Faulhuber、Petersen、van Velthoven和Voigtlaender四位研究人员解决了长期开放的Heil-Ramanathan-Topiwala（HRT）数学猜想[1]。该团队证明存在满足特定线性关系的非零Schwartz函数的反例[1]，通过AI辅助的初期策略和数值计算，结合向量化Zak变换和收缩映射论证完成了证明[1]。
研究过程中，团队获得的关键算子范数界达到0.996，略低于理论阈值1[1]。具体的无理数移位使用了α = (√5-1)/2[1]。论文虽然采用了AI辅助工具，但保持了手写的最终论证和可读的概述部分[1]。
A team of mathematicians comprising Faulhuber, Petersen, van Velthoven, and Voigtlaender has resolved the Heil-Ramanathan-Topiwala (HRT) conjecture, a problem that had remained open for years [1]. The researchers proved the existence of a counterexample by demonstrating a non-zero Schwartz function satisfying specific linear relationships [1].
The proof employed a combination of AI-assisted initial strategies and numerical computation to establish the result [1]. A critical technical achievement involved deriving an operator norm bound of 0.996, a value marginally below the unit threshold necessary for the counterexample [1]. The team utilized vectorized Zak transforms and contraction mapping arguments in their demonstration [1]. Their approach incorporated an irrational shift parameter of α = (√5−1)/2 [1]. Despite leveraging computational assistance in the exploratory phase, the final proof and conceptual overview were maintained in a hand-written, readable form [1].