Hilbert transforms along Lipschitz directions

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Proves a uniform strong L² bound for the planar Hilbert transform along any Lipschitz unit vector field, at integration lengths bounded by an absolute multiple of its reciprocal Lipschitz constant. The estimate is uniform over inner truncations and yields an L²-bounded principal-value operator, establishing Stein’s weak-type conjecture at this short scale. (OpenAI’s Lean note)

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The paper states

Theorem 1.1

In A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026

Possible match to OpenAI’s description, selected from the papers’ main statements because it shares the most wording. No review confirming the selection is recorded.

Six other possible matches are listed below the card, among them Corollary 1.8 in the same paper and Proposition 1.15 in the same paper.

There are absolute constants a_*\in(0,1/2) and C_*<\infty such that every Lipschitz field v:\R^2\to S^1 with \Lip(v)\le1 satisfies

\sup_{0<\epsilon<a_*} \norm{H_{v,a_*}^{\epsilon}f}_{L^2(\R^2)} \le C_*\norm{f}_{L^2(\R^2)} \qquad(f\in\mathcal S(\R^2)).

Its symmetric principal value therefore satisfies

\norm{H_{v,a_*}f}_2\le C_*\norm f_2, \qquad \big|\{x:|H_{v,a_*}f(x)|>\lambda\}\big| \le C_*^2\lambda^{-2}\norm f_2^2 \quad(\lambda>0).

Six more possible matches

Corollary 1.8 in A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026.
Proposition 1.15 in A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026.
Theorem 1.4 (Restricted estimate on a vertical band) in A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026.
Proposition 1.9 in A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026.
Theorem 1.2 (Lacey–Li) in A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026.
Proposition 1.14 (Packet representation) in A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026.

Reported proof verification: One Lean statement in the release; one independent group reports that Comparator accepted a run for this file at the version shown here.
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Papers

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A uniform Hilbert transform estimate for Lipschitz directions

25 September 2026, current edition.

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We prove a uniform L² bound for the Hilbert transform along a Lipschitz unit vector field in the plane, with integration restricted to a fixed absolute multiple of the reciprocal Lipschitz seminorm. The bound is uniform in the inner truncation and holds for fields depending on both coordinates. This gives an affirmative answer to Stein’s weak-$(2,2)$ conjecture.

One Lean statement in this family

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`LipschitzHilbert`

The paper states

Theorem 1.1

In A uniform Hilbert transform estimate for Lipschitz directions, 25 September 2026

There are absolute constants a_*\in(0,1/2) and C_*<\infty such that every Lipschitz field v:\R^2\to S^1 with \Lip(v)\le1 satisfies

\sup_{0<\epsilon<a_*} \norm{H_{v,a_*}^{\epsilon}f}_{L^2(\R^2)} \le C_*\norm{f}_{L^2(\R^2)} \qquad(f\in\mathcal S(\R^2)).

Its symmetric principal value therefore satisfies

\norm{H_{v,a_*}f}_2\le C_*\norm f_2, \qquad \big|\{x:|H_{v,a_*}f(x)|>\lambda\}\big| \le C_*^2\lambda^{-2}\norm f_2^2 \quad(\lambda>0).

The words they share: principal, value, symmetric, 1/2, field, lipschitz, epsilon, norm, directions, transform, uniform, hilbert, estimate, constants.

Possible Lean counterpart

Selected from shared wording. No review of whether the statements mean the same thing is recorded.

Uniform short Hilbert-transform bounds for Lipschitz directions

LipschitzHilbert.lean, the release lists it as a supporting statement.

theorem main :    ∃ a : ℝ, a∈Ioo 0 (1/2) ∧ ∃ C : ℝ≥0, 0<C ∧ ShortHilbertConclusion a C := by  sorry

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The file states one more theorem: uniform_truncation.

One independent group reports that Comparator accepted a run for this Lean statement file: cert-machine. Who ran the proof-verification program

cert-machine records it as “CERTIFIED”.

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Independent comparisons and proof-verification reports

Does the Lean express the advertised result?

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MathVet and Carlos Toledo’s cert-machine independently assessed whether this family’s Lean statements express the whole result OpenAI describes in its catalogue. These are assessments of the family as a whole, not of each paper theorem separately.

MathVet finds that the Lean statements express or imply the whole advertised result (their term: “full”).

Their note: None; statement is exactly the short-scale uniform bound plus principal-value/weak-type consequences (not the full-scale Stein conjecture).

Definitions the group asks readers to compare with the paper: smoothPrincipalValue extends the symmetric quotient at 0 by Function.update... 0 (deriv g 0) (custom, correct for Schwartz f); file temporarily removes/re-adds the instance instCommCStarAlgebraComplex to fix scalar-multiplication elaboration (harmless hack)

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cert-machine finds that the Lean statements express or imply the whole advertised result (their term: “MATCHES”).

“MATCHES”, in their words: The challenge statements state the headline as OpenAI wrote it.

Their method: Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule.

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OpenAI’s own note on what its Lean covers

The formalization proves uniform $L^2$ bounds for short Hilbert transforms along Lipschitz unit vector fields in the plane, including fields depending on both coordinates. For Lipschitz constant $K>0$, every hard truncation with $0<\varepsilon\le R\le1/(10^6K)$ has one universal $L^2$ bound on Schwartz functions, independent of the inner cutoff.
At a fixed short scale for $1$-Lipschitz fields, the formalization also gives principal-value and weak-$(2,2)$ estimates and bounded extensions to all of $L^2$. These results establish the stated short-scale form of Stein’s question.

AI-drafted comparisons, with no recorded human review

Uniform short Hilbert-transform bounds for Lipschitz directions LipschitzHilbert
Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: uniform L² bounds for short hard truncations of Hilbert transforms along Lipschitz unit vector fields, with the principal-value and weak-(2,2) consequences stated in the theorem.

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Uniform short Hilbert-transform bounds for Lipschitz directions `LipschitzHilbert`

One report, at the release’s recorded version.

Reported by	Their word	Their account	Date	Original report
cert-machine	“CERTIFIED”	Re-run through Comparator with the independent nanoda kernel switched on: Lean’s kernel and nanoda both accept OpenAI’s solution.	10 October 2026	Original report

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29 references in A uniform Hilbert transform estimate for Lipschitz directions

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