Tencent Hunyuan's Hyra Agent + Hy3 Pin the Sumset-Difference Exponent at 2, Ending a 50-Year-Old Problem

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 finite integer sets proving the optimal exponent relating sumset growth |A+A| and difference set growth |A-A| is exactly 2. This settles an extremal problem open since 1969, when a classical theorem established the bound sigma^(1/2) <= delta <= sigma^2; the upper exponent 2 was known optimal, but the best lower-bound constructions had only reached about 1.1 in over five decades. Hyra autonomously surveyed literature, built candidate sets, guessed ratios, wrote the proof, and translated it into formally verifiable Lean 4 code, with all 2,212 compilation tasks passing and zero 'sorry' placeholders. The paper, 'Settling the Optimal Exponent Relating Sumsets and Difference Sets' (arXiv:2607.27199), lists Shanda Li as first author. Hyra uses a dual-loop architecture, with a Context Agent directing Proposal Agents and continuously refining evaluation criteria. The result marks a shift from AI as proof assistant to AI as principal investigator, potentially structurally compressing discovery cycles in foundational mathematics.

The Announcement

On July 30, Tencent Hunyuan announced that its research agent Hyra, in collaboration with the open-weight model Hy3, constructed a family of integer sets proving that the optimal exponent relating sumset and difference set growth on finite integer sets is exactly 2 — settling an extremal problem that had been open since 1969.

Paper: *Settling the Optimal Exponent Relating Sumsets and Difference Sets* (arXiv:2607.27199), first author Shanda Li
A companion Lean 4 formalization is open-sourced: all 2,212 compilation tasks pass, with zero sorry placeholders

The Problem

Given a finite integer set 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 usually two different values while a+b = b+a, the difference set |A-A| typically grows faster than the sumset |A+A|. Mathematicians asked the converse: can one construct exotic sets where the sumset expands far faster than the difference set?

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

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

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

From 1.1 to 2 in One Step

Marica (1969): 1.029
Freiman–Pigarev: 1.059
Penman–Wells: 1.125
Codex + human guidance (2026): 1.8
Hyra + Hy3: 2.0 (optimal)

Hyra produced an explicit construction with the exponent exactly equal to 2 — compressing 50 years of incremental progress into a single result.

Hyra Is Not a Model, It's a Research Organization

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

The Context Agent dispatches tasks; multiple Proposal Agents work asynchronously
Trial-and-error feedback flows into an experience bank, which informs the lead
A dual-loop structure: the inner loop optimizes solutions; the outer loop simultaneously improves *what counts as better* — preventing the AI from gaming the metric instead of truly solving the problem. This is the key distinction between Hyra and ordinary agents.

In this breakthrough, Hyra autonomously surveyed literature, constructed candidates, guessed ratios, wrote the proof, and then translated the proof into formally verifiable code — with human mathematicians acting more as gatekeepers.

Three Application Tracks

AI for AI: NanoChat BPB 0.9015, NanoGPT 76.4s, SOL 0.771 — beating Recursive records
AI for Science: best results on 29 of 55 open problems in EinsteinArena; a 15-parameter Transformer for sunspot recurrence — originality in basic research
AI for Fun: ranked 3rd of ~730 humans on Othello Botzone

Takeaway

This is not merely "AI solved a problem" — it is an AI research agent completing a full publishable proof as principal investigator, upgrading from proof assistant to proof partner. Problems where the upper bound is known but a matching construction is missing are especially suited to such agents: searching a vast possibility space for counterexamples is precisely where large models + agents excel. Once this long-horizon autonomous research workflow becomes reusable, discovery cycles in foundational 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, 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 in-depth analysis