2V<=beta I, eta>0 with beta eta<=1.
The Gibbs-Gaussian augmentation is (X,X+sqrt(eta)Z) with independent Gibbs X and standard Gaussian Z; S_y is the actual reflected conditional law. The Bochner prerequisite uses smooth compactly supported f; extending to the noncompact score test remains separate.

Bochner identity and curvature energy

\[\int(L_yf)^2\,dS_y=\int H_f\,dS_y+\int D^2W_y[\nabla f,\nabla f]\,dS_y\ge\frac{\alpha+\eta^{-1}}4\int\|\nabla f\|^2\,dS_y.\]

Read the formalized proofs

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

ASTIS mathematical exposition

Bochner energy under the actual reflected conditional Gibbs kernel

AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalBochner.conditional_bochner_energy · theorem · Teaching coverage

Statement

For the actual Gibbs-Gaussian augmentation there are common Markov kernels R,S: R disintegrates the swapped joint law and S_y is the reflected pushforward of R_y. Every S_y is normalized exp(-W_y) times volume, W_y(u)=V((y+u)/2)+norm(u-y)^2/(8 eta), with W_y C2 and finite strictly positive normalizer. For every smooth compactly supported f, the square of L_y f, squared gradient, coordinate Hessian-square term and genuine conditional curvature term are integrable under this same S_y. The exact normalized Bochner identity holds, and the square-generator energy bounds ((alpha+eta inverse)/4) times the gradient energy.

\[\int(L_yf)^2\,dS_y=\int H_f\,dS_y+\int D^2W_y[\nabla f,\nabla f]\,dS_y\ge\frac{\alpha+\eta^{-1}}4\int\|\nabla f\|^2\,dS_y.\]

All objects and hypotheses

E is finite-dimensional real inner-product with Borel structure and canonical volume, including dimension zero. V:E->R is C2.
Alpha,beta are finite nonnegative real parameters with explicit 0<alpha<=beta and genuine alpha/beta Hessian quadratic-form bounds on V.
Eta>0 and beta eta<=1 are retained from the ConditionalScoreDomain parent interface. The upper scale and upper curvature restrictions are not needed by the abstract Bochner identity; they are not removed from this conditional consumer.
Mu is volume tilted by -V; J is the law of (X,X+sqrt(eta)Z) from independent X~mu and standard Gaussian Z. The same constructed R,S work for every y and every smooth compactly supported f.
No probability, conditional density, global normalizer, integrability, integration-by-parts or Poincare premise is silently added; they are derived where claimed.

Mathematical proof

1. Keep one actual conditional kernel

Use ConditionalScoreDomain to obtain common R,S, their Markov properties, disintegration and reflected pushforward. For each y it provides exactly S_y=volume tilted by -W_y, W_y C2 and the genuine lower curvature (alpha+eta inverse)/4. Do not replace S_y by an unrelated measure with an assumed curvature inequality.

\[S_y=(x\mapsto2x-y)_\#R_y=Z_y^{-1}e^{-W_y(u)}\,du.\]

Corresponding Lean step

conditional_curvature_and_score_domain supplies the same existential witnesses; the consumer retains hR,hS,hcond,hSR,hSy and hcurv.

2. Prove the global normalizer is finite and positive

The shared compact-test identity alone cannot prove global weight integrability. Instead, S_y is already an actual probability measure and equals the tilt. If exp(-W_y) were not integrable, Mathlib tilt would be the zero measure, contradicting S_y(univ)=1. Thus the weight is integrable. Its pointwise strict positivity and nonzero volume make Z_y strictly positive. This rules out a totalized zero-measure normalization.

\[0<Z_y:=\int e^{-W_y(u)}\,du<\infty.\]

Corresponding Lean step

hweight contradicts tilted_of_not_integrable using the Markov probability mass. integral_exp_pos gives hZ before any normalized inequality is transferred.

3. Transport the full identity and every integral domain

Apply the actual shared Bochner theorem to W_y and f. It supplies all four weighted Integrable facts and the raw identity. The tilted-integrability equivalence transfers each fact to this same S_y. For every integrand g, the exact tilted integral equals Z_y inverse times the weighted volume integral, so multiplying the raw identity gives the normalized identity.

\[\int g\,dS_y=Z_y^{-1}\int e^{-W_y}g,\qquad\int(L_yf)^2\,dS_y=\int H_f\,dS_y+\int C_f\,dS_y.\]

Corresponding Lean step

integrated_bochner_identity is a genuine public shared theorem call. hI uses integrable_tilted_iff only after hweight; htilt uses integral_tilted and scalar integral algebra.

4. Use the actual conditional curvature constant

Apply the shared curvature-energy consequence to the inherited genuine Hessian lower bound, then multiply by positive Z_y inverse. This yields precisely (alpha+eta inverse)/4 times the S_y gradient energy. It remains a compact-test square-generator estimate. Constructing a weighted closed core/range or resolvent and extending to noncompact scores are still necessary before claiming conditional Poincare or score variance.

\[\frac{\alpha+\eta^{-1}}4\int\|\nabla f\|^2\,dS_y\le\int(L_yf)^2\,dS_y.\]

Corresponding Lean step

The final calc applies hbound to hcurv and uses inv_nonneg.mpr hZ.le; no variance or spectral-gap theorem is assumed.

Lean statement · conditional_bochner_energy

Common actual kernels, density and finite positive normalizer precede the universal compact-test integrability, identity and exact energy inequality.

theorem conditional_bochner_energy {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ Integrable (fun u => Real.exp (-W y u)) ∧
        0 < (∫ u, Real.exp (-W y u)) ∧
        ∀ f : E → ℝ, ContDiff ℝ ∞ f → HasCompactSupport f →
          let L := fun u => Laplacian.laplacian f u - inner ℝ (gradient (W y) u) (gradient f u)
          let H := fun u => ∑ i, ‖gradient (fun z => fderiv ℝ f z ((stdOrthonormalBasis ℝ E) i)) u‖^2
          let C := fun u => fderiv ℝ (fderiv ℝ (W y)) u (gradient f u) (gradient f u)
          Integrable (fun u => (L u)^2) (S y) ∧
          Integrable (fun u => ‖gradient f u‖^2) (S y) ∧
          Integrable H (S y) ∧ Integrable C (S y) ∧
          (∫ u, (L u)^2 ∂S y) = (∫ u, H u ∂S y) + ∫ u, C u ∂S y ∧
          (((α:ℝ)+1/η)/4) * (∫ u, ‖gradient f u‖^2 ∂S y) ≤ ∫ u, (L u)^2 ∂S y

Exact module and namespace context

Lean proof · conditional_bochner_energy

Reuse the genuine conditional construction and the full shared Bochner identity; derive the normalizer from actual probability and transfer all integrals exactly.

theorem conditional_bochner_energy {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ Integrable (fun u => Real.exp (-W y u)) ∧
        0 < (∫ u, Real.exp (-W y u)) ∧
        ∀ f : E → ℝ, ContDiff ℝ ∞ f → HasCompactSupport f →
          let L := fun u => Laplacian.laplacian f u - inner ℝ (gradient (W y) u) (gradient f u)
          let H := fun u => ∑ i, ‖gradient (fun z => fderiv ℝ f z ((stdOrthonormalBasis ℝ E) i)) u‖^2
          let C := fun u => fderiv ℝ (fderiv ℝ (W y)) u (gradient f u) (gradient f u)
          Integrable (fun u => (L u)^2) (S y) ∧
          Integrable (fun u => ‖gradient f u‖^2) (S y) ∧
          Integrable H (S y) ∧ Integrable C (S y) ∧
          (∫ u, (L u)^2 ∂S y) = (∫ u, H u ∂S y) + ∫ u, C u ∂S y ∧
          (((α:ℝ)+1/η)/4) * (∫ u, ‖gradient f u‖^2 ∂S y) ≤ ∫ u, (L u)^2 ∂S y := by
  obtain ⟨R,S,hR,hS,hcond,hSR,hfiber⟩ :=
    ConditionalScoreDomain.conditional_curvature_and_score_domain hα hαβ hV hH hη hβη
  let _ : IsMarkovKernel S := hS
  dsimp only
  refine ⟨R,S,hR,hS,hcond,hSR,?_⟩
  intro y
  let W := fun u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
  obtain ⟨hSy,hWC,_,_,hcurv,_,_⟩ := hfiber y
  change S y = (volume : Measure E).tilted (fun u => -W u) at hSy
  have hweight : Integrable (fun u => Real.exp (-W u)) := by
    by_contra hn
    have hz : S y = 0 := hSy.trans (tilted_of_not_integrable hn)
    have hu := measure_univ (μ := S y)
    rw [hz] at hu
    norm_num at hu
  have hZ : 0 < ∫ u, Real.exp (-W u) := integral_exp_pos hweight
  refine ⟨hSy,hWC,hweight,hZ,?_⟩
  intro f hf hc
  obtain ⟨hL,hG,hHess,hC,hidentity,hbound⟩ :=
    WeightedBochner.integrated_bochner_identity W f hWC hf hc
  have hI (g : E → ℝ) (hg : Integrable (fun u => Real.exp (-W u)*g u)) :
      Integrable g (S y) := by
    rw [hSy]
    exact (integrable_tilted_iff hweight g).mpr (by simpa only [smul_eq_mul] using hg)
  have htilt (g : E → ℝ) : (∫ u, g u ∂S y) =
      (∫ u, Real.exp (-W u))⁻¹ * (∫ u, Real.exp (-W u)*g u) := by
    rw [hSy, integral_tilted]
    rw [← integral_const_mul]
    apply integral_congr_ae
    filter_upwards [] with u
    change (Real.exp (-W u) / (∫ z, Real.exp (-W z))) • g u =
      (∫ z, Real.exp (-W z))⁻¹ * (Real.exp (-W u) * g u)
    simp only [smul_eq_mul, div_eq_mul_inv]
    ring
  refine ⟨hI _ hL,hI _ hG,hI _ hHess,hI _ hC,?_,?_⟩
  · rw [htilt,htilt,htilt,hidentity,mul_add]
  · rw [htilt,htilt]
    calc
      _ = (∫ u, Real.exp (-W u))⁻¹ *
          ((((α:ℝ)+1/η)/4) * (∫ u, Real.exp (-W u)*‖gradient f u‖^2)) := by ring
      _ ≤ _ := mul_le_mul_of_nonneg_left (hbound _ hcurv) (inv_nonneg.mpr hZ.le)

end AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalBochner

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Paper setting: Euclidean d>=1, V C2, 0<alpha<=beta and actual Hessian bounds alpha I<=D2V<=beta I, eta>0 with beta eta<=1.	E is finite-dimensional real inner-product with Borel structure and canonical volume, including dimension zero. V:E->R is C2.	generalization	Arbitrary finite-dimensional real inner-product spaces and zero dimension are allowed. The positive-dimensional Euclidean source is recovered by specialization; no full manifold result is claimed.
Appendix C.1 invokes conditional Poincare.	Actual conditional Bochner identity and square-generator energy lower bound on smooth compact tests.	API-limitation	This is an explicitly labeled analytic prerequisite, not proof of the source Poincare or variance conclusion.
Source strong-convexity/smoothness and step-size setting.	Explicit 0<alpha<=beta, C2 V, genuine Hessian bounds and beta eta<=1 retained via ConditionalScoreDomain.	same	The Bochner argument itself needs only the lower curvature; inherited upper restrictions are disclosed as parent-interface restrictions.
Actual conditional normalized Gibbs law.	One pair of common Markov R,S with disintegration and reflected pushforward, finite strictly positive normalizer, four integrability conclusions and normalized identity for every y.	source-implicit	Probability of the constructed S rules out the zero tilt before the exact normalized integral transfer.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Disclosed abstract finite-dimensional and zero-dimensional extension. — Explicitly disclosed domain extension.
scopes: Authored analytic prerequisite, not an explicit PBPS C.1 theorem or proof of its Poincare conclusion. — No closed generator, noncompact score extension, variance or macro coercivity; missing core/range/resolvent explicit.
assumptions: Inherited beta eta<=1 preserved although abstract Bochner does not need it. — Upper scale restriction retained from parent interface and distinguished from abstract Bochner needs.

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

Compact-test Bochner identity and curvature-energy bound only. No curvature-to-Poincare theorem, closed weighted operator/core, dense range or resolvent construction, noncompact score extension, variance estimate, macro coercivity, process/mixing/error/cost or complete-paper result.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

MeasureTheory.tilted_of_not_integrable
MeasureTheory.integral_exp_pos
MeasureTheory.integrable_tilted_iff
MeasureTheory.integral_tilted

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS v1 Appendix C.1 — Analytic prerequisite for the invoked conditional Poincare inequality. The Bochner edge does not discharge that inequality.
Kolesnikov and Milman, weighted Reilly identity, Theorem 1.1 (1.3) — Euclidean compact-test localized version proved directly; not the full compact manifold/boundary theorem. Zero dimension is an additional extension.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Genuine compact-test weighted Bochner identity and integrability
TODO — not closed by these contributions Actual curvature square-generator energy estimate

A dense closable gradient for a genuine Gibbs measure

Kolesnikov and Milman background; Chen, Chewi, Lu and Zhang PBPS consumer; ASTIS expanded mathematical proof.

arXiv:1310.2526v7 · section2.5 weighted Sobolev construction; direct Euclidean prerequisite · faithful paraphrase

Complete source statement (ASTIS restatement)

Section2.5 of Kolesnikov-Milman describes weighted H1 and H10 by completing smooth test functions in the graph norm on its compact weighted manifold. The selected ASTIS analytic prerequisite is a directly proved Euclidean full-space construction, not that source spectral conclusion: for d>=1, C1 W with integrable exp(-W), the graph of smooth compact f and their genuine gradients determines a densely defined closable scalar-to-vector L2 operator under the normalized Gibbs measure. Its graph closure remains single-valued. This explicit construction is needed before a weighted resolvent route to PBPS conditional Poincare. No manifold boundary theorem or D*D operator-core assertion is included.

Selected full-space prerequisite: d>=1, W:R^d->R C1 and exp(-W) integrable, actual normalized Gibbs measure and smooth compact tests.
The background discussion is on compact weighted manifolds with smoother potential; full-space construction and weaker C1 regularity are explicit scope changes, not silent specialization.

Actual gradient graph and closure

\[\operatorname{Graph}D=\{([f],[\nabla f]):f\in C_c^\infty(E)\},\qquad \overline{\operatorname{Dom}D}=L^2(\mu),\qquad D\ \text{is closable}.\]

Read the formalized proofs

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

ASTIS mathematical exposition

A dense closable gradient for a genuine Gibbs measure

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradient.compact_gradient_closable · theorem · Teaching coverage

Statement

Let E be a finite-dimensional real inner-product Borel space, W:E->R be C1, and exp(-W) be integrable against canonical volume. Set mu=Z^{-1}exp(-W)dx. There is a partial real-linear operator D from scalar L2(mu) to vector L2(mu) whose domain is dense, whose graph is closable, and whose closure is closed. For any scalar L2 class u and vector L2 class v, (u,v) belongs to its graph exactly when there exists a smooth compactly supported scalar f with u=f and v=gradient f almost everywhere. This equivalence fixes the actual differential operator and its exact initial domain.

\[\operatorname{Graph}D=\{([f],[\nabla f]):f\in C_c^\infty(E)\},\qquad \overline{\operatorname{Dom}D}=L^2(\mu),\qquad D\ \text{is closable}.\]

All objects and hypotheses

E is finite-dimensional real inner-product with its Borel sigma algebra and canonical volume; dimension zero is included.
W is C1 and the actual positive function exp(-W) is volume-integrable. No curvature, Poincare, gradient coercivity or operator-closure assumption is made.
Mu is the actual normalized exponential tilt. Positive exponential and nonzero canonical volume give a finite positive normalizer and hence a probability; the zero-tilt fallback is not used.
Lp objects are equivalence classes modulo mu-almost-everywhere equality. D is a partial linear map; closable means the topological closure of its graph is still the graph of a partial linear map, not that D is bounded or everywhere defined.
Background weighted H1 completion in Kolesnikov-Milman section2.5 is on compact manifolds. This directly proved full-space Euclidean construction is a selected analytic prerequisite, not that source spectral theorem.

Mathematical proof

1. Establish weighted directional integration by parts

Take F=exp(-W)g with g C1 compact and f C1. F, its directional derivative, and all three products needed by Haar IBP are continuous and compactly supported. Apply the actual volume theorem and expand the genuine derivative of F. The drift term has the minus sign inside the derivative of exp(-W).

\[\int e^{-W}gD_vf=-\int e^{-W}(D_vg-gD_vW)f.\]

Corresponding Lean step

The local weighted_directional proof derives all three Integrable premises; it never applies a Haar theorem to mu.

2. Construct the weighted negative divergence test

For a C1 compact vector field P, project onto each orthonormal basis vector and sum the directional formula. This constructs a continuous compact scalar q, namely minus divergence P plus gradient W dot P. The public theorem does not separately export a divergence formula; this is a proved internal construction.

\[q=-\operatorname{div}P+\langle\nabla W,P\rangle,\qquad \int\langle\nabla f,P\rangle\,d\mu=\int fq\,d\mu.\]

Corresponding Lean step

raw_vector_ibp proves the finite sum and integrability; tilted_vector_ibp transfers the exact identity and proves q in L2(mu).

3. Build the exact smooth gradient graph

Pairs of scalar and vector L2 classes belong to the graph when they have representatives f and gradient f for a smooth compact f. Genuine derivative addition and scalar multiplication prove this set is a linear submodule. This defines a relation before any representative-dependent operator is selected.

\[G=\{(u,v):\exists f\in C_c^\infty(E),\ u=f\ \mu\text{-a.e.},\ v=\nabla f\ \mu\text{-a.e.}\}.\]

Corresponding Lean step

The local smoothGradientGraph submodule proves zero, addition and scalar closure using Lp coeFn almost-everywhere identities and genuine fderiv rules.

4. Extend the integration-by-parts pairing to the graph closure

For fixed smooth compact P and its scalar q, the equality between the two L2 inner products is a closed condition on a scalar/vector pair. Every graph pair satisfies it by the proved weighted IBP and representative identities, so every pair in the topological closure also satisfies it.

\[(u,v)\in\overline G\ \Longrightarrow\ \langle v,[P]\rangle_{L^2(\mu;E)}=\langle u,[q]\rangle_{L^2(\mu)}.\]

Corresponding Lean step

closure_unique uses continuous L2 inner products and closure_minimal; no closedness of the derivative is assumed.

5. Use vector-valued smooth density to prove single-valuedness

If (0,v) belongs to the closed graph relation, the pairing identity makes v orthogonal to every smooth compact vector test. The pinned Mathlib theorem makes these tests dense in vector L2 for every measure finite on compacts, including this probability. A continuous zero pairing therefore vanishes on all vector L2; pairing with v gives v=0.

\[(0,v)\in\overline G\ \Longrightarrow\ \forall P\in C_c^\infty(E;E),\ \langle v,[P]\rangle=0\ \Longrightarrow\ \|v\|_2^2=0.\]

Corresponding Lean step

Lp.dense_hasCompactSupport_contDiff, Dense.induction and inner_self_eq_zero prove the zero-fiber statement. This implies both G and its closure are single-valued.

6. Construct the partial operator and its dense closed extension

Only after single-valuedness is proved does toLinearPMap construct D from G. Its graph is proved equal to G, and the analogous construction from the closed relation proves IsClosable. Scalar smooth density and L2 integrability of each compact continuous gradient prove the initial domain dense. Mathlib then supplies the closed graph of D.closure.

\[\operatorname{Graph}D=G,\qquad \operatorname{Graph}\overline D=\overline G,\qquad \overline{\operatorname{Dom}D}=L^2(\mu).\]

Corresponding Lean step

Submodule.toLinearPMap_graph_eq is used with the proved zero-fiber witnesses; IsClosable.closure_isClosed is applied only after closability is derived.

Lean statement · compact_gradient_closable

Exact graph equivalence, dense domain, IsClosable and closed closure, for the actual normalized Gibbs L2 spaces.

theorem compact_gradient_closable (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∃ D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ,
      Dense (D.domain : Set (Lp ℝ 2 μ)) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
      ∀ (u : Lp ℝ 2 μ) (v : Lp E 2 μ), (u,v) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[μ] f ∧ v =ᵐ[μ] gradient f

Exact module and namespace context

Lean proof · compact_gradient_closable

Every helper and the graph submodule are local haves/lets in the public proof; its folded Lean contains the complete argument.

theorem compact_gradient_closable (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∃ D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ,
      Dense (D.domain : Set (Lp ℝ 2 μ)) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
      ∀ (u : Lp ℝ 2 μ) (v : Lp E 2 μ), (u,v) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[μ] f ∧ v =ᵐ[μ] gradient f := by
  have weighted_directional (W f g : E → ℝ) (hW : ContDiff ℝ 1 W)
      (hf : ContDiff ℝ 1 f) (hg : ContDiff ℝ 1 g)
      (hgc : HasCompactSupport g) (v : E) :
      (∫ x, Real.exp (-W x) * g x * fderiv ℝ f x v) =
        - ∫ x, Real.exp (-W x) *
          (fderiv ℝ g x v - g x * fderiv ℝ W x v) * f x := by
    let F := fun x => Real.exp (-W x) * g x
    have hF : ContDiff ℝ 1 F := hW.neg.exp.mul hg
    have hFc : HasCompactSupport F := hgc.mul_left
    have hDF : Continuous (fun x => fderiv ℝ F x v) :=
      (hF.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
    have hDf : Continuous (fun x => fderiv ℝ f x v) :=
      (hf.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
    have hDFc : HasCompactSupport (fun x => fderiv ℝ F x v) := by
      refine HasCompactSupport.of_support_subset_isCompact hFc.isCompact ?_
      intro x hx
      by_contra hn
      exact hx (by simp [fderiv_of_notMem_tsupport ℝ hn])
    have h1 : Integrable (fun x => fderiv ℝ F x v * f x) :=
      (hDF.mul hf.continuous).integrable_of_hasCompactSupport hDFc.mul_right
    have h2 : Integrable (fun x => F x * fderiv ℝ f x v) :=
      (hF.continuous.mul hDf).integrable_of_hasCompactSupport hFc.mul_right
    have h3 : Integrable (fun x => F x * f x) :=
      (hF.continuous.mul hf.continuous).integrable_of_hasCompactSupport hFc.mul_right
    have hi := integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable h1 h2 h3
      (fun x _ => hF.differentiable one_ne_zero x)
      (fun x _ => hf.differentiable one_ne_zero x)
    change (∫ x, F x * fderiv ℝ f x v) = _
    rw [hi]
    congr 1
    apply integral_congr_ae
    filter_upwards [] with x
    have hd := ((hW.differentiable one_ne_zero x).hasFDerivAt.neg.exp).mul
      (hg.differentiable one_ne_zero x).hasFDerivAt
    have he : fderiv ℝ F x v = Real.exp (-W x) *
        (fderiv ℝ g x v - g x * fderiv ℝ W x v) := by
      rw [show fderiv ℝ F x = _ from hd.fderiv]
      simp only [add_apply, smul_apply, neg_apply, smul_eq_mul, Pi.neg_apply]
      ring
    rw [he]

  have gradient_cont (f : E → ℝ) (hf : ContDiff ℝ 1 f) : Continuous (gradient f) := by
    exact (toDual ℝ E).symm.continuous.comp (hf.fderiv_right (m := 0) (by norm_num)).continuous

  have gradient_compact (f : E → ℝ) (hc : HasCompactSupport f) :
      HasCompactSupport (gradient f) := by
    refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
    intro x hx
    by_contra hn
    exact hx (by simp [gradient, fderiv_of_notMem_tsupport ℝ hn])

  have raw_vector_ibp (W : E → ℝ) (hW : ContDiff ℝ 1 W)
      (P : E → E) (hP : ContDiff ℝ 1 P) (hPc : HasCompactSupport P) :
      ∃ q : E → ℝ, Continuous q ∧ HasCompactSupport q ∧
        ∀ (f : E → ℝ), ContDiff ℝ 1 f →
          (∫ x, Real.exp (-W x) * inner ℝ (gradient f x) (P x)) =
            ∫ x, Real.exp (-W x) * f x * q x := by
    let b := stdOrthonormalBasis ℝ E
    let g := fun i x => inner ℝ (P x) (b i)
    have hg (i) : ContDiff ℝ 1 (g i) := hP.inner ℝ contDiff_const
    have hgc (i) : HasCompactSupport (g i) := by
      refine HasCompactSupport.of_support_subset_isCompact hPc.isCompact ?_
      intro x hx
      by_contra hn
      exact hx (by simp [g, image_eq_zero_of_notMem_tsupport hn])
    have hdg (i) : Continuous (fun x => fderiv ℝ (g i) x (b i)) :=
      ((hg i).fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
    have hdgc (i) : HasCompactSupport (fun x => fderiv ℝ (g i) x (b i)) := by
      refine HasCompactSupport.of_support_subset_isCompact (hgc i).isCompact ?_
      intro x hx
      by_contra hn
      exact hx (by simp [fderiv_of_notMem_tsupport ℝ hn])
    have hdW (i) : Continuous (fun x => fderiv ℝ W x (b i)) :=
      (hW.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const
    let Q := fun i x => -(fderiv ℝ (g i) x (b i) - g i x * fderiv ℝ W x (b i))
    have hQ (i) : Continuous (Q i) := ((hdg i).sub ((hg i).continuous.mul (hdW i))).neg
    have hQc (i) : HasCompactSupport (Q i) := ((hdgc i).sub (hgc i).mul_right).neg
    let q := fun x => ∑ i, Q i x
    refine ⟨q,continuous_finsetSum _ (fun i _ => hQ i),?_,?_⟩
    · have hfun : q = ∑ i, Q i := by funext x; simp [q]
      rw [hfun]
      exact HasCompactSupport.finset_sum (s := Finset.univ) (f := Q) fun i _ => hQc i
    intro f hf
    have hA (i) : Integrable (fun x => Real.exp (-W x) * g i x * fderiv ℝ f x (b i)) := by
      apply (((hW.neg.exp).continuous.mul (hg i).continuous).mul
        ((hf.fderiv_right (m := 0) (by norm_num)).continuous.clm_apply continuous_const)).integrable_of_hasCompactSupport
      exact (hgc i).mul_left.mul_right
    have hB (i) : Integrable (fun x => Real.exp (-W x) * f x * Q i x) :=
      (((hW.neg.exp).continuous.mul hf.continuous).mul (hQ i)).integrable_of_hasCompactSupport (hQc i).mul_left
    have hpoint (x : E) : inner ℝ (gradient f x) (P x) =
        ∑ i, g i x * fderiv ℝ f x (b i) := by
      rw [← b.sum_inner_mul_inner (gradient f x) (P x)]
      apply Finset.sum_congr rfl
      intro i _
      rw [AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt (hf.differentiable one_ne_zero x)]
      simp only [g, real_inner_comm (b i) (P x), mul_comm]
    simp_rw [hpoint, Finset.mul_sum]
    rw [integral_finsetSum _ (fun i _ => by simpa [mul_assoc] using hA i)]
    change _ = ∫ x, Real.exp (-W x) * f x * ∑ i, Q i x
    simp_rw [Finset.mul_sum]
    rw [integral_finsetSum _ (fun i _ => hB i)]
    apply Finset.sum_congr rfl
    intro i _
    rw [show (∫ x, Real.exp (-W x) * (g i x * fderiv ℝ f x (b i))) =
      ∫ x, Real.exp (-W x) * g i x * fderiv ℝ f x (b i) by congr 1; funext x; ring]
    rw [weighted_directional W f (g i) hW hf (hg i) (hgc i) (b i), ← integral_neg]
    apply integral_congr_ae
    filter_upwards [] with x
    dsimp only [Q]
    ring

  have tilted_vector_ibp (W : E → ℝ) (hW : ContDiff ℝ 1 W)
      (hI : Integrable (fun x => Real.exp (-W x)))
      (P : E → E) (hP : ContDiff ℝ 1 P) (hPc : HasCompactSupport P) :
      let μ := (volume : Measure E).tilted (fun x => -W x)
      ∃ q : E → ℝ, Continuous q ∧ HasCompactSupport q ∧ MemLp q 2 μ ∧
        ∀ (f : E → ℝ), ContDiff ℝ 1 f →
          (∫ x, inner ℝ (gradient f x) (P x) ∂μ) = ∫ x, f x * q x ∂μ := by
    let μ := (volume : Measure E).tilted (fun x => -W x)
    let : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hI
    obtain ⟨q,hq,hqc,hi⟩ := raw_vector_ibp W hW P hP hPc
    refine ⟨q,hq,hqc,hq.memLp_of_hasCompactSupport hqc,?_⟩
    have ht (g : E → ℝ) : (∫ x, g x ∂μ) =
        (∫ x, Real.exp (-W x))⁻¹ * ∫ x, Real.exp (-W x) * g x := by
      rw [show μ = (volume : Measure E).tilted (fun x => -W x) from rfl, integral_tilted]
      rw [← integral_const_mul]
      apply integral_congr_ae
      filter_upwards [] with x
      change (Real.exp (-W x) / (∫ z, Real.exp (-W z))) • g x =
        (∫ z, Real.exp (-W z))⁻¹ * (Real.exp (-W x) * g x)
      simp only [smul_eq_mul, div_eq_mul_inv]
      ring
    intro f hf
    rw [ht,ht,hi f hf]
    congr 1
    apply integral_congr_ae
    filter_upwards [] with x
    ring

  let smoothGradientGraph (μ : Measure E) : Submodule ℝ (Lp ℝ 2 μ × Lp E 2 μ) :=
    {
      carrier := {u | ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
        u.1 =ᵐ[μ] f ∧ u.2 =ᵐ[μ] gradient f}
      zero_mem' := by
        refine ⟨0,contDiff_const,HasCompactSupport.zero,?_,?_⟩
        · exact Lp.coeFn_zero _ _ _
        · have hz : gradient (0 : E → ℝ) = (0 : E → E) := by
            funext x
            exact gradient_fun_const x (0 : ℝ)
          change (0 : Lp E 2 μ) =ᵐ[μ] gradient (0 : E → ℝ)
          rw [hz]
          exact Lp.coeFn_zero E 2 μ
      add_mem' := by
        rintro u v ⟨f,hf,hfc,hfu,hfg⟩ ⟨g,hg,hgc,hgu,hgg⟩
        refine ⟨f+g,hf.add hg,hfc.add hgc,?_,?_⟩
        · exact (Lp.coeFn_add u.1 v.1).trans (hfu.add hgu)
        have he : gradient (f+g) = gradient f + gradient g := by
          funext x
          simp only [gradient, fderiv_add (hf.differentiable (by simp) x)
            (hg.differentiable (by simp) x), map_add, Pi.add_apply]
        rw [he]
        exact (Lp.coeFn_add u.2 v.2).trans (hfg.add hgg)
      smul_mem' := by
        rintro a u ⟨f,hf,hfc,hfu,hfg⟩
        refine ⟨a • f,contDiff_const.smul hf,hfc.smul_left,?_,?_⟩
        · exact (Lp.coeFn_smul a u.1).trans (hfu.const_smul a)
        have he : gradient (a • f) = a • gradient f := by
          funext x
          simp only [gradient, fderiv_const_smul (hf.differentiable (by simp) x),
            map_smul, Pi.smul_apply]
        rw [he]
        exact (Lp.coeFn_smul a u.2).trans (hfg.const_smul a)
    }

  have closure_unique (W : E → ℝ) (hW : ContDiff ℝ 1 W)
      (hI : Integrable (fun x => Real.exp (-W x))) :
      let μ := (volume : Measure E).tilted (fun x => -W x)
      ∀ u ∈ (smoothGradientGraph μ).topologicalClosure, u.1 = 0 → u.2 = 0 := by
    let μ := (volume : Measure E).tilted (fun x => -W x)
    let : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hI
    dsimp only
    intro u hu huz
    have hd : Dense {v : Lp E 2 μ | ∃ P : E → E,
        v =ᵐ[μ] P ∧ HasCompactSupport P ∧ ContDiff ℝ ∞ P} :=
      Lp.dense_hasCompactSupport_contDiff (by norm_num)
    have horth : ∀ v ∈ {v : Lp E 2 μ | ∃ P : E → E,
        v =ᵐ[μ] P ∧ HasCompactSupport P ∧ ContDiff ℝ ∞ P}, inner ℝ u.2 v = 0 := by
      rintro v ⟨P,hv,hPc,hP⟩
      obtain ⟨q,hq,hqc,hqL,hqi⟩ := tilted_vector_ibp W hW hI P (contDiff_infty.mp hP 1) hPc
      let qv : Lp ℝ 2 μ := hqL.toLp q
      have hc : IsClosed {z : Lp ℝ 2 μ × Lp E 2 μ |
          inner ℝ z.2 v = inner ℝ z.1 qv} :=
        isClosed_eq (continuous_snd.inner continuous_const) (continuous_fst.inner continuous_const)
      have hs : (smoothGradientGraph μ : Set (Lp ℝ 2 μ × Lp E 2 μ)) ⊆
          {z | inner ℝ z.2 v = inner ℝ z.1 qv} := by
        rintro z ⟨f,hf,hfc,hfz,hfg⟩
        change inner ℝ z.2 v = inner ℝ z.1 qv
        calc
          inner ℝ z.2 v = ∫ x, inner ℝ (gradient f x) (P x) ∂μ := by
            rw [L2.inner_def]
            apply integral_congr_ae
            filter_upwards [hfg,hv] with x hx hy
            rw [hx,hy]
          _ = ∫ x, f x * q x ∂μ := hqi f (contDiff_infty.mp hf 1)
          _ = inner ℝ z.1 qv := by
            rw [L2.inner_def]
            apply integral_congr_ae
            filter_upwards [hfz,hqL.coeFn_toLp] with x hx hy
            rw [hx,show qv x = q x from hy]
            simp [mul_comm]
      have he := (closure_minimal hs hc) hu
      change inner ℝ u.2 v = inner ℝ u.1 qv at he
      simpa [huz] using he
    have hall : ∀ v : Lp E 2 μ, inner ℝ u.2 v = 0 :=
      hd.induction horth (isClosed_eq (continuous_const.inner continuous_id) continuous_const)
    exact inner_self_eq_zero.mp (hall u.2)

  let μ := (volume : Measure E).tilted (fun x => -W x)
  let : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hI
  let G := smoothGradientGraph μ
  have hu : ∀ z ∈ G.topologicalClosure, z.1 = 0 → z.2 = 0 := closure_unique W hW hI
  let D := G.toLinearPMap
  have hgraph : D.graph = G :=
    G.toLinearPMap_graph_eq (fun z hz hzero => hu z (subset_closure hz) hzero)
  have hc : D.IsClosable := by
    refine ⟨G.topologicalClosure.toLinearPMap,?_⟩
    rw [hgraph]
    exact (G.topologicalClosure.toLinearPMap_graph_eq hu).symm
  have hd : Dense {u : Lp ℝ 2 μ | ∃ f : E → ℝ,
      u =ᵐ[μ] f ∧ HasCompactSupport f ∧ ContDiff ℝ ∞ f} :=
    Lp.dense_hasCompactSupport_contDiff (by norm_num)
  have hD : Dense (D.domain : Set (Lp ℝ 2 μ)) := by
    apply hd.mono
    rintro u ⟨f,huf,hfc,hf⟩
    have hgf : MemLp (gradient f) 2 μ :=
      (gradient_cont f (contDiff_infty.mp hf 1)).memLp_of_hasCompactSupport (gradient_compact f hfc)
    change u ∈ G.map (LinearMap.fst ℝ (Lp ℝ 2 μ) (Lp E 2 μ))
    exact ⟨(u,hgf.toLp (gradient f)),⟨f,hf,hfc,huf,hgf.coeFn_toLp⟩,rfl⟩
  refine ⟨D,hD,hc,hc.closure_isClosed,?_⟩
  intro u v
  rw [hgraph]
  rfl

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradient

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Selected full-space prerequisite: d>=1, W:R^d->R C1 and exp(-W) integrable, actual normalized Gibbs measure and smooth compact tests.	E is finite-dimensional real inner-product with its Borel sigma algebra and canonical volume; dimension zero is included.	generalization	Finite-dimensional real inner-product Borel E includes dimension zero. This extends the source positive-dimensional Euclidean setting and is explicitly disclosed.
Background section2.5 compact-manifold weighted Sobolev completion and spectral discussion.	Direct Euclidean full-space graph construction under C1 W and actual exponential integrability.	generalization	No claim of automatic compact-manifold specialization or spectral conclusion; only the analytic prerequisite is supplied, with weaker C1 regularity explicitly stated.
Sobolev gradient completion.	Exact L2 almost-everywhere gradient graph equivalence, dense partial-linear domain, IsClosable and closed closure.	source-implicit	Representative independence and closed-graph construction are derived from weighted vector IBP and smooth density, never assumed.

Dense compact-test gradient operator, exact genuine-gradient graph and closed extension only. No operator core of D*D, dense mean-zero generator range, resolvent regularity, Poincare, noncompact score variance, macro coercivity, process/mixing/error/cost or complete-paper result.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Explicit abstract finite-dimensional and zero-dimensional extension. — Canonical volume is nonzero in zero dimension; vector space and gradients then trivial. Extension disclosed.
scopes: Direct full-space C1 prerequisite, not compact-manifold Sobolev/spectral source theorem verbatim. — No self-adjoint generator, D*D core, spectral gap, resolvent or PI. Full-space/C1 changes disclosed.

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

Dense compact-test gradient operator, exact genuine-gradient graph and closed extension only. No operator core of D*D, dense mean-zero generator range, resolvent regularity, Poincare, noncompact score variance, macro coercivity, process/mixing/error/cost or complete-paper result.

Source and reuse

ASTIS parents called

AutoSamplingTheory.TechnicalLemmas.Analysis.Calculus.Gradient.fderiv_apply_eq_inner_gradient_of_differentiableAt

Mathlib API called (external library)

integral_mul_fderiv_eq_neg_fderiv_mul_of_integrable
MeasureTheory.integral_tilted
MeasureTheory.Lp.dense_hasCompactSupport_contDiff
Dense.induction
Submodule.toLinearPMap_graph_eq
LinearPMap.IsClosable.closure_isClosed

Mathematical sources

Kolesnikov and Milman, section2.5 weighted Sobolev completion — Direct Euclidean full-space analytic prerequisite; compact manifold spectral conclusions are outside this result.
Chen, Chewi, Lu and Zhang, PBPS Appendix C.1 — Weighted-space construction needed by the conditional Poincare route; does not prove PI.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Exact genuine smooth compact gradient graph
TODO — not closed by these contributions Dense domain and single-valued closed graph extension

The actual conditional PBPS gradient is densely defined and closable

Kolesnikov and Milman background; Chen, Chewi, Lu and Zhang PBPS consumer; ASTIS expanded mathematical proof.

arXiv:2609.06905v1 · Appendix C.1 weighted-space analytic prerequisite · faithful paraphrase

Complete source statement (ASTIS restatement)

PBPS Appendix C.1 uses conditional Poincare for the actual reflected Gibbs-Gaussian conditional law. The selected ASTIS prerequisite constructs the actual weighted gradient domain and closure on that law: with W_y(u)=V((y+u)/2)+norm(u-y)^2/(8eta), S_y is normalized exp(-W_y) volume, and the smooth compact gradient graph in L2(S_y) is densely defined and closable. The original source PI/variance application is not proved by this prerequisite. Kernels, normalization and potential regularity must be those of the actual augmentation.

Paper setting: Euclidean d>=1, V C2, 0<alpha<=beta, genuine alpha/beta Hessian bounds, eta>0 and beta eta<=1.
The actual joint law is (X,X+sqrt(eta)Z) with independent Gibbs X and standard Gaussian Z; the reflected conditional law is S_y.

Actual gradient graph and closure

\[S_y(du)=Z_y^{-1}e^{-W_y(u)}du,\quad W_y(u)=V((y+u)/2)+\frac{\|u-y\|^2}{8\eta},\quad \operatorname{Graph}D_y=\{([f],[\nabla f]):f\in C_c^\infty(E)\}.\]

Read the formalized proofs

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

ASTIS mathematical exposition

The actual conditional PBPS gradient is densely defined and closable

AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradient.conditional_gradient_closable · theorem · Teaching coverage

Statement

For the actual Gibbs-Gaussian augmentation, choose common Markov kernels R,S with R disintegrating the swapped joint law and S_y the reflected pushforward of R_y. For every y, S_y is normalized exp(-W_y) volume, W_y(u)=V((y+u)/2)+norm(u-y)^2/(8eta), with W_y C2 and finite positive normalizer. On the scalar and vector L2 spaces of this same S_y there exists a partial real-linear gradient D_y with dense domain, IsClosable and closed closure. Its graph consists exactly of pairs represented by f and the genuine gradient f for smooth compact scalar f.

\[S_y(du)=Z_y^{-1}e^{-W_y(u)}du,\quad W_y(u)=V((y+u)/2)+\frac{\|u-y\|^2}{8\eta},\quad \operatorname{Graph}D_y=\{([f],[\nabla f]):f\in C_c^\infty(E)\}.\]

All objects and hypotheses

E is finite-dimensional real inner-product Borel, including dimension zero; V is C2.
Alpha,beta are nonnegative real parameters with explicit 0<alpha<=beta and genuine alpha/beta Hessian quadratic-form bounds. Eta>0 and beta eta<=1 are retained from the parent conditional-kernel interface.
Mu is volume tilted by -V and J is the law of (X,X+sqrt(eta)Z) under independent Gibbs X and standard Gaussian Z. The constructed R,S are shared before the universal quantifier over y.
Weight integrability and positive normalization are conclusions inherited from the proved actual conditional law, not new hypotheses. The abstract gradient closure itself requires no curvature restrictions.

Mathematical proof

1. Preserve the actual conditional law

Use the already proved ConditionalBochner construction once. Keep its same R,S, swapped disintegration and pointwise reflected pushforward, then fix y. No new abstract measure or surrogate score is substituted.

\[R=\operatorname{Law}(X\mid Y),\qquad S_y=(x\mapsto2x-y)_\#R_y.\]

Corresponding Lean step

The consumer destructures one conditional_bochner_energy witness and retains hR,hS,hcond,hSR.

2. Supply the genuine weight and regularity

The parent provides S_y as the actual tilt, W_y C2, integrable exp(-W_y) and a positive normalizer. Lower the regularity only to C1 for the shared construction. The finite positive normalizer remains explicitly stated under the same fiber law.

\[W_y\in C^2(E)\subset C^1(E),\qquad 0<Z_y=\int e^{-W_y}<\infty.\]

Corresponding Lean step

hSy,hW,hI,hZ are obtained from the same hfiber y; no new integrability assumption is added.

3. Instantiate the exact gradient graph on that fiber

Rewrite the scalar/vector L2 spaces using the proved equality of measures and apply compact_gradient_closable with W_y. This gives the dense domain and closed extension together with the exact smooth-gradient graph. It is still an analytic prerequisite and supplies neither the generator core nor the conditional PI estimate.

\[\overline{\operatorname{Dom}D_y}=L^2(S_y),\qquad D_y\ \text{closable},\qquad \overline D_y\ \text{closed}.\]

Corresponding Lean step

The final application uses the actual hI and hW.of_le; graph membership remains an iff for both directions.

Lean statement · conditional_gradient_closable

One actual pair R,S before all y; for each y, actual density, C2 potential, positive finite normalizer and an operator with exact gradient graph.

theorem conditional_gradient_closable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ Integrable (fun u => Real.exp (-W y u)) ∧
        0 < (∫ u, Real.exp (-W y u)) ∧
        ∃ D : Lp ℝ 2 (S y) →ₗ.[ℝ] Lp E 2 (S y),
          Dense (D.domain : Set (Lp ℝ 2 (S y))) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
          ∀ (u : Lp ℝ 2 (S y)) (v : Lp E 2 (S y)), (u,v) ∈ D.graph ↔
            ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
              u =ᵐ[S y] f ∧ v =ᵐ[S y] gradient f

Exact module and namespace context

Lean proof · conditional_gradient_closable

Apply the proved conditional law and the genuine weighted-gradient construction; no PI, variance or core conclusion is assumed.

theorem conditional_gradient_closable {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ Integrable (fun u => Real.exp (-W y u)) ∧
        0 < (∫ u, Real.exp (-W y u)) ∧
        ∃ D : Lp ℝ 2 (S y) →ₗ.[ℝ] Lp E 2 (S y),
          Dense (D.domain : Set (Lp ℝ 2 (S y))) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
          ∀ (u : Lp ℝ 2 (S y)) (v : Lp E 2 (S y)), (u,v) ∈ D.graph ↔
            ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
              u =ᵐ[S y] f ∧ v =ᵐ[S y] gradient f := by
  obtain ⟨R,S,hR,hS,hcond,hSR,hfiber⟩ :=
    ConditionalBochner.conditional_bochner_energy hα hαβ hV hH hη hβη
  dsimp only
  refine ⟨R,S,hR,hS,hcond,hSR,?_⟩
  intro y
  obtain ⟨hSy,hW,hI,hZ,_⟩ := hfiber y
  refine ⟨hSy,hW,hI,hZ,?_⟩
  rw [hSy]
  exact WeightedGradient.compact_gradient_closable _ (hW.of_le (by norm_num)) hI

end AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalGradient

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Paper setting: Euclidean d>=1, V C2, 0<alpha<=beta, genuine alpha/beta Hessian bounds, eta>0 and beta eta<=1.	E is finite-dimensional real inner-product Borel, including dimension zero; V is C2.	generalization	Finite-dimensional real inner-product Borel E includes dimension zero. This extends the source positive-dimensional Euclidean setting and is explicitly disclosed.
Appendix C.1 conditional Poincare application.	Actual conditional dense closable gradient construction only.	API-limitation	This prerequisite does not prove the source variance estimate, PI, a D*D core or resolvent regularity.
Source Hessian and scale assumptions.	Explicit 0<alpha<=beta, V C2, genuine Hessian bounds and beta eta<=1 retained.	same	The conditional parent interface retains these restrictions even though the abstract gradient result has no curvature hypothesis.
Actual conditional Gibbs normalization.	One common R,S with disintegration and reflected pushforward; every fiber has actual density, C2 potential, integrable weight and positive normalizer.	source-implicit	The shared exponential integrability premise is supplied by the proved actual conditional law, not introduced as a new paper assumption.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Explicit abstract finite-dimensional and zero-dimensional extension. — Public domain and lesson explicitly allow zero dimension.
scopes: Authored analytic prerequisite, not the PI assertion in C.1. — Operator core, resolvent regularity and PI remain open; pointwise properties concern the chosen conditional version.
assumptions: Upper curvature and beta eta<=1 are inherited interface restrictions, absent from abstract gradient construction. — hSy,hW,hI,hZ from same hfiber y; shared hI and hW.of_le are supplied, not assumed anew.

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

Dense compact-test gradient operator, exact genuine-gradient graph and closed extension only. No operator core of D*D, dense mean-zero generator range, resolvent regularity, Poincare, noncompact score variance, macro coercivity, process/mixing/error/cost or complete-paper result.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

ContDiff.of_le

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS Appendix C.1 — Actual conditional weighted-space analytic prerequisite to PI, not PI or variance.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Exact genuine smooth compact gradient graph
TODO — not closed by these contributions Dense domain and single-valued closed graph extension

A unique weak resolvent from a closed Hilbert graph

Kolesnikov-Milman background; Chen-Chewi-Lu-Zhang PBPS consumer; ASTIS expanded prerequisite proof.

arXiv:1310.2526v7 · sections2.4-2.5; abstract weak-form prerequisite · faithful paraphrase

Complete source statement (ASTIS restatement)

Kolesnikov-Milman sections2.4-2.5 discuss weighted Poisson solvability and Sobolev spaces on compact manifolds. The selected ASTIS prerequisite is a separate abstract weak-form construction, not a formalization of that classical compact-manifold regularity theorem: for real Hilbert spaces H,K and a closed partial linear A, every epsilon>0 and f in H admit a unique u in Dom(A) satisfying epsilon inner(u,v)+inner(Au,Av)=inner(f,v) for every domain v. The exact energy is epsilon norm(u)^2+norm(Au)^2=inner(f,u), with norm(u)<=epsilon^-1 norm(f) and energy<=epsilon^-1 norm(f)^2. Domain norms here are ambient H norms. Density and PI are not hypotheses. This supplies the actual conditional closed-gradient consumer, not a compact Bochner extension or epsilon-to-zero PI argument.

Selected abstract prerequisite: H,K real Hilbert spaces, A closed partial real-linear, epsilon>0, f arbitrary in H; no dense-domain or PI hypothesis.
Background source uses a compact weighted manifold. Abstract Hilbert weak existence and full-space conditional application are explicit changes of scope, not source Poisson regularity.

Weak equation and energy

\[\varepsilon\langle u,v\rangle_H+\langle Au,Av\rangle_K=\langle f,v\rangle_H\quad(v\in\operatorname{Dom}A),\qquad \|u\|_H\le\varepsilon^{-1}\|f\|_H.\]

Read the formalized proofs

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

ASTIS mathematical exposition

A unique weak resolvent from a closed Hilbert graph

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.ClosedGraphResolvent.weak_resolvent · theorem · Teaching coverage

Statement

Let H,K be real Hilbert spaces and A a partial real-linear operator with closed graph. For each epsilon>0 and f in H, there is a unique domain element u such that epsilon inner(u,v)+inner(Au,Av)=inner(f,v) for every v in the domain. It satisfies the exact energy identity epsilon norm(u)^2+norm(Au)^2=inner(f,u), the ambient bound norm(u)<=epsilon^-1 norm(f), and the energy bound <=epsilon^-1 norm(f)^2. No density, boundedness, Poincare or prior solvability is assumed.

\[\varepsilon\langle u,v\rangle_H+\langle Au,Av\rangle_K=\langle f,v\rangle_H\quad(v\in\operatorname{Dom}A),\qquad \|u\|_H\le\varepsilon^{-1}\|f\|_H.\]

All objects and hypotheses

H and K are complete real inner-product spaces; they may be infinite dimensional.
A is a partial real-linear map with a linear-subspace domain, and its graph is closed in H times K. The domain is not assumed dense or all of H, and A is not assumed bounded.
Epsilon is strictly positive and f is an arbitrary H element. Domain-element norms in the statement are ambient H norms, not graph norms.
This is an abstract Hilbert weak-form prerequisite inspired by the Poisson/Sobolev discussion in Kolesnikov-Milman sections2.4-2.5. It is not that source compact-manifold Poisson existence/regularity theorem, nor PBPS conditional Poincare.

Mathematical proof

1. Place the scaled graph in a Hilbert product

Set s=sqrt(epsilon)>0. On H direct-sum K with its L2 product norm use T(a,b)=(s^-1 a,b). Define G as the inverse image under T of the graph of A. T is continuous linear, so G is a closed subspace and therefore complete. This avoids assuming the unbounded operator is continuous.

\[G=T^{-1}(\operatorname{Graph}A)=\{(s u,Au):u\in\operatorname{Dom}A\},\qquad s=\sqrt\varepsilon.\]

Corresponding Lean step

The local hex proof uses WithLp.fstL/sndL, Submodule.comap, hA.preimage T.continuous and hG.completeSpace_coe.

2. Project and recover a genuine domain element

Project z=(s^-1 f,0) orthogonally onto G. Membership of p in G gives an actual u in the domain with u=s^-1 p_1 and Au=p_2. Since s is nonzero, p_1=s u. No totalized inverse or arbitrary differential representative is used.

\[p=P_G(s^{-1}f,0)=(s u,Au),\qquad u\in\operatorname{Dom}A.\]

Corresponding Lean step

starProjection_apply_mem and LinearPMap.mem_graph_iff provide u; scalar cancellation establishes hu'.

3. Orthogonality is exactly the weak equation

For every domain v, the vector w=(s v,Av) lies in G. Orthogonality of z-p to w gives inner(s^-1 f-s u,s v)-inner(Au,Av)=0. Expanding both inner products and using s^2=epsilon yields the weak equation for every test in the closed domain.

\[0=\langle z-p,(s v,Av)\rangle=\langle f,v\rangle-\varepsilon\langle u,v\rangle-\langle Au,Av\rangle.\]

Corresponding Lean step

Submodule.starProjection_inner_eq_zero, the L2 product inner product and explicit nonzero-s scalar simplification close the local existence proof.

4. Derive the exact energy and explicit bounds

Use v=u in the weak equation. Cauchy-Schwarz bounds its right side by norm(f) norm(u). When norm(u)=0 the norm bound is immediate; otherwise divide its positive value to obtain epsilon norm(u)<=norm(f). Substitute this estimate back into the energy identity. The constants are explicit and deteriorate as epsilon approaches zero.

\[\varepsilon\|u\|_H^2+\|Au\|_K^2=\langle f,u\rangle_H\le\|f\|_H\|u\|_H\le\varepsilon^{-1}\|f\|_H^2.\]

Corresponding Lean step

he is the self-test identity; hn handles the zero/nonzero norm branches, hb divides by positive epsilon, heb combines Cauchy-Schwarz and hb.

5. Prove uniqueness by testing the difference

If w also solves the same equation, subtract the two equations tested at d=w-u. Domain linearity and A(w-u)=Aw-Au give epsilon norm(d)^2+norm(Ad)^2=0. Positivity of epsilon forces the ambient norm of d to vanish, hence the two domain elements are equal.

\[\varepsilon\|w-u\|_H^2+\|A(w-u)\|_K^2=0\quad\Longrightarrow\quad w=u.\]

Corresponding Lean step

The last quantified clause assumes only the weak equation for w, not its norm/energy bounds; Subtype.ext concludes equality from ambient norm zero.

Lean statement · weak_resolvent

Existence, exact energy, ambient H norm bound, energy bound, and uniqueness among all weak solutions on A.domain.

theorem weak_resolvent {H K : Type*}
    [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
    [NormedAddCommGroup K] [InnerProductSpace ℝ K] [CompleteSpace K]
    (A : H →ₗ.[ℝ] K) (hA : A.IsClosed) (ε : ℝ) (hε : 0 < ε) (f : H) :
    ∃ u : A.domain,
      (∀ v : A.domain, ε * ⟪(u : H), (v : H)⟫ + ⟪A u, A v⟫ = ⟪f, (v : H)⟫) ∧
      ε * ‖(u : H)‖^2 + ‖A u‖^2 = ⟪f, (u : H)⟫ ∧
      ‖(u : H)‖ ≤ ε⁻¹ * ‖f‖ ∧
      ε * ‖(u : H)‖^2 + ‖A u‖^2 ≤ ε⁻¹ * ‖f‖^2 ∧
      ∀ w : A.domain,
        (∀ v : A.domain, ε * ⟪(w : H), (v : H)⟫ + ⟪A w, A v⟫ = ⟪f, (v : H)⟫) → w = u

Exact module and namespace context

Lean proof · weak_resolvent

The scaled-graph existence proof is local to the single public theorem; the complete folded proof includes it and all bounds and uniqueness.

theorem weak_resolvent {H K : Type*}
    [NormedAddCommGroup H] [InnerProductSpace ℝ H] [CompleteSpace H]
    [NormedAddCommGroup K] [InnerProductSpace ℝ K] [CompleteSpace K]
    (A : H →ₗ.[ℝ] K) (hA : A.IsClosed) (ε : ℝ) (hε : 0 < ε) (f : H) :
    ∃ u : A.domain,
      (∀ v : A.domain, ε * ⟪(u : H), (v : H)⟫ + ⟪A u, A v⟫ = ⟪f, (v : H)⟫) ∧
      ε * ‖(u : H)‖^2 + ‖A u‖^2 = ⟪f, (u : H)⟫ ∧
      ‖(u : H)‖ ≤ ε⁻¹ * ‖f‖ ∧
      ε * ‖(u : H)‖^2 + ‖A u‖^2 ≤ ε⁻¹ * ‖f‖^2 ∧
      ∀ w : A.domain,
        (∀ v : A.domain, ε * ⟪(w : H), (v : H)⟫ + ⟪A w, A v⟫ = ⟪f, (v : H)⟫) → w = u := by
  have hex : ∃ u : A.domain, ∀ v : A.domain,
      ε * ⟪(u : H), (v : H)⟫ + ⟪A u, A v⟫ = ⟪f, (v : H)⟫ := by
    let s := Real.sqrt ε
    have hs : 0 < s := Real.sqrt_pos.2 hε
    have hs0 : s ≠ 0 := ne_of_gt hs
    have hss : s * s = ε := Real.mul_self_sqrt hε.le
    let T : WithLp 2 (H × K) →L[ℝ] H × K :=
      (s⁻¹ • (WithLp.fstL 2 ℝ H K)).prod (WithLp.sndL 2 ℝ H K)
    let G : Submodule ℝ (WithLp 2 (H × K)) := A.graph.comap T.toLinearMap
    have hG : IsClosed (G : Set (WithLp 2 (H × K))) := hA.preimage T.continuous
    let : CompleteSpace G := hG.completeSpace_coe
    let z : WithLp 2 (H × K) := WithLp.toLp 2 (s⁻¹ • f, 0)
    let p := G.starProjection z
    have hp : T p ∈ A.graph := G.starProjection_apply_mem z
    obtain ⟨u,hu,hAu⟩ := A.mem_graph_iff.mp hp
    have hu' : s • (u : H) = p.fst := by
      rw [hu]
      change s • (s⁻¹ • p.fst) = p.fst
      simp [smul_smul, hs0]
    refine ⟨u, ?_⟩
    intro v
    let w : WithLp 2 (H × K) := WithLp.toLp 2 (s • (v : H), A v)
    have hw : w ∈ G := by
      change T w ∈ A.graph
      apply A.mem_graph_iff.mpr
      refine ⟨v, ?_, ?_⟩
      · change (v : H) = s⁻¹ • (s • (v : H))
        simp [smul_smul, hs0]
      · rfl
    have ho := G.starProjection_inner_eq_zero z w hw
    change ⟪s⁻¹ • f - p.fst, s • (v : H)⟫ + ⟪(0 : K) - p.snd, A v⟫ = 0 at ho
    have hAu' : A u = p.snd := hAu
    rw [← hu', ← hAu'] at ho
    simp only [inner_sub_left, real_inner_smul_left, real_inner_smul_right, zero_sub,
      inner_neg_left] at ho
    simp only [← mul_assoc, mul_inv_cancel₀ hs0, hss, one_mul] at ho
    linarith
  obtain ⟨u,hu⟩ := hex
  have he : ε * ‖(u : H)‖^2 + ‖A u‖^2 = ⟪f, (u : H)⟫ := by
    simpa only [real_inner_self_eq_norm_sq] using hu u
  have hn : ε * ‖(u : H)‖ ≤ ‖f‖ := by
    have hcs := real_inner_le_norm f (u : H)
    by_cases hz : ‖(u : H)‖ = 0
    · simp [hz]
    · have hp : 0 < ‖(u : H)‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hz)
      nlinarith [sq_nonneg ‖A u‖]
  have hb : ‖(u : H)‖ ≤ ε⁻¹ * ‖f‖ := by
    have hn' : ‖(u : H)‖ ≤ ‖f‖ / ε := (le_div_iff₀ hε).2 (by simpa [mul_comm] using hn)
    simpa [div_eq_mul_inv, mul_comm] using hn'
  have heb : ε * ‖(u : H)‖^2 + ‖A u‖^2 ≤ ε⁻¹ * ‖f‖^2 := by
    rw [he]
    calc
      ⟪f, (u : H)⟫ ≤ ‖f‖ * ‖(u : H)‖ := real_inner_le_norm _ _
      _ ≤ ‖f‖ * (ε⁻¹ * ‖f‖) := mul_le_mul_of_nonneg_left hb (norm_nonneg _)
      _ = _ := by ring
  refine ⟨u,hu,he,hb,heb,?_⟩
  intro w hw
  let d : A.domain := w-u
  have hd : ε * ⟪(d : H), (d : H)⟫ + ⟪A d, A d⟫ = 0 := by
    have heq := sub_eq_zero.mpr ((hw d).trans (hu d).symm)
    have hAd : A d = A w - A u := A.toFun.map_sub w u
    rw [hAd, inner_sub_left]
    change ε * ⟪(w : H) - (u : H), (d : H)⟫ + _ = 0
    rw [inner_sub_left]
    rw [hAd] at heq
    nlinarith
  rw [real_inner_self_eq_norm_sq, real_inner_self_eq_norm_sq] at hd
  have hz : ‖(d : H)‖ = 0 := by
    by_contra hz
    have hp : 0 < ‖(d : H)‖ := lt_of_le_of_ne (norm_nonneg _) (Ne.symm hz)
    have hpos := mul_pos hε (sq_pos_of_pos hp)
    nlinarith [sq_nonneg ‖A d‖]
  apply Subtype.ext
  exact sub_eq_zero.mp (norm_eq_zero.mp hz)

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.ClosedGraphResolvent

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Compact weighted-manifold Poisson existence/regularity and Sobolev background.	Arbitrary real Hilbert H,K and closed partial-linear A; positive-epsilon weak existence, energy and uniqueness.	generalization	Explicit abstract prerequisite, not a claim of equivalence to the source classical elliptic theorem. No regularity or boundary result.
Full-space conditional Poincare route needs a weak equation.	No domain density assumed, ambient H norm and epsilon^-1 bounds proved.	source-implicit	Orthogonal projection proves the weak equation without assuming PI or bounded A; this supplies only one analytic input.

Weak equation on the closed gradient domain only. No classical PDE regularity, identified maximal differential domain, D*D operator core, compact-test Bochner extension, epsilon-uniform estimate, Poincare, noncompact score variance, measurable parameter selection, process/error/cost or full-paper conclusion.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

objects: Separately proved abstract weak resolvent, not background differential-operator theorem transcription. — A is arbitrary, not identified with the manifold differential operator; source contract and lesson disclose the change.
domains: Compact-manifold setting replaced by arbitrary Hilbert spaces and potentially nondense domains. — No manifold, finite-dimension, boundary or density assumption; these are disclosed changes, not source equivalence.
scopes: No classical Poisson regularity or spectral conclusion; selected prerequisite only. — Module, source contract and lesson consistently retain these boundaries.

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

Weak equation on the closed gradient domain only. No classical PDE regularity, identified maximal differential domain, D*D operator core, compact-test Bochner extension, epsilon-uniform estimate, Poincare, noncompact score variance, measurable parameter selection, process/error/cost or full-paper conclusion.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

WithLp.fstL
WithLp.sndL
Submodule.comap
IsClosed.completeSpace_coe
Submodule.starProjection_apply_mem
Submodule.starProjection_inner_eq_zero
LinearPMap.mem_graph_iff
real_inner_le_norm

Mathematical sources

Kolesnikov and Milman, sections2.4-2.5 — Poisson/Sobolev background; this directly proved abstract weak-form prerequisite does not claim compact-manifold classical regularity.
Chen, Chewi, Lu and Zhang, PBPS Appendix C.1 — Actual conditional Poincare route requires further regularity/core and limiting arguments.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Actual domain weak solution and uniqueness
TODO — not closed by these contributions Exact energy and positive-epsilon norm and energy constants

Solve the weak equation on the actual conditional gradient closure

Kolesnikov-Milman background; Chen-Chewi-Lu-Zhang PBPS consumer; ASTIS expanded prerequisite proof.

arXiv:2609.06905v1 · Appendix C.1; actual conditional weak-form prerequisite · faithful paraphrase

Complete source statement (ASTIS restatement)

PBPS Appendix C.1 applies conditional Poincare to the actual reflected Gibbs-Gaussian conditional law. The selected ASTIS analytic prerequisite retains that law with W_y(u)=V((y+u)/2)+norm(u-y)^2/(8eta), its finite positive normalizer and the genuine dense closable compact gradient D_y. For each y, the same D_y is chosen before all positive epsilon and f in scalar L2(S_y); A=D_y.closure then has a unique weak solution to epsilon inner(u,v)+inner(Au,Av)=inner(f,v) on its domain, with exact energy identity and epsilon^-1 norm/energy bounds. The source PI or score variance is not proved here. No measurable solution family or classical regularity is asserted.

Source Euclidean d>=1, V C2, 0<alpha<=beta, genuine Hessian bounds, eta>0 and beta eta<=1.
Actual independent Gibbs-Gaussian augmentation (X,X+sqrt(eta)Z) and reflected conditional kernel. Normalization and closed genuine gradient are derived prerequisites, not additional paper assumptions.

Weak equation and energy

\[\varepsilon\langle u,v\rangle_{L^2(S_y)}+\langle\overline D_yu,\overline D_yv\rangle_{L^2(S_y;E)}=\langle f,v\rangle_{L^2(S_y)},\quad v\in\operatorname{Dom}\overline D_y.\]

Read the formalized proofs

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

ASTIS mathematical exposition

Solve the weak equation on the actual conditional gradient closure

AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalResolvent.conditional_weak_resolvent · theorem · Teaching coverage

Statement

For the actual Gibbs-Gaussian augmentation, retain common Markov kernels R,S with swapped-joint disintegration and reflected pushforward. Every S_y is the normalized exp(-W_y) law with W_y(u)=V((y+u)/2)+norm(u-y)^2/(8eta), C2 potential and finite positive normalizer. Choose one genuine compact-test gradient D_y on scalar/vector L2(S_y), dense and closable with the exact almost-everywhere smooth-gradient graph. For this same D_y, every epsilon>0 and scalar L2 input f has a unique weak solution u in the domain of D_y.closure with the exact energy identity and epsilon^-1 norm and energy bounds.

\[\varepsilon\langle u,v\rangle_{L^2(S_y)}+\langle\overline D_yu,\overline D_yv\rangle_{L^2(S_y;E)}=\langle f,v\rangle_{L^2(S_y)},\quad v\in\operatorname{Dom}\overline D_y.\]

All objects and hypotheses

E is finite-dimensional real inner-product Borel, including dimension zero, and V is C2.
Explicit 0<alpha<=beta, genuine lower/upper Hessian quadratic-form bounds, eta>0 and beta eta<=1 are inherited from the actual kernel interface. They are not needed by the abstract closed-graph theorem.
Mu is the actual normalized tilt by -V. J is the pushforward of independent Gibbs X and standard Gaussian Z under (X,X+sqrt(eta)Z).
Common R,S precede every y; each D_y precedes every epsilon and f. The exact graph iff characterizes D_y, not every element of its closure.
Integrable exp(-W_y) and its positive normalizer are proved conclusions supplied by the same actual conditional law, not additional paper assumptions. No measurability of y-to-D_y or y-to-solution is asserted.

Mathematical proof

1. Preserve the same conditional probability and gradient

Use conditional_gradient_closable to obtain common R,S and, at each y, the actual normalized potential and one dense closable partial gradient. Retain its exact graph equivalence with smooth compact f and their genuine gradients. The operator is chosen before epsilon and f.

\[\operatorname{Graph}D_y=\{([g],[\nabla g]):g\in C_c^\infty(E)\},\qquad S_y=Z_y^{-1}e^{-W_y}\,du.\]

Corresponding Lean step

hfiber supplies hSy,hW,hI,hZ,D,hDense,hClose,hClosed,hgraph; every one is retained in the result.

2. Use the proved closed extension on the actual fiber

Take A=D_y.closure in the Hilbert spaces of scalar and vector L2(S_y). The parent theorem already proves that this closure has a closed graph. Thus the abstract theorem applies without assuming PI, domain coercivity or a classical differential equation.

\[H=L^2(S_y),\quad K=L^2(S_y;E),\quad A=\overline D_y,\quad\operatorname{Graph}A\ \text{closed}.\]

Corresponding Lean step

Instantiate ClosedGraphResolvent.weak_resolvent with D.closure and hClosed for each positive epsilon and each input f.

3. Retain uniqueness and constants without claiming regularity

The abstract result supplies the exact weak equation over every closed-domain test, energy identity, both explicit bounds and uniqueness on that same domain. This is an L2 domain statement. It does not justify substitution into compact-test Bochner or a pointwise PDE identity.

\[\|u\|_{L^2(S_y)}\le\varepsilon^{-1}\|f\|_{L^2(S_y)},\quad\varepsilon\|u\|^2+\|\overline D_yu\|^2\le\varepsilon^{-1}\|f\|^2.\]

Corresponding Lean step

All clauses are returned directly for the same D and actual S_y; generator core/regularity and the epsilon-to-zero step remain separate obligations.

Lean statement · conditional_weak_resolvent

Actual common conditional kernels, normalized C2 potential, dense closable exact gradient graph, then every positive-epsilon weak solution with exact energy, bounds and uniqueness.

theorem conditional_weak_resolvent {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ Integrable (fun u => Real.exp (-W y u)) ∧
        0 < (∫ u, Real.exp (-W y u)) ∧
        ∃ D : Lp ℝ 2 (S y) →ₗ.[ℝ] Lp E 2 (S y),
          Dense (D.domain : Set (Lp ℝ 2 (S y))) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
          (∀ (u : Lp ℝ 2 (S y)) (v : Lp E 2 (S y)), (u,v) ∈ D.graph ↔
            ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
              u =ᵐ[S y] f ∧ v =ᵐ[S y] gradient f) ∧
          ∀ (ε : ℝ), 0 < ε → ∀ f : Lp ℝ 2 (S y),
            ∃ u : D.closure.domain,
              (∀ v : D.closure.domain, ε * ⟪(u : Lp ℝ 2 (S y)), (v : Lp ℝ 2 (S y))⟫ +
                ⟪D.closure u, D.closure v⟫ = ⟪f, (v : Lp ℝ 2 (S y))⟫) ∧
              ε * ‖(u : Lp ℝ 2 (S y))‖^2 + ‖D.closure u‖^2 = ⟪f, (u : Lp ℝ 2 (S y))⟫ ∧
              ‖(u : Lp ℝ 2 (S y))‖ ≤ ε⁻¹ * ‖f‖ ∧
              ε * ‖(u : Lp ℝ 2 (S y))‖^2 + ‖D.closure u‖^2 ≤ ε⁻¹ * ‖f‖^2 ∧
              ∀ w : D.closure.domain,
                (∀ v : D.closure.domain, ε * ⟪(w : Lp ℝ 2 (S y)), (v : Lp ℝ 2 (S y))⟫ +
                  ⟪D.closure w, D.closure v⟫ = ⟪f, (v : Lp ℝ 2 (S y))⟫) → w = u

Exact module and namespace context

Lean proof · conditional_weak_resolvent

The consumer uses the same parent kernels and operator; its only analytic application is the proved closed-graph weak resolvent.

theorem conditional_weak_resolvent {E : Type*} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ Integrable (fun u => Real.exp (-W y u)) ∧
        0 < (∫ u, Real.exp (-W y u)) ∧
        ∃ D : Lp ℝ 2 (S y) →ₗ.[ℝ] Lp E 2 (S y),
          Dense (D.domain : Set (Lp ℝ 2 (S y))) ∧ D.IsClosable ∧ D.closure.IsClosed ∧
          (∀ (u : Lp ℝ 2 (S y)) (v : Lp E 2 (S y)), (u,v) ∈ D.graph ↔
            ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
              u =ᵐ[S y] f ∧ v =ᵐ[S y] gradient f) ∧
          ∀ (ε : ℝ), 0 < ε → ∀ f : Lp ℝ 2 (S y),
            ∃ u : D.closure.domain,
              (∀ v : D.closure.domain, ε * ⟪(u : Lp ℝ 2 (S y)), (v : Lp ℝ 2 (S y))⟫ +
                ⟪D.closure u, D.closure v⟫ = ⟪f, (v : Lp ℝ 2 (S y))⟫) ∧
              ε * ‖(u : Lp ℝ 2 (S y))‖^2 + ‖D.closure u‖^2 = ⟪f, (u : Lp ℝ 2 (S y))⟫ ∧
              ‖(u : Lp ℝ 2 (S y))‖ ≤ ε⁻¹ * ‖f‖ ∧
              ε * ‖(u : Lp ℝ 2 (S y))‖^2 + ‖D.closure u‖^2 ≤ ε⁻¹ * ‖f‖^2 ∧
              ∀ w : D.closure.domain,
                (∀ v : D.closure.domain, ε * ⟪(w : Lp ℝ 2 (S y)), (v : Lp ℝ 2 (S y))⟫ +
                  ⟪D.closure w, D.closure v⟫ = ⟪f, (v : Lp ℝ 2 (S y))⟫) → w = u := by
  obtain ⟨R,S,hR,hS,hcond,hSR,hfiber⟩ :=
    ConditionalGradient.conditional_gradient_closable hα hαβ hV hH hη hβη
  dsimp only
  refine ⟨R,S,hR,hS,hcond,hSR,?_⟩
  intro y
  obtain ⟨hSy,hW,hI,hZ,D,hDense,hClose,hClosed,hgraph⟩ := hfiber y
  refine ⟨hSy,hW,hI,hZ,D,hDense,hClose,hClosed,hgraph,?_⟩
  intro ε hε f
  exact AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.ClosedGraphResolvent.weak_resolvent D.closure hClosed ε hε f

end AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalResolvent

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Euclidean d>=1.	Arbitrary finite-dimensional real inner-product Borel E, including zero dimension.	generalization	Explicit dimension-zero extension retained.
Appendix C.1 conditional Poincare/variance.	Actual conditional closed-gradient epsilon weak equation only.	API-limitation	Generator core/regularity, compact Bochner extension, mean-zero limit and score extension remain open.
Actual augmentation and conditional Gibbs law.	Same common R,S; every actual normalized C2 fiber; one exact D_y before all epsilon,f.	source-implicit	No unrelated law or arbitrary surrogate operator and no extra integrability premise.
Source curvature and scale hypotheses.	0<alpha<=beta, C2, true Hessian bounds, eta>0 and beta eta<=1 retained.	same	Inherited parent restrictions, not required by the abstract weak theorem.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Explicit finite-dimensional and zero-dimensional extension. — No positive-dimension premise; zero dimension explicitly disclosed.
scopes: Authored weak-resolvent prerequisite, not C.1 PI/variance. — Graph iff describes D, not closure; uniqueness is of domain L2 classes, not pointwise representatives.
assumptions: Curvature and beta eta<=1 retained parent restrictions, not abstract weak-existence premises. — hSy,hW,hI,hZ,hClosed from same parent, not added premises.

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

Weak equation on the closed gradient domain only. No classical PDE regularity, identified maximal differential domain, D*D operator core, compact-test Bochner extension, epsilon-uniform estimate, Poincare, noncompact score variance, measurable parameter selection, process/error/cost or full-paper conclusion.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

No direct Mathlib call recorded; see the ASTIS parents.

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS Appendix C.1 — Actual conditional weak-form prerequisite, not the conditional PI or variance estimate.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Actual domain weak solution and uniqueness
TODO — not closed by these contributions Exact energy and positive-epsilon norm and energy constants

Actual conditional curvature and the noncompact directional-score domain

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; ASTIS expanded mathematical proof.

arXiv:2609.06905v1 · Appendix C.1 after C.1: conditional Hessian, sharp score derivative and noncompact input before conditional Poincare · faithful paraphrase

Complete source statement (ASTIS restatement)

The conditional negative log-density has Hessian (D2V((y+u)/2)+eta^-1 I)/4, bounded below by (alpha+eta^-1)I/4. The score spatial derivative is (eta^-1 I-D2V((y+u)/2))/4 and has operator norm at most (eta^-1-alpha)/4 under beta eta<=1. The conditional Poincare step uses directional scores a dot s_y, for unit a. This selected input packet makes the noncompact directional-score L2 and finite gradient-energy domain explicit; it does not prove the curvature-to-Poincare inequality or the resulting variance bound.

The paper works on R^d, d>=1, with C2 potential V and genuine Hessian bounds alpha I<=D2V<=beta I, explicit 0<alpha<=beta, eta>0 and beta eta<=1.
The actual conditional law of Y_minus given Y_plus=y, with Y_plus/minus=X plus/minus sqrt(eta)Z and X Gibbs, has density proportional to exp(-W_y(u)), W_y(u)=V((y+u)/2)+norm(y-u)^2/(8eta). The score is s_y(u)=-grad V((y+u)/2)/2-(y-u)/(4eta).

conditional curvature and directional score domain

\[D^2W_y\succeq\frac{\alpha+\eta^{-1}}4I,\qquad\|D_us_y\|_{\mathrm{op}}\le\frac{\eta^{-1}-\alpha}4,\qquad q_a\in L^2(S_y).\]

Read the formalized proofs

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

ASTIS mathematical exposition

Actual conditional curvature and the noncompact directional-score domain

AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScoreDomain.conditional_curvature_and_score_domain · theorem · Teaching coverage

Statement

For the actual Gibbs-Gaussian augmentation J, there exist common Markov kernels R,S with R disintegrating the swapped joint law and S_y its reflected pushforward. Every S_y is the normalized tilt by -W_y, with W_y(u)=V((y+u)/2)+norm(u-y)^2/(8 eta). The theorem computes the genuine Hessian of W_y and derivative of the dual-valued score s_y(u)=-DV((y+u)/2)/2-inner(y-u,.)/(4 eta), proves the lower curvature (alpha+eta inverse)/4 and sharp score derivative norm bound L=(eta inverse-alpha)/4. Every directional score q_a(u)=s_y(u)(a) is C1 and lies in L2(S_y); its mean, centered square and gradient square are integrable, and its gradient norm is bounded by L norm(a), for every y,a,u.

\[D^2W_y\succeq\frac{\alpha+\eta^{-1}}4I,\qquad\|D_us_y\|_{\mathrm{op}}\le\frac{\eta^{-1}-\alpha}4,\qquad q_a\in L^2(S_y).\]

All objects and hypotheses

E is a finite-dimensional real inner-product space with its Borel structure and canonical volume. Dimension zero is allowed. V:E->R is C2.
Alpha and beta are finite nonnegative reals with explicit 0<alpha<=beta. For every x,v the genuine second derivative satisfies alpha norm(v)^2<=D2V(x)[v,v]<=beta norm(v)^2.
Eta is positive and beta eta<=1. The explicit alpha<=beta condition is retained even in dimension zero, where the Hessian inequalities alone do not imply it.
Mu is normalized exponential weighting of volume by -V; J is the law of (X,X+sqrt(eta)Z) with X~mu and independent standard Gaussian Z. Kernels, normalization, tails and moment conditions are derived.
The same R,S work for every y and every direction a. No compact support is imposed on q_a. All regularity, pointwise derivative bounds and fiber integrability conclusions are for the selected everywhere-defined conditional version.

Mathematical proof

1. Fix the actual reflected conditional law

Use the existing conditional-score construction to obtain Markov R and S. R disintegrates the actual swapped Gibbs-Gaussian augmentation; every S_y is the pushforward of R_y through x to 2x-y. The same S_y is volume tilted by minus W_y, where W_y(u)=V((y+u)/2)+norm(u-y)^2/(8 eta). Symmetry of the norm identifies this exponent with the earlier density. No abstract measure with assumed curvature or moments replaces this conditional law.

\[S_y=(x\mapsto2x-y)_\#R_y\propto e^{-W_y(u)}\,du.\]

Corresponding Lean step

Final assembly retains R,S from reflected_conditional_covariance and rewrites hSnu using norm_sub_rev. All following estimates use that S.

2. Compute both genuine derivatives

The midpoint map has derivative one half times identity. Apply the chain rule twice to V composed with the midpoint, and differentiate the quadratic term. Its Hessian is the inner-product map divided by 4 eta. For the score, differentiating minus one half DV gives minus one quarter D2V, while differentiating minus inner(y-u,.)/(4 eta) gives plus the inner-product map divided by 4 eta. V being C2 yields W_y C2 and s_y C1. The Hessian lower bound of V therefore gives the displayed positive lower curvature for W_y.

\[D^2W_y(u)=\tfrac14\bigl(D^2V((y+u)/2)+\eta^{-1}I\bigr),\qquad D_us_y(u)=\tfrac14\bigl(\eta^{-1}I-D^2V((y+u)/2)\bigr).\]

Corresponding Lean step

conditional_derivatives proves both Frechet derivative identities and regularity by chain rules. The final lower bound substitutes the genuine hH lower inequality.

3. Obtain the sharp score derivative norm

The explicit inequalities alpha<=beta and beta eta<=1 imply beta<=eta inverse and L=(eta inverse-alpha)/4>=0. Use the real Riesz isometry to view D_us_y as an endomorphism of E. Symmetry follows from the symmetry of the genuine second derivative of V. Its quadratic forms lie between zero and L norm(v)^2; the Rayleigh quotient norm formula therefore gives operator norm at most L. The operator is positive semidefinite and may vanish at the endpoint. Transfer the bound back through the Riesz isometry. Keeping alpha<=beta explicitly is necessary when dimension zero is allowed.

\[0\preceq D_us_y\preceq L I,\qquad \|D_us_y\|_{\mathrm{op}}\le L:=\frac{\eta^{-1}-\alpha}{4}.\]

Corresponding Lean step

sharp_score_norm uses ContDiffAt.isSymmSndFDerivAt and norm_eq_iSup_rayleighQuotient on the Riesz endomorphism, then opNorm_le_bound on the dual-valued map. No Hessian operator-norm premise is assumed.

4. Derive Gaussian tails and linear score growth

The existing quadratic-regularization theorem at zero regularization gives strong convexity and beta-Lipschitz continuity of the actual gradient. Write G=norm(gradient V(0)). The first-order lower bound and completing the square imply V>=m, with m=V(0)-G^2/(2 alpha), and norm(DV(x))<=G+beta norm(x). For fixed y, put R=norm(y), k=1/(16 eta), C=exp(-m+R^2/(8 eta)), and b=(beta+eta inverse)/4. The triangle inequality gives exp(-W_y(u))<=C exp(-k norm(u)^2) and norm(s_y(u))<=G/2+bR+b norm(u). For direction a, evaluation yields norm(q_a(u))<=A+B norm(u), where A=(G/2+bR)norm(a), B=b norm(a).

\[e^{-W_y(u)}\le C e^{-k\|u\|^2},\qquad |q_a(u)|\le A+B\|u\|.\]

Corresponding Lean step

potential_controls, weight_envelope and score_envelope are local proved adapters, reusing public QuadraticRegularization and StrongConvexFirstOrder. They are not treated as publicly available helpers of ConditionalScore.

5. Prove second moments by absorbing the quadratic factor

The linear bound gives q_a(u)^2<=2A^2+2B^2 norm(u)^2. The elementary exponential inequality t<=exp(t), with t=k r^2/2, gives r^2 exp(-k r^2)<=(2/k) exp(-k r^2/2). Consequently the unnormalized squared-score integrand is bounded by C(2A^2+4B^2/k) exp(-k norm(u)^2/2), which is integrable on finite-dimensional E. The original Gaussian envelope also proves integrability of the normalizing weight. The tilted-integral equivalence then proves q_a in L2(S_y). No second-moment assumption is introduced.

\[e^{-W_y(u)}q_a(u)^2\le C\left(2A^2+\frac{4B^2}{k}\right)e^{-k\|u\|^2/2}.\]

Corresponding Lean step

absorb_quadratic and weighted_linear_memLp use the exponential lower bound, finite-dimensional Gaussian integrability, integrable_tilted_iff and memLp_two_iff_integrable_sq.

6. Verify the noncompact input domain

Evaluation at a is a continuous linear map, so q_a is C1 and its gradient has norm at most L norm(a). Its gradient is continuous and uniformly bounded, hence has integrable square under the probability S_y. The already proved L2 membership gives L1 integrability; subtracting the constant expectation preserves L2 and proves the centered square integrable. These are precisely the three clauses of the existing Poincare.Admissible domain. This supplies a valid noncompact directional-score input, but the curvature-to-Poincare theorem and its applicable test-domain extension are separate missing proofs.

\[q_a\in L^2(S_y),\quad \int(q_a-\mathbb E q_a)^2\,dS_y<\infty,\quad \|\nabla q_a\|\le L\|a\|.\]

Corresponding Lean step

directional_score proves C1 and the norm bound through continuous-linear evaluation. admissible_of_memLp_gradient_bound derives the Admissible clauses from MemLp, MemLp subtraction and bounded continuous gradient; it does not use Poincare.Satisfies or variance_le.

Lean statement · conditional_curvature_and_score_domain

One public theorem constructs actual compatible kernels and proves conditional curvature, the sharp score derivative bound and the full directional-score admissibility domain.

theorem conditional_curvature_and_score_domain {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) - (1/(4*η)) • innerSL ℝ (y-u)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ ContDiff ℝ 1 (s y) ∧
        (∀ u v w,
          (fderiv ℝ (fderiv ℝ (W y)) u v) w =
            (1/4:ℝ)*(fderiv ℝ (fderiv ℝ V) ((1/2:ℝ) • (y+u)) v) w + (1/(4*η))*inner ℝ v w ∧
          (fderiv ℝ (s y) u v) w = (1/(4*η))*inner ℝ v w -
            (1/4:ℝ)*(fderiv ℝ (fderiv ℝ V) ((1/2:ℝ) • (y+u)) v) w) ∧
        (∀ u v, (((α:ℝ)+1/η)/4)*‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ (W y)) u v) v) ∧
        (∀ u, ‖fderiv ℝ (s y) u‖ ≤ (1/η-(α:ℝ))/4) ∧
        ∀ a : E, ContDiff ℝ 1 (fun u => s y u a) ∧
          MemLp (fun u => s y u a) 2 (S y) ∧
          AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.Admissible (S y) (fun u => s y u a) ∧
          ∀ u, ‖gradient (fun z => s y z a) u‖ ≤ ((1/η-(α:ℝ))/4)*‖a‖

Exact module and namespace context

Lean proof · conditional_curvature_and_score_domain

Chain rules and symmetric Rayleigh quotients give exact derivatives and sharp norms; Gaussian domination derives L2 moments; continuous evaluation and bounded gradients give the noncompact input domain.

theorem conditional_curvature_and_score_domain {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {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 := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2)) (μ.prod (stdGaussian E))
    let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
    let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) - (1/(4*η)) • innerSL ℝ (y-u)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      ∀ y, S y = (volume : Measure E).tilted (fun u => -W y u) ∧
        ContDiff ℝ 2 (W y) ∧ ContDiff ℝ 1 (s y) ∧
        (∀ u v w,
          (fderiv ℝ (fderiv ℝ (W y)) u v) w =
            (1/4:ℝ)*(fderiv ℝ (fderiv ℝ V) ((1/2:ℝ) • (y+u)) v) w + (1/(4*η))*inner ℝ v w ∧
          (fderiv ℝ (s y) u v) w = (1/(4*η))*inner ℝ v w -
            (1/4:ℝ)*(fderiv ℝ (fderiv ℝ V) ((1/2:ℝ) • (y+u)) v) w) ∧
        (∀ u v, (((α:ℝ)+1/η)/4)*‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ (W y)) u v) v) ∧
        (∀ u, ‖fderiv ℝ (s y) u‖ ≤ (1/η-(α:ℝ))/4) ∧
        ∀ a : E, ContDiff ℝ 1 (fun u => s y u a) ∧
          MemLp (fun u => s y u a) 2 (S y) ∧
          AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.Admissible (S y) (fun u => s y u a) ∧
          ∀ u, ‖gradient (fun z => s y z a) u‖ ≤ ((1/η-(α:ℝ))/4)*‖a‖ := by
  have conditional_derivatives {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
      {V : E → ℝ} (hV : ContDiff ℝ 2 V) (η : ℝ) (y : E) :
      let W := fun u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
      let s := fun u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
        (1/(4*η)) • innerSL ℝ (y-u)
      ContDiff ℝ 2 W ∧ ContDiff ℝ 1 s ∧
        ∀ u v w : E,
          (fderiv ℝ (fderiv ℝ W) u v) w =
            (1/4:ℝ) * (fderiv ℝ (fderiv ℝ V) ((1/2:ℝ) • (y+u)) v) w +
              (1/(4*η)) * inner ℝ v w ∧
          (fderiv ℝ s u v) w =
            (1/(4*η)) * inner ℝ v w -
              (1/4:ℝ) * (fderiv ℝ (fderiv ℝ V) ((1/2:ℝ) • (y+u)) v) w := by
    let mid := fun u : E => (1/2:ℝ) • (y+u)
    let W := fun u : E => V (mid u) + ‖u-y‖^2/(8*η)
    let s := fun u : E => -(1/2:ℝ) • fderiv ℝ V (mid u) -
        (1/(4*η)) • innerSL ℝ (y-u)
    let J : E →L[ℝ] (E →L[ℝ] ℝ) :=
      { toFun := fun v => innerSL ℝ v
        map_add' := by intros; ext; simp
        map_smul' := by intros; ext; simp
        cont := (innerSL ℝ (E := E)).continuous }
    have hmid (u : E) : HasFDerivAt mid ((1/2:ℝ) • ContinuousLinearMap.id ℝ E) u := by
      convert ((hasFDerivAt_const (𝕜 := ℝ) y u).add (hasFDerivAt_id u)).const_smul (1/2:ℝ) using 1 <;> simp [mid] <;> rfl
    have hVd := hV.differentiable (by norm_num)
    have hVdd := (hV.fderiv_right (m := 1) (by norm_num)).differentiable_one
    have hmidC : ContDiff ℝ 2 mid := by fun_prop
    have hWC : ContDiff ℝ 2 W := hV.comp hmidC |>.add (((contDiff_id.sub contDiff_const).norm_sq (𝕜 := ℝ)).div_const _)
    have hsC : ContDiff ℝ 1 s := by
      apply ContDiff.sub
      · exact ((hV.fderiv_right (m := 1) (by norm_num)).comp (hmidC.of_le (by norm_num))).const_smul _
      · exact (J.contDiff.comp (contDiff_const.sub contDiff_id)).const_smul _
    have hWfd (u : E) : fderiv ℝ W u =
        (1/2:ℝ) • fderiv ℝ V (mid u) + (1/(4*η)) • J (u-y) := by
      have hv := (hVd (mid u)).hasFDerivAt.comp u (hmid u)
      have hq := (((hasFDerivAt_id u).sub_const y).norm_sq).const_mul (1/(8*η))
      have hw : HasFDerivAt W ((1/2:ℝ) • fderiv ℝ V (mid u) + (1/(4*η)) • J (u-y)) u := by
        convert hv.add hq using 1 <;> first | rfl | (ext v; simp [W,J,Function.comp_def]; ring)
      exact hw.fderiv
    have hWdd (u : E) : HasFDerivAt (fderiv ℝ W)
        ((1/4:ℝ) • fderiv ℝ (fderiv ℝ V) (mid u) + (1/(4*η)) • J) u := by
      rw [show fderiv ℝ W = _ from funext hWfd]
      convert (((hVdd (mid u)).hasFDerivAt.comp u (hmid u)).const_smul (1/2:ℝ)).add
        ((J.hasFDerivAt.comp u ((hasFDerivAt_id u).sub_const y)).const_smul (1/(4*η))) using 1 <;>
        first | rfl | (ext v w; simp; ring)
    have hsD (u : E) : HasFDerivAt s
        ((1/(4*η)) • J - (1/4:ℝ) • fderiv ℝ (fderiv ℝ V) (mid u)) u := by
      convert (((hVdd (mid u)).hasFDerivAt.comp u (hmid u)).const_smul (-(1/2:ℝ))).sub
        ((J.hasFDerivAt.comp u ((hasFDerivAt_const (𝕜 := ℝ) y u).sub (hasFDerivAt_id u))).const_smul (1/(4*η))) using 1 <;>
        first | rfl | (ext v w; simp; ring)
    refine ⟨hWC,hsC,?_⟩
    intro u v w
    rw [(hWdd u).fderiv,(hsD u).fderiv]
    constructor <;> rfl

  have sharp_score_norm {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
      {V : E → ℝ} (hV : ContDiff ℝ 2 V) {α β η : ℝ}
      (hαβ : α ≤ β) (hη : 0 < η) (hβη : β*η ≤ 1)
      (hH : ∀ x v : E, α*‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v ∧
        (fderiv ℝ (fderiv ℝ V) x v) v ≤ β*‖v‖^2)
      (x : E) (D : E →L[ℝ] (E →L[ℝ] ℝ))
      (hD : ∀ v w, D v w = (1/(4*η))*inner ℝ v w -
        (1/4:ℝ)*(fderiv ℝ (fderiv ℝ V) x v) w) :
      ‖D‖ ≤ (1/η-α)/4 := by
    let R : (E →L[ℝ] ℝ) →L[ℝ] E :=
      { toFun := (toDual ℝ E).symm
        map_add' := (toDual ℝ E).symm.map_add
        map_smul' := by intros; simp
        cont := (toDual ℝ E).symm.continuous }
    let T := R.comp D
    have hinner (v w : E) : inner ℝ (T v) w = D v w := toDual_symm_apply
    have hsym : T.IsSymmetric := by
      intro v w
      change inner ℝ (T v) w = inner ℝ v (T w)
      calc
        _ = D v w := hinner v w
        _ = D w v := by
          rw [hD,hD,hV.contDiffAt.isSymmSndFDerivAt (by norm_num) v w,real_inner_comm w v]
        _ = inner ℝ (T w) v := (hinner w v).symm
        _ = inner ℝ v (T w) := real_inner_comm _ _
    have hβ : β ≤ 1/η := (le_div_iff₀ hη).2 hβη
    have hc : 0 ≤ (1/η-α)/4 := div_nonneg (sub_nonneg.mpr (hαβ.trans hβ)) (by norm_num)
    have hdiag (v : E) : D v v = ((1/η)*‖v‖^2-(fderiv ℝ (fderiv ℝ V) x v) v)/4 := by
      rw [hD,real_inner_self_eq_norm_sq]
      ring
    have hbounds (v : E) : 0 ≤ D v v ∧ D v v ≤ ((1/η-α)/4)*‖v‖^2 := by
      rw [hdiag]
      have hh := hH x v
      have hb := mul_le_mul_of_nonneg_right hβ (sq_nonneg ‖v‖)
      constructor <;> nlinarith
    have hnormT : ‖T‖ ≤ (1/η-α)/4 := by
      rw [T.norm_eq_iSup_rayleighQuotient hsym]
      apply ciSup_le
      intro v
      change |inner ℝ (T v) v / ‖v‖^2| ≤ (1/η-α)/4
      rw [hinner,abs_of_nonneg (div_nonneg (hbounds v).1 (sq_nonneg _))]
      by_cases hv : v=0
      · simpa [hv] using hc
      · exact (div_le_iff₀ (sq_pos_of_pos (norm_pos_iff.mpr hv))).2 (hbounds v).2
    apply D.opNorm_le_bound hc
    intro v
    calc
      ‖D v‖ = ‖T v‖ := ((toDual ℝ E).symm.norm_map (D v)).symm
      _ ≤ ‖T‖ * ‖v‖ := T.le_opNorm v
      _ ≤ ((1/η-α)/4)*‖v‖ := mul_le_mul_of_nonneg_right hnormT (norm_nonneg _)

  have absorb_quadratic {a r : ℝ} (ha : 0 < a) :
      r^2 * Real.exp (-a*r^2) ≤ (2/a)*Real.exp (-(a/2)*r^2) := by
    have hx : (a/2)*r^2 ≤ Real.exp ((a/2)*r^2) := by
      linarith [Real.add_one_le_exp ((a/2)*r^2)]
    have hm := mul_le_mul_of_nonneg_right hx (Real.exp_nonneg (-a*r^2))
    have he : Real.exp ((a/2)*r^2)*Real.exp (-a*r^2) = Real.exp (-(a/2)*r^2) := by
      rw [← Real.exp_add]
      congr 1
      ring
    rw [he] at hm
    calc
      r^2 * Real.exp (-a*r^2) ≤ Real.exp (-(a/2)*r^2)/(a/2) :=
        (le_div_iff₀ (by positivity : 0 < a/2)).2 (by nlinarith [hm])
      _ = (2/a)*Real.exp (-(a/2)*r^2) := by ring

  have weighted_linear_memLp {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (w q : E → ℝ) (hw : Continuous w) (hq : Continuous q)
      {a C A B : ℝ} (ha : 0 < a) (hC : 0 ≤ C) (hA : 0 ≤ A) (hB : 0 ≤ B)
      (hweight : ∀ u, Real.exp (w u) ≤ C*Real.exp (-a*‖u‖^2))
      (hgrowth : ∀ u, ‖q u‖ ≤ A+B*‖u‖) :
      Integrable (fun u => Real.exp (w u)) (volume : Measure E) ∧
        MemLp q 2 ((volume : Measure E).tilted w) := by
    have hwi : Integrable (fun u => Real.exp (w u)) (volume : Measure E) := by
      apply ((AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_mul_norm_sq
        (E := E) ha).const_mul C).mono' hw.rexp.aestronglyMeasurable
      filter_upwards with u
      simpa only [Real.norm_eq_abs,abs_of_pos (Real.exp_pos _)] using hweight u
    refine ⟨hwi,?_⟩
    apply (memLp_two_iff_integrable_sq hq.aestronglyMeasurable).2
    apply (integrable_tilted_iff hwi (fun u => q u ^ 2)).2
    have hdom := (AutoSamplingTheory.TechnicalLemmas.Analysis.Integrability.integrable_exp_neg_mul_norm_sq
        (E := E) (show 0 < a/2 by positivity)).const_mul (C*(2*A^2+4*B^2/a))
    apply hdom.mono' (hw.rexp.smul (hq.pow 2)).aestronglyMeasurable
    filter_upwards with u
    change ‖Real.exp (w u)*(q u)^2‖ ≤ C*(2*A^2+4*B^2/a)*Real.exp (-(a/2)*‖u‖^2)
    simp only [smul_eq_mul,Real.norm_eq_abs,abs_of_nonneg (mul_nonneg (Real.exp_nonneg _) (sq_nonneg _))]
    have hs : (q u)^2 ≤ 2*A^2+2*B^2*‖u‖^2 := by
      have hh := (sq_le_sq₀ (norm_nonneg (q u)) (by positivity)).2 (hgrowth u)
      rw [Real.norm_eq_abs,sq_abs] at hh
      nlinarith [sq_nonneg (A-B*‖u‖)]
    have he : Real.exp (-a*‖u‖^2) ≤ Real.exp (-(a/2)*‖u‖^2) := by
      apply Real.exp_le_exp.mpr
      nlinarith [mul_nonneg ha.le (sq_nonneg ‖u‖)]
    calc
      Real.exp (w u)*(q u)^2 ≤ (C*Real.exp (-a*‖u‖^2))*(2*A^2+2*B^2*‖u‖^2) :=
        mul_le_mul (hweight u) hs (sq_nonneg _) (mul_nonneg hC (Real.exp_nonneg _))
      _ = C*(2*A^2*Real.exp (-a*‖u‖^2)+2*B^2*(‖u‖^2*Real.exp (-a*‖u‖^2))) := by ring
      _ ≤ C*(2*A^2*Real.exp (-(a/2)*‖u‖^2)+2*B^2*((2/a)*Real.exp (-(a/2)*‖u‖^2))) := by
        apply mul_le_mul_of_nonneg_left _ hC
        exact add_le_add (mul_le_mul_of_nonneg_left he (by positivity))
          (mul_le_mul_of_nonneg_left (absorb_quadratic (r := ‖u‖) ha) (by positivity))
      _ = C*(2*A^2+4*B^2/a)*Real.exp (-(a/2)*‖u‖^2) := by ring

  have admissible_of_memLp_gradient_bound {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (μ : Measure E) [IsFiniteMeasure μ] (q : E → ℝ)
      (hq : ContDiff ℝ 1 q) (hLp : MemLp q 2 μ)
      {M : ℝ} (hM : ∀ u, ‖gradient q u‖ ≤ M) :
      AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.Poincare.Admissible μ q := by
    have hgc : Continuous (gradient q) :=
      (toDual ℝ E).symm.continuous.comp (hq.continuous_fderiv (by norm_num))
    have hgLp : MemLp (gradient q) 2 μ :=
      MemLp.of_bound hgc.aestronglyMeasurable M (Filter.Eventually.of_forall hM)
    refine ⟨hLp.integrable (by norm_num),?_,?_⟩
    · exact (hLp.sub (memLp_const _)).integrable_sq
    · exact (memLp_two_iff_integrable_sq_norm hgc.aestronglyMeasurable).1 hgLp

  have directional_score {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E] [CompleteSpace E]
      (s : E → (E →L[ℝ] ℝ)) (hs : ContDiff ℝ 1 s) {c : ℝ} (hc : 0 ≤ c)
      (hD : ∀ u, ‖fderiv ℝ s u‖ ≤ c) (a : E) :
      ContDiff ℝ 1 (fun u => s u a) ∧ ∀ u,
        ‖gradient (fun z => s z a) u‖ ≤ c*‖a‖ := by
    let ev := ContinuousLinearMap.apply' ℝ (RingHom.id ℝ) a
    have hq : ContDiff ℝ 1 (fun u => s u a) := ev.contDiff.comp hs
    refine ⟨hq,?_⟩
    intro u
    have hqD : HasFDerivAt (fun z => s z a) (ev.comp (fderiv ℝ s u)) u :=
      ev.hasFDerivAt.comp u ((hs.differentiable (by norm_num)) u).hasFDerivAt
    have hdn : ‖fderiv ℝ (fun z => s z a) u‖ ≤ c*‖a‖ := by
      rw [hqD.fderiv]
      apply ContinuousLinearMap.opNorm_le_bound _ (mul_nonneg hc (norm_nonneg _))
      intro v
      change ‖(fderiv ℝ s u v) a‖ ≤ (c*‖a‖)*‖v‖
      calc
        ‖(fderiv ℝ s u v) a‖ ≤ ‖fderiv ℝ s u v‖*‖a‖ := (fderiv ℝ s u v).le_opNorm a
        _ ≤ (‖fderiv ℝ s u‖*‖v‖)*‖a‖ := mul_le_mul_of_nonneg_right ((fderiv ℝ s u).le_opNorm v) (norm_nonneg _)
        _ ≤ (c*‖v‖)*‖a‖ := mul_le_mul_of_nonneg_right (mul_le_mul_of_nonneg_right (hD u) (norm_nonneg _)) (norm_nonneg _)
        _ = (c*‖a‖)*‖v‖ := by ring
    rw [← toDual_gradient,(toDual ℝ E).norm_map] at hdn
    exact hdn

  have potential_controls {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] {V : E → ℝ} {α β : ℝ≥0}
      (hα : 0 < (α : ℝ)) (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) :
      (∀ x, V 0 - (α : ℝ)⁻¹/2*‖gradient V 0‖^2 ≤ V x) ∧
      (∀ x, ‖fderiv ℝ V x‖ ≤ ‖gradient V 0‖ + (β : ℝ)*‖x‖) := by
    have hd : Differentiable ℝ V := hV.differentiable (by norm_num)
    have hreg := QuadraticRegularization.strongConvexOn_and_lipschitzWith_gradient_add_quadratic
      hV hH (r := 0) (0 : E)
    simp only [NNReal.coe_zero, zero_div, zero_mul, add_zero] at hreg
    constructor
    · intro x
      have hfirst := StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn hreg.1
          (fun z _ => (hd z).hasGradientAt) (x := 0) (y := x)
          (Set.mem_univ _) (Set.mem_univ _)
      simp only [sub_zero] at hfirst
      have hinner := (abs_le.mp (abs_real_inner_le_norm (gradient V 0) x)).1
      have hyoung := two_mul_le_add_mul_sq (a := ‖x‖) (b := ‖gradient V 0‖) hα
      nlinarith
    · intro x
      rw [← toDual_gradient]
      rw [(toDual ℝ E).norm_map]
      calc
        ‖gradient V x‖ ≤ ‖gradient V x - gradient V 0‖ + ‖gradient V 0‖ :=
          norm_le_norm_sub_add _ _
        _ ≤ (β : ℝ)*‖x‖ + ‖gradient V 0‖ := by
          apply add_le_add _ (le_refl _)
          simpa using hreg.2.norm_sub_le x 0
        _ = _ := add_comm _ _

  have weight_envelope {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      (V : E → ℝ) {m η R : ℝ} (hV : ∀ x, m ≤ V x) (hη : 0 < η)
      (y u : E) (hy : ‖y‖ ≤ R) :
      Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η)) ≤
        Real.exp (-m+R^2/(8*η)) * Real.exp (-(1/(16*η))*‖u‖^2) := by
    have hR : 0 ≤ R := (norm_nonneg y).trans hy
    have htri : ‖u‖ ≤ ‖y-u‖ + R := by
      have h := norm_le_norm_sub_add u y
      rw [norm_sub_rev u y] at h
      linarith
    have hsq : ‖u‖^2 ≤ 2*‖y-u‖^2 + 2*R^2 := by
      have ht := (sq_le_sq₀ (norm_nonneg u) (by positivity)).mpr htri
      nlinarith [sq_nonneg (‖y-u‖-R)]
    have hq := div_le_div_of_nonneg_right hsq (by positivity : 0 ≤ 16*η)
    have hdiv : ‖u‖^2/(16*η) ≤ ‖y-u‖^2/(8*η) + R^2/(8*η) := by
      have heq : (2*‖y-u‖^2+2*R^2)/(16*η) = ‖y-u‖^2/(8*η)+R^2/(8*η) := by
        field_simp
        ring
      rw [heq] at hq
      exact hq
    rw [← Real.exp_add]
    apply Real.exp_le_exp.mpr
    have hv := hV ((1/2:ℝ) • (y+u))
    simp only [div_eq_mul_inv, one_mul] at hdiv ⊢
    nlinarith

  have score_envelope {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [CompleteSpace E] (V : E → ℝ) {G β η R : ℝ}
      (hG : ∀ x, ‖fderiv ℝ V x‖ ≤ G+β*‖x‖) (hβ : 0 ≤ β) (hη : 0 < η)
      (y u : E) (hy : ‖y‖ ≤ R) :
      ‖-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
        (1/(4*η)) • innerSL ℝ (y-u)‖ ≤
        G/2 + ((β+η⁻¹)/4)*R + ((β+η⁻¹)/4)*‖u‖ := by
    have hmid : ‖(1/2:ℝ) • (y+u)‖ ≤ (R+‖u‖)/2 := by
      rw [norm_smul, Real.norm_eq_abs]
      norm_num
      have h := norm_add_le y u
      linarith
    have hfd : ‖fderiv ℝ V ((1/2:ℝ) • (y+u))‖ ≤ G+β*((R+‖u‖)/2) :=
      (hG _).trans (add_le_add (le_refl _) (mul_le_mul_of_nonneg_left hmid hβ))
    have hdiff : ‖y-u‖ ≤ R+‖u‖ := (norm_sub_le _ _).trans (add_le_add hy (le_refl _))
    calc
      _ ≤ ‖-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u))‖ +
          ‖(1/(4*η)) • innerSL ℝ (y-u)‖ := norm_sub_le _ _
      _ = (1/2:ℝ)*‖fderiv ℝ V ((1/2:ℝ) • (y+u))‖ + (1/(4*η))*‖y-u‖ := by
        simp only [norm_smul, Real.norm_eq_abs, innerSL_apply_norm]
        rw [abs_of_pos (by positivity : 0 < 1/(4*η))]
        norm_num
      _ ≤ (1/2:ℝ)*(G+β*((R+‖u‖)/2)) + (1/(4*η))*(R+‖u‖) := by
        gcongr
      _ = _ := by simp only [div_eq_mul_inv, mul_inv_rev]; ring
  let W := fun y u : E => V ((1/2:ℝ) • (y+u)) + ‖u-y‖^2/(8*η)
  let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) - (1/(4*η)) • innerSL ℝ (y-u)
  obtain ⟨R,S,hR,hS,hcond,hSR,hSν,hder⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScore.reflected_conditional_covariance hα hV hH hη
  let _ : IsMarkovKernel S := hS
  obtain ⟨hmin,hgrad⟩ := potential_controls hα hV hH
  have hαβr : (α:ℝ) ≤ β := hαβ
  have hβInv : (β:ℝ) ≤ 1/η := (le_div_iff₀ hη).2 hβη
  have hc : 0 ≤ (1/η-(α:ℝ))/4 := div_nonneg (sub_nonneg.mpr (hαβr.trans hβInv)) (by norm_num)
  dsimp only
  refine ⟨R,S,hR,hS,hcond,hSR,?_⟩
  intro y
  have hSy : S y = (volume : Measure E).tilted (fun u => -W y u) := by
    rw [hSν y]
    congr 1
    funext u
    dsimp [W]
    rw [norm_sub_rev u y]
    ring
  obtain ⟨hWC,hsC,hcalc⟩ := conditional_derivatives hV η y
  have hDn (u : E) : ‖fderiv ℝ (s y) u‖ ≤ (1/η-(α:ℝ))/4 :=
    sharp_score_norm hV hαβr hη hβη hH ((1/2:ℝ) • (y+u)) (fderiv ℝ (s y) u)
      (fun v w => (hcalc u v w).2)
  refine ⟨hSy,hWC,hsC,hcalc,?_,hDn,?_⟩
  · intro u v
    rw [(hcalc u v v).1,real_inner_self_eq_norm_sq]
    have hco : 1/(4*η) = (1/η)/4 := by ring
    rw [hco]
    nlinarith [(hH ((1/2:ℝ) • (y+u)) v).1]
  · intro a
    obtain ⟨hqC,hqG⟩ := directional_score (s y) hsC hc hDn a
    let b := ((β:ℝ)+η⁻¹)/4
    let A := ‖gradient V 0‖/2+b*‖y‖
    let m := V 0 - (α:ℝ)⁻¹/2*‖gradient V 0‖^2
    have hb : 0 ≤ b := by dsimp [b]; positivity
    have hA : 0 ≤ A := by dsimp [A]; positivity
    have hqgrowth (u : E) : ‖s y u a‖ ≤ (A*‖a‖)+(b*‖a‖)*‖u‖ := by
      have hsg := score_envelope V hgrad (NNReal.coe_nonneg β) hη y u (le_refl ‖y‖)
      calc
        ‖s y u a‖ ≤ ‖s y u‖*‖a‖ := (s y u).le_opNorm a
        _ ≤ (A+b*‖u‖)*‖a‖ := mul_le_mul_of_nonneg_right hsg (norm_nonneg _)
        _ = _ := by ring
    have hweight (u : E) : Real.exp (-W y u) ≤
        Real.exp (-m+‖y‖^2/(8*η))*Real.exp (-(1/(16*η))*‖u‖^2) := by
      have hw := weight_envelope V hmin hη y u (le_refl ‖y‖)
      convert hw using 1
      dsimp [W]
      rw [norm_sub_rev u y]
      congr 1
      ring
    have hLp : MemLp (fun u => s y u a) 2 (S y) := by
      rw [hSy]
      exact (weighted_linear_memLp (fun u => -W y u) (fun u => s y u a)
        hWC.continuous.neg hqC.continuous (by positivity) (Real.exp_nonneg _)
        (mul_nonneg hA (norm_nonneg _)) (mul_nonneg hb (norm_nonneg _)) hweight hqgrowth).2
    exact ⟨hqC,hLp,admissible_of_memLp_gradient_bound (S y) (fun u => s y u a) hqC hLp hqG,hqG⟩

end AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScoreDomain

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
The paper works on R^d, d>=1, with C2 potential V and genuine Hessian bounds alpha I<=D2V<=beta I, explicit 0<alpha<=beta, eta>0 and beta eta<=1.	E is a finite-dimensional real inner-product space with its Borel structure and canonical volume. Dimension zero is allowed. V:E->R is C2. Alpha and beta are finite nonnegative reals with explicit 0<alpha<=beta. For every x,v the genuine second derivative satisfies alpha norm(v)^2<=D2V(x)[v,v]<=beta norm(v)^2. Eta is positive and beta eta<=1. The explicit alpha<=beta condition is retained even in dimension zero, where the Hessian inequalities alone do not imply it.	generalization	The theorem allows any finite-dimensional real inner-product space including zero dimension. Explicit alpha<=beta and the original upper scale restriction remain in place; the former is needed to keep the sharp constant nonnegative in zero dimension.
Unit direction a in the score-variance input.	Every direction a, with gradient bound L norm(a).	generalization	Evaluation is linear in a; retaining norm(a) specializes exactly to the unit-direction bound.
Gradient and symmetric score derivative in Euclidean coordinates.	Continuous-dual score with Frechet derivative; real Riesz identification supplies the symmetric endomorphism used in the norm proof.	same	The Riesz map preserves norms and inner products; signs and all factors are retained.
The conditional Poincare application uses noncompact directional scores.	For the same actual conditional version, directional scores are C1, MemLp 2 and Poincare.Admissible, derived from Gaussian tails.	source-implicit	Normalization, measurability, second moments, centered square and gradient square integrability are proved. Admissible is only the input domain; no inequality or its extension is assumed or claimed.

Actual conditional curvature, sharp score derivative norm and noncompact directional-score domain only. No curvature-to-Poincare theorem, noncompact test extension of that inequality, conditional variance bound, general L2/H1 differentiation, macroscopic coercivity, process/mixing/implementation/error/query-cost or complete-paper theorem.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Abstract finite-dimensional and zero-dimensional extension, disclosed rather than exact source-domain equivalence. — No positive-dimension hypothesis; lesson discloses generalization. Explicit alpha<=beta retains nonnegative bound in zero dimension.
conclusion: Detailed noncompact moment/domain proof elaborates implicit source conditions without new assumptions. — conditional_derivatives retains minus quarter Hessian in Ds and plus quadratic term. Rayleigh/Riesz supplies sharp norm. Weighted Gaussian moments and gradient bounds establish the domain.
scopes: Accepted pre-Poincare edge only; Admissible alone does not put a function in an arbitrary future Poincare test class. — Admissible is exactly L1, centered-square and gradient-square integrability, not Satisfies. Its membership does not imply membership in an arbitrary future test class; extension remains open.

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

Actual conditional curvature, sharp score derivative norm and noncompact directional-score domain only. No curvature-to-Poincare theorem, noncompact test extension of that inequality, conditional variance bound, general L2/H1 differentiation, macroscopic coercivity, process/mixing/implementation/error/query-cost or complete-paper theorem.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

ContDiffAt.isSymmSndFDerivAt
ContinuousLinearMap.norm_eq_iSup_rayleighQuotient
Real.add_one_le_exp
MeasureTheory.integrable_tilted_iff
MeasureTheory.memLp_two_iff_integrable_sq
MeasureTheory.MemLp.integrable_sq

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS v1 Appendix C.1 — Conditional Hessian and score derivative estimates before conditional Poincare, with explicit noncompact test-domain proof.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Genuine conditional-potential Hessian and positive lower curvature
Local proof component; source adapter/review separate Genuine C1 score derivative and sharp operator norm bound
Local proof component; source adapter/review separate Actual directional-score L2 and finite centered-square/gradient-square domain

Actual reflected conditional expectation and its score derivative

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; ASTIS expanded mathematical proof.

arXiv:2609.06905v1 · Appendix C.1, conditional density and differentiation in C.1; selected proof step only · faithful paraphrase

Complete source statement (ASTIS restatement)

The conditional density of Y_- given Y_+=y is proportional to exp(-V((y+u)/2)-norm(y-u)^2/(8eta)). Differentiating the conditional expectation of f gives its covariance with the unnormalized logarithmic derivative -grad V((y+u)/2)/2-(y-u)/(4eta). This is the differentiation input to Appendix C.1; the subsequent conditional variance estimate and extension to general L2 inputs are separate obligations.

The paper works on R^d, d>=1, with C2 potential V and genuine Hessian bounds 0<alpha I<=D2V<=beta I, and 0<eta<=1/beta.
Let X have normalized density proportional to exp(-V), and let Y_+=X+sqrt(eta)Z and Y_-=X-sqrt(eta)Z for independent standard Gaussian Z. The selected differentiation step uses smooth compactly supported real f.

conditional covariance derivative

\[DT_f(y)=\mathbb E_{S_y}[f\,s_y]-\mathbb E_{S_y}[f]\,\mathbb E_{S_y}[s_y].\]

Read the formalized proofs

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

ASTIS mathematical exposition

Actual reflected conditional expectation and its score derivative

AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScore.reflected_conditional_covariance · theorem · Teaching coverage

Statement

There exist Markov kernels R and S such that R disintegrates the actual law of (Y,X), S_y is the pushforward of R_y by x↦2x−y, and every S_y has the normalized density proportional to exp(−V((y+u)/2)−‖y−u‖²/(8η)). For every smooth compactly supported real f and every y, both s_y and f s_y are S_y-integrable, and the Fréchet derivative of T_f(y)=∫f dS_y is ∫f s_y dS_y−T_f(y)∫s_y dS_y. Here s_y(u)=−DV((y+u)/2)/2−⟨y−u,·⟩/(4η) is a continuous linear functional, the derivative of the unnormalized log weight.

\[DT_f(y)=\mathbb E_{S_y}[f\,s_y]-\mathbb E_{S_y}[f]\,\mathbb E_{S_y}[s_y].\]

All objects and hypotheses

E is a finite-dimensional real inner-product space with its Borel measurable structure; dimension zero is allowed. The potential V:E→R is C².
Constants α and β are finite nonnegative reals, α>0, and the actual second Fréchet derivative satisfies α‖v‖²≤D²V(x)[v,v]≤β‖v‖² for every x,v. The scale η is positive. No upper bound βη≤1 is needed for this differentiation step.
μ is volume tilted by −V and J is the law of (X,X+sqrt(η)Z), where X has law μ and Z is an independent standard Gaussian. Normalization and all integrability used in the proof are derived from the curvature assumptions.
The public conclusion quantifies over every C∞ compactly supported real test f and every parameter y. The internally proved differentiation argument works for bounded continuous f, including the noncompact test f≡1 needed for the normalizer.

Mathematical proof

1. Push the actual backward conditional kernel through reflection

Positive curvature gives a positive finite Gibbs normalizer. The existing Gaussian conditional-kernel theorem therefore supplies a Markov kernel R disintegrating (Y,X), with R_y equal to μ tilted by −‖x−y‖²/(2η). Tilting twice adds the exponents. The affine change of variables u=2x−y has inverse x=(u+y)/2 and sends volume to 2^(−dim E) times volume. A constant positive factor cancels from a normalized tilt. Substituting the inverse yields W(y,u)=−V((y+u)/2)−‖y−u‖²/(8η). To obtain joint measurability, define S by mapping the product kernel id×R through (y,x)↦2x−y; its fibers are exactly these pushforwards.

\[S_y=(x\mapsto2x-y)_\#R_y=\frac{e^{W(y,u)}\,du}{Z(y)},\qquad Z(y)=\int e^{W(y,u)}\,du.\]

Corresponding Lean step

actual_reflected_density combines normalized_augmentation_density, exists_tilted_isCondKernel, tilted_tilted, an affine measurable equivalence and the Haar scaling formula. reflected_kernel uses Kernel.prod and Kernel.map to construct the measurable Markov kernel.

2. Extract a potential lower bound and linear gradient growth

Apply the existing quadratic-regularization theorem with regularization parameter zero. Its strong-convexity conclusion gives V(x)≥V(0)+⟨∇V(0),x⟩+α‖x‖²/2. Completing the square gives the global lower bound m=V(0)−‖∇V(0)‖²/(2α). The same theorem's β-Lipschitz gradient gives ‖DV(x)‖≤G+β‖x‖, where G=‖∇V(0)‖. These estimates are consequences of the supplied genuine Hessian inequalities, not new premises.

\[V(x)\ge m:=V(0)-\frac{\|\nabla V(0)\|^2}{2\alpha},\qquad\|DV(x)\|\le G+\beta\|x\|.\]

Corresponding Lean step

potential_controls specializes strongConvexOn_and_lipschitzWith_gradient_add_quadratic at r=0, uses its first-order lower bound, and identifies gradient and derivative norms through the Riesz isometry.

3. Derive an integrable local envelope, including the normalizer

Fix y₀ and put R=‖y₀‖+1. On the unit ball around y₀, ‖y‖≤R. The triangle inequality gives ‖u‖²≤2‖y−u‖²+2R². Thus w=e^W is bounded by C exp(−a‖u‖²), where a=1/(16η) and C=exp(−m+R²/(8η)). The derivative score obeys ‖s_y(u)‖≤A+b‖u‖ with b=(β+η⁻¹)/4 and A=G/2+bR. Absorb this linear factor into a weaker Gaussian: (A+br)exp(−ar²)≤(A+b(1+2/a))exp(−ar²/2). Consequently C(A+b(1+2/a))exp(−a‖u‖²/2) dominates ‖D_yw(y,u)‖ uniformly in that ball and is integrable on E. A fixed-y version also proves integrability of w itself.

\[\|D_yw(y,u)\|\le C\bigl(A+b(1+2/a)\bigr)e^{-a\|u\|^2/2}.\]

Corresponding Lean step

weight_envelope, score_envelope and absorb_linear feed local_controls. Gaussian integrability is reused from integrable_exp_neg_mul_norm_sq; no domination hypothesis is accepted by the public theorem.

4. Differentiate the numerator and the noncompact normalizer

The ordinary chain rule gives D_yw=w s_y. For a continuous test f with |f|≤M, multiply the local envelope by M. Continuity of f, V and DV provides the required measurability, so dominated differentiation proves DN_f(y)=∫f(u)w(y,u)s_y(u)du for N_f=∫fw. Apply the same argument separately to f≡1 to obtain DZ(y)=∫w(y,u)s_y(u)du. This second use is essential: compact support of the original test cannot justify differentiating Z. Positivity of the exponential and its already proved integrability give Z(y)>0.

\[DN_f(y)=\int fws_y\,du,\qquad DZ(y)=\int ws_y\,du,\qquad Z(y)>0.\]

Corresponding Lean step

weight_derivative proves the chain-rule formula. unnormalized_derivative applies hasFDerivAt_integral_of_dominated_of_fderiv_le with the derived neighborhood envelope. normalized_covariance invokes it both for f and for the constant one.

5. Normalize and center the derivative

Integration against a tilt is division by Z. The quotient rule gives D(N_f/Z)=Z⁻¹DN_f−(N_f/Z)Z⁻¹DZ. Each weighted volume integral becomes an expectation under S_y. This gives the stated covariance and proves that the score and f times the score are integrable under S_y. The unnormalized score s_y is not itself the derivative of log of the normalized density: that derivative is s_y−E_{S_y}s_y. Under the real Riesz identification, the continuous-linear-functional formula is the source's vector gradient formula.

\[D(N_f/Z)=Z^{-1}DN_f-(N_f/Z)Z^{-1}DZ.\]

Corresponding Lean step

quotient_derivative uses the derivative of inversion and multiplication. integral_tilted and integrable_tilted_iff convert the derivative and its integrability to the actual normalized conditional measure.

6. Recover the smooth compactly supported source inputs

A continuous compactly supported f has bounded norm, so every source test satisfies the bounded-continuous premise used internally. Substitute the everywhere fiber identity for S into both the parameterized expectation and the derivative. This uses the same constructed R and S for all tests and all y. The result supplies the differentiation input to Appendix C.1's conditional gradient–variance estimate; the quantitative variance estimate and its subsequent Sobolev extension are separate obligations.

\[f\in C_c^\infty(E)\Longrightarrow\sup_u|f(u)|<\infty.\]

Corresponding Lean step

The final assembly obtains the norm bound from bddAbove_range_of_hasCompactSupport and rewrites normalized_covariance through hSν.

Lean statement · reflected_conditional_covariance

One public theorem constructs both Markov kernels from the genuine Gibbs/Gaussian law, identifies every reflected fiber and differentiates every smooth compactly supported test expectation at every parameter.

theorem reflected_conditional_covariance {E : Type u} [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)) (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) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
      (1/(4*η)) • innerSL ℝ (y-u)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      (∀ y, S y = (volume : Measure E).tilted
        (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))) ∧
      ∀ (f : E → ℝ), ContDiff ℝ ∞ f → HasCompactSupport f →
        ∀ y, Integrable (s y) (S y) ∧ Integrable (fun u => f u • s y u) (S y) ∧
          HasFDerivAt (fun z => ∫ u, f u ∂S z)
            ((∫ u, f u • s y u ∂S y) -
              (∫ u, f u ∂S y) • (∫ u, s y u ∂S y)) y

Exact module and namespace context

Lean proof · reflected_conditional_covariance

Local helpers derive normalization, the affine pushforward density, Gaussian domination, both unnormalized derivatives and the normalized covariance. The public theorem assumes none of those conclusions.

theorem reflected_conditional_covariance {E : Type u} [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)) (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) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
      (1/(4*η)) • innerSL ℝ (y-u)
    ∃ R S : Kernel E E, IsMarkovKernel R ∧ IsMarkovKernel S ∧
      (J.map Prod.swap).IsCondKernel R ∧
      (∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) ∧
      (∀ y, S y = (volume : Measure E).tilted
        (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))) ∧
      ∀ (f : E → ℝ), ContDiff ℝ ∞ f → HasCompactSupport f →
        ∀ y, Integrable (s y) (S y) ∧ Integrable (fun u => f u • s y u) (S y) ∧
          HasFDerivAt (fun z => ∫ u, f u ∂S z)
            ((∫ u, f u • s y u ∂S y) -
              (∫ u, f u ∂S y) • (∫ u, s y u ∂S y)) y := by
  have map_tilt {E F : Type u} [MeasurableSpace E] [MeasurableSpace F]
      (μ : Measure E) (e : E ≃ᵐ F) (f : E → ℝ) (hf : Measurable f) :
      (μ.tilted f).map e = (μ.map e).tilted (f ∘ e.symm) := by
    unfold Measure.tilted
    rw [AutoSamplingTheory.TechnicalLemmas.Measure.RadonNikodym.measurableEquiv_map_withDensity e _ (by fun_prop)]
    congr 1
    funext y
    have hi : (∫ y, Real.exp (f (e.symm y)) ∂(μ.map e)) = ∫ x, Real.exp (f x) ∂μ := by
      rw [integral_map_equiv e]
      simp only [e.symm_apply_apply]
    simp only [Function.comp_apply, hi]

  have smul_tilt {E : Type u} [MeasurableSpace E] (μ : Measure E) (f : E → ℝ)
      (hf : Measurable f) {c : ℝ} (hc : 0 < c) :
      (ENNReal.ofReal c • μ).tilted f = μ.tilted f := by
    unfold Measure.tilted
    rw [integral_smul_measure, ENNReal.toReal_ofReal hc.le, smul_eq_mul,
      withDensity_smul_measure, ← withDensity_smul _ (by fun_prop)]
    congr 1
    funext x
    simp only [Pi.smul_apply, smul_eq_mul]
    rw [← ENNReal.ofReal_mul hc.le]
    congr 1
    by_cases hz : (∫ x, Real.exp (f x) ∂μ) = 0
    · simp [hz]
    · field_simp

  have reflection_affine {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (y : E) :
      ∃ e : E ≃ᵐ E,
        (∀ x, e x = (2:ℝ) • x-y) ∧
        (∀ u, e.symm u = (1/2:ℝ) • (u+y)) ∧
        Measure.map e (volume : Measure E) =
          ENNReal.ofReal (abs ((2:ℝ)^Module.finrank ℝ E)⁻¹) • volume := by
    let e : E ≃ᵐ E :=
      { toFun := fun x => (2:ℝ) • x-y
        invFun := fun u => (1/2:ℝ) • (u+y)
        left_inv := by intro x; simp [smul_smul]
        right_inv := by intro u; simp [smul_smul]
        measurable_toFun := by
          change Measurable (fun x : E => (2:ℝ) • x-y)
          fun_prop
        measurable_invFun := by
          change Measurable (fun u : E => (1/2:ℝ) • (u+y))
          fun_prop }
    refine ⟨e, fun _ => rfl, fun _ => rfl, ?_⟩
    change Measure.map (fun x : E => (2:ℝ) • x-y) volume = _
    simp only [sub_eq_add_neg]
    change Measure.map ((fun x : E => x + -y) ∘ (fun x : E => (2:ℝ) • x)) volume = _
    rw [← Measure.map_map (by fun_prop) (by fun_prop),
      Measure.map_addHaar_smul volume (by norm_num : (2:ℝ) ≠ 0),
      Measure.map_smul, map_add_right_eq_self]

  have reflected_exponent {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      (V : E → ℝ) (η : ℝ) (y u : E) :
      -V ((1/2:ℝ) • (u+y)) - ‖(1/2:ℝ) • (u+y)-y‖^2/(2*η) =
        -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η) := by
    rw [show (1/2:ℝ) • (u+y)-y = (1/2:ℝ) • (u-y) by module]
    rw [norm_smul, Real.norm_eq_abs, norm_sub_rev u y, add_comm u y]
    norm_num
    simp only [div_eq_mul_inv, mul_inv_rev]
    ring

  have actual_reflected_density {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      {V : E → ℝ} {α η : ℝ} (hα : 0 < α) (hV : ContDiff ℝ 2 V)
      (hH : ∀ x v : E, α*‖v‖^2 ≤ (fderiv ℝ (fderiv ℝ V) x v) v)
      (hη : 0 < η) :
      let μ := (volume : Measure E).tilted (fun x => -V x)
      let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
        (μ.prod (stdGaussian E))
      ∃ R : Kernel E E, IsMarkovKernel R ∧ (J.map Prod.swap).IsCondKernel R ∧
        ∀ y, Measure.map (fun x => (2:ℝ) • x-y) (R y) =
          (volume : Measure E).tilted
            (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η)) := by
    have density := AutoSamplingTheory.ExampleCases.ProximalBPS.GibbsAugmentation.normalized_augmentation_density
       hα hV hH hη
    have hi : Integrable (fun x => Real.exp (-V x)) (volume : Measure E) := by
      by_contra hn
      exact (ne_of_gt density.1) (integral_undef hn)
    have : IsProbabilityMeasure ((volume : Measure E).tilted (fun x => -V x)) :=
      isProbabilityMeasure_tilted hi
    have hVm : Measurable V := hV.continuous.measurable
    obtain ⟨R,hR,hRf,hcond⟩ :=
      AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel.exists_tilted_isCondKernel
        ((volume : Measure E).tilted (fun x => -V x)) hη
    refine ⟨R,hR,hcond,?_⟩
    intro y
    rw [hRf y, tilted_tilted hi]
    obtain ⟨e,he,heinv,hemap⟩ := reflection_affine y
    rw [show (fun x => (2:ℝ) • x-y) = e from (funext he).symm]
    rw [map_tilt _ e _ (by fun_prop), hemap,
      smul_tilt _ _ (by fun_prop) (by positivity)]
    congr 1
    funext u
    simp only [Function.comp_apply, Pi.add_apply, heinv]
    simpa only [neg_div, sub_eq_add_neg] using reflected_exponent V η y u

  have reflected_kernel {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E] (R : Kernel E E) [IsMarkovKernel R] :
      ∃ S : Kernel E E, IsMarkovKernel S ∧
        ∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y) := by
    let F := fun p : E × E => (2:ℝ) • p.2-p.1
    have hF : Measurable F := by fun_prop
    let S := (Kernel.id ×ₖ R).map F
    have : IsMarkovKernel S := Kernel.IsMarkovKernel.map _ hF
    refine ⟨S, inferInstance, ?_⟩
    intro y
    change ((Kernel.id ×ₖ R).map F) y = _
    rw [Kernel.map_apply _ hF, Kernel.prod_apply, Kernel.id_apply,
      Measure.dirac_prod, Measure.map_map hF (by fun_prop)]
    rfl



  have potential_controls {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] {V : E → ℝ} {α β : ℝ≥0}
      (hα : 0 < (α : ℝ)) (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) :
      (∀ x, V 0 - (α : ℝ)⁻¹/2*‖gradient V 0‖^2 ≤ V x) ∧
      (∀ x, ‖fderiv ℝ V x‖ ≤ ‖gradient V 0‖ + (β : ℝ)*‖x‖) := by
    have hd : Differentiable ℝ V := hV.differentiable (by norm_num)
    have hreg := QuadraticRegularization.strongConvexOn_and_lipschitzWith_gradient_add_quadratic
      hV hH (r := 0) (0 : E)
    simp only [NNReal.coe_zero, zero_div, zero_mul, add_zero] at hreg
    constructor
    · intro x
      have hfirst := StrongConvexFirstOrder.firstOrder_lower_bound_of_strongConvexOn hreg.1
          (fun z _ => (hd z).hasGradientAt) (x := 0) (y := x)
          (Set.mem_univ _) (Set.mem_univ _)
      simp only [sub_zero] at hfirst
      have hinner := (abs_le.mp (abs_real_inner_le_norm (gradient V 0) x)).1
      have hyoung := two_mul_le_add_mul_sq (a := ‖x‖) (b := ‖gradient V 0‖) hα
      nlinarith
    · intro x
      rw [← toDual_gradient]
      rw [(toDual ℝ E).norm_map]
      calc
        ‖gradient V x‖ ≤ ‖gradient V x - gradient V 0‖ + ‖gradient V 0‖ :=
          norm_le_norm_sub_add _ _
        _ ≤ (β : ℝ)*‖x‖ + ‖gradient V 0‖ := by
          apply add_le_add _ (le_refl _)
          simpa using hreg.2.norm_sub_le x 0
        _ = _ := add_comm _ _

  have weight_envelope {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      (V : E → ℝ) {m η R : ℝ} (hV : ∀ x, m ≤ V x) (hη : 0 < η)
      (y u : E) (hy : ‖y‖ ≤ R) :
      Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η)) ≤
        Real.exp (-m+R^2/(8*η)) * Real.exp (-(1/(16*η))*‖u‖^2) := by
    have hR : 0 ≤ R := (norm_nonneg y).trans hy
    have htri : ‖u‖ ≤ ‖y-u‖ + R := by
      have h := norm_le_norm_sub_add u y
      rw [norm_sub_rev u y] at h
      linarith
    have hsq : ‖u‖^2 ≤ 2*‖y-u‖^2 + 2*R^2 := by
      have ht := (sq_le_sq₀ (norm_nonneg u) (by positivity)).mpr htri
      nlinarith [sq_nonneg (‖y-u‖-R)]
    have hq := div_le_div_of_nonneg_right hsq (by positivity : 0 ≤ 16*η)
    have hdiv : ‖u‖^2/(16*η) ≤ ‖y-u‖^2/(8*η) + R^2/(8*η) := by
      have heq : (2*‖y-u‖^2+2*R^2)/(16*η) = ‖y-u‖^2/(8*η)+R^2/(8*η) := by
        field_simp
        ring
      rw [heq] at hq
      exact hq
    rw [← Real.exp_add]
    apply Real.exp_le_exp.mpr
    have hv := hV ((1/2:ℝ) • (y+u))
    simp only [div_eq_mul_inv, one_mul] at hdiv ⊢
    nlinarith

  have score_envelope {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [CompleteSpace E] (V : E → ℝ) {G β η R : ℝ}
      (hG : ∀ x, ‖fderiv ℝ V x‖ ≤ G+β*‖x‖) (hβ : 0 ≤ β) (hη : 0 < η)
      (y u : E) (hy : ‖y‖ ≤ R) :
      ‖-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
        (1/(4*η)) • innerSL ℝ (y-u)‖ ≤
        G/2 + ((β+η⁻¹)/4)*R + ((β+η⁻¹)/4)*‖u‖ := by
    have hmid : ‖(1/2:ℝ) • (y+u)‖ ≤ (R+‖u‖)/2 := by
      rw [norm_smul, Real.norm_eq_abs]
      norm_num
      have h := norm_add_le y u
      linarith
    have hfd : ‖fderiv ℝ V ((1/2:ℝ) • (y+u))‖ ≤ G+β*((R+‖u‖)/2) :=
      (hG _).trans (add_le_add (le_refl _) (mul_le_mul_of_nonneg_left hmid hβ))
    have hdiff : ‖y-u‖ ≤ R+‖u‖ := (norm_sub_le _ _).trans (add_le_add hy (le_refl _))
    calc
      _ ≤ ‖-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u))‖ +
          ‖(1/(4*η)) • innerSL ℝ (y-u)‖ := norm_sub_le _ _
      _ = (1/2:ℝ)*‖fderiv ℝ V ((1/2:ℝ) • (y+u))‖ + (1/(4*η))*‖y-u‖ := by
        simp only [norm_smul, Real.norm_eq_abs, innerSL_apply_norm]
        rw [abs_of_pos (by positivity : 0 < 1/(4*η))]
        norm_num
      _ ≤ (1/2:ℝ)*(G+β*((R+‖u‖)/2)) + (1/(4*η))*(R+‖u‖) := by
        gcongr
      _ = _ := by simp only [div_eq_mul_inv, mul_inv_rev]; ring

  have absorb_linear {a A b r : ℝ} (ha : 0 < a) (hA : 0 ≤ A) (hb : 0 ≤ b) (hr : 0 ≤ r) :
      (A+b*r)*Real.exp (-a*r^2) ≤
        (A+b*(1+2/a))*Real.exp (-(a/2)*r^2) := by
    let t := a/2*r^2
    have ht : 0 ≤ t := by dsimp [t]; positivity
    have he1 : 1 ≤ Real.exp t := Real.one_le_exp_iff.mpr ht
    have he2 : t ≤ Real.exp t := by linarith [Real.add_one_le_exp t]
    have hr2 : r^2 ≤ (2/a)*Real.exp t := by
      have hh : r^2*a ≤ 2*Real.exp t := by dsimp [t] at he2; nlinarith
      calc
        r^2 ≤ (2*Real.exp t)/a := (le_div_iff₀ ha).2 hh
        _ = _ := by ring
    have hrb : r ≤ (1+2/a)*Real.exp t := by
      nlinarith [sq_nonneg (r-1)]
    have hcoef : A+b*r ≤ (A+b*(1+2/a))*Real.exp t := by
      have h1 := mul_le_mul_of_nonneg_left he1 hA
      have h2 := mul_le_mul_of_nonneg_left hrb hb
      nlinarith
    have h := mul_le_mul_of_nonneg_right hcoef (Real.exp_nonneg (-a*r^2))
    rw [mul_assoc, ← Real.exp_add] at h
    have heq : t + -a*r^2 = -(a/2)*r^2 := by dsimp [t]; ring
    rw [heq] at h
    exact h



  have weight_derivative {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] {V : E → ℝ} (hV : Differentiable ℝ V)
      (η : ℝ) (y u : E) :
      HasFDerivAt
        (fun z => Real.exp (-V ((1/2:ℝ) • (z+u)) - ‖z-u‖^2/(8*η)))
        (Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η)) •
          (-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
            (1/(4*η)) • innerSL ℝ (y-u))) y := by
    have hv := (hV ((1/2:ℝ) • (y+u))).hasFDerivAt.comp y
      (((hasFDerivAt_id y).add_const u).const_smul (1/2:ℝ))
    have hq := (((hasFDerivAt_id y).sub_const u).norm_sq).const_mul (8*η)⁻¹
    convert (hv.neg.sub hq).exp using 1 <;>
      first | (congr 1; ext z; simp [div_eq_mul_inv]; ring) |
        (ext v; simp [div_eq_mul_inv, mul_inv_rev]; ring)



  have quotient_derivative {E : Type u} [NormedAddCommGroup E] [NormedSpace ℝ E]
      {N Z : E → ℝ} {DN DZ : E →L[ℝ] ℝ} {y : E}
      (hN : HasFDerivAt N DN y) (hZ : HasFDerivAt Z DZ y) (hz : Z y ≠ 0) :
      HasFDerivAt (fun z => N z/Z z)
        ((Z y)⁻¹ • DN - (N y/Z y) • ((Z y)⁻¹ • DZ)) y := by
    have hi := (hasFDerivAt_inv hz).comp y hZ
    have hp := hN.mul hi
    convert! hp using 1
    ext v
    simp [div_eq_mul_inv, pow_two]
    ring


  have local_controls {E : Type u} [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)) (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) :
      let w := fun y u : E => Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
      let dw := fun y u : E => w y u •
        (-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) - (1/(4*η)) • innerSL ℝ (y-u))
      (∀ y, Integrable (w y) (volume : Measure E)) ∧
        ∀ y₀, ∃ bound : E → ℝ, Integrable bound (volume : Measure E) ∧
          (∀ u, 0 ≤ bound u) ∧
          ∀ y ∈ Metric.ball y₀ 1, ∀ u, ‖dw y u‖ ≤ bound u := by
    let w := fun y u : E => Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
    let dw := fun y u : E => w y u •
      (-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) - (1/(4*η)) • innerSL ℝ (y-u))
    let m := V 0 - (α : ℝ)⁻¹/2*‖gradient V 0‖^2
    let G := ‖gradient V 0‖
    let a := 1/(16*η)
    have ha : 0 < a := by dsimp [a]; positivity
    obtain ⟨hlower,hgrowth⟩ := potential_controls hα hV hH
    have hVc : Continuous V := hV.continuous
    have hgauss : Integrable (fun u : E => Real.exp (-a*‖u‖^2)) volume :=
      Integrability.integrable_exp_neg_mul_norm_sq ha
    constructor
    · intro y
      refine (hgauss.const_mul (Real.exp (-m+‖y‖^2/(8*η)))).mono' (by dsimp [w]; fun_prop) ?_
      filter_upwards with u
      rw [Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
      exact weight_envelope V hlower hη y u (le_refl _)
    · intro y₀
      let R := ‖y₀‖+1
      let b := ((β : ℝ)+η⁻¹)/4
      let A := G/2+b*R
      let C := Real.exp (-m+R^2/(8*η))
      let bound := fun u : E => C*(A+b*(1+2/a))*Real.exp (-(a/2)*‖u‖^2)
      have hR : 0 ≤ R := by dsimp [R]; positivity
      have hb : 0 ≤ b := by dsimp [b]; positivity
      have hA : 0 ≤ A := by dsimp [A,G]; positivity
      have hC : 0 ≤ C := Real.exp_nonneg _
      refine ⟨bound, (Integrability.integrable_exp_neg_mul_norm_sq
        (E := E) (a := a/2) (by positivity)).const_mul _, ?_, ?_⟩
      · intro u
        dsimp [bound]
        positivity
      · intro y hy u
        have hyR : ‖y‖ ≤ R := by
          have ht := norm_le_norm_sub_add y y₀
          have hd : ‖y-y₀‖ < 1 := by simpa only [Metric.mem_ball, dist_eq_norm] using hy
          dsimp [R]
          linarith
        have hw : w y u ≤ C*Real.exp (-a*‖u‖^2) := weight_envelope V hlower hη y u hyR
        have hs : ‖-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
            (1/(4*η)) • innerSL ℝ (y-u)‖ ≤ A+b*‖u‖ :=
          score_envelope V hgrowth β.coe_nonneg hη y u hyR
        have hp := absorb_linear ha hA hb (norm_nonneg u)
        change ‖dw y u‖ ≤ bound u
        dsimp only [dw]
        rw [norm_smul, Real.norm_eq_abs, abs_of_pos (Real.exp_pos _)]
        calc
          _ ≤ w y u*(A+b*‖u‖) := mul_le_mul_of_nonneg_left hs (Real.exp_nonneg _)
          _ ≤ (C*Real.exp (-a*‖u‖^2))*(A+b*‖u‖) :=
            mul_le_mul_of_nonneg_right hw (by positivity)
          _ = C*((A+b*‖u‖)*Real.exp (-a*‖u‖^2)) := by ring
          _ ≤ C*((A+b*(1+2/a))*Real.exp (-(a/2)*‖u‖^2)) :=
            mul_le_mul_of_nonneg_left hp hC
          _ = bound u := by dsimp [bound]; ring



  have unnormalized_derivative {E : Type u} [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)) (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)
      (f : E → ℝ) (hf : Continuous f) {M : ℝ} (hM : 0 ≤ M) (hMf : ∀ u, ‖f u‖ ≤ M) :
      let w := fun y u : E => Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
      let dw := fun y u : E => w y u •
        (-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) - (1/(4*η)) • innerSL ℝ (y-u))
      ∀ y, Integrable (fun u => f u*w y u) (volume : Measure E) ∧
        Integrable (fun u => f u • dw y u) (volume : Measure E) ∧
        HasFDerivAt (fun z => ∫ u, f u*w z u ∂(volume : Measure E))
          (∫ u, f u • dw y u ∂(volume : Measure E)) y := by
    let w := fun y u : E => Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
    let dw := fun y u : E => w y u •
      (-(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) - (1/(4*η)) • innerSL ℝ (y-u))
    have hVc : Continuous V := hV.continuous
    have hVdc : Continuous (fderiv ℝ V) := (hV.fderiv_right (m := 1) (by norm_num)).continuous
    have hVd : Differentiable ℝ V := hV.differentiable (by norm_num)
    obtain ⟨hwI,hcontrols⟩ := local_controls hα hV hH hη
    have hfwI (y : E) : Integrable (fun u => f u*w y u) (volume : Measure E) := by
      refine ((hwI y).const_mul M).mono' (by dsimp [w]; fun_prop) ?_
      filter_upwards with u
      rw [norm_mul, Real.norm_eq_abs (w y u), abs_of_pos (Real.exp_pos _)]
      exact mul_le_mul_of_nonneg_right (hMf u) (Real.exp_nonneg _)
    dsimp only
    intro y
    obtain ⟨B,hBI,hB0,hB⟩ := hcontrols y
    have hbound : ∀ u, ∀ z ∈ Metric.ball y 1, ‖f u • dw z u‖ ≤ M*B u := by
      intro u z hz
      rw [norm_smul]
      exact mul_le_mul (hMf u) (hB z hz u) (norm_nonneg _) hM
    have hfdI : Integrable (fun u => f u • dw y u) (volume : Measure E) := by
      refine (hBI.const_mul M).mono' (by dsimp [dw,w]; fun_prop) ?_
      exact ae_of_all _ (fun u => hbound u y (Metric.mem_ball_self zero_lt_one))
    refine ⟨hfwI y,hfdI,?_⟩
    apply hasFDerivAt_integral_of_dominated_of_fderiv_le
      (F' := fun z u => f u • dw z u) (bound := fun u => M*B u)
      (Metric.ball_mem_nhds y zero_lt_one)
    · exact Filter.Eventually.of_forall (fun z => by fun_prop)
    · exact hfwI y
    · exact hfdI.aestronglyMeasurable
    · exact ae_of_all _ hbound
    · exact hBI.const_mul M
    · filter_upwards with u
      intro z _
      exact (weight_derivative hVd η z u).const_mul (f u)



  have normalized_tilt_integral {E F : Type u} [MeasurableSpace E]
      [NormedAddCommGroup F] [NormedSpace ℝ F] (μ : Measure E) (W : E → ℝ) (g : E → F) :
      ∫ u, g u ∂(μ.tilted W) = (∫ u, Real.exp (W u) ∂μ)⁻¹ •
        ∫ u, Real.exp (W u) • g u ∂μ := by
    rw [integral_tilted]
    simp only [div_eq_inv_mul, mul_smul]
    rw [integral_smul]

  have normalized_covariance {E : Type u} [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)) (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)
      (f : E → ℝ) (hf : Continuous f) {M : ℝ} (hM : 0 ≤ M) (hMf : ∀ u, ‖f u‖ ≤ M) :
      let ν := fun y : E => (volume : Measure E).tilted
        (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
      let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
        (1/(4*η)) • innerSL ℝ (y-u)
      ∀ y, Integrable (s y) (ν y) ∧ Integrable (fun u => f u • s y u) (ν y) ∧
        HasFDerivAt (fun z => ∫ u, f u ∂ν z)
          ((∫ u, f u • s y u ∂ν y) - (∫ u, f u ∂ν y) • (∫ u, s y u ∂ν y)) y := by
    let ν := fun y : E => (volume : Measure E).tilted
      (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
    let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
      (1/(4*η)) • innerSL ℝ (y-u)
    let w := fun y u : E => Real.exp (-V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η))
    let dw := fun y u : E => w y u • s y u
    let Z := fun y => ∫ u, w y u ∂(volume : Measure E)
    let N := fun y => ∫ u, f u*w y u ∂(volume : Measure E)
    let DZ := fun y => ∫ u, dw y u ∂(volume : Measure E)
    let DN := fun y => ∫ u, f u • dw y u ∂(volume : Measure E)
    have hN := unnormalized_derivative hα hV hH hη f hf hM hMf
    have hZraw := unnormalized_derivative hα hV hH hη (fun _ : E => (1:ℝ))
      continuous_const (M := 1) (by norm_num) (fun _ => by norm_num)
    have hZ : ∀ y, Integrable (w y) (volume : Measure E) ∧
        Integrable (dw y) (volume : Measure E) ∧ HasFDerivAt Z (DZ y) y := by
      intro y
      simpa only [one_mul, one_smul] using hZraw y
    have hTf (y : E) : (∫ u, f u ∂ν y) = N y/Z y := by
      have hscalar (μ : Measure E) (W g : E → ℝ) :
          ∫ u, g u ∂(μ.tilted W) = (∫ u, Real.exp (W u) ∂μ)⁻¹ •
            ∫ u, Real.exp (W u) • g u ∂μ := by
        rw [integral_tilted]
        simp only [div_eq_inv_mul, mul_smul]
        rw [integral_smul]
      rw [hscalar]
      change (Z y)⁻¹ * (∫ u, w y u * f u ∂(volume : Measure E)) = N y / Z y
      rw [div_eq_mul_inv, mul_comm (N y)]
      congr 1
      apply integral_congr_ae
      filter_upwards with u
      exact mul_comm _ _
    have hTs (y : E) : (∫ u, s y u ∂ν y) = (Z y)⁻¹ • DZ y := by
      exact normalized_tilt_integral _ _ _
    have hTfs (y : E) : (∫ u, f u • s y u ∂ν y) = (Z y)⁻¹ • DN y := by
      rw [normalized_tilt_integral]
      congr 1
      apply integral_congr_ae
      filter_upwards with u
      simp only [dw, smul_smul]
      rw [mul_comm]
    dsimp only
    intro y
    have hsI : Integrable (s y) (ν y) := by
      rw [integrable_tilted_iff (hZ y).1]
      exact (hZ y).2.1
    have hfsI : Integrable (fun u => f u • s y u) (ν y) := by
      rw [integrable_tilted_iff (hZ y).1]
      apply (hN y).2.1.congr
      filter_upwards with u
      simp only [smul_smul]
      rw [mul_comm]
    refine ⟨hsI,hfsI,?_⟩
    change HasFDerivAt (fun z => ∫ u, f u ∂ν z)
      ((∫ u, f u • s y u ∂ν y) - (∫ u, f u ∂ν y) • (∫ u, s y u ∂ν y)) y
    rw [show (fun z => ∫ u, f u ∂ν z) = (fun z => N z/Z z) from funext hTf,
      hTf y,hTs y,hTfs y]
    exact quotient_derivative (hN y).2.2 (hZ y).2.2 (ne_of_gt (integral_exp_pos (hZ y).1))
  dsimp only
  obtain ⟨R,hR,hcond,hν⟩ := actual_reflected_density hα hV (fun x v => (hH x v).1) hη
  let : IsMarkovKernel R := hR
  obtain ⟨S,hS,hSR⟩ := reflected_kernel R
  have hSν (y : E) : S y = (volume : Measure E).tilted
      (fun u => -V ((1/2:ℝ) • (y+u)) - ‖y-u‖^2/(8*η)) :=
    (hSR y).trans (hν y)
  refine ⟨R,S,hR,hS,hcond,hSR,hSν,?_⟩
  intro f hf hfc y
  obtain ⟨M,hM⟩ := hf.continuous.norm.bddAbove_range_of_hasCompactSupport hfc.norm
  have hMf (u : E) : ‖f u‖ ≤ M := hM (Set.mem_range_self u)
  have hM0 : 0 ≤ M := (norm_nonneg (f 0)).trans (hMf 0)
  simpa only [hSν] using normalized_covariance hα hV hH hη f hf.continuous hM0 hMf y


end AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScore

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
The paper works on R^d, d>=1, with C2 potential V and genuine Hessian bounds 0<alpha I<=D2V<=beta I, and 0<eta<=1/beta.	E is a finite-dimensional real inner-product space with its Borel measurable structure; dimension zero is allowed. The potential V:E→R is C². Constants α and β are finite nonnegative reals, α>0, and the actual second Fréchet derivative satisfies α‖v‖²≤D²V(x)[v,v]≤β‖v‖² for every x,v. The scale η is positive. No upper bound βη≤1 is needed for this differentiation step.	generalization	The density and differentiation identity are valid in any finite-dimensional real inner-product space including dimension zero and for every eta>0. This does not relax the scale hypotheses of subsequent quantitative estimates.
Vector gradient of the normalized conditional expectation.	HasFDerivAt with a continuous linear functional equal to the centered integral of the unnormalized log-weight derivative.	same	The real Riesz identification converts the covector derivative to a gradient; subtracting the expectation of the log-weight derivative accounts for the normalizer.
Conditional density and differentiation are expressed analytically.	R disintegrates the actual swapped Gaussian augmentation; S is its jointly measurable reflected pushforward with an everywhere tilted-density identity. Normalization and local domination are derived.	source-implicit	The proof makes conditional-kernel measurability, positive normalizers and integral interchange explicit. It does not yet state a transformed-joint IsCondKernel interface or identify the old L2 PUP representative.
Smooth compactly supported real observable f.	The public conclusion quantifies over every C∞ compactly supported real test f and every parameter y. The internally proved differentiation argument works for bounded continuous f, including the noncompact test f≡1 needed for the normalizer.	same	The public test class is unchanged. The stronger bounded-continuous internal helper is used also for f=1 to differentiate the noncompact normalizer.

Actual reflected conditional-kernel density and smooth-test derivative only. The transformed joint law's IsCondKernel interface and its identification with the existing L² PUP representative are not yet exposed. Conditional Poincare, score variance, the L²/H¹ extension, macroscopic coercivity, half-turn/process semantics, invariance/nonexplosion, mixing/error/cost and full-paper completion remain open.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Generalization to finite-dimensional inner-product spaces including zero dimension. — The broader ambient and scale domains are disclosed in the lesson and candidate assumptions; public test class matches the selected source input.
assumptions: Upper scale bound removed only for the selected differentiation component. — Potential lower bound, linear derivative growth, normalizers and local domination are derived, not assumed. Removing the scale upper bound is valid only for this differentiation edge.
conclusion: Vector gradient represented by its real continuous-dual Frechet derivative. — Both numerator and noncompact normalizer are differentiated; quotient-rule centering is retained. Riesz identification converts the functional to the source vector gradient; f is fixed in the parameter differentiation.
quantifiers: Everywhere formulas belong to the constructed conditional version. — The constructed suitable version satisfies the everywhere formulas; the theorem does not assert them for arbitrary versions. Compact support supplies the bound for each test.
scopes: No completed PUP Sobolev, Poincare, variance or coercivity theorem. — The module, binding and lesson preserve this scope. No operator-norm differentiability or identification with the old PUP L² representative is asserted.

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

Actual reflected conditional-kernel density and smooth-test derivative only. The transformed joint law's IsCondKernel interface and its identification with the existing L² PUP representative are not yet exposed. Conditional Poincare, score variance, the L²/H¹ extension, macroscopic coercivity, half-turn/process semantics, invariance/nonexplosion, mixing/error/cost and full-paper completion remain open.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

Measure.map_addHaar_smul
tilted_tilted
integral_tilted
integrable_tilted_iff
hasFDerivAt_integral_of_dominated_of_fderiv_le
Continuous.bddAbove_range_of_hasCompactSupport

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS v1 Appendix C.1 — Conditional density and differentiated conditional expectation only; expanded normalization and domination proof.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Actual reflected conditional kernel and everywhere normalized density
Local proof component; source adapter/review separate Smooth-test conditional expectation derivative, with normalization and local domination proved

From the closed Gibbs gradient to ordinary weak derivatives

Samplinglib's explicit Euclidean analytic prerequisite to the proof route of Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; not a quotation or a claim to have proved Lemma B.1.

arXiv:2609.06905v1 · Appendix C.1 closed-gradient/density and conditional Poincare route; background Kolesnikov–Milman1310.2526v7§2.5 · faithful paraphrase

Complete source statement (ASTIS restatement)

The scalar representative u and vector representative G are locally integrable for ordinary Euclidean volume. For every ψ and v as above, both ψ⟨G,v⟩ and uD_vψ are volume-integrable, and their integrals have opposite signs. Thus G is the ordinary distributional gradient of u in this test sense; the converse graph-domain characterization is not claimed.

E is any finite-dimensional real inner-product space with its Borel sigma algebra, including dimension zero. W:E→R is C1 and exp(-W) is integrable for Euclidean volume; μ=volume.tilted(-W) is the actual normalized Gibbs law.
D is a given closable partial real-linear map from scalar L²(μ) to vector L²(μ). Its original graph is exactly the pairs represented μ-almost everywhere by (f,∇f) for smooth compactly supported f. This is the interface already proved by WeightedGradient, not an assumed integration-by-parts formula.
Fix (u,G) in the graph of this same D.closure. Tests ψ are arbitrary C1 compact scalar functions, and v is any constant vector. The L2 coercions select measurable representatives; no differentiability, global volume integrability or tail bound for these representatives is assumed.

Ordinary weak derivative

\[u,G\in L^1_{\mathrm{loc}}(dx),\qquad\int\psi\langle G,v\rangle\,dx=-\int uD_v\psi\,dx.\]

Read the formalized proofs

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

ASTIS mathematical exposition

From the closed Gibbs gradient to ordinary weak derivatives

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientDistribution.closed_gradient_distributional · theorem · Teaching coverage

Statement

\[u,G\in L^1_{\mathrm{loc}}(dx),\qquad\int\psi\langle G,v\rangle\,dx=-\int uD_v\psi\,dx.\]

All objects and hypotheses

E is any finite-dimensional real inner-product space with its Borel sigma algebra, including dimension zero. W:E→R is C1 and exp(-W) is integrable for Euclidean volume; μ=volume.tilted(-W) is the actual normalized Gibbs law.
D is a given closable partial real-linear map from scalar L²(μ) to vector L²(μ). Its original graph is exactly the pairs represented μ-almost everywhere by (f,∇f) for smooth compactly supported f. This is the interface already proved by WeightedGradient, not an assumed integration-by-parts formula.
Fix (u,G) in the graph of this same D.closure. Tests ψ are arbitrary C1 compact scalar functions, and v is any constant vector. The L2 coercions select measurable representatives; no differentiability, global volume integrability or tail bound for these representatives is assumed.

Mathematical proof

1. Start with integrability for the Gibbs law

For either scalar or vector representative h, finite μ makes L² membership imply L¹ membership. The actual tilted-measure integrability equivalence transfers this to integrability of e^(−W)h for volume. Positivity and finiteness of normalization follow from the assumed integrable exponential density.

\[h\in L^2(\mu)\Longrightarrow h\in L^1(\mu)\Longleftrightarrow e^{-W}h\in L^1(dx).\]

Corresponding Lean step

private lp_locallyIntegrable_volume: MemLp.integrable, Lp.memLp, integrable_tilted_iff

2. Remove the weight locally

Multiplication by a continuous scalar preserves local integrability. Apply this to e^W and the locally integrable e^(−W)h, then cancel exponentials. Doing this for both u and G establishes their local volume integrability without asserting global volume integrability.

\[h=e^W(e^{-W}h)\in L^1_{\mathrm{loc}}(dx).\]

Corresponding Lean step

LocallyIntegrable.continuous_smul (Real.continuous_exp.comp hW); smul_smul, Real.exp_add

3. Choose the inverse-weighted test

Given compact C1 ψ, set φ=e^Wψ. Continuity at infinity is irrelevant: its support stays inside the compact support of ψ, and φ is genuinely C1. Apply the existing weighted identity to the same D,u,G; no operator is replaced.

\[\varphi=e^W\psi,\qquad\int\varphi\langle G,v\rangle\,d\mu=-\int u(D_v\varphi-\varphi D_vW)\,d\mu.\]

Corresponding Lean step

hφ,hφc; WeightedGradientWeak.closed_gradient_weighted_ibp

4. Use the genuine first-order product rule

Differentiate e^Wψ using the C1 hypotheses. Subtract φD_vW and cancel its two drift contributions. No classical derivative of u is used.

\[D_v(e^W\psi)=e^W\psi D_vW+e^WD_v\psi,\qquad D_v\varphi-\varphi D_vW=e^WD_v\psi.\]

Corresponding Lean step

hq; HasFDerivAt.exp.mul, hd.fderiv, ring

5. Transfer both test-product integrabilities

The parent proves both weighted products integrable under μ. Apply integrable_tilted_iff to each separately; multiply by e^(−W) and cancel e^(−W)e^W=1. These are actual volume-integrability proofs, not an appeal to the convention for undefined integrals.

\[e^{-W}\varphi\langle G,v\rangle=\psi\langle G,v\rangle,\qquad e^{-W}u(D_v\varphi-\varphi D_vW)=uD_v\psi.\]

Corresponding Lean step

hcancel,hl,hr; hli,hri from integrable_tilted_iff

6. Cancel the positive normalization

The tilted integral is Z⁻¹ times its density-weighted volume integral. Substitute the two pointwise cancellations into the parent identity and cancel Z⁻¹, which is nonzero. Together with local integrability this is the ordinary weak derivative identity, in one direction only.

\[Z^{-1}\int\psi\langle G,v\rangle\,dx=-Z^{-1}\int uD_v\psi\,dx,\qquad Z=\int e^{-W}dx>0.\]

Corresponding Lean step

ht, integral_tilted; mul_left_cancel₀ (inv_ne_zero (integral_exp_pos hI).ne')

Lean statement · closed_gradient_distributional

All same-operator graph assumptions, local-volume and product-integrability conclusions are explicit below.

theorem closed_gradient_distributional (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∀ (D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ), D.IsClosable →
      (∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[μ] f ∧ G =ᵐ[μ] gradient f) →
      ∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.closure.graph →
        LocallyIntegrable (fun x => u x) ∧ LocallyIntegrable (fun x => G x) ∧
        ∀ (ψ : E → ℝ), ContDiff ℝ 1 ψ → HasCompactSupport ψ → ∀ v : E,
          Integrable (fun x => ψ x * inner ℝ (G x) v) ∧
          Integrable (fun x => u x * fderiv ℝ ψ x v) ∧
          (∫ x, ψ x * inner ℝ (G x) v) = - ∫ x, u x * fderiv ℝ ψ x v

Exact module and namespace context

Lean proof · closed_gradient_distributional

Invert the positive density locally; insert exp(W)ψ in the weighted identity; cancel genuine derivative terms and positive normalization.

theorem closed_gradient_distributional (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∀ (D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ), D.IsClosable →
      (∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.graph ↔
        ∃ f : E → ℝ, ContDiff ℝ ∞ f ∧ HasCompactSupport f ∧
          u =ᵐ[μ] f ∧ G =ᵐ[μ] gradient f) →
      ∀ (u : Lp ℝ 2 μ) (G : Lp E 2 μ), (u,G) ∈ D.closure.graph →
        LocallyIntegrable (fun x => u x) ∧ LocallyIntegrable (fun x => G x) ∧
        ∀ (ψ : E → ℝ), ContDiff ℝ 1 ψ → HasCompactSupport ψ → ∀ v : E,
          Integrable (fun x => ψ x * inner ℝ (G x) v) ∧
          Integrable (fun x => u x * fderiv ℝ ψ x v) ∧
          (∫ x, ψ x * inner ℝ (G x) v) = - ∫ x, u x * fderiv ℝ ψ x v := by
  let μ := (volume : Measure E).tilted (fun x => -W x)
  dsimp only
  intro D hD hgraph u G hu
  refine ⟨lp_locallyIntegrable_volume W hW.continuous hI u,
    lp_locallyIntegrable_volume W hW.continuous hI G, ?_⟩
  intro ψ hψ hc v
  let φ := fun x => Real.exp (W x) * ψ x
  have hφ : ContDiff ℝ 1 φ := hW.exp.mul hψ
  have hφc : HasCompactSupport φ := hc.mul_left
  have hq (x : E) : fderiv ℝ φ x v - φ x * fderiv ℝ W x v =
      Real.exp (W x) * fderiv ℝ ψ x v := by
    have hd := ((hW.differentiable one_ne_zero x).hasFDerivAt.exp).mul
      (hψ.differentiable one_ne_zero x).hasFDerivAt
    rw [show fderiv ℝ φ x = _ from hd.fderiv]
    simp only [add_apply, smul_apply, smul_eq_mul]
    dsimp [φ]
    ring
  have hcancel (x : E) : Real.exp (-W x) * Real.exp (W x) = 1 := by
    rw [← Real.exp_add]
    simp
  have hl (x : E) : Real.exp (-W x) * (φ x * inner ℝ (G x) v) =
      ψ x * inner ℝ (G x) v := by
    dsimp only [φ]
    rw [← mul_assoc, ← mul_assoc, hcancel, one_mul]
  have hr (x : E) : Real.exp (-W x) *
      (u x * (fderiv ℝ φ x v - φ x * fderiv ℝ W x v)) = u x * fderiv ℝ ψ x v := by
    rw [hq]
    calc
      _ = (Real.exp (-W x) * Real.exp (W x)) * (u x * fderiv ℝ ψ x v) := by ring
      _ = _ := by rw [hcancel, one_mul]
  obtain ⟨hleft,hright,he⟩ := WeightedGradientWeak.closed_gradient_weighted_ibp
    W hW hI D hD hgraph u G hu φ hφ hφc v
  have hli := (integrable_tilted_iff hI _).mp hleft
  have hri := (integrable_tilted_iff hI _).mp hright
  simp only [smul_eq_mul, hl, hr] at hli hri
  refine ⟨hli,hri,?_⟩
  have ht (g : E → ℝ) : (∫ x, g x ∂μ) =
      (∫ x, Real.exp (-W x))⁻¹ * ∫ x, Real.exp (-W x) * g x := by
    rw [show μ = (volume : Measure E).tilted (fun x => -W x) from rfl, integral_tilted]
    rw [← integral_const_mul]
    apply integral_congr_ae
    filter_upwards [] with x
    change (Real.exp (-W x) / (∫ z, Real.exp (-W z))) • g x = _
    simp only [smul_eq_mul, div_eq_mul_inv]
    ring
  change (∫ x, φ x * inner ℝ (G x) v ∂μ) =
    -∫ x, u x * (fderiv ℝ φ x v - φ x * fderiv ℝ W x v) ∂μ at he
  rw [ht, ht] at he
  simp only [hl, hr, ← mul_neg] at he
  exact mul_left_cancel₀ (inv_ne_zero (integral_exp_pos hI).ne') he

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientDistribution

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Source uses a conditional Euclidean Sobolev gradient and its closedness in the Poincare/density route.	Any given closable exact smooth compact gradient graph for normalized exp(-W) volume.	source-implicit	The same operator is supplied by an existing construction; its distributional meaning is now proved using the compiled weighted test identity.
Actual conditional potential comes from a C2 strongly convex V with proved normalization.	C1 W and integrable exp(-W), no curvature bound.	generalization	Only the local product rule and compact tests enter; no curvature-dependent coercivity is claimed.
Kolesnikov–Milman background discusses compact weighted manifolds, boundary conditions and spectral properties.	Ordinary weak derivative direction only: no converse domain characterization, H2, resolvent PDE adapter, D*D operator core, Poincare or complete paper.	API-limitation	Domain and conclusion boundary deliberately narrower than a full background regularity/spectral result; this is a prerequisite, not that theorem.
Representative choice, compact approximation and integrability are implicit in the analytic route.	L2 representatives become locally volume-integrable, and both ordinary weak-test products are volume-integrable.	source-implicit	Positive continuous inverse Gibbs weight and actual tilted-integrability equivalence justify the transfer. No derivatives of the representatives or tail bounds are assumed.

Ordinary weak derivative direction only: no converse domain characterization, H2, resolvent PDE adapter, D*D operator core, Poincare or complete paper.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

assumptions: Explicit closable-operator and exact smooth-graph hypotheses elaborate the scoped source; the distributional identity is not assumed. — D.IsClosable and the graph biconditional are inputs; exp(W)ψ substitution and cancellation prove the desired identity.
scopes: The inverse-weight lemma is an authored prerequisite, not a quoted paper theorem. — companion:pbps-distributional-gradient-prerequisite; attribution denies Lemma B.1; binding role prerequisite.

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

Ordinary weak derivative direction only: no converse domain characterization, H2, resolvent PDE adapter, D*D operator core, Poincare or complete paper.

Source and reuse

ASTIS parents called

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedGradientWeak.closed_gradient_weighted_ibp

Mathlib API called (external library)

MeasureTheory.MemLp.integrable
MeasureTheory.integrable_tilted_iff
MeasureTheory.LocallyIntegrable.continuous_smul
MeasureTheory.integral_tilted
MeasureTheory.integral_exp_pos
HasFDerivAt.exp
HasFDerivAt.mul

Mathematical sources

PBPS Appendix C.1: analytic prerequisite — Ordinary weak derivative direction only: no converse domain characterization, H2, resolvent PDE adapter, D*D operator core, Poincare or complete paper.
Kolesnikov–Milman §2.5: weighted Sobolev background — Direct full-space proof, not the compact-manifold spectral theorem.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Local volume integrability and ordinary weak identity

Joint augmentation density relative to the input law times volume

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang. ASTIS mathematical restatement of the Gaussian joint-law component, not copied source prose or a claim of complete source-density formalization.

arXiv:2609.06905v1 · Section 2.2, equations (2.6)-(2.7): joint-law density before inserting the Gibbs density of mu · faithful paraphrase

Complete source statement (ASTIS restatement)

For independent X with probability law mu and standard Gaussian Z on R^d, and eta>0, the joint law of (X,X+sqrt(eta) Z) equals mu(dx) q_eta(y-x) dy, where q_eta is the centered isotropic Gaussian density with variance eta. Inserting the paper's Gibbs density mu(dx)=Z_V^{-1} exp(-V(x)) dx is a separate final substitution to obtain the displayed joint Gibbs density.

The paper works on R^d with d>=1 and a twice continuously differentiable potential with positive lower and upper Hessian bounds. Its target mu is the normalized Gibbs probability measure.
The source takes 0<eta<=1/beta and states the independent Gaussian augmentation X~mu, Z~N(0,I), Y=X+sqrt(eta) Z.
The selected intermediate identity is a density relative to mu tensor volume, rather than volume tensor volume. It needs only probability of mu and eta>0.

joint-density intermediate between (2.7) and (2.6)

\[\mathcal L(X,X+\sqrt\eta Z)(dx,dy)=\mu(dx)q_\eta(y-x)dy,\qquad q_\eta(w)=((\sqrt{2\pi\eta})^{-1})^d\exp(-\|w\|^2/(2\eta)).\]

Read the formalized proofs

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

ASTIS mathematical exposition

The joint density of a Gaussian augmentation for any input law

AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianAugmentation.augmentation_eq_withDensity · theorem · Teaching coverage

Statement

Let E be a finite-dimensional real inner-product space with its Borel sigma algebra and canonical volume, mu a probability measure on E, and eta>0. Independently draw X with law mu and Z with the standard Gaussian law, and put Y=X+sqrt(eta) Z. Then the entire joint law of (X,Y) has density q_eta(y-x) relative to mu tensor volume. The input mu need not have a density relative to volume.

\[\Phi_\eta(x,z)=(x,x+\sqrt\eta\,z),\quad (\Phi_\eta)_\#(\mu\otimes\gamma_E)=(\mu\otimes\mathrm{vol}_E)\,q_\eta(y-x),\quad q_\eta(w)=((\sqrt{2\pi\eta})^{-1})^d\exp(-\|w\|^2/(2\eta)),\quad d=\dim_{\mathbb R}E.\]

All objects and hypotheses

E is a normed additive commutative group with a real inner product, finite real dimension, and its Borel measurable structure. Volume is the canonical inner-product-space volume, not an arbitrary Haar normalization.
mu is any probability measure on E. Independence is represented by the product measure mu.prod(stdGaussian E). Singular inputs, including point masses, are allowed.
eta is a positive real number. There is no upper scale bound and no potential, curvature, or input-density assumption. Finite dimension zero is included.
The reference measure is mu tensor volume. A density relative to volume tensor volume requires an additional density for mu, such as the source's Gibbs representation. That substitution is outside this theorem.

Mathematical proof

1. Scale the independent Gaussian noise

The existing isotropic Gaussian density theorem identifies the law of sqrt(eta) Z with volume weighted by q_eta. Mapping the two independent coordinates separately leaves the first law unchanged and applies this identity to the second law.

\[(x,z)\mapsto(x,\sqrt\eta\,z),\qquad (\mathrm{id}\times(\sqrt\eta\,\cdot))_\#(\mu\otimes\gamma_E)=\mu\otimes(\mathrm{vol}_E q_\eta).\]

Corresponding Lean step

The new local ASTIS IsotropicGaussianDensity.map_sqrt_smul_stdGaussian_eq_withDensity parent supplies the actual noise-law identity. Measure.map_prod_map and Measure.map_id then establish hprod. The density parent is compiled local work, not asserted to be Registry-admitted merely because this module imports it.

2. Express the independent product by a joint density

The product-density theorem moves the weight in the second coordinate to a density on the product reference measure. The Gaussian density is measurable, and the first probability measure needs no Lebesgue density.

\[\mu\otimes(\mathrm{vol}_E q_\eta)=(\mu\otimes\mathrm{vol}_E)\,[q_\eta(w)].\]

Corresponding Lean step

MeasureTheory.prod_withDensity_right is applied with the proved measurability of q. The formula is with respect to mu.prod volume, so it remains valid for a Dirac input.

3. The additive shear preserves the reference measure

The shear S(x,w)=(x,x+w) is a measurable equivalence, with inverse S inverse(x,y)=(x,y-x). It leaves the first coordinate fixed and translates volume in the second coordinate. Therefore it preserves mu tensor volume even when mu is not translation-invariant.

\[S_\#(\mu\otimes\mathrm{vol}_E)=\mu\otimes\mathrm{vol}_E,\qquad S^{-1}(x,y)=(x,y-x).\]

Corresponding Lean step

MeasurableEquiv.shearAddRight supplies the equivalence, and measurePreserving_prod_add mu volume supplies hvol. Probability of mu and canonical-volume sigma-finiteness discharge the product-measure prerequisites; no Jacobian proof is introduced.

4. Transport the density through the shear

Augmentation is exactly the composition of independent noise scaling and the shear. The existing ASTIS measurable-equivalence density theorem composes the density with the inverse shear, changing q_eta(w) into q_eta(y-x). This gives equality of the entire joint measures, not only equality of their total mass.

\[(\Phi_\eta)_\#(\mu\otimes\gamma_E)=S_\#((\mu\otimes\mathrm{vol}_E)q_\eta(w))=(\mu\otimes\mathrm{vol}_E)q_\eta(y-x).\]

Corresponding Lean step

Measure.map_map uses the proved factorization hPhi and measurable maps. RadonNikodym.measurableEquiv_map_withDensity transports the explicit product density. The inverse shear's second coordinate is -x+y; additive commutativity rewrites it to y-x.

Lean statement · augmentation_eq_withDensity

Measure.map denotes pushforward; Measure.prod is the independent product measure. withDensity takes an ENNReal-valued function obtained by ENNReal.ofReal from the nonnegative displayed real density. Equality is equality of measures, hence holds on every measurable event. Module.finrank is a natural dimension, and the normalizer uses a natural power of an inverse square root.

theorem augmentation_eq_withDensity (μ : Measure E) [IsProbabilityMeasure μ]
    (η : ℝ) (hη : 0 < η) :
    Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
        (μ.prod (stdGaussian E)) =
      (μ.prod (volume : Measure E)).withDensity (fun p =>
        ENNReal.ofReal
          (((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E *
            Real.exp (-‖p.2 - p.1‖ ^ 2 / (2 * η))))

Exact module and namespace context

Lean proof · augmentation_eq_withDensity

One theorem composes the newly compiled isotropic Gaussian density with existing product-density and measurable-equivalence transport. All abbreviations are local to the proof. No desired joint-law equality, density of mu, conditional representative, or volume-preservation claim is assumed as a hypothesis. The source-specific Gibbs density remains a separate consumer-level substitution.

theorem augmentation_eq_withDensity (μ : Measure E) [IsProbabilityMeasure μ]
    (η : ℝ) (hη : 0 < η) :
    Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
        (μ.prod (stdGaussian E)) =
      (μ.prod (volume : Measure E)).withDensity (fun p =>
        ENNReal.ofReal
          (((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E *
            Real.exp (-‖p.2 - p.1‖ ^ 2 / (2 * η)))) := by
  let σ : E → E := fun z => Real.sqrt η • z
  let q : E → ℝ≥0∞ := fun z => ENNReal.ofReal
    (((Real.sqrt (2 * Real.pi * η))⁻¹) ^ Module.finrank ℝ E *
      Real.exp (-‖z‖ ^ 2 / (2 * η)))
  let S : E × E ≃ᵐ E × E := MeasurableEquiv.shearAddRight E
  have hq : Measurable q := by fun_prop
  have hnoise : (stdGaussian E).map σ = (volume : Measure E).withDensity q :=
    TechnicalLemmas.Measure.IsotropicGaussianDensity.map_sqrt_smul_stdGaussian_eq_withDensity
      η hη
  have hprod : Measure.map (Prod.map id σ) (μ.prod (stdGaussian E)) =
      μ.prod ((volume : Measure E).withDensity q) := by
    rw [← Measure.map_prod_map μ (stdGaussian E) measurable_id (by fun_prop),
      hnoise, Measure.map_id]
  have hΦ : (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2)) =
      S ∘ Prod.map id σ := rfl
  have hvol : (μ.prod (volume : Measure E)).map S = μ.prod volume :=
    (measurePreserving_prod_add μ volume).map_eq
  rw [hΦ, ← Measure.map_map S.measurable (by fun_prop), hprod,
    prod_withDensity_right hq,
    TechnicalLemmas.Measure.RadonNikodym.measurableEquiv_map_withDensity
      S _ (f := fun p : E × E => q p.2) (by fun_prop), hvol]
  congr 1
  funext p
  change q (-p.1 + p.2) = q (p.2 - p.1)
  rw [neg_add_eq_sub]

end AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianAugmentation

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
The input is the paper's normalized smooth strongly log-concave Gibbs law on R^d	Any probability measure mu on a finite-dimensional real inner-product Borel space E	generalization	The selected density is relative to mu tensor volume. Only the independent Gaussian coordinate is given a volume density, so mu may even be a Dirac measure. The missing Gibbs substitution is a separate obligation, not a hidden hypothesis.
d>=1 and Euclidean Lebesgue volume	Every finite real dimension, including zero, with the canonical inner-product-space volume	generalization	The proved noise-density parent and shear identity apply in zero dimension too. No arbitrary Haar normalization or extra measure-space parameter is introduced.
0<eta<=1/beta	eta>0	generalization	The upper bound is not used for the joint Gaussian-density identity; it belongs to other curvature and algorithmic statements.
Joint Gibbs density relative to dx dy, with the source's target-density definition	Density relative to mu.prod volume, with factor q_eta(y-x)	unresolved	The actual Gaussian augmentation adapter is proved, but the final target-density and Gibbs-normalizer substitution is still outside this theorem and remains an unbound source obligation.
Independent sampling and translation of the Gaussian auxiliary coordinate	Product probability measure, proved measurable factorization, explicit density transport, and shear preserving mu.prod volume	source-implicit	The proof supplies the required measurable maps and product-measure contracts. Translation invariance is required only of volume, not of mu.

Actual joint-law density relative to mu tensor volume only; source Gibbs-density substitution remains unbound and open. No reflection or PBPS-process claim.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

domains: The formal component explicitly includes arbitrary probability inputs, abstract finite-dimensional inner-product spaces and zero dimension rather than only the source's Gibbs sampling setting. — The fresh extraction and lesson openly state this generalization. The proof leaves μ untouched in the first coordinate; a Dirac input is allowed, and dimension zero gives density one on the one-point space.
assumptions: The isolated Gaussian identity uses η>0 without the source's sampler upper bound η≤1/β. — No step of scaling, product-density conversion or shear transport uses β. This wider range is appropriate for the selected intermediate theorem and does not modify any sampler guarantee.
conclusion: The theorem's reference measure is μ⊗volume; the full displayed source Gibbs density uses volume⊗volume. — The missing Gibbs-density insertion is explicitly retained as gibbs-density-substitution. Consequently this component may be accepted without certifying full equation (2.6); arbitrary singular μ prevents silently treating the two reference measures as interchangeable.
scopes: No conditional representative, reflection property, PBPS-process invariance, convergence or query-cost result is supplied. — Those conclusions occur nowhere in the current formal statement or selected binding, and the lesson expressly excludes them. The metadata correction gives exact dependency targets without enlarging mathematical completion credit.

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

This is an actual joint density relative to mu tensor canonical volume. It is not yet the source's exp(-V(x)-||x-y||^2/(2 eta)) density relative to volume tensor volume; substituting mu's Gibbs density and normalizer is the remaining edge.
Generic finite-dimensional E, dimension zero, arbitrary probability mu and all eta>0 are explicit generalizations of the paper's Euclidean Gibbs setting with 0<eta<=1/beta.
No reflection theorem, conditional-law representative, PBPS transition/process invariance, convergence, or query complexity is asserted. The density-defined reflection result is an intended downstream consumer.
The isotropic-density parent is a frozen compiled local result. Independent source review and shared Registry/publication admission are separately required.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

MeasureTheory.Measure.map_prod_map, map_id and map_map: independent-product and composed pushforwards.
MeasureTheory.prod_withDensity_right: second-coordinate density on a product reference measure.
MeasurableEquiv.shearAddRight and MeasureTheory.measurePreserving_prod_add: the additive shear and invariance of mu tensor volume.
Measurability of the explicit Gaussian density and continuous affine maps; neg_add_eq_sub.

Mathematical sources

Chen, Chewi, Lu and Zhang, Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler v1 — Joint-law density component of equations (2.6)-(2.7), with reference measure mu tensor volume kept explicit. Original ASTIS exposition, not reproduced paper prose.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Identify the generative joint law with the explicit density relative to mu tensor volume
TODO — not closed by these contributions Insert the target's normalized Gibbs density to recover the density relative to volume tensor volume in (2.6)

The actual macroscopic reflection has a differentiable conditional representative

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; ASTIS expanded mathematical proof.

arXiv:2609.06905v1 · Appendix B.1 compression and Appendix C.1 conditional representation/differentiation; selected proof step · faithful paraphrase

Complete source statement (ASTIS restatement)

The macroscopic compressed reflection acts by the conditional expectation T_f(y)=E[f(Y_minus)|Y_plus=y]. The conditional law has density proportional to exp(-V((y+u)/2)-norm(y-u)^2/(8eta)), and the derivative of T_f is its covariance with the unnormalized logarithmic derivative -DV((y+u)/2)/2-inner(y-u,.)/(4eta). This is the conditional-representative and differentiation step connecting Appendix B.1 to Appendix C.1; quantitative variance estimates and the extension to general L2 inputs remain separate.

The paper works on R^d, d>=1, with C2 potential V, genuine Hessian bounds 0<alpha I<=D2V<=beta I, and 0<eta<=1/beta.
X has normalized Gibbs law proportional to exp(-V); Y_plus=X+sqrt(eta)Z and Y_minus=X-sqrt(eta)Z for independent standard Gaussian Z. P projects in L2 of the augmented joint law onto the Y coordinate, and U is pullback by the actual reflection. Select smooth compactly supported real f for the differentiation step.

actual compressed representative

\[PUP[f\circ\mathrm{snd}]=[T_f\circ\mathrm{snd}],\quad T_f\in L^2(\nu),\quad DT_f=\operatorname{Cov}_{S_y}(f,s_y).\]

Read the formalized proofs

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

ASTIS mathematical exposition

The actual macroscopic reflection has a differentiable conditional representative

AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicRepresentative.macroscopic_reflection_smooth_representative · theorem · Teaching coverage

Statement

Let nu=J.snd, Lambda be the law of (Y_plus,Y_minus), and P be the L2(J) orthogonal projection onto functions measurable in the second coordinate. There are one Markov kernel S disintegrating Lambda and an involutive self-adjoint reflection isometry U, acting by composition with F(x,y)=(x,2x-y). For every smooth compactly supported real f, its macroscopic class g=[f composed with snd] satisfies Pg=g and PUPg=[T_f composed with snd], where T_f(y)=integral f dS_y belongs to L2(nu). For every y, s_y and f s_y are S_y-integrable and DT_f(y)=E[f s_y]-E[f]E[s_y], with s_y(u)=-DV((y+u)/2)/2-inner(y-u,.)/(4 eta).

\[PUP[f\circ\mathrm{snd}]=[T_f\circ\mathrm{snd}],\quad T_f\in L^2(\nu),\quad DT_f=\operatorname{Cov}_{S_y}(f,s_y).\]

All objects and hypotheses

E is a finite-dimensional real inner-product space with its Borel measurable structure; dimension zero is allowed. The potential V:E→R is C².
Constants α and β are finite nonnegative reals, α>0, and the actual second Fréchet derivative satisfies α‖v‖²≤D²V(x)[v,v]≤β‖v‖² for every x,v. The scale η is positive. No upper bound βη≤1 is needed for this differentiation step.
μ is volume tilted by −V and J is the law of (X,X+sqrt(η)Z), where X has law μ and Z is an independent standard Gaussian. Normalization and all integrability used in the proof are derived from the curvature assumptions.
The conclusion quantifies over every smooth compactly supported real f and every parameter y. P is the actual conditional orthogonal projection, not an arbitrary operator assumed to have the desired integral representation.

Mathematical proof

1. Use one compatible pair of conditional kernels

The previously proved conditional-score theorem constructs R disintegrating the actual law of (Y,X), and S_y as the reflected pushforward of R_y. Its normalization, density and derivative estimates follow from the genuine Hessian bounds. Keep these same R and S throughout the proof. The reflection theorem also supplies an isometry U and an unrelated conditional kernel witness; only its U is used. No equality of independently chosen versions is assumed.

\[S_y=(x\mapsto 2x-y)_\#R_y.\]

Corresponding Lean step

The final assembly obtains R,S from reflected_conditional_covariance and U from actual_reflection_block_identities; it discards the latter theorem's kernel witness.

2. Disintegrate the actual reflected joint law

Write the swapped augmentation as J_swap=nu tensor R, with nu=J.snd. Push this measure through G(y,x)=(y,2x-y). For every measurable set, the composition-product formula and the fiber pushforward identity show that this pushforward is nu tensor S. Its first coordinate is unchanged. Composing G with the swap gives precisely the joint law Lambda of (Y_plus,Y_minus).

\[\Lambda=\nu\otimes S,\qquad \Lambda_{1}=\nu.\]

Corresponding Lean step

reflected_disintegration proves the measure equality by compProd_apply and map_apply; hLambda identifies the composed measurable maps.

3. Construct the macroscopic L2 class

A smooth compactly supported real f is bounded. Since J is a probability measure, f composed with the second coordinate belongs to L2(J). Let g be its Lp class. The representative f composed with snd is measurable for the sigma-algebra generated by snd, so g lies in that closed measurable subspace. Orthogonal projection onto this subspace therefore fixes g.

\[g=[f\circ\mathrm{snd}],\qquad Pg=g.\]

Corresponding Lean step

macroscopic_class uses MemLp.of_bound, toLp, mem_lpMeas_iff_aestronglyMeasurable and starProjection_eq_self_iff. It does not assert compact support on the product space.

4. Identify conditional projection using the selected kernel

Every v in L2(J) is integrable because J is finite. The conditional expectation formula identifies Pv with the conditional integral of v(x,y). The selected R agrees almost everywhere with the canonical disintegration kernel by uniqueness; this is enough to obtain the formula J-almost everywhere. This step is proved for the R already used to construct S.

\[(Pv)(x,y)=\int v(x\prime,y)\,R_y(dx\prime)\quad J\text{-a.e.}\]

Corresponding Lean step

projection_kernel uses condExp_prod_ae_eq_integral_condDistrib', eq_condKernel_of_measure_eq_compProd and Lp.condExpL2_ae_eq_condExp.

5. Transport representatives through reflection and disintegration

The actual reflection F(x,y)=(x,2x-y) preserves J, so composing a J-almost-everywhere representative identity with F remains valid almost everywhere. Thus Ug equals f(2x-y) J-almost everywhere. Disintegrating this equality yields R_y-almost-everywhere equality for nu-almost every y, sufficient for equality of conditional integrals. Pushforward integration then identifies these integrals with T_f(y). Combining with Pg=g gives the actual compressed operator representative.

\[PUPg=[T_f\circ\mathrm{snd}],\qquad T_f(y)=\int f(u)\,S_y(du).\]

Corresponding Lean step

compressed_representative combines reflection_preserves_augmentation, fiber_ae, projection_kernel and expectation_map. Fiber representative identities are not asserted for every y.

6. Join L2 membership with the everywhere classical derivative

Measurability of a kernel integral and the bound |T_f(y)|<=sup|f| give T_f in L2(nu). For this same everywhere-defined kernel S, the conditional-score theorem supplies integrability of s_y and f s_y and the derivative at every y. The operator identity is an almost-everywhere identity of representatives, whereas this derivative statement is everywhere for the selected representative. Neither statement alone supplies a quantitative variance estimate or a Sobolev extension.

\[DT_f(y)=\mathbb E_{S_y}[f s_y]-\mathbb E_{S_y}[f]\,\mathbb E_{S_y}[s_y].\]

Corresponding Lean step

expectation_memLp uses measurable kernel integration and MemLp.of_bound; the final derivative and both integrability clauses use hder for the same S.

Lean statement · macroscopic_reflection_smooth_representative

One compatible reflected kernel supplies the actual joint disintegration, actual compressed L2 representative and its everywhere classical derivative.

theorem macroscopic_reflection_smooth_representative {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {η : ℝ}
    (hα : 0 < (α : ℝ)) (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) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    let Λ := J.map (fun p : E × E => (p.2,(2:ℝ) • p.1-p.2))
    let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
      (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
        ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
    let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
      (1/(4*η)) • innerSL ℝ (y-u)
    ∃ S : Kernel E E, IsMarkovKernel S ∧ Λ.IsCondKernel S ∧ Λ.fst = J.snd ∧
      ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
        (∀ g : Lp ℝ 2 J, (U g : E × E → ℝ) =ᵐ[J]
          (fun p => g (p.1,(2:ℝ) • p.1-p.2))) ∧
        Function.Involutive U ∧ IsSelfAdjoint U.toContinuousLinearMap ∧
        ∀ (f : E → ℝ), ContDiff ℝ ∞ f → HasCompactSupport f →
          ∃ g : Lp ℝ 2 J,
            (g : E × E → ℝ) =ᵐ[J] (fun p => f p.2) ∧ P g = g ∧
            ((P * U.toContinuousLinearMap * P) g : E × E → ℝ) =ᵐ[J]
              (fun p => ∫ u, f u ∂S p.2) ∧
            MemLp (fun y => ∫ u, f u ∂S y) 2 J.snd ∧
            ∀ y, Integrable (s y) (S y) ∧ Integrable (fun u => f u • s y u) (S y) ∧
              HasFDerivAt (fun z => ∫ u, f u ∂S z)
              ((∫ u, f u • s y u ∂S y) -
                (∫ u, f u ∂S y) • (∫ u, s y u ∂S y)) y

Exact module and namespace context

Lean proof · macroscopic_reflection_smooth_representative

Local measure and conditional-expectation adapters join the existing actual reflection isometry and conditional-score construction, discharging every adapter premise from the public Hessian hypotheses.

theorem macroscopic_reflection_smooth_representative {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    {V : E → ℝ} {α β : NNReal} {η : ℝ}
    (hα : 0 < (α : ℝ)) (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) :
    let μ := (volume : Measure E).tilted (fun x => -V x)
    let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    let Λ := J.map (fun p : E × E => (p.2,(2:ℝ) • p.1-p.2))
    let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
      (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
        ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
    let s := fun y u : E => -(1/2:ℝ) • fderiv ℝ V ((1/2:ℝ) • (y+u)) -
      (1/(4*η)) • innerSL ℝ (y-u)
    ∃ S : Kernel E E, IsMarkovKernel S ∧ Λ.IsCondKernel S ∧ Λ.fst = J.snd ∧
      ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
        (∀ g : Lp ℝ 2 J, (U g : E × E → ℝ) =ᵐ[J]
          (fun p => g (p.1,(2:ℝ) • p.1-p.2))) ∧
        Function.Involutive U ∧ IsSelfAdjoint U.toContinuousLinearMap ∧
        ∀ (f : E → ℝ), ContDiff ℝ ∞ f → HasCompactSupport f →
          ∃ g : Lp ℝ 2 J,
            (g : E × E → ℝ) =ᵐ[J] (fun p => f p.2) ∧ P g = g ∧
            ((P * U.toContinuousLinearMap * P) g : E × E → ℝ) =ᵐ[J]
              (fun p => ∫ u, f u ∂S p.2) ∧
            MemLp (fun y => ∫ u, f u ∂S y) 2 J.snd ∧
            ∀ y, Integrable (s y) (S y) ∧ Integrable (fun u => f u • s y u) (S y) ∧
              HasFDerivAt (fun z => ∫ u, f u ∂S z)
              ((∫ u, f u • s y u ∂S y) -
                (∫ u, f u ∂S y) • (∫ u, s y u ∂S y)) y := by
  have reflected_disintegration {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (ρ : Measure (E × E)) [IsFiniteMeasure ρ]
      (R S : Kernel E E) [IsMarkovKernel R] [IsMarkovKernel S]
      [ρ.IsCondKernel R]
      (hS : ∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y)) :
      let G := fun p : E × E => (p.1,(2:ℝ) • p.2-p.1)
      (ρ.map G).fst = ρ.fst ∧ (ρ.map G).IsCondKernel S := by
    let G := fun p : E × E => (p.1,(2:ℝ) • p.2-p.1)
    have hG : Measurable G := by fun_prop
    have hfst : (ρ.map G).fst = ρ.fst := by
      exact Measure.fst_map_prodMk (by fun_prop)
    have hmap : ρ.fst ⊗ₘ S = (ρ.fst ⊗ₘ R).map G := by
      ext t ht
      rw [Measure.compProd_apply ht, Measure.map_apply hG ht,
        Measure.compProd_apply (hG ht)]
      apply lintegral_congr_ae
      filter_upwards with y
      rw [hS y, Measure.map_apply (by fun_prop) (measurable_prodMk_left ht)]
      rfl
    refine ⟨hfst,⟨?_⟩⟩
    rw [hfst,hmap,Measure.disintegrate ρ R]



  have macroscopic_class {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (J : Measure (E × E)) [IsProbabilityMeasure J]
      (f : E → ℝ) (hf : Continuous f) {M : ℝ} (hMf : ∀ u, ‖f u‖ ≤ M) :
      let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
        (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
          ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
      ∃ g : Lp ℝ 2 J, (g : E × E → ℝ) =ᵐ[J] (fun p => f p.2) ∧ P g = g := by
    let mY : MeasurableSpace (E × E) := MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)
    let _ : MeasurableSpace (E × E) := Prod.instMeasurableSpace
    have hm : mY ≤ Prod.instMeasurableSpace := measurable_snd.comap_le
    let : Fact (mY ≤ Prod.instMeasurableSpace) := ⟨hm⟩
    have hLp : MemLp (fun p : E × E => f p.2) 2 J :=
      MemLp.of_bound (hf.comp continuous_snd).aestronglyMeasurable M
        (Filter.Eventually.of_forall fun p => hMf p.2)
    let g := hLp.toLp (fun p : E × E => f p.2)
    have hg : (g : E × E → ℝ) =ᵐ[J] (fun p => f p.2) := hLp.coeFn_toLp
    have hSnd : Measurable[mY] (Prod.snd : E × E → E) := measurable_iff_comap_le.mpr le_rfl
    have hgm : AEStronglyMeasurable[mY] (g : E × E → ℝ) J :=
      (hf.stronglyMeasurable.comp_measurable hSnd).aestronglyMeasurable.congr hg.symm
    refine ⟨g,hg,?_⟩
    change (lpMeas ℝ ℝ mY 2 J).starProjection g = g
    exact Submodule.starProjection_eq_self_iff.mpr (mem_lpMeas_iff_aestronglyMeasurable.mpr hgm)



  have projection_kernel {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (J : Measure (E × E)) [IsProbabilityMeasure J]
      (R : Kernel E E) [IsMarkovKernel R] [(J.map Prod.swap).IsCondKernel R]
      (v : Lp ℝ 2 J) :
      let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
        (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
          ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
      (P v : E × E → ℝ) =ᵐ[J] (fun p => ∫ x, v (x,p.2) ∂R p.2) := by
    have hvI := (Lp.memLp v).integrable (by norm_num)
    have hswap : Integrable (fun p : E × E => v p.swap) (J.map Prod.swap) := by
      apply (integrable_map_equiv (MeasurableEquiv.prodComm : E × E ≃ᵐ E × E) _).2
      exact hvI
    have h := condExp_prod_ae_eq_integral_condDistrib'
      (μ := J) (X := Prod.snd) (Y := Prod.fst)
      (f := fun p : E × E => v p.swap) measurable_snd measurable_fst.aemeasurable hswap
    have heq : R =ᵐ[(J.map Prod.swap).fst] (J.map Prod.swap).condKernel :=
      eq_condKernel_of_measure_eq_compProd R (Measure.disintegrate _ _).symm
    rw [Measure.fst_map_swap] at heq
    have heq' : ∀ᵐ p ∂J, R p.2 = (J.map Prod.swap).condKernel p.2 :=
      ae_of_ae_map measurable_snd.aemeasurable heq
    have hP := (Lp.memLp v).condExpL2_ae_eq_condExp (𝕜 := ℝ) measurable_snd.comap_le
    rw [Lp.toLp_coeFn] at hP
    apply hP.trans
    filter_upwards [h,heq'] with p hp he
    rw [he]
    simp only [condDistrib] at hp
    exact hp

  have fiber_ae {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (J : Measure (E × E)) [IsProbabilityMeasure J]
      (R : Kernel E E) [IsMarkovKernel R] [(J.map Prod.swap).IsCondKernel R]
      {a b : E × E → ℝ} (hab : a =ᵐ[J] b) :
      ∀ᵐ y ∂J.snd, (fun x => a (x,y)) =ᵐ[R y] (fun x => b (x,y)) := by
    have hswap : (fun p : E × E => a p.swap) =ᵐ[J.map Prod.swap] (fun p => b p.swap) := by
      apply (MeasurableEquiv.prodComm.measurableEmbedding.ae_map_iff).2
      change ∀ᵐ p ∂J, a p.swap.swap = b p.swap.swap
      simp only [Prod.swap_swap]
      exact hab
    rw [← Measure.disintegrate (J.map Prod.swap) R] at hswap
    have h := Measure.ae_ae_of_ae_compProd hswap
    rw [Measure.fst_map_swap] at h
    exact h



  have expectation_memLp {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (ν : Measure E) [IsFiniteMeasure ν] (S : Kernel E E) [IsMarkovKernel S]
      (f : E → ℝ) (hf : Continuous f) {M : ℝ} (hMf : ∀ u, ‖f u‖ ≤ M) :
      MemLp (fun y => ∫ u, f u ∂S y) 2 ν := by
    refine MemLp.of_bound hf.stronglyMeasurable.integral_kernel.aestronglyMeasurable M ?_
    filter_upwards with y
    simpa using (norm_integral_le_of_norm_le_const (μ := S y)
      (Filter.Eventually.of_forall hMf))

  have expectation_map {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (R S : Kernel E E) (hS : ∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y))
      (f : E → ℝ) (hf : Continuous f) (y : E) :
      ∫ u, f u ∂S y = ∫ x, f ((2:ℝ) • x-y) ∂R y := by
    rw [hS y, integral_map (by fun_prop) hf.aestronglyMeasurable]


  have compressed_representative {E : Type u}
      [NormedAddCommGroup E] [InnerProductSpace ℝ E]
      [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
      (J : Measure (E × E)) [IsProbabilityMeasure J]
      (R S : Kernel E E) [IsMarkovKernel R] [IsMarkovKernel S]
      [(J.map Prod.swap).IsCondKernel R]
      (hS : ∀ y, S y = (R y).map (fun x => (2:ℝ) • x-y))
      (U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J)
      (hF : MeasurePreserving (fun p : E × E => (p.1,(2:ℝ) • p.1-p.2)) J J)
      (hU : ∀ g : Lp ℝ 2 J, (U g : E × E → ℝ) =ᵐ[J]
        (fun p => g (p.1,(2:ℝ) • p.1-p.2)))
      (f : E → ℝ) (hf : Continuous f) (g : Lp ℝ 2 J)
      (hg : (g : E × E → ℝ) =ᵐ[J] (fun p => f p.2)) :
      let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
        (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
          ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
      (P (U g) : E × E → ℝ) =ᵐ[J] (fun p => ∫ u, f u ∂S p.2) := by
    have hUg : (U g : E × E → ℝ) =ᵐ[J] (fun p => f ((2:ℝ) • p.1-p.2)) :=
      (hU g).trans (hF.quasiMeasurePreserving.ae_eq hg)
    have hs := fiber_ae J R hUg
    have hs' : ∀ᵐ p ∂J, (fun x => (U g) (x,p.2)) =ᵐ[R p.2]
        (fun x => f ((2:ℝ) • x-p.2)) :=
      ae_of_ae_map measurable_snd.aemeasurable hs
    apply (projection_kernel J R (U g)).trans
    filter_upwards [hs'] with p hp
    calc
      (∫ x, (U g) (x,p.2) ∂R p.2) = ∫ x, f ((2:ℝ) • x-p.2) ∂R p.2 :=
        integral_congr_ae hp
      _ = ∫ u, f u ∂S p.2 := (expectation_map R S hS f hf p.2).symm
  let μ := (volume : Measure E).tilted (fun x => -V x)
  let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
    (μ.prod (stdGaussian E))
  let Λ := J.map (fun p : E × E => (p.2,(2:ℝ) • p.1-p.2))
  have density := AutoSamplingTheory.ExampleCases.ProximalBPS.GibbsAugmentation.normalized_augmentation_density
    hα hV (fun x v => (hH x v).1) hη
  have hi : Integrable (fun x => Real.exp (-V x)) (volume : Measure E) := by
    by_contra hn
    exact (ne_of_gt density.1) (integral_undef hn)
  have : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hi
  have : IsProbabilityMeasure J := Measure.isProbabilityMeasure_map (by fun_prop)
  obtain ⟨R,S,hR,hS,hcond,hSR,hSν,hder⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ConditionalScore.reflected_conditional_covariance hα hV hH hη
  let _ : IsMarkovKernel R := hR
  let _ : IsMarkovKernel S := hS
  let _ : (J.map Prod.swap).IsCondKernel R := hcond
  have hΛ : (J.map Prod.swap).map (fun p : E × E => (p.1,(2:ℝ) • p.2-p.1)) = Λ := by
    rw [Measure.map_map (by fun_prop) (by fun_prop)]
    rfl
  obtain ⟨hfst,hΛcond⟩ := reflected_disintegration (J.map Prod.swap) R S hSR
  rw [hΛ,Measure.fst_map_swap] at hfst
  rw [hΛ] at hΛcond
  obtain ⟨R₀,hR₀,hRf₀,U,hU,hUi,hUs,hrest⟩ :=
    AutoSamplingTheory.ExampleCases.ProximalBPS.ReflectionL2.actual_reflection_block_identities μ hη
  have hF : MeasurePreserving (fun p : E × E => (p.1,(2:ℝ) • p.1-p.2)) J J :=
    ⟨by fun_prop,
      (AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianReflection.reflection_preserves_augmentation μ η hη).2⟩
  dsimp only
  refine ⟨S,hS,hΛcond,hfst,U,hU,hUi,hUs,?_⟩
  intro f hf hfc
  obtain ⟨M,hM⟩ := hf.continuous.norm.bddAbove_range_of_hasCompactSupport hfc.norm
  have hMf (u : E) : ‖f u‖ ≤ M := hM (Set.mem_range_self u)
  obtain ⟨g,hg,hPg⟩ := macroscopic_class J f hf.continuous hMf
  refine ⟨g,hg,hPg,?_,expectation_memLp J.snd S f hf.continuous hMf,?_⟩
  · change ((lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
      ∘L condExpL2 ℝ ℝ measurable_snd.comap_le) (U
        (((lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
          ∘L condExpL2 ℝ ℝ measurable_snd.comap_le) g)) =ᵐ[J] _
    rw [hPg]
    exact compressed_representative J R S hSR U hF hU f hf.continuous g hg
  · intro y
    exact hder f hf hfc y


end AutoSamplingTheory.ExampleCases.ProximalBPS.MacroscopicRepresentative

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
The paper works on R^d, d>=1, with C2 potential V, genuine Hessian bounds 0<alpha I<=D2V<=beta I, and 0<eta<=1/beta.	E is a finite-dimensional real inner-product space with its Borel measurable structure; dimension zero is allowed. The potential V:E→R is C². Constants α and β are finite nonnegative reals, α>0, and the actual second Fréchet derivative satisfies α‖v‖²≤D²V(x)[v,v]≤β‖v‖² for every x,v. The scale η is positive. No upper bound βη≤1 is needed for this differentiation step.	generalization	This selected identity holds in finite-dimensional real inner-product spaces including dimension zero and for every eta>0. Subsequent quantitative bounds retain their own scale restrictions.
Conditional expectation expressed as a macroscopic operator.	One Markov S disintegrates Lambda; actual PUPg equals T_f composed with snd J-almost everywhere.	source-implicit	Joint disintegration and a.e. representative transport are made explicit. Conditional versions are compared only almost everywhere where uniqueness permits.
Gradient covariance for smooth compactly supported real f.	Everywhere HasFDerivAt with a continuous-dual-valued covariance and integrable score and f score.	same	The Riesz identification converts the covector to a gradient. The statement does not extend differentiation to arbitrary L2 inputs.

Actual joint disintegration and the differentiable representative of PUP on smooth compactly supported macroscopic tests only. No conditional Poincare or score variance bound, general L2/H1 extension, macroscopic coercivity, process semantics, invariance/nonexplosion, mixing, implementation error, query cost or full-paper completion.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Finite-dimensional inner-product generalization including dimension zero. — No positive-dimension hypothesis; lesson explicitly discloses zero dimension. Public test class preserved.
assumptions: Positive scale without source upper bound for this selected identity. — Disclosed omission of beta eta<=1 is valid for this selected edge. Gibbs probability derived from normalized_augmentation_density and isProbabilityMeasure_tilted; no target identity or derivative assumed.
scopes: Everywhere classical derivative for selected representative; L2 equality remains a.e. — Reflection measure preservation transports a.e. equality, then ae_ae_of_ae_compProd transfers it before conditional integration. No promotion to everywhere fibers. Variance, Sobolev and coercivity remain open.
conclusion: Gradient encoded by continuous-dual Frechet derivative. — Actual pushforward, macroscopic_class and compressed_representative prove all clauses; final assembly rewrites Pg. Riesz identifies dual derivative with source gradient.
scopes: Conditional density is inherited internally and not re-exported by this integration theorem. — hSnu is obtained from ConditionalScore; binding lists actual disintegration, representative and derivative, not a new density obligation.

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

Actual joint disintegration and the differentiable representative of PUP on smooth compactly supported macroscopic tests only. No conditional Poincare or score variance bound, general L2/H1 extension, macroscopic coercivity, process semantics, invariance/nonexplosion, mixing, implementation error, query cost or full-paper completion.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

Measure.compProd_apply
Measure.disintegrate
Measure.ae_ae_of_ae_compProd
ProbabilityTheory.condExp_prod_ae_eq_integral_condDistrib'
MeasureTheory.Lp.condExpL2_ae_eq_condExp
Submodule.starProjection_eq_self_iff
MeasureTheory.MemLp.of_bound

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS v1 Appendix C.1 and Appendix B.1 — Conditional expectation representation and differentiation joined to the compressed reflection; selected proof step only.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Reflected kernel disintegrates the actual two-output joint law
Local proof component; source adapter/review separate Actual PUP on smooth macroscopic tests has an L2 conditional-expectation representative
Local proof component; source adapter/review separate Same representative has the score-covariance derivative everywhere

Actual reflection and conditional-projection blocks

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; ASTIS expanded mathematical proof.

arXiv:2609.06905v1 · Appendix B.1, equations B.1 and B.3–B.5; joint density 2.6 · faithful paraphrase

Complete source statement (ASTIS restatement)

The conditional resampling operator is the orthogonal projection P onto functions of Y. Reflection pullback U is a self-adjoint unitary. With A=PUP, B=(I−P)UP and D=(I−P)U(I−P), the block identities are B*B=I−A² on the range of P and B*D=−AB*. Consequently ‖Bf‖²=‖f‖²−‖Af‖² for f in that range. The full-space extension of the first identity has P in place of I.

The paper uses the normalized Gibbs target on R^d, d≥1, with C² potential V and Hessian bounds 0<αI≤∇²V≤βI, and 0<η≤1/β. Its joint law has density proportional to exp(−V(x)−‖y−x‖²/(2η)).
The selected block-algebra component acts on real L² of this joint law. P is conditional expectation given Y and U is pullback by (x,y)↦(x,2x−y).

whole-space block extension

\[B^*B=P-A^2,\qquad B^*D=-AB^*,\qquad Pf=f\Longrightarrow\|Bf\|^2=\|f\|^2-\|Af\|^2.\]

Read the formalized proofs

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

ASTIS mathematical exposition

Actual reflection and conditional-projection blocks

AutoSamplingTheory.ExampleCases.ProximalBPS.ReflectionL2.actual_reflection_block_identities · theorem · Teaching coverage

Statement

There is a Markov kernel R with R_y=μ tilted by −‖x−y‖²/(2η), and a real linear isometry U with Uf=f∘F almost everywhere, U²=I and U*=U. For every L² class f, Pf(x,y)=∫f(x′,y)R_y(dx′) J-almost everywhere. The actual blocks obey B*B=P−A² and B*D=−AB*. For every Pf=f, ‖Bf‖²=‖f‖²−‖Af‖².

\[B^*B=P-A^2,\qquad B^*D=-AB^*,\qquad Pf=f\Longrightarrow\|Bf\|^2=\|f\|^2-\|Af\|^2.\]

All objects and hypotheses

E is a finite-dimensional real inner-product space with its Borel measurable structure; dimension zero is allowed. μ is any probability measure on E and η>0.
J is the law of (X,X+sqrt(η)Z), with independent X of law μ and standard Gaussian Z. All operators act on the real Hilbert space L²(J).
F(x,y)=(x,2x−y). P is the inclusion into L²(J) of conditional expectation onto the sub-sigma-algebra generated by Y; P is defined, not assumed.
Let A=PUP, B=(I−P)UP and D=(I−P)U(I−P), with multiplication denoting composition. The energy identity quantifies over every f satisfying Pf=f.

Mathematical proof

1. Lift the actual reflection

The Gaussian augmentation is preserved by F, and F∘F=id. Pullback therefore preserves L² norms and defines a linear isometry U on equivalence classes. Applying the involution twice gives U²=I. Inner-product preservation implies ⟨Uf,g⟩=⟨f,Ug⟩, hence U*=U.

\[U^2=I,\quad\langle Uf,g\rangle=\langle Uf,U^2g\rangle=\langle f,Ug\rangle.\]

Corresponding Lean step

reflection_lift uses the proved measure-preserving reflection and Lp.compMeasurePreservingₗᵢ.

2. Identify the conditional projection

The proved quadratic-tilt kernel disintegrates the swapped joint law (Y,X). Conditional-distribution integration and a.e. uniqueness of disintegration identify the conditional expectation given Y. Since J is a probability measure, each L² representative is integrable. The L² conditional expectation agrees a.e. with this integral, so the actual orthogonal projection has the stated kernel semantics.

\[(Pf)(x,y)=\int f(x^{\prime},y)R_y(dx^{\prime})\quad J\text{-a.e.}\]

Corresponding Lean step

kernel_condExp and projection_kernel use condExp_prod_ae_eq_integral_condDistrib', eq_condKernel_of_measure_eq_compProd and MemLp.condExpL2_ae_eq_condExp. No everywhere assertion for arbitrary representatives is made.

3. Expand the diagonal and off-diagonal blocks

Write Q=I−P. Orthogonal projection gives P*=P and P²=P. Thus B*B=PUQUP=PU²P−PUPUP=P−A². Also B*D=PUQUQ=PU²Q−PUPUQ=−AB*, using PQ=0. These are identities on the whole Hilbert space; P becomes the identity only on its range.

\[B^*B=PU(I-P)UP=P-A^2,\qquad B^*D=PU(I-P)U(I-P)=-AB^*.\]

Corresponding Lean step

block_algebra proves both identities from the established projection and reflection equalities by noncommutative ring normalization.

4. Recover the energy transferred from the macroscopic space

The compression A is self-adjoint. For Pf=f, take the inner product of B*B=P−A² with f. The P term is ‖f‖² and self-adjointness changes ⟨f,A²f⟩ into ‖Af‖². No centering or spectral-gap premise is needed for this algebraic identity.

\[\|Bf\|^2=\langle f,(P-A^2)f\rangle=\|f\|^2-\|Af\|^2.\]

Corresponding Lean step

block_energy uses the adjoint norm identity and the actual compression self-adjointness.

Lean statement · actual_reflection_block_identities

All ambient spaces, μ and η are quantified. The joint law, reflection and conditional projection are local definitions. R and U are constructed witnesses with actual-law semantics.

theorem actual_reflection_block_identities
    {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (μ : Measure E) [IsProbabilityMeasure μ] {η : ℝ} (hη : 0 < η) :
    let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    let F := fun p : E × E => (p.1,(2:ℝ) • p.1-p.2)
    let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
      (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
        ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
    ∃ R : Kernel E E, IsMarkovKernel R ∧
      (∀ y, R y = μ.tilted (fun x => -‖x-y‖^2/(2*η))) ∧
      ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
        (∀ f, U f =ᵐ[J] f ∘ F) ∧ Function.Involutive U ∧
        IsSelfAdjoint U.toContinuousLinearMap ∧
        (∀ f : Lp ℝ 2 J, (P f : E × E → ℝ) =ᵐ[J]
          fun p => ∫ x, f (x,p.2) ∂R p.2) ∧
        let A := P * U.toContinuousLinearMap * P
        let B := (1-P) * U.toContinuousLinearMap * P
        let D := (1-P) * U.toContinuousLinearMap * (1-P)
        star B*B=P-A^2 ∧ star B*D= -(A*star B) ∧
          (∀ f, P f=f → ‖B f‖^2 = ‖f‖^2-‖A f‖^2)

Exact module and namespace context

Lean proof · actual_reflection_block_identities

Five local proof helpers feed one public theorem. The test rewrites the actual Gibbs normalized density to its generative augmentation using normalized_augmentation_density; positive normalization supplies integrability and the probability instance.

theorem actual_reflection_block_identities
    {E : Type u} [NormedAddCommGroup E] [InnerProductSpace ℝ E]
    [FiniteDimensional ℝ E] [MeasurableSpace E] [BorelSpace E]
    (μ : Measure E) [IsProbabilityMeasure μ] {η : ℝ} (hη : 0 < η) :
    let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    let F := fun p : E × E => (p.1,(2:ℝ) • p.1-p.2)
    let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J :=
      (lpMeas ℝ ℝ (MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)) 2 J).subtypeL
        ∘L condExpL2 ℝ ℝ measurable_snd.comap_le
    ∃ R : Kernel E E, IsMarkovKernel R ∧
      (∀ y, R y = μ.tilted (fun x => -‖x-y‖^2/(2*η))) ∧
      ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
        (∀ f, U f =ᵐ[J] f ∘ F) ∧ Function.Involutive U ∧
        IsSelfAdjoint U.toContinuousLinearMap ∧
        (∀ f : Lp ℝ 2 J, (P f : E × E → ℝ) =ᵐ[J]
          fun p => ∫ x, f (x,p.2) ∂R p.2) ∧
        let A := P * U.toContinuousLinearMap * P
        let B := (1-P) * U.toContinuousLinearMap * P
        let D := (1-P) * U.toContinuousLinearMap * (1-P)
        star B*B=P-A^2 ∧ star B*D= -(A*star B) ∧
          (∀ f, P f=f → ‖B f‖^2 = ‖f‖^2-‖A f‖^2) := by
  let J₀ := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
    (μ.prod (stdGaussian E))
  let H := Lp ℝ 2 J₀
  have reflection_lift (μ : Measure E) [IsProbabilityMeasure μ] (η : ℝ) (hη : 0 < η) :
      let J := Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
        (μ.prod (stdGaussian E))
      let R := fun p : E × E => (p.1, (2:ℝ) • p.1-p.2)
      ∃ U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J,
        (∀ f, U f =ᵐ[J] f ∘ R) ∧ Function.Involutive U ∧ IsSelfAdjoint U.toContinuousLinearMap := by
    dsimp only
    let J := Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    let R := fun p : E × E => (p.1, (2:ℝ) • p.1-p.2)
    obtain ⟨hinv, hmap⟩ :=
      AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianReflection.reflection_preserves_augmentation μ η hη
    have hp : MeasurePreserving R J J := ⟨by fun_prop, hmap⟩
    let U : Lp ℝ 2 J →ₗᵢ[ℝ] Lp ℝ 2 J := Lp.compMeasurePreservingₗᵢ ℝ R hp
    have hU : Function.Involutive U := by
      intro f
      change Lp.compMeasurePreserving R hp (Lp.compMeasurePreserving R hp f) = f
      rw [← Lp.compMeasurePreserving_comp_apply]
      have hRR : R ∘ R = id := funext hinv
      simp only [hRR, Lp.compMeasurePreserving_id_apply]
    refine ⟨U, fun f => Lp.coeFn_compMeasurePreserving f hp, hU, ?_⟩
    apply ContinuousLinearMap.isSelfAdjoint_iff_isSymmetric.2
    intro f g
    change inner ℝ (U f) g = inner ℝ f (U g)
    calc
      inner ℝ (U f) g = inner ℝ (U f) (U (U g)) := by rw [hU g]
      _ = inner ℝ f (U g) := U.inner_map_map f (U g)

  have kernel_condExp (J : Measure (E × E)) [IsProbabilityMeasure J]
      (R : Kernel E E) [IsMarkovKernel R] [(J.map Prod.swap).IsCondKernel R]
      (f : E × E → ℝ) (hf : Integrable f J) :
      J[f | MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)] =ᵐ[J]
        fun p => ∫ x, f (x,p.2) ∂R p.2 := by
    have hswap : Integrable (fun p : E × E => f p.swap) (J.map Prod.swap) := by
      apply (integrable_map_equiv (MeasurableEquiv.prodComm : E × E ≃ᵐ E × E) _).2
      change Integrable f J
      exact hf
    have h := condExp_prod_ae_eq_integral_condDistrib'
      (μ := J) (X := Prod.snd) (Y := Prod.fst)
      (f := fun p : E × E => f p.swap) measurable_snd measurable_fst.aemeasurable hswap
    have heq : R =ᵐ[(J.map Prod.swap).fst] (J.map Prod.swap).condKernel :=
      eq_condKernel_of_measure_eq_compProd R (Measure.disintegrate _ _).symm
    rw [Measure.fst_map_swap] at heq
    have heq' : ∀ᵐ p ∂J, R p.2 = (J.map Prod.swap).condKernel p.2 := by
      exact ae_of_ae_map measurable_snd.aemeasurable heq
    filter_upwards [h, heq'] with p hp he
    rw [he]
    simp only [condDistrib] at hp
    exact hp

  have projection_kernel (μ : Measure E) [IsProbabilityMeasure μ] {η : ℝ} (hη : 0 < η) :
      let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
        (μ.prod (stdGaussian E))
      ∃ R : Kernel E E, IsMarkovKernel R ∧
        (∀ y, R y = μ.tilted (fun x => -‖x-y‖^2/(2*η))) ∧
        (∀ f : Lp ℝ 2 J,
          (condExpL2 ℝ ℝ (μ := J) measurable_snd.comap_le f : E × E → ℝ) =ᵐ[J]
            fun p => ∫ x, f (x,p.2) ∂R p.2) := by
    dsimp only
    let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
      (μ.prod (stdGaussian E))
    have : IsProbabilityMeasure J := Measure.isProbabilityMeasure_map (by fun_prop)
    obtain ⟨R,hR,hformula,hcond⟩ :=
      AutoSamplingTheory.TechnicalLemmas.Probability.GaussianConditionalKernel.exists_tilted_isCondKernel μ hη
    let _ := hR
    let _ : (J.map Prod.swap).IsCondKernel R := hcond
    refine ⟨R,hR,hformula,?_⟩
    intro f
    have hL2 := (Lp.memLp f).condExpL2_ae_eq_condExp (𝕜 := ℝ) measurable_snd.comap_le
    rw [Lp.toLp_coeFn] at hL2
    exact hL2.trans (kernel_condExp J R f ((Lp.memLp f).integrable (by norm_num)))

  have block_algebra {A : Type u} [Ring A] [StarRing A] (U P : A)
      (hU : star U = U) (hP : star P = P) (hUU : U*U=1) (hPP : P*P=P) :
      let B := (1-P)*U*P
      let D := (1-P)*U*(1-P)
      let A₀ := P*U*P
      star B*B=P-A₀^2 ∧ star B*D= -(A₀*star B) := by
    have hpt (X : A) : P*(P*X)=P*X := by rw [← mul_assoc,hPP]
    have hut (X : A) : U*(U*X)=X := by rw [← mul_assoc,hUU,one_mul]
    dsimp only
    simp only [star_mul,star_sub,star_one,hU,hP]
    constructor <;> noncomm_ring [hPP,hUU,hpt,hut]

  have block_energy (P A B : H →L[ℝ] H) (hA : IsSelfAdjoint A)
      (hBB : star B * B = P-A^2) (f : H) (hf : P f=f) :
      ‖B f‖^2 = ‖f‖^2-‖A f‖^2 := by
    have h := B.apply_norm_sq_eq_inner_adjoint_right f
    have he : B.adjoint.comp B = P-A^2 := hBB
    rw [he] at h
    have hAf : inner ℝ f ((A^2) f) = ‖A f‖^2 := by
      rw [pow_two]
      change inner ℝ f (A (A f)) = _
      calc
        inner ℝ f (A (A f)) = inner ℝ f (A.adjoint (A f)) := by rw [hA.adjoint_eq]
        _ = inner ℝ (A f) (A f) := A.adjoint_inner_right f (A f)
        _ = ‖A f‖^2 := real_inner_self_eq_norm_sq (A f)
    simpa only [sub_apply, hf, inner_sub_right,
      hAf, real_inner_self_eq_norm_sq, RCLike.re_to_real] using h

  dsimp only
  let J := Measure.map (fun p : E × E => (p.1,p.1+Real.sqrt η • p.2))
    (μ.prod (stdGaussian E))
  let mY : MeasurableSpace (E × E) := MeasurableSpace.comap Prod.snd (inferInstance : MeasurableSpace E)
  let _ : MeasurableSpace (E × E) := Prod.instMeasurableSpace
  have hmY : mY ≤ Prod.instMeasurableSpace := measurable_snd.comap_le
  let S := lpMeas ℝ ℝ mY 2 J
  have : Fact (mY ≤ Prod.instMeasurableSpace) := ⟨hmY⟩
  let P : Lp ℝ 2 J →L[ℝ] Lp ℝ 2 J := S.subtypeL ∘L condExpL2 ℝ ℝ (μ := J) hmY
  have hPdef : P = S.starProjection := rfl
  have hP : IsSelfAdjoint P := by rw [hPdef]; exact isSelfAdjoint_starProjection S
  have hPP : P*P=P := by rw [hPdef]; exact S.isIdempotentElem_starProjection
  obtain ⟨U,hUae,hUi,hUs⟩ := reflection_lift μ η hη
  obtain ⟨R,hR,hRf,hRp⟩ := projection_kernel μ hη
  have hUU : U.toContinuousLinearMap * U.toContinuousLinearMap = 1 := by
    apply ContinuousLinearMap.ext
    intro f
    exact hUi f
  obtain ⟨hBB,hBD⟩ := block_algebra U.toContinuousLinearMap P hUs.star_eq hP.star_eq hUU hPP
  have hA : IsSelfAdjoint (P*U.toContinuousLinearMap*P) := by
    change star (P*U.toContinuousLinearMap*P) = P*U.toContinuousLinearMap*P
    simp only [star_mul,hP.star_eq,hUs.star_eq,mul_assoc]
  refine ⟨R,hR,hRf,U,hUae,hUi,hUs,hRp,hBB,hBD,?_⟩
  intro f hf
  exact block_energy P (P*U.toContinuousLinearMap*P) ((1-P)*U.toContinuousLinearMap*P) hA hBB f hf

end AutoSamplingTheory.ExampleCases.ProximalBPS.ReflectionL2

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
The paper uses the normalized Gibbs target on R^d, d≥1, with C² potential V and Hessian bounds 0<αI≤∇²V≤βI, and 0<η≤1/β. Its joint law has density proportional to exp(−V(x)−‖y−x‖²/(2η)).	E is a finite-dimensional real inner-product space with its Borel measurable structure; dimension zero is allowed. μ is any probability measure on E and η>0.	generalization	These algebraic identities need only an input probability and positive Gaussian scale, not curvature or an upper scale bound. The separately compiled Gibbs-density consumer retains C² and positive Hessian lower bound, derives normalization and specializes every conclusion on the explicit source density. Zero dimension is also valid.
Conditional projection is written with conditional-kernel representatives.	Integral representation is J-a.e. for every L² class.	source-implicit	Conditional expectation and L² classes are defined modulo null sets; arbitrary representatives need not have an everywhere-defined integrable section. Finite probability provides L²-to-L¹ integrability.
B*B=I−A² on the macroscopic subspace.	B*B=P−A² on the full Hilbert space, with Pf=f explicitly required in the energy clause.	same	Compressed blocks are extended by zero on the orthogonal complement; this retains the subspace identity without falsely replacing P by I everywhere.

Actual reflection and conditional-projection block identities only. No macro coercivity, Sobolev regularity, square roots or inverses, half-turn process, invariance or nonexplosion, modified-energy contraction, mixing, implementation error, expected cost or full-paper closure.

Encoder–denoiser: accepted · domain-mismatch

Detected semantic differences

domains: Disclosed generalization to arbitrary probability input, including zero dimension. — Lesson and candidate assumptions disclose it; Gibbs consumer applies full result on exact source density.
assumptions: Curvature and upper scale restrictions unnecessary for this algebra; no extension of full mixing claim. — Actual preservation, conditional kernel and orthogonal projection support proof; consumer supplies Gibbs normalization.
scopes: Macroscopic identity represented as B*B=P−A² on full space. — Actual projection; block_energy retains Pf=f.
quantifiers: A.e. representative formulas for each f, everywhere fiber formula for R. — Theorem, authored proof and anonymous reconstruction distinguish ∀y from ∀f a.e.

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

Actual reflection and conditional-projection block identities only. No macro coercivity, Sobolev regularity, square roots or inverses, half-turn process, invariance or nonexplosion, modified-energy contraction, mixing, implementation error, expected cost or full-paper closure.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

Lp.compMeasurePreservingₗᵢ
condExpL2
condExp_prod_ae_eq_integral_condDistrib'
eq_condKernel_of_measure_eq_compProd
isSelfAdjoint_starProjection
Submodule.isIdempotentElem_starProjection

Mathematical sources

Chen, Chewi, Lu and Zhang, PBPS v1 Appendix B.1 — Equations B.1 and B.3–B.5, expanded actual-law proof.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Actual pullback is a self-adjoint isometric involution
Local proof component; source adapter/review separate Actual conditional projection agrees a.e. with quadratic-tilt kernel
Local proof component; source adapter/review separate Whole-space block identities and macroscopic energy identity

The closed-gradient resolvent is a genuine weak PDE solution

Samplinglib's explicit Euclidean weak-resolvent prerequisite to the analytic route for Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang; not a quoted theorem or full Lemma B.1.

arXiv:2609.06905v1 · Appendix C.1 analytic Poincare prerequisite; background1310.2526v7 sections2.4-2.5 · faithful paraphrase

Complete source statement (ASTIS restatement)

There exists u in the domain of the same closed gradient such that ε⟨u,v⟩L²(μ)+⟨G,D.closure v⟩L²(μ)=⟨f,v⟩L²(μ) for every domain test v, with G=D.closure u. The representatives u and G are locally volume-integrable and satisfy the ordinary weak gradient identity against every C1 compact scalar test and constant direction, including both integrable test products. For every smooth compact ψ, each of e^(−W)uψ, e^(−W)⟨G,∇ψ⟩ and e^(−W)fψ is volume-integrable and satisfies the displayed equation. This is εe^(−W)u−div(e^(−W)G)=e^(−W)f in the distributional sense, not a classical second-order equation.

E is any finite-dimensional real inner-product space with a Borel measurable structure, including dimension zero. W:E→R is C1, exp(-W) is volume-integrable, and μ is its normalized exponential tilt.
D is a given closable partial real-linear operator from scalar L²(μ) to vector L²(μ). Its graph is exactly the pairs represented μ-almost everywhere by (φ,∇φ) for smooth compact φ; WeightedGradient constructs such an operator.
ε is positive and f is an arbitrary scalar L²(μ) forcing. The theorem constructs one u in this same D.closure.domain and uses G=D.closure u everywhere. No weak equation, differentiability of u, or higher regularity is assumed.
The ordinary gradient identity uses all C1 compact tests and constant directions. The resolvent PDE uses smooth compact scalar tests ψ, embedded in the original gradient graph. Coercions select L2 representatives; all asserted integrabilities specify μ or volume.

Weighted weak equation

\[\varepsilon\!\int e^{-W}u\psi\,dx+\int e^{-W}\langle G,\nabla\psi\rangle\,dx=\int e^{-W}f\psi\,dx,\qquad G=\overline D u.\]

Read the formalized proofs

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

ASTIS mathematical exposition

The closed-gradient resolvent is a genuine weak PDE solution

AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedResolvent.weak_resolvent_distributional · theorem · Teaching coverage

Statement

\[\varepsilon\!\int e^{-W}u\psi\,dx+\int e^{-W}\langle G,\nabla\psi\rangle\,dx=\int e^{-W}f\psi\,dx,\qquad G=\overline D u.\]

All objects and hypotheses

E is any finite-dimensional real inner-product space with a Borel measurable structure, including dimension zero. W:E→R is C1, exp(-W) is volume-integrable, and μ is its normalized exponential tilt.
D is a given closable partial real-linear operator from scalar L²(μ) to vector L²(μ). Its graph is exactly the pairs represented μ-almost everywhere by (φ,∇φ) for smooth compact φ; WeightedGradient constructs such an operator.
ε is positive and f is an arbitrary scalar L²(μ) forcing. The theorem constructs one u in this same D.closure.domain and uses G=D.closure u everywhere. No weak equation, differentiability of u, or higher regularity is assumed.
The ordinary gradient identity uses all C1 compact tests and constant directions. The resolvent PDE uses smooth compact scalar tests ψ, embedded in the original gradient graph. Coercions select L2 representatives; all asserted integrabilities specify μ or volume.

Mathematical proof

1. Construct the actual same-domain solution

The existing closed-graph resolvent theorem applies to D.closure, whose closedness follows from closability of D. For positive ε it supplies a domain element u and an equation against every element of that domain. This step constructs a solution; it does not postulate the weak equation. All later uses refer to this one witness.

\[\exists u\in\operatorname{Dom}(\overline D),\quad\varepsilon\langle u,v\rangle_\mu+\langle\overline D u,\overline D v\rangle_\mu=\langle f,v\rangle_\mu\quad(\forall v\in\operatorname{Dom}(\overline D)).\]

Corresponding Lean step

ClosedGraphResolvent.weak_resolvent D.closure hD.closure_isClosed ε hε f

2. Identify its ordinary weak gradient

Apply the previously compiled distributional-gradient theorem to (u,G), where G=D.closure u and the pair belongs to that exact closure graph. It yields local volume integrability and the directional weak identity, with both integrable products. It never differentiates u pointwise.

\[u,G\in L^1_{\mathrm{loc}}(dx),\qquad\int\chi\langle G,a\rangle\,dx=-\int uD_a\chi\,dx\quad(\chi\in C_c^1,\ a\in E).\]

Corresponding Lean step

WeightedGradientDistribution.closed_gradient_distributional; D.closure.mem_graph u

3. Embed a genuine smooth compact test

For ψ smooth and compactly supported, both ψ and ∇ψ are continuous and compactly supported, hence in L²(μ). Choose their Lp classes p and q. The exact original graph places (p,q) in D.graph and graph closure places it in D.closure.graph. Thus there is a domain test v whose scalar class is p and whose image is q. No inverse-weighted test is presumed smooth.

\[p=[\psi]_\mu,\quad q=[\nabla\psi]_\mu,\qquad(p,q)\in\operatorname{graph}(D)\subseteq\operatorname{graph}(\overline D).\]

Corresponding Lean step

hgCont,hgComp,hp,hg,hpq,hpqc; MemLp.toLp; LinearPMap.mem_graph_iff

4. Expand three integrable L2 pairings

Test the abstract equation at this v. The L2 inner-product integrability theorem proves each product integrable for μ. Almost-everywhere representative equalities replace p by ψ and q by ∇ψ, both in the integrability proofs and in the integral formula. The scalar inner product is ordinary real multiplication.

\[\varepsilon\int u\psi\,d\mu+\int\langle G,\nabla\psi\rangle\,d\mu=\int f\psi\,d\mu.\]

Corresponding Lean step

hu v; hpe,hge; L2.integrable_inner.congr; L2.inner_def, integral_congr_ae

5. Transfer all three products to volume

For this actual normalized Gibbs tilt, integrable_tilted_iff converts each of the three μ-integrability facts to volume-integrability after multiplication by e^(−W). In particular, the displayed PDE integrals are not assigned values by a totalized fallback for nonintegrable functions.

\[u\psi,\ \langle G,\nabla\psi\rangle,\ f\psi\in L^1(\mu)\Longrightarrow e^{-W}u\psi,\ e^{-W}\langle G,\nabla\psi\rangle,\ e^{-W}f\psi\in L^1(dx).\]

Corresponding Lean step

hwu,hwG,hwf via integrable_tilted_iff hI

6. Cancel only the positive normalization

Write μ=Z⁻¹e^(−W)dx. Its finite normalization Z is strictly positive. Rewrite the three tilted integrals and cancel the common nonzero factor Z⁻¹. The plus sign on the gradient pairing is precisely the distributional minus-divergence convention. No Hessian, Laplacian or operator-core conclusion is used.

\[Z=\int e^{-W}dx>0,\qquad\varepsilon e^{-W}u-\operatorname{div}(e^{-W}G)=e^{-W}f\quad\text{in distributions}.\]

Corresponding Lean step

ht, integral_tilted; mul_left_cancel₀ (inv_ne_zero (integral_exp_pos hI).ne'); ring

Lean statement · weak_resolvent_distributional

The exact quantifiers, same-operator domain and all integrabilities appear below.

theorem weak_resolvent_distributional {E : Type*}
    [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
    [MeasurableSpace E] [BorelSpace E]
    (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∀ (D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ), D.IsClosable →
      (∀ (a : Lp ℝ 2 μ) (H : Lp E 2 μ), (a,H) ∈ D.graph ↔
        ∃ φ : E → ℝ, ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧
          a =ᵐ[μ] φ ∧ H =ᵐ[μ] gradient φ) →
      ∀ (ε : ℝ), 0 < ε → ∀ f : Lp ℝ 2 μ,
      ∃ u : D.closure.domain,
        (∀ v : D.closure.domain,
          ε * ⟪(u : Lp ℝ 2 μ), (v : Lp ℝ 2 μ)⟫ +
            ⟪D.closure u, D.closure v⟫ = ⟪f, (v : Lp ℝ 2 μ)⟫) ∧
        LocallyIntegrable (fun x => (u : Lp ℝ 2 μ) x) ∧
        LocallyIntegrable (fun x => D.closure u x) ∧
        (∀ ψ : E → ℝ, ContDiff ℝ 1 ψ → HasCompactSupport ψ → ∀ v : E,
          Integrable (fun x => ψ x * inner ℝ (D.closure u x) v) ∧
          Integrable (fun x => (u : Lp ℝ 2 μ) x * fderiv ℝ ψ x v) ∧
          (∫ x, ψ x * inner ℝ (D.closure u x) v) =
            - ∫ x, (u : Lp ℝ 2 μ) x * fderiv ℝ ψ x v) ∧
        ∀ ψ : E → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ →
          Integrable (fun x => Real.exp (-W x) * ((u : Lp ℝ 2 μ) x * ψ x)) ∧
          Integrable (fun x => Real.exp (-W x) * inner ℝ (D.closure u x) (gradient ψ x)) ∧
          Integrable (fun x => Real.exp (-W x) * (f x * ψ x)) ∧
          ε * (∫ x, Real.exp (-W x) * ((u : Lp ℝ 2 μ) x * ψ x)) +
            (∫ x, Real.exp (-W x) * inner ℝ (D.closure u x) (gradient ψ x)) =
            ∫ x, Real.exp (-W x) * (f x * ψ x)

Exact module and namespace context

Lean proof · weak_resolvent_distributional

Construct u once; identify its weak gradient; embed smooth compact tests; expand L2 pairings and cancel normalization.

theorem weak_resolvent_distributional {E : Type*}
    [NormedAddCommGroup E] [InnerProductSpace ℝ E] [FiniteDimensional ℝ E]
    [MeasurableSpace E] [BorelSpace E]
    (W : E → ℝ) (hW : ContDiff ℝ 1 W)
    (hI : Integrable (fun x => Real.exp (-W x))) :
    let μ := (volume : Measure E).tilted (fun x => -W x)
    ∀ (D : Lp ℝ 2 μ →ₗ.[ℝ] Lp E 2 μ), D.IsClosable →
      (∀ (a : Lp ℝ 2 μ) (H : Lp E 2 μ), (a,H) ∈ D.graph ↔
        ∃ φ : E → ℝ, ContDiff ℝ ∞ φ ∧ HasCompactSupport φ ∧
          a =ᵐ[μ] φ ∧ H =ᵐ[μ] gradient φ) →
      ∀ (ε : ℝ), 0 < ε → ∀ f : Lp ℝ 2 μ,
      ∃ u : D.closure.domain,
        (∀ v : D.closure.domain,
          ε * ⟪(u : Lp ℝ 2 μ), (v : Lp ℝ 2 μ)⟫ +
            ⟪D.closure u, D.closure v⟫ = ⟪f, (v : Lp ℝ 2 μ)⟫) ∧
        LocallyIntegrable (fun x => (u : Lp ℝ 2 μ) x) ∧
        LocallyIntegrable (fun x => D.closure u x) ∧
        (∀ ψ : E → ℝ, ContDiff ℝ 1 ψ → HasCompactSupport ψ → ∀ v : E,
          Integrable (fun x => ψ x * inner ℝ (D.closure u x) v) ∧
          Integrable (fun x => (u : Lp ℝ 2 μ) x * fderiv ℝ ψ x v) ∧
          (∫ x, ψ x * inner ℝ (D.closure u x) v) =
            - ∫ x, (u : Lp ℝ 2 μ) x * fderiv ℝ ψ x v) ∧
        ∀ ψ : E → ℝ, ContDiff ℝ ∞ ψ → HasCompactSupport ψ →
          Integrable (fun x => Real.exp (-W x) * ((u : Lp ℝ 2 μ) x * ψ x)) ∧
          Integrable (fun x => Real.exp (-W x) * inner ℝ (D.closure u x) (gradient ψ x)) ∧
          Integrable (fun x => Real.exp (-W x) * (f x * ψ x)) ∧
          ε * (∫ x, Real.exp (-W x) * ((u : Lp ℝ 2 μ) x * ψ x)) +
            (∫ x, Real.exp (-W x) * inner ℝ (D.closure u x) (gradient ψ x)) =
            ∫ x, Real.exp (-W x) * (f x * ψ x) := by
  let μ := (volume : Measure E).tilted (fun x => -W x)
  dsimp only
  intro D hD hgraph ε hε f
  let : IsProbabilityMeasure μ := isProbabilityMeasure_tilted hI
  obtain ⟨u,hu,_⟩ := ClosedGraphResolvent.weak_resolvent D.closure hD.closure_isClosed ε hε f
  obtain ⟨huL,hGL,hderiv⟩ := WeightedGradientDistribution.closed_gradient_distributional
    W hW hI D hD hgraph u (D.closure u) (D.closure.mem_graph u)
  refine ⟨u,hu,huL,hGL,hderiv,?_⟩
  intro ψ hψ hc
  have hgCont : Continuous (gradient ψ) :=
    (toDual ℝ E).symm.continuous.comp
      (((contDiff_infty.mp hψ 1).fderiv_right (m := 0) (by norm_num)).continuous)
  have hgComp : HasCompactSupport (gradient ψ) := by
    refine HasCompactSupport.of_support_subset_isCompact hc.isCompact ?_
    intro x hx
    by_contra hn
    exact hx (by simp [gradient, fderiv_of_notMem_tsupport ℝ hn])
  have hp : MemLp ψ 2 μ := hψ.continuous.memLp_of_hasCompactSupport hc
  have hg : MemLp (gradient ψ) 2 μ := hgCont.memLp_of_hasCompactSupport hgComp
  let p : Lp ℝ 2 μ := hp.toLp ψ
  let q : Lp E 2 μ := hg.toLp (gradient ψ)
  have hpq : (p,q) ∈ D.graph :=
    (hgraph p q).mpr ⟨ψ,hψ,hc,hp.coeFn_toLp,hg.coeFn_toLp⟩
  have hpqc : (p,q) ∈ D.closure.graph := by
    rw [← hD.graph_closure_eq_closure_graph]
    exact D.graph.le_topologicalClosure hpq
  obtain ⟨v,hv,hDv⟩ := D.closure.mem_graph_iff.mp hpqc
  have he := hu v
  rw [hv,hDv] at he
  have hpe (a : Lp ℝ 2 μ) :
      (fun x => inner ℝ (a x) (p x)) =ᵐ[μ] (fun x => a x * ψ x) := by
    filter_upwards [hp.coeFn_toLp] with x hx
    rw [show p x = ψ x from hx]
    simp [mul_comm]
  have hge :
      (fun x => inner ℝ (D.closure u x) (q x)) =ᵐ[μ]
        (fun x => inner ℝ (D.closure u x) (gradient ψ x)) := by
    filter_upwards [hg.coeFn_toLp] with x hx
    rw [show q x = gradient ψ x from hx]
  have hiu := (L2.integrable_inner (𝕜 := ℝ) (u : Lp ℝ 2 μ) p).congr (hpe u)
  have hiG := (L2.integrable_inner (𝕜 := ℝ) (D.closure u) q).congr hge
  have hif := (L2.integrable_inner (𝕜 := ℝ) f p).congr (hpe f)
  have hwu := (integrable_tilted_iff hI _).mp hiu
  have hwG := (integrable_tilted_iff hI _).mp hiG
  have hwf := (integrable_tilted_iff hI _).mp hif
  simp only [smul_eq_mul] at hwu hwG hwf
  refine ⟨hwu,hwG,hwf,?_⟩
  rw [L2.inner_def, L2.inner_def, L2.inner_def,
    integral_congr_ae (hpe u), integral_congr_ae hge,
    integral_congr_ae (hpe f)] at he
  have ht (g : E → ℝ) : (∫ x, g x ∂μ) =
      (∫ x, Real.exp (-W x))⁻¹ * ∫ x, Real.exp (-W x) * g x := by
    rw [show μ = (volume : Measure E).tilted (fun x => -W x) from rfl, integral_tilted]
    rw [← integral_const_mul]
    apply integral_congr_ae
    filter_upwards [] with x
    change (Real.exp (-W x) / (∫ z, Real.exp (-W z))) • g x = _
    simp only [smul_eq_mul, div_eq_mul_inv]
    ring
  rw [ht,ht,ht] at he
  apply mul_left_cancel₀ (inv_ne_zero (integral_exp_pos hI).ne')
  convert he using 1
  ring

end AutoSamplingTheory.TechnicalLemmas.FunctionalInequalities.WeightedResolvent

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
Conditional Poincare is used in PBPS C.1; weighted Poisson/Sobolev arguments require actual weak solutions and admissible tests.	C1 W, finite Gibbs normalization, same closable exact smooth gradient graph, positive epsilon, arbitrary L2 forcing.	source-implicit	The closed-graph resolvent is constructed, and its PDE is derived by proving the test belongs to its domain; the desired equation is not assumed.
Actual PBPS conditional potentials are C2 with Hessian bounds.	C1 W suffices for this weak prerequisite, with no curvature estimate.	generalization	No quantitative Poincare or second-order regularity is claimed.
Background Poisson results on compact weighted manifolds carry regularity and boundary conditions.	Weighted distributional PDE existence only. No H2, converse domain characterization, D*D operator core, Poincare, measurable fiberwise solution selection, or complete paper. Separately chosen resolvent witnesses are not definitionally identified.	API-limitation	This is the direct full-space weak prerequisite, not the compact-manifold regularity theorem or a hidden transfer of it.
Use of density, representatives and integration in the analytic route is implicit.	Same chosen resolvent witness throughout; C∞ compact domain tests; three weighted volume product integrabilities explicit.	source-implicit	C1 inverse-weighted functions are not inserted into a smooth core. Separate chosen conditional solutions would need uniqueness to identify.

Weighted distributional PDE existence only. No H2, converse domain characterization, D*D operator core, Poincare, measurable fiberwise solution selection, or complete paper. Separately chosen resolvent witnesses are not definitionally identified.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

objects: The anonymous reconstruction makes equivalence classes and their chosen representatives explicit without changing the theorem. — Lean uses Lp elements in the variational equation and their coercions in volume integrals.
scopes: Source acceptance is for the expressly authored prerequisite, not equivalence to a theorem stated in either cited paper. — The packet identifies the result as an explicit Euclidean prerequisite and excludes a full Lemma B.1 attribution. The metadata phrase 'faithful paraphrase' is acceptable only with that adjacent restriction.

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

Weighted distributional PDE existence only. No H2, converse domain characterization, D*D operator core, Poincare, measurable fiberwise solution selection, or complete paper. Separately chosen resolvent witnesses are not definitionally identified.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

LinearPMap.IsClosable.closure_isClosed
LinearPMap.IsClosable.graph_closure_eq_closure_graph
LinearPMap.mem_graph_iff
MeasureTheory.MemLp.coeFn_toLp
MeasureTheory.L2.integrable_inner
MeasureTheory.L2.inner_def
MeasureTheory.integrable_tilted_iff
MeasureTheory.integral_tilted

Mathematical sources

PBPS Appendix C.1: analytic prerequisite — Weighted distributional PDE existence only. No H2, converse domain characterization, D*D operator core, Poincare, measurable fiberwise solution selection, or complete paper. Separately chosen resolvent witnesses are not definitionally identified.
Kolesnikov–Milman §§2.4–2.5: weighted Poisson/Sobolev background — Direct Euclidean weak equation only; no automatic compact-manifold regularity transfer.

ASTIS 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 edges are actually covered?

TODO — not closed by these contributions Constructed weak solution, ordinary gradient and integrable weighted PDE tests

Proposition 2.1(iii): reflection symmetry of the augmented law

Fan Chen, Sinho Chewi, Jianfeng Lu and Matthew S. Zhang. ASTIS mathematical restatement, not original prose; no endorsement implied.

arXiv:2609.06905v1 · Section 2.2, Proposition 2.1(iii), (2.6), (2.7), (2.14) · faithful paraphrase

Complete source statement (ASTIS restatement)

For the paper's augmented Gibbs law, reflecting the auxiliary point through the current position leaves the joint distribution unchanged. Applying this reflection twice gives the original state.

The source works on Euclidean space with a twice continuously differentiable potential V satisfying alpha I <= Hessian V <= beta I for 0<alpha<=beta, and mu(dx)=Z^{-1} exp(-V(x)) dx.
The source takes 0<eta<=1/beta and defines the joint density proportional to exp(-V(x)-||x-y||^2/(2 eta)).
The paper also displays the generative representation: independent X~mu and Z~N(0,I), then Y=X+sqrt(eta) Z. Equality of the normalized density and this generative law is a separate formal obligation.

(2.14) and Proposition 2.1(iii)

\[R(x,y)=(x,2x-y),\qquad R\circ R=\mathrm{id},\qquad R_\#\pi_\eta=\pi_\eta.\]

generative representation (2.7)

\[\Phi_\eta(x,z)=(x,x+\sqrt\eta\,z),\qquad \pi_\eta=(\Phi_\eta)_\#(\mu\otimes\gamma).\]

Read the formalized proofs

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

ASTIS mathematical exposition

Why reflecting the Gaussian auxiliary point preserves its joint law

AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianReflection.reflection_preserves_augmentation · theorem · Teaching coverage

Statement

Let E be a finite-dimensional real inner-product space with its Borel sigma algebra, let mu be a probability measure on E, and let eta>0. Independently draw X with law mu and a standard Gaussian Z, and put Y=X+sqrt(eta) Z. The map R(x,y)=(x,2x-y) is an involution, and (X,2X-Y) has exactly the same joint law as (X,Y).

\[\Phi_\eta(x,z)=(x,x+\sqrt\eta\,z),\quad \pi_\eta=(\Phi_\eta)_\#(\mu\otimes\gamma),\quad R\circ R=\mathrm{id},\quad R_\#\pi_\eta=\pi_\eta.\]

All objects and hypotheses

E is a normed additive commutative group with a real inner product and finite real dimension; its measurable sets are the Borel sets.
mu is a probability measure. The product mu tensor gamma encodes independence; gamma is Mathlib's standard Gaussian on E.
eta is a positive real scale. Positivity retains the sampling interpretation; the reflection calculation itself uses no division by eta.
This selected proof component is more general than the paper's Gibbs/curvature setting. It neither assumes nor proves that its pushforward law equals the density displayed in (2.6).

Mathematical proof

1. Check the reflection is an involution

The first coordinate stays fixed. Reflecting the second coordinate twice cancels the two subtractions, so no exceptional point or almost-everywhere qualification is needed.

\[R(R(x,y))=(x,2x-(2x-y))=(x,y).\]

Corresponding Lean step

The first conjunction branch proves equality of the two product coordinates. The second coordinate is sub_sub_cancel; it holds pointwise for every state.

2. Reverse the Gaussian displacement

Negation is an orthogonal linear isometry. Hence the standard Gaussian is unchanged by Z becoming -Z. Product pushforward preserves the first law and changes only the independent auxiliary variable.

\[(-\mathrm{id})_\#\gamma=\gamma,\qquad S(x,z)=(x,-z),\qquad S_\#(\mu\otimes\gamma)=\mu\otimes\gamma.\]

Corresponding Lean step

ProbabilityTheory.stdGaussian_map is instantiated with LinearIsometryEquiv.neg. Measure.map_prod_map and Measure.map_id then transport this equality to the product measure. These are Mathlib results, not new ASTIS Gaussian lemmas.

3. Relate noise reversal to state reflection

Reflecting Y around X reverses its displacement from X. This is an equality of actual measurable functions, not merely equality in distribution.

\[R(\Phi_\eta(x,z))=(x,2x-(x+\sqrt\eta\,z))=(x,x-\sqrt\eta\,z)=\Phi_\eta(S(x,z)).\]

Corresponding Lean step

The proof establishes the composition identity pointwise with scalar/additive algebra. Measurability of the reflection, augmentation and sign map is supplied before using Measure.map_map.

4. Transport the invariant product law

Pushforward composition and the preceding symmetry now give equality of the entire joint measures, hence equality on every measurable event. No density argument, integral interchange or conditional representative is hidden in this calculation.

\[R_\#\pi_\eta=(R\circ\Phi_\eta)_\#(\mu\otimes\gamma)=(\Phi_\eta\circ S)_\#(\mu\otimes\gamma)=(\Phi_\eta)_\#S_\#(\mu\otimes\gamma)=\pi_\eta.\]

Corresponding Lean step

The final rewrite uses Measure.map_map twice, the proved composition identity and the proved product symmetry. It does not assume the desired invariance as a hypothesis.

Lean statement · reflection_preserves_augmentation

Measure.map is pushforward and Measure.prod is product measure. The IsProbabilityMeasure typeclass supplies finiteness for the product transport theorem; BorelSpace ties measurable sets to the topology. Function.Involutive is a pointwise statement that applying a function twice is the identity. The conjunction packages both parts of the source's reflection claim without adding a duplicate wrapper.

theorem reflection_preserves_augmentation (μ : Measure E) [IsProbabilityMeasure μ]
    (η : ℝ) (_hη : 0 < η) :
    Function.Involutive (fun p : E × E => (p.1, (2 : ℝ) • p.1 - p.2)) ∧
      Measure.map (fun p : E × E => (p.1, (2 : ℝ) • p.1 - p.2))
        (Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
          (μ.prod (stdGaussian E))) =
        Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
          (μ.prod (stdGaussian E))

Exact module and namespace context

Lean proof · reflection_preserves_augmentation

Local names abbreviate the three displayed affine functions inside the proof only. Every map needed by pushforward composition is proved measurable. ASTIS composes Mathlib's Gaussian-isometry theorem and product/map transport with the source-specific reflection algebra. The source hypotheses on potential smoothness are not silently used or claimed: they belong to the still-open density and algorithmic adapters.

theorem reflection_preserves_augmentation (μ : Measure E) [IsProbabilityMeasure μ]
    (η : ℝ) (_hη : 0 < η) :
    Function.Involutive (fun p : E × E => (p.1, (2 : ℝ) • p.1 - p.2)) ∧
      Measure.map (fun p : E × E => (p.1, (2 : ℝ) • p.1 - p.2))
        (Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
          (μ.prod (stdGaussian E))) =
        Measure.map (fun p : E × E => (p.1, p.1 + Real.sqrt η • p.2))
          (μ.prod (stdGaussian E)) := by
  let Φ : E × E → E × E := fun p => (p.1, p.1 + Real.sqrt η • p.2)
  let R : E × E → E × E := fun p => (p.1, (2 : ℝ) • p.1 - p.2)
  let S : E × E → E × E := Prod.map id (fun z => -z)
  have hΦ : Measurable Φ := by fun_prop
  have hR : Measurable R := by fun_prop
  have hS : Measurable S := by fun_prop
  have hneg : Measure.map (fun z : E => -z) (stdGaussian E) = stdGaussian E := by
    simpa using (stdGaussian_map (LinearIsometryEquiv.neg ℝ (E := E)))
  have hprod : Measure.map S (μ.prod (stdGaussian E)) = μ.prod (stdGaussian E) := by
    dsimp [S]
    rw [← Measure.map_prod_map μ (stdGaussian E) measurable_id (by fun_prop),
      hneg, Measure.map_id]
  have hcomp : R ∘ Φ = Φ ∘ S := by
    funext p
    apply Prod.ext
    · rfl
    · dsimp [R, Φ, S]
      simp only [smul_neg, two_smul]
      abel
  constructor
  · intro p
    apply Prod.ext
    · rfl
    · exact sub_sub_cancel ((2 : ℝ) • p.1) p.2
  · change Measure.map R (Measure.map Φ (μ.prod (stdGaussian E))) =
      Measure.map Φ (μ.prod (stdGaussian E))
    rw [Measure.map_map hR hΦ, hcomp, ← Measure.map_map hΦ hS, hprod]

end AutoSamplingTheory.ExampleCases.ProximalBPS.GaussianReflection

Exact module and namespace context

Source assumptions versus formal assumptions

Source	Actual Lean	Difference kind	Why it matters
The target is a strongly log-concave smooth Gibbs probability law on R^d	Any probability measure on a finite-dimensional real inner-product space with its Borel sigma algebra	generalization	The selected symmetry calculation only uses independence and symmetry of the auxiliary standard Gaussian. Neither a density nor potential regularity enters this proof component; the density adapter remains open.
0<eta<=1/beta	0<eta	generalization	The upper step-size restriction controls other curvature and algorithmic bounds. Gaussian sign symmetry works at every positive scale; it does not establish those bounds.
The joint law is written by both a normalized density and a generative representation	The joint law is explicitly a pushforward of mu.prod(stdGaussian E)	unresolved	The formalized measure is the generative representation. Identifying it with the density requires a separate normalization/change-of-variables proof.
The state-space maps are affine maps on Euclidean space	Measurability is proved from BorelSpace, finite-dimensional topology, addition and scalar multiplication	source-implicit	The pushforward composition rules require measurable maps. The proof supplies this prerequisite rather than treating the totalized map API as mathematical invariance.

The generative-law proof component, not the density identification or any PBPS transition/process invariance theorem.

Encoder–denoiser: accepted · equivalent-after-elaboration

Detected semantic differences

domains: Compared with the full paper setting, the selected result allows arbitrary finite-dimensional real inner-product spaces. — Explicitly disclosed in extraction, module and lesson; a valid broader calculation.
assumptions: The actual theorem allows any probability μ and all η>0, without Gibbs, smoothness, curvature or upper-step-size assumptions. — Explicit in lesson and candidate assumptions; not presented as proving those source assumptions.
scopes: The source proposition concerns its density-defined joint law; the Lean law is a generative pushforward. — The missing density-to-generative identification is expressly a separate obligation. Acceptance covers only generative-reflection.
scopes: The generative proof route does not formalize the source's direct density proof or its operator/process consequences. — No such closure is claimed. L² and algorithmic adapters remain outside this declaration.

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

Only the generative-law symmetry component of Proposition 2.1(iii) is formalized here; source assimilation still needs (2.6)=(2.7).
A measure-preserving deterministic reflection is not an invariance proof for the conditional half-turn process or the complete PBPS Markov chain.
No conditional density, non-explosion, reversibility, L2 operator domain, mixing rate or query complexity is asserted.
The claimed downstream uses in Section 3.1 and Appendix B are intended consumers, not already compiled Lean callers.

Source and reuse

ASTIS parents called

Mathlib API called (external library)

ProbabilityTheory.stdGaussian_map: invariance of the standard Gaussian under linear isometric equivalence.
LinearIsometryEquiv.neg: negation as a real linear isometry.
MeasureTheory.Measure.map_prod_map, map_id and map_map: measurable product and composition pushforwards.
Measurability of continuous affine maps; elementary additive-group cancellation.

Mathematical sources

Chen, Chewi, Lu and Zhang, Accelerated High-Accuracy Sampling from a Warm Start via the Proximal Bouncy Particle Sampler v1 — Proposition 2.1(iii), with the generative law (2.7) and reflection (2.14). ASTIS exposition, not copied source prose.

ASTIS 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 edges are actually covered?

Local proof component; source adapter/review separate Involution and reflection invariance of the displayed generative Gaussian augmentation
TODO — not closed by these contributions Identify the normalized density (2.6) with the generative probability law (2.7) under the source hypotheses

Common setting and conventions

Optimized expected gradient complexity

Statement

Proof architecture and calculations

Augment, then reflect the auxiliary variable

Use discrete hypocoercivity, not a reversible gap shortcut

Keep implementation mismatch separate from ideal mixing

Balance localization against block length

Strict boundary

The Gaussian posterior is an actual measurable conditional kernel

Complete source statement (ASTIS restatement)

source Gibbs specialization

normalized generic conditional kernel

Read the formalized proofs

The Gaussian posterior is an actual measurable conditional kernel

Statement

All objects and hypotheses

Mathematical proof

1. A bounded positive likelihood has a genuine normalizer

2. Make the normalized fibers into one measurable Markov kernel

3. Express the actual joint law in observation-first order

4. Cancel only the proven positive normalizer

5. Identify the observation marginal and certify disintegration

Source assumptions versus formal assumptions

Detected semantic differences

Scope and omitted-condition boundaries

Source and reuse

ASTIS parents called

Mathlib API called (external library)

Mathematical sources

Which proof edges are actually covered?

The actual normalized Gibbs augmentation law

Complete source statement (ASTIS restatement)

(2.6)-(2.7), explicit normalizer

Read the formalized proofs

From a genuine Hessian bound to the normalized augmented Gibbs law

Statement

All objects and hypotheses

Mathematical proof

1. Establish the Gibbs probability from curvature

2. Use the generative Gaussian joint law

3. Change the reference measure to product volume

4. Combine the exponents and retain nonvacuous normalization

Source assumptions versus formal assumptions

Detected semantic differences

Scope and omitted-condition boundaries

Source and reuse

ASTIS parents called

Mathlib API called (external library)

Mathematical sources

Which proof edges are actually covered?

A Gibbs L2 representative is locally Lebesgue L2

Complete source statement (ASTIS restatement)

Local L2

Read the formalized proofs

A Gibbs L2 representative is locally Lebesgue L2

Statement

All objects and hypotheses

Mathematical proof

1. Use the squared-norm characterization of L2

2. Remove only the Gibbs normalization

3. Invert the weight locally

4. Transfer measurability to volume

5. Restrict to an arbitrary compact set

Source assumptions versus formal assumptions

Scope and omitted-condition boundaries

Source and reuse

ASTIS parents called

Mathlib API called (external library)

Mathematical sources

Which proof edges are actually covered?

Gaussian-noise density prerequisite for the proximal augmentation

Complete source statement (ASTIS restatement)

isotropic-noise density prerequisite of (2.6)-(2.7)

Read the formalized proofs

The explicit density of isotropic Gaussian noise

Statement

All objects and hypotheses

Mathematical proof

1. Choose orthonormal coordinates