Tencent's Hyra agent cracks a 50-year combinatorics problem

The research agent Hyra, developed at Tencent and running on the open-weights model Hy3, has produced a general construction for a question that has lingered in additive combinatorics for decades: how much larger can the sumset of a set of integers be than its difference set? Mathematicians had long proved the relevant exponent cannot exceed 2, but no one knew whether that upper bound was tight — whether the exponent could approach 2 arbitrarily closely. Hyra found the construction in a single 24-hour compute run; the human researchers were left to verify the logic, clean up edge cases, and write the proof.

Open weights, not open source

Hy3 is released as open weights — downloadable, runnable, fine-tunable — but the training data and full pipeline remain unpublished. That distinction matters: the community received the artifact, not the recipe. Hyra was programmed to search for specific sets and extract a general pattern, a task standard language models tend to miss because it demands guided search rather than statistical completion. The fact that the run finished in 24 hours speaks to inference efficiency, not merely model scale.

Verification in Lean 4 via GPT-5.6 Sol

The final translation into the formal language Lean 4 was performed by GPT-5.6 Sol. This is not a convenience; Lean 4 forces every proof step to be mechanically checkable, leaving no gaps for the human eye to gloss over. Using a separate model for formal verification has become a de facto standard in papers emerging from AI labs — the model that discovers the idea is not necessarily the one that can write clean, hallucination-free Lean code.

The upper bound is optimal, period

The paper confirms the exponent can approach 2 without reaching it, making the decades-old bound optimal. In additive-combinatorics terms, this closes a chapter opened in the seventies. The proof itself rests on a recursive construction that Hyra identified and then generalized — a pattern recognizably human, compressed into one day of GPU cycles.

What this means for practice

The result will not change any product tomorrow, but it signals a shift in the value chain of mathematical research: an agent that finds the kernel, a second model that translates it to formal code, humans who code-review the logic. Hy3 is available for download, yet without the training dataset its capabilities cannot be reproduced or audited — worth remembering when headlines proclaim an “open model solved an unsolvable problem.”