ACTIVE GALACTIC retweeted
OpenAIが、ミレニアム問題の一部であるヤン-ミルズ理論の構成に成功したようです
openai.com/index/building-sc…
ACTIVE GALACTIC retweeted
「箱を揺さぶればもっと入るはず」という直感的なアプローチで立方体充填問題の記録が更新されるなんて。今まで誰もやったことがなかったのか疑問に思ったけれど、直感的だからこそ計算が簡単ではなかったのか。
new cube packing record for n=12 at 2.931514578 found with this method. beats Haowei Lin's 2026 record (2.93277+).
s=2.931514577965
# i x y z w qx qy qz
# cube=c+R(w,qx,qy,qz)[-1/2,1/2]^3, Hamilton
1 1.5243879843 2.4315145780 1.3320162706 0.5838886869 -0.5838886869 -0.3988408220 -0.3988408220
2 2.4315145780 2.4315145780 0.5000000000 0.7071067812 0.7071067812 0 0
3 0.5000000000 1.4369539605 1.4377198700 0.5395289954 0.4570650535 0.4570650535 -0.5395289954
4 0.5098735814 0.5769311229 2.3545747284 0.4560511351 -0.5403863020 -0.4560472394 0.5403896022
5 0.6086093613 2.2929283270 2.3887741023 0.0258332833 0.7609821302 -0.6482073268 -0.0081302199
6 2.4315145780 1.4315145780 0.5000000000 0 0 0 -1
7 2.4298401750 2.4215249296 2.2620514795 0.0000269579 0.0016621483 -0.9999712786 0.0073944948
8 1.5800388387 1.4806909324 2.4315145780 0 -0.9765112947 -0.2154662186 0
9 1.6870669871 0.7046048868 1.3931621797 0.1630483719 -0.8266039691 -0.3023204668 0.4458065074
10 0.6714192000 0.5125505069 0.5042784926 0.0019279356 0.0017767559 -0.7100527881 -0.7041435680
11 2.4315145780 0.5648562958 2.4315145780 0.5 -0.5 -0.5 0.5
12 0.5 2.3777488832 0.5 0.7071067812 -0.7071067812 0 0
ACTIVE GALACTIC retweeted
BREAKING NEWS
The Norwegian Nobel Committee has decided to award the 2026 #NobelPeacePrize to Navanethem “Navi” Pillay for her efforts to promote peace and international law.
#NobelPrize
ACTIVE GALACTIC retweeted
BREAKING NEWS
The Royal Swedish Academy of Sciences has decided to award the 2026 #NobelPrize in Chemistry to Henri B. Kagan and Kenso Soai “for the discovery of non-linear effects and autocatalysis in asymmetric organic synthesis.”
ACTIVE GALACTIC retweeted
BREAKING NEWS
The Royal Swedish Academy of Sciences has decided to award the 2026 #NobelPrize in Physics to Francis Halzen “for decisive contributions to the IceCube Neutrino Observatory and the discovery of high-energy neutrinos of astrophysical origin.”
ACTIVE GALACTIC retweeted
BREAKING NEWS
The 2026 #NobelPrize in Physiology or Medicine has been awarded to Karl Deisseroth, Peter Hegemann and Georg Nagel “for their discoveries concerning light-gated ion channels and optogenetics.”
ACTIVE GALACTIC retweeted
drama summary for those confused:
- Aug 15th: Tristan Buckmaster & Levent Alpöge make progress on a few important math problems
- they do NOT have a proof for the $1,000,000 Millenium Prize problem. BUT, they do claim to have a proof for a similar (non-Millenium) Navier Stokes problem that could help lead the way there
- Levent works at Anthropic, but this research was independent of his work there, with a mix of GPT and Claude models. Tristan is not related to Anthropic.
- Early Sep: Rumor spreads to OpenAI that Anthropic has solved a major problem. Tristan emails OpenAI to clarify. without revealing the problem they solved or how they did it.
- After hearing of the rumor, OpenAI started researching Navier Stokes with a new internal model.
- Sep 6th: OpenAI's Sebastien Bubeck tells Tristan that they solved the $1,000,000 Millenium Prize Navier Stokes problem. The approach is very similar to Tristan & Levent's approach to the non-Millenium problem.
- Tristan is suspicious of the timing, as only few others were trying this approach. OpenAI says the model didn't access his user data directly, but leaves unanswered whether Tristan's chat conversations were part of the training.
- OpenAI says they would partially credit Tristan for the $1,000,000 discovery (even though Tristan did not solve the $1,000,000 problem) — but only if they remove Levent as an author, as he works for Anthropic.
- Sep 8th: Tristan refuses to remove Levent, and rushes to publish their results independently.
Currently unclear is whether Anthropic had a separate solution for the $1,000,000 problem, or whether the rumor was about Tristan & Levent's independent research.
ACTIVE GALACTIC retweeted
ここに出てくる Alpöge 氏、S^6 の複素構造をAIで発見したと発表した人と同じ。
S^6 のときの本人のコメントを見る限り、証明を自分で十分に理解・検証してから公表したというより、まずAIを走らせてインパクトの強い結果を出し、数学的な検証はその後専門家に委ねる、と見える。
AIで証明を生成するのは自由だが、その検証コストを数学界に外部化したまま「解けた」と発表するのは、研究者としてあまり健全な態度には思えない。そしてそのような状況でありながら、現在はNSに取り組んでいたと。
It appears OpenAI is claiming its AI just solved Navier-Stokes, the 3rd most famous unsolved math problem in the world.
It's one of the 7 Millennium Prize Problems, with a $1 million bounty.
(The only two more famous ones are the Riemann Hypothesis and P vs NP)
DRAMA: The mathematician who spent a year on this (with AI) says OpenAI raced to the finish after his work leaked to them, then pressured him to drop his co-author, who works at Anthropic
To be clear, what's actually verified is the mathematician's result on Navier-Stokes next-door neighbor problem. OpenAI's proof of the real thing hasn't been revealed yet, and OpenAI hasn't responded to the accusations
Here's Fable 5.1's summary:
ACTIVE GALACTIC retweeted
This image helps explain what happened along the Nepal–Tibet border. According to reports, a magnitude 4.4 earthquake on the Tibetan side triggered a massive ice avalanche, which then led to the flash flooding of the Bhotekoshi River in Nepal. Around 1,000 people are missing, and many have been killed.
Readers added context they thought people might want to know
Image is AI-generated and has China/Nepal labels reversed. USGS later clarified there was no earthquake — the M4.4 signal was produced by the glacial collapse and debris flow itself (now listed as M5.2 landslide).
earthquake.usgs.gov/earthquakes/ev…
bbc.co.uk/news/articles/…
ACTIVE GALACTIC retweeted
BREAKING: At least 380 people missing after flash flood hits near Nepal-Tibet border
ACTIVE GALACTIC retweeted
ヒューマノイドの歩行プログラム開発経験者であれば遊脚を前に振り出した際に発生するヨー軸回りのモーメントを腕を振ることでキャンセルするのは常識というか本能ですが一方でダチョウなどはお尻を振ってそのヨー軸回りモーメントをキャンセルしているのでこのヒューマノイドの走行はそれらのハイブリッドな貴ガス
ACTIVE GALACTIC retweeted
ゲームでは昔は8bitで角度を表す256°系の三角関数を作ったりしてた
角度をラジアンで表すのをやめ、「1回転=1」で表せば、コードからπを消せる。しかも速く、正確になる――という提案。
記事でCasey Muratoriが推すのは、角度を turn(回転) で扱う方法だ。0.25 turn=90度、0.5=180度、1=360度となる。ゲームなどでは、もともと0〜1の周期値を sin に渡すためだけに 2π を掛けるコードがよくある。しかし高速な三角関数の内部では、計算しやすい範囲へ変換するため、結局πで割るような処理が行われる。つまり「ラジアンに変換して、すぐ別の表現へ戻す」という余計な往復が発生している。
turnをそのまま受け取る三角関数を用意すれば、この変換を省ける。さらに浮動小数点では、π/2のような値は厳密に表現できない一方、0.25、0.5、0.75は2進数で正確に表現できる。つまりturnは、典型的な角度についてコードが読みやすいだけでなく、計算量と数値精度でも有利だという。
重要なのは、これは単なる「度 vs ラジアン」の好みの話ではないことだ。プログラマーがラジアンを当然の前提としているのは、数学教育や標準ライブラリのAPIがそうなっているからにすぎない。円を0〜1でパラメータ化するturnも正当な数学的表現であり、三角関数そのものがラジアンを要求しているわけではない。
しかも発想自体はすでに実用化されている。たとえばCUDAには、入力 x に対して sin(πx) などを計算する sincospi があり、これは実質的に「半回転」を単位にしたAPIだ。既存ライブラリでも似た最適化が存在することから、著者は「πをtau(2π)に置き換えるより、そもそもπをコードから追い出した方がいい」と主張している。
computerenhance.com/p/turns-…






































