Annular variation and dyadic absolute bounds for the triangular Hilbert transform
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Proves maximal and annular r-variation bounds, for every r > 2, from complex \(L^3(\mathbb R^2)\times L^3(\mathbb R^2)\) to \(L^{3/2}(\mathbb R^2)\). The maximal estimate controls both hard truncation endpoints and gives almost-everywhere and norm convergence. Pairing with a third input settles the triangular Hilbert transform estimate at the symmetric \(L^3\times L^3\times L^3\) point. (OpenAI’s Lean note)
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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.
Three other possible matches are listed below the card, among them Theorem 1.1(Full annular variation) in Annular variation of the triangular Hilbert transform at the symmetric point and Theorem 1.2(Disjoint-annulus count estimate) in Annular variation of the triangular Hilbert transform at the symmetric point.
There is an absolute constant C such that, for every pair of complex functions F,G\in L^3(\R^2),
We prove the annular r-variation estimate for the triangular Hilbert transform from complex \(L^3\times L^3\) to L3/2 for every r > 2. The partitions may depend on the output point and range over all positive scales. The estimate yields the two-endpoint maximal bound and joint almost-everywhere and norm principal values, and resolves the symmetric scalar triangular Hilbert transform problem.
We prove a uniform \(L^3\times L^3\times L^3\) estimate for the dyadic triangular Hilbert form with unrestricted real inputs. The sum of the absolute local contributions over any finite set of scales is bounded by forty times the product of the input norms. In particular, the bound allows coefficients of modulus at most one to vary independently among admissible interval triples.
For arbitrary complex inputs in \(L^3(\mathbb R^2)\), we prove the pointwise maximal \(L^3\times L^3\to L^{3/2}\) estimate for the triangular Hilbert transform, with the supremum over both hard truncation endpoints. The estimate yields joint almost-everywhere and L3/2 convergence as the lower endpoint tends to zero and the upper endpoint tends to infinity. This also proves the conjectured scalar estimate at the symmetric point.
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TriangularHilbert.lean, the release lists it as a main result.
defMainEstimate : Prop := ∃ C : ℝ≥0, ∀ F G : Plane → ℂ, MemLp F 3 volume → MemLp G 3 volume → (∀ᵐ z ∂volume, GoodPoint F G z) ∧ AEMeasurable (maximal F G) volume ∧ maximalNorm F G ≤ (C : ℝ≥0∞) * eLpNorm F 3 volume * eLpNorm G 3 volume⋮theoremmain_estimate : MainEstimate := bysorry
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MathVet finds that the Lean statements express part of the advertised result (their term: “partial”).
Their note:Maximal L^3 x L^3 -> L^{3/2} bound only; annular r-variation (r > 2), a.e./norm convergence and the trilinear symmetric-point estimate are not stated.
Definitions the group asks readers to compare with the paper:maximal F G z := if GoodPoint F G z then ⨆ E : Endpoints, ENNReal.ofReal ‖truncation F G E z‖ else 0 (junk value 0 off the good set; the statement itself asserts ∀ᵐ z, GoodPoint F G z)
“partial”, in their words: Only a special case is stated, or a co-equal headline claim has no challenge statement.
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cert-machine finds that the Lean statements express a narrower statement than the advertised result (their term: “NARROWER — UNDECLARED”).
The gap they name:Not stated: the annular r-variation estimate (every r > 2), the a.e. and norm convergence of the truncations, and the trilinear L^3 x L^3 x L^3 estimate. Also not stated: the dyadic triangular Hilbert form bound of the family’s second manuscript, which the family title names but the headline text does not.
“NARROWER — UNDECLARED”, in their words: The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
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The formalized result proves the symmetric maximal estimate for the triangular Hilbert transform: for arbitrary complex $F,G\in L^3(\mathbb R^2)$, the $L^{3/2}$ norm of the supremum over all finite hard-truncation intervals is at most $C\|F\|_3\|G\|_3$ for one absolute constant $C$. It also establishes a common full-measure set on which the truncated integrals are defined and almost-everywhere measurability of the maximal output.
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Unreviewed AI draft: states a paper’s theorem as the paper does. States the main theorem: the L^3 × L^3 → L^{3/2} bound for the maximal triangular Hilbert transform at the symmetric point, with one absolute constant.
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Only a special case is stated, or a co-equal headline claim has no challenge statement.
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An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge. Its protocol describes a planned error-rate study; this import does not establish that the study has been completed.
The challenge statements state less than the headline, and OpenAI’s Lean note does not say so.
Their method
Every Comparator challenge re-run on Linux with nanoda switched on, decided by a preregistered rule; every family’s challenge statements read against OpenAI’s headline, clause by clause. Decider: Comparator (leanprover/comparator d03acab1 = tag v4.34.0, built with the release’s toolchain v4.34.1), lean4export 076e8e57, landrun 811cfff5, nanoda_lib 3a240721 — on GitHub’s Linux runners…
Re-run through Comparator in a sandboxed virtual machine: accepted.
Date
8 October 2026
Version
at the release’s recorded version
Their method
An independent, model-assisted, pre-referee reading of each family’s challenge statements against its headline claim under a published rubric (PROTOCOL.md), with its own Comparator runs per challenge.