Tencent Hunyuan's Hyra Agent and Hy3 Settle 50-Year-Old Sumset-Difference Set Problem at Exponent 2

Forum topic · QianXun · 2026-08-24

Summary

On July 30, Tencent Hunyuan announced that its recursive self-improving research agent Hyra, working with the open-weight model Hy3, constructed a family of integer sets proving that the optimal exponent relating |A+A| and |A-A| on finite integer sets is exactly 2, resolving an extremal problem open since 1969. The paper 'Settling the Optimal Exponent Relating Sumsets and Difference Sets' (arXiv:2607.27199), first-authored by Shanda Li, comes with an open-sourced Lean 4 formalization in which all 2,212 compilation tasks pass with zero 'sorry' statements. The problem asks whether sumsets can grow faster than difference sets; a 1969 theorem bounded the exponent C(A) = log σ / log δ above by 2, and the best known constructions over five decades only slightly exceeded 1.1. Hyra operates as a miniature AI research organization with a Context Agent (research lead) and Proposal Agents (researchers), using a dual-loop optimization that refines both solutions and evaluation criteria. Beyond this proof, Hyra set records in AI-for-AI, AI-for-Science, and AI-for-Fun benchmarks, marking a shift from proof assistant to research partner.

Background in one line

On July 30, Tencent Hunyuan announced that its research agent Hyra, together with the open-weight model Hy3, constructed a family of integer sets proving that the optimal exponent relating |A+A| and |A-A| on finite integer sets is exactly 2 — ending an extremal problem unsolved since 1969. The paper, *Settling the Optimal Exponent Relating Sumsets and Difference Sets* (arXiv:2607.27199), lists Shanda Li as first author, and an accompanying Lean 4 formalized proof is open-sourced, with all 2,212 compilation tasks passing and zero sorry.

What the problem asks

Given a finite set of integers A:

A+A: how many distinct results can pairwise sums produce
A-A: how many distinct results can pairwise differences produce

Since a-b and b-a are typically two different values while a+b=b+a, the difference set |A-A| usually outgrows the sumset |A+A|. Mathematicians asked the reverse: can one construct an exotic set where the sumset inflates far faster than the difference set?

Define σ(A)=|A+A|/|A| (sumset inflation), δ(A)=|A-A|/|A| (difference set inflation), and the exponent:

> C(A) = log σ / log δ, measuring how aggressively the sumset inflates relative to the difference set. The larger, the more anomalous.

A classic 1969 theorem gives the upper bound C(A) < 2, i.e. σ^{1/2} ≤ δ ≤ σ². The right-hand exponent 2 was already known to be optimal, but whether the left-hand 1/2 could be improved stayed open for over 50 years. The best constructions worldwide only slightly exceeded 1.1.

From 1.1 to 2 in half a century

Progress trajectory:

1. Marica 1969 — exponent ~1.029 2. Freiman–Pigarev — ~1.059 3. Penman–Wells — ~1.125 4. 2026 (Codex + human guidance) — ~1.8 5. Hyra + Hy3 — exactly 2.0, optimal

Hyra directly produced an explicit construction with exponent precisely equal to 2 — closing 50 years of incremental progress in one leap.

Hyra is not a model, it is a research organization

Hyra is a recursive self-improving research agent released by Tencent Hunyuan on July 21. Its harness defines two roles — a Context Agent (research lead) and Proposal Agents (researchers) — simulating a miniature AI research organization:

The Context Agent delegates tasks to multiple Proposal Agents working asynchronously.
Agents iterate, experiment, and feed results back into an experience bank that in turn inspires new proposals.
A dual loop: the inner loop optimizes solutions, while the outer loop simultaneously improves the evaluation criteria themselves — preventing the AI from gaming scoring loopholes instead of genuinely solving problems. This is the key distinction between Hyra and ordinary agents.

In this breakthrough, Hyra autonomously surveyed literature, constructed counterexamples, guessed ratios, wrote proofs, and translated them into formally verifiable code; human mathematicians acted more as gatekeepers.

Three-scenario preview

AI for AI: NanoChat BPB 0.9015; NanoGPT 76.4s, SOL 0.771 — beating the previous Recursive record.
AI for Science: best results on 29 of 55 EinsteinArena open problems; sunspot recurrence modeling with a 15-parameter Transformer — original fundamental research.
AI for Fun: ranked 3rd among 730 humans on Othello Botzone.

Assessment

This is not merely "AI solved a problem" — it is an AI research agent completing a full publishable proof in the role of principal investigator, upgrading from proof assistant to proof partner. Classic bottlenecks of the form "upper bound established, only the matching construction is missing" are especially friendly to agents: searching a vast possibility space for counterexamples is exactly where large models plus agent scaffolding excel. Once such long-horizon autonomous research workflows become reusable, the discovery cycle in fundamental mathematics could be structurally compressed.

References

1. arXiv:2607.27199, *Settling the Optimal Exponent Relating Sumsets and Difference Sets* 2. Tencent Hunyuan Hyra blog / @TencentHunyuan X post, 7/30/2026 3. GitHub linhaowei1/sum-diff-proof (Lean 4 formalization, 2,212 tasks, zero sorry) 4. aiproducthub.cn news brief 5. freeai.help deep-dive