# R-equivalence on Cubic Surfaces I: Existing Cases with Non-Trivial Universal Equivalence

arXiv: https://arxiv.org/abs/2603.19215 · HTML: https://mlsanity.com/paper/2603.19215/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 2 of its 25 references that are in the map.
- First posted on arXiv 19 Mar 2026.

## Builds on (most cited first, 2 of 2)

- [AlphaEvolve: A coding agent for scientific and algorithmic discovery](https://mlsanity.com/paper/2506.13131/) (2025)
- [Towards Autonomous Mathematics Research](https://mlsanity.com/paper/2602.10177/) (2026)

## Abstract (arXiv)

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-Th'el`ene 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 X3+Y3+Z3+zeta3 T3=0 over mathbbQ2(zeta3)--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).