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Precise statement
For fixed $1<p<\infty$ and a Banach space $X$, Burkholder and Bourgain showed in 1983 that the UMD property is equivalent to boundedness of the Hilbert transform on $L^p(\mathbb{R};X)$, with the quadratic comparisons $\hbar_{p,X}\lesssim\beta_{p,X}^{2}$ and $\beta_{p,X}\lesssim\hbar_{p,X}^{2}$ between the Hilbert transform constant and the UMD constant. What is the optimal dependence between $\hbar_{p,X}$ and $\beta_{p,X}$, uniformly over all UMD spaces $X$: can either quadratic bound be improved, in the best case to a linear one?
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fidelity, correctness, priority, or novelty.
What AI did
GPT-6 Astra
The paper's AI disclosure: "The examples proving the quadratic dependencies were found in conversations with OpenAI's Astra model. All statements and proofs in the final manuscript have been reviewed by the authors and subsequently proof-checked in Lean 4 using OpenAI's Astra model." The repository README adds that the Lean code, documentation, scripts and tests were generated with GPT-6 Astra under human direction. Co-developed: the decisive constructions came out of conversations with the model, and two named authors reviewed and take responsibility for the manuscript.
Checked here on 12 September 2026 by reading the repository at commit 055d95e. Main/PaperStatement.lean states the theorem as the two dimensions and all eight inequalities, with $n$ and $\sqrt n$ literally as in the paper and universal constants. Mathlib has no UMD constant, so the two constants are project definitions and are the trust surface; both were read. umdConstant is the infimum of $C$ over all $\sigma$-finite sample spaces, filtrations, $L^p$ martingales and unimodular coefficients for which the martingale transform is bounded by $C$ times the difference sum, and hilbertConstant is the infimum of $C$ over all $C^1$ compactly supported $f$ admitting a principal-value Hilbert transform in $L^p$ with norm at most $C\,\|f\|_p$; these are the standard notions. No sorry, no axiom declarations, no native_decide anywhere; tests/AxiomAudit.lean walks every project declaration transitively and fails on any admission or any axiom beyond propext, Classical.choice and Quot.sound. GitHub Actions ran the build and that audit on Ubuntu and Windows at the reviewed commit, both green, which puts the kernel check on machines other than the authors'. The build was not repeated here. Not formalised: Corollary 1.3 and the exact coefficients of Remark 1.2.
Neither exponent can be lowered. Theorem 1.1 constructs explicit $2^{n}$-dimensional spaces $X_n$ and $Y_n$ with $\hbar_{2,X_n}\asymp n$, $\beta_{2,X_n}\asymp\sqrt n$ and $\beta_{2,Y_n}\asymp n$, $\hbar_{2,Y_n}\asymp\sqrt n$, with universal comparison constants, so both quadratic comparisons are sharp at $p=2$; Corollary 1.3 extends the growth rates to every fixed $1<p<\infty$ by extrapolation, with constants depending on $p$. The spaces are built from the summation operators Wenzel had proposed as candidates for exactly this separation. The Lean formalisation covers Theorem 1.1 at $p=2$ over both real and complex scalars; the extrapolation to other $p$ and the exact coefficients of Remark 1.2 are outside it.
Known method families
construction (source-reported)
Source-reported tools: construction.
Independent: unknown · difference confidence: 0
What remains uncertain
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.
VibeMath has not independently verified the mathematical claim.
AI-attempt independence and training-data exposure are unknown unless explicitly documented.
VibeMath has not independently audited the mathematical statement, proof, or novelty claim.