Let $V$ be a smooth cubic surface over a $p$-adic field $k$ with good reduction. Swinnerton-Dyer (1981) proved that $R$-equivalence is trivial on $V(k)$ except perhaps if $V$ is one of three special types--those whose $R$-equivalence he could not bound by proving the universal (admissible) equivalence is trivial. We consider all surfaces $V$ currently known to have non-trivial universal equivalence. Beyond being intractable to Swinnerton-Dyer's approach, we observe that if these surfaces also had non-trivial $R$-equivalence, they would contradict Colliot-Thélène and Sansuc's conjecture regarding the $k$-rationality of universal torsors for geometrically rational surfaces. By devising new methods to study $R$-equivalence, we prove that for 2-adic surfaces with all-Eckardt reductions (the third special type, which contains every existing case of non-trivial universal equivalence), $R$-equivalence is trivial or of exponent 2. For the explicit cases, we confirm triviality: the diagonal cubic $X^3+Y^3+Z^3+ζ_3 T^3=0$ over $\mathbb{Q}_2(ζ_3)$--answering a long-standing question of Manin's (Cubic Forms, 1972)--and the cubic with universal equivalence of exponent 2 (Kanevsky, 1982). This is the first in a series of works derived from a year of interactions with generative AI models such as AlphaEvolve and Gemini 3 Deep Think, with the latter proving many of our lemmas. We disclose the timeline and nature of their use towards this paper, and describe our broader AI-assisted research program in a companion report (in preparation).

翻译：设 $V$ 为 $p$-进域 $k$ 上的光滑三次曲面，且具有好约化。Swinnerton-Dyer (1981) 证明了 $V(k)$ 上的 $R$-等价是平凡的，除非 $V$ 属于三种特殊类型之一——他未能通过证明泛函（容许）等价是平凡来界定这些曲面的 $R$-等价。我们考虑目前已知具有非平凡泛函等价的所有曲面 $V$。除了这些曲面对于 Swinnerton-Dyer 的方法难以处理之外，我们观察到，如果这些曲面还具有非平凡的 $R$-等价，那么它们将与 Colliot-Thélène 和 Sansuc 关于几何有理曲面上泛函扭子的 $k$-有理性的猜想相矛盾。通过设计研究 $R$-等价的新方法，我们证明了对于具有全 Eckardt 约化的 2-进曲面（即第三种特殊类型，包含了所有已知的非平凡泛函等价情形），$R$-等价是平凡或指数为 2 的。针对具体情形，我们确认了平凡性：对角三次曲面 $X^3+Y^3+Z^3+ζ_3 T^3=0$ 在 $\mathbb{Q}_2(ζ_3)$ 上——回答了 Manin 长期悬而未决的问题（《三次形式》，1972 年）——以及泛函等价指数为 2 的三次曲面（Kanevsky, 1982）。本文是一系列工作的开篇，这些工作源于与诸如 AlphaEvolve 和 Gemini 3 Deep Think 等生成式 AI 模型一年互动的成果，其中后一种模型证明了我们的许多引理。我们披露了这些模型在此论文中使用的时间线和性质，并在另一份配套报告中（正在准备中）描述了我们更广泛的 AI 辅助研究计划。