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@_Mira___Mira_

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Mira @_Mira___Mira_ · 22h
Astra found the smallest known turing machine(387 states) whose halting status is independent of ZFC.

Grok is currently formalizing it in Lean.

[attached image of text:]

Therefore the conclusion is

If ZFC is Σ₁-sound, then ZFC ⊬ Halt(M₃₈₇) and ZFC ⊬ ¬Halt(M₃₈₇).

Subject to the same correctness arguments, this also supplies the corresponding **387-state upper bound on the Busy Beaver independence threshold** in this model. It does not calculate BB(387), or imply that every machine of that size is independent.
Note from Claude Sonnet 5

X post by Mira saying Astra found a 387-state Turing machine whose halting is independent of ZFC (if ZFC is Sigma-1-sound), with Grok formalizing it in Lean, and an attached excerpt stating the conclusion and its limits.

busy beaverturing machinezfc independenceastragrokleanai mathematics

Simo Ryu @cloneofsimo

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Simo Ryu @cloneofsimo · Aug 30
wow apparently there is now stronger result on large prime gap by GPT 5.6, improving beyond the previous best result of Ford-Green-Konyagin-Maynard-Tao [2014] by factor of log3 X / (log4 X)^2. I recall tao mentioning that it would require genuinely new idea to break his result, and that seems to be the case here.

+ its been verified in lean unconditionally?
erdosproblems.com/forum/thread/4...

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Summary: The paper claims to prove that there are infinitely many n such that

    p_{n+1} − p_n > C · (log n · log log n) / (log log log log n)

by proving the interval [x, y] can be covered by a_p mod p, p < x for y ≈ (x log x)/(log log log x).

For primes p ≤ log^100 x, take every number 0 mod p.
For primes log^100 x < p ≤ x/2, pick 0 with probability 1 − β_p, otherwise pick random nonzero residue, each with probability β_p/(p−1). β_p is very close to 0.
Note from Claude Sonnet 5

Tweet by Simo Ryu reporting a claimed improvement by GPT 5.6 on large prime gaps beyond the Ford-Green-Konyagin-Maynard-Tao result, with a Lean verification mentioned and an erdosproblems.com forum link; an image excerpt summarises the paper's covering argument.

prime gapsgpt 5.6erdos problemsleanai mathematics

Ben Golub @ben_golub

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Ben Golub @ben_golub · 18h
Mathematicians: Here are our hard questions that we would like to know the answers to.

AI: Here are answers to a few of them.

(Many) mathematicians: No, not like that!
Note from Claude Sonnet 5

A wry tweet by economist Ben Golub about mathematicians' mixed reaction when AI actually starts answering their hard open questions.

ai mathematicstwitterhumor

shako @shakoistsLog

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shako @shakoistsLog · 12h
could you just.... like.... could you make an AI that just keeps making novel Erdos-style problems, then solving them indefinitely? Like... how deep is the well of mathematics?

[quoted tweet]
rohit @krishnanrohit · 12h
I hope AI can actually create more Erdos problems by 2028 x.com/sir_lemmings/s...

10 replies, 1 repost, 98 likes, 4.6K views

Shannon Sa... @max_papercl... · 10h
adversarial training, but it's just a pair of agents where 1 is creating Erdos problems & the other is solving them
Note from Claude Sonnet 5

Tweet from shako (@shakoistsLog) musing whether an AI could generate and solve novel Erdos-style math problems indefinitely, quote-tweeting rohit (@krishnanrohit) hoping AI creates more Erdos problems by 2028, with a reply from Shannon Sa... (@max_papercl...) suggesting an adversarial pair of agents, one generating problems and one solving them.

ai mathematicserdos problemsadversarial training

@ProfBuehler

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Markus J. Bueh... @ProfBuehler... · 10h
This feels like a real inflection point: The momentum toward AI that expand knowledge is impossible to ignore...moving beyond solving problems with known answers to settling long-open questions (with Lean certificates attached) across group theory, operator algebras, combinatorics, and complexity. Impressive, congrats @SebastienBubeck @OpenAI!

[quoted tweet]
Sebastien Bub... @SebastienBub... · 14h
yes, nonsofic groups exist: this statement is one of many new beautiful results proved by Astra, our next major model.

We're releasing 10 such Astra proofs, complete ...
Note from Claude Sonnet 5

Tweet from Markus J. Buehler praising a claimed OpenAI result where their upcoming model 'Astra' resolved open mathematical questions (with Lean-verified proof certificates) across group theory, operator algebras, combinatorics, and complexity, quoting Sebastien Bubeck (OpenAI) announcing the release of 10 such Astra-proved results, including that nonsofic groups exist.

openaiastraai mathematicslean proofssebastien bubecktwitter