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The trilinear Hilbert transform conjecture at the L3 x L3 x L3 to L1 point (slopes 1, 2, 3)

The trilinear Hilbert transform with slopes 1, 2, 3 is T(f1,f2,f3)(x)=p.v.∫Rf1(x−t)f2(x−2t)f3(x−3t) dttT(f_1,f_2,f_3)(x)=\mathrm{p.v.}\int_{\mathbb R}f_1(x-t)f_2(x-2t)f_3(x-3t)\,\frac{dt}{t}. Its bilinear predecessor was proved bounded by Lacey and Thiele (1997-1999). In the trilinear case quadratic modulations survive in the associated four-linear form (the coefficients (−1,3,−3,1)(-1,3,-3,1) annihilate polynomials of degree two), so time-frequency analysis does not close the argument; Tao (2015) obtained only sublogarithmic growth for truncations, and Hu and Lie record the straight-line boundedness problem as open. The standard conjecture asks for bounds Lp1×Lp2×Lp3→LrL^{p_1}\times L^{p_2}\times L^{p_3}\to L^r over a range of exponents. In particular, is TT bounded from L3(R)×L3(R)×L3(R)L^3(\mathbb R)\times L^3(\mathbb R)\times L^3(\mathbb R) to L1(R)L^1(\mathbb R)?

Result
Proved(see note)
Status
Partial result
AI contribution
AI-discovered
Method
Argument
Field
Harmonic analysis: multilinear singular integrals
Posed by
Standard conjecture after Lacey and Thiele's bilinear theorem; the manuscript cites it as Conjecture 1.7 of B. Hu and V. Lie, On the curved trilinear Hilbert transform (arXiv:2308.10706, 2023)
Year posed
—
Years open
—
Solved
2026-10-05
Model
Unreleased internal OpenAI model
Vendor
OpenAI
Collaborators
—
Verification
Unreviewed
Publication
Announced
Collection
OpenAI math release (October 2026), version adc7f12
Significance
50 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Theorem 1.1: there is a finite CC with ∥T(f1,f2,f3)∥L1≤C∏j∥fj∥L3\|T(f_1,f_2,f_3)\|_{L^1}\le C\prod_{j}\|f_j\|_{L^3} for Schwartz functions, so TT extends uniquely to a bounded trilinear map L3×L3×L3→L1L^3\times L^3\times L^3\to L^1. The proof combines the Leng-Sah-Sawhney inverse theorem with new compression, curvature-localization and sparse counting lemmas that sum the scales without loss. It concerns only the fixed slopes 1, 2, 3 and the single exponent point (3,3,3;1)(3,3,3;1): the rest of the conjectured range, other slopes, maximal or truncation-uniform versions and higher-degree multilinear Hilbert transforms are not treated, as the manuscript itself says.

What the AI did

The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result, across roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. This result is not among the README's stated exceptions (the Riemann zeta zero-free region work and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author. The README also cautions that unformalized results could have issues.

Verification

No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source, read against the conjecture as the manuscript cites it; the proof (roughly 400 KB of TeX across ten sections) was not refereed. The family has no Lean formalization at the pinned commit (lean/docs/086.md does not exist). The argument uses the degree-two case of the Leng-Sah-Sawhney inverse theorem for Gowers norms as an external input.

Sources

Changelog1 change

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