Recently came across a news item: AI has pushed a century-old math problem forward to 97.9%. Sounds like it’s almost solved, right? But in the math world, 97.9% isn’t like a game where grinding out the last 2.1% gets you through.

This time, the model is from Tencent Hunyuan @TencentHunyuan ’s Hy4 preview and Hyra, tackling a problem that’s stumped mathematicians for over a century: What’s the smallest possible volume of a 3D shape with the same width when measured from any direction? Mathematicians previously guessed the answer was around 0.41986. The strictest proven lower bound before was 0.380799; this time, it’s been pushed to 0.41104. 0.41104 ÷ 0.41986 is about 97.9%. So, this percentage represents numerical progress, not that 97.9% of the proof work is done—in fact, the problem still isn’t fully solved.

What’s even more important behind this news is what Hy4 preview has delivered: The full proof PDF has been made public The LaTeX source code has also been made public Key calculations were re-verified using exact rational numbers Peers can jump right in to check it, even hunt for bugs. Next time you see “AI cracks a scientific puzzle,” I suggest asking four questions first: What exactly has it delivered? Is the evidence public? Can the results be reproduced? How far along is it?

For Hy4 preview this time, the first three questions have verifiable answers. The last one? We’ll have to wait for the math community to keep checking.

Full proof: https://github.com/Tencent-Hunyuan/Hyra-results/blob/main/AI4Science/3d_blaschke_lebesgue/3d_Blaschke_Lebesgue.pdf… Hy4 preview model:

Tencent-Hunyuan/Hyra-results

Source: https://github.com/Tencent-Hunyuan/Hyra-results

Hyra Results

Companion artifacts for Hyra: Hunyuan Research Agent.

This repository collects concrete solutions that Hyra produced across a range of open problems in science, mathematics, engineering, and creative design, released alongside the Hyra launch post. Each folder holds the final solution artifact and, where relevant, the self-contained scripts that reproduce it.

📝 Launch post: hy.tencent.com/research/hyra

News

• 2026-08-19: 🧾 Three new results, each with a Lean 4 formalization of its finite and numerical core.

  • Beurling–Ahlfors transform. Iwaniec’s 1982 conjecture predicts the sharp Lᵖ operator norm p*−1; the best proven uniform coefficient drops from 1.575 (Bañuelos and Janakiraman, 2008) to 1.523958. Paper: AI4Science/beurling_ahlfors_bellman/.
  • Partial Hadamard matrices. Their asymptotic count is now known throughout every power-law range above n², not just the cubic range m ≫ n³, lowering the critical exponent from 3 (arXiv:2603.30013) to 2. Paper: AI4Science/partial_hadamard_counting/.
  • Commutators close to the identity. The identity is never a commutator, but comes within ε at a price: the least ‖D‖·‖X‖ needed falls from Tao’s O(log⁵(1/ε)), since refined to O(log⁴), to O(log³(1/ε)), against Popa’s log(1/ε) lower bound. Paper: AI4Science/commutator_log_cubed/.

• 2026-08-17: 📐 Blaschke–Lebesgue in three dimensions. The least volume of a convex body of constant width w is conjectured to be Meissner’s ≈ 0.419860 w³; the best certified lower bound rises from 4π/33 · w³ ≈ 0.380799 w³ (Nishioka, arXiv:2606.01754) to > 0.411040 w³, closing 77.4% of the remaining gap. Paper: AI4Science/3d_blaschke_lebesgue/.

• 2026-07-29: 🎉 The sum-vs-difference problem is settled. For finite A ⊆ ℤ, the optimal exponent relating |A+A| to |A−A| has supremum exactly 2, approached arbitrarily closely but never attained: a complete, machine-checked resolution going beyond the sums_diffs record below. Proof: sum-diff-proof.

Results

Each row compares the best prior published result (Prev best, from the cited system or leaderboard) against Hyra. Arrows mark the better direction (↓ lower is better, ↑ higher is better); the Hyra-winning value is in bold.

Prev-best sources.

• ^a TTT-Discover: Learning to Discover at Test Time (arXiv:2601.16175).
• ^b SimpleTES: Evaluation-driven Scaling for Scientific Discovery (arXiv:2604.19341).
• ^c AdderBoard trained-weights leaderboard (github.com/anadim/AdderBoard).
• ^d Olaparib, an approved PARP1 inhibitor (drug baseline).
• ^e Recursive: First Steps Toward Automated AI Research (github.com/recursive-org/first-steps-toward-automated-ai-research); AI4AI baselines: nanoGPT-speedrun, nanochat, and SOL-ExecBench.
• ^f Erich Friedman’s Packing Center (erich-friedman.github.io/packing). Hyra’s record-improving packings are credited there as “Found by Haowei Lin”: 100 across 28 shape-in-shape families, each beating the previously listed best.
• ^g Baseline: a “copy last frame” (persistence) forecast. Hyra’s score is the forecast R² on a fully-held-out, half-century-long segment of the record.
• ^h Nishioka: An improved lower bound for the three-dimensional Blaschke–Lebesgue problem from spectral and dual perspectives (arXiv:2606.01754), which gives 4π/33 ≈ 0.380799109526. Hyra’s certified bound is (130838246407123/10¹⁵)·π > 0.411040473721188.
• ⁱ R. Bañuelos and P. Janakiraman, Lᵖ-bounds for the Beurling–Ahlfors transform (2008), which gives the uniform coefficient 1.575. Iwaniec’s conjectured sharp value is 1.
• ^j D. Davis: Counting Partial Hadamard Matrices in the Cubic Regime (arXiv:2603.30013).
• ^k B. Bilich: An O(log⁴(1/ε)) refinement of Tao’s construction of commutators close to the identity (github.com/bilichboris/TaoCommutators); the published exponent is Tao’s 5, since refined to 4.

Metrics are as defined by each benchmark; comparisons are against the cited published results.

Metric notes.

• C₁: first autoconvolution/autocorrelation constant, max(f∗f)/(∫f)² (minimize).
• R: second autocorrelation ratio, ‖f∗f‖₂² / (‖f∗f‖₁·‖f∗f‖∞) (maximize).
• C₅: Erdős minimum-overlap constant (minimize).
• C(A): sum-vs-difference exponent, log(|A+A|/|A|) / log(|A−A|/|A|) (maximize).
• params: unique trainable parameters of a transformer that adds two 10-digit integers at ≥ 0.99 accuracy (minimize).
• objective: PARP1 docking objective, Vina score + 10·(1 − QED) (minimize).
• CNOTs added: extra CNOTs inserted by SWAP routing (minimize; 1 SWAP = 3 CNOTs).
• forecast R²: rolling-origin, free-running (24-month) forecast R² for monthly sunspot numbers (maximize).
• Vol/w³ bound: largest proven universal lower bound on Vol(K)/w³ over all convex bodies K ⊂ ℝ³ of constant width w (maximize; the conjectured optimum is ≈ 0.419860).
• C_BA: smallest proven uniform coefficient in ‖B‖_{Lᵖ→Lᵖ} ≤ C_BA·(p*−1) for the Beurling–Ahlfors transform, over all 1 < p < ∞ (minimize; the conjectured optimum is 1).
• regime exponent: infimum of the power-law exponents u for which m/nᵘ → ∞ is known to force the asymptotic formula for N(n,m) (minimize; the endpoint value itself is not claimed).
• log exponent: smallest proven k in m(ε) = O(logᵏ(1/ε)) for commutators within ε of the identity (minimize; the matching lower bound is k = 1).

Note. The results above were current as of 2026-07-10, except 3d_blaschke_lebesgue, added 2026-08-17, and the three entries added 2026-08-19. Several of these problems live on public, continuously-updated leaderboards; later entries there may warm-start from Hyra’s published solutions to reach still-better numbers.

AI4Fun

Creative and game-playing demos (no leaderboard comparison):

• AI4Fun/reversi/: a AlphaZero-style Reversi (Othello) bot for the Botzone 8×8 arena (C++ pattern/n-tuple net + PUCT-MCTS + exact endgame solver).
• AI4Fun/music/: a five-part arrangement of 望春風 (Bāng-chhun-hong, 1933, 鄧雨賢).
• AI4Fun/3d_penguin/: a procedural 3-D QQ-penguin built by a single Blender bpy script.
• AI4Fun/3d_hunyuan/: a procedural 3-D Tencent Hunyuan (混元) logo orb.

Citation

If you use these results, please cite:

@misc{hyra2026,
  title        = {Hyra: Hunyuan Research Agent},
  author       = {{Hyra Team}},
  year         = {2026},
  howpublished = {\url{https://hy.tencent.com/research/hyra}},
}

License

This repository is licensed under the Apache License, Version 2.0; see LICENSE.