OpenAI's Astra cuts the bounded prime gap record to 186
Two things worth your attention this evening: OpenAI quietly posted a paper it says was proved by GPT-6 Astra, and New York City decided that 600,000 children should not touch generative AI at all. One is a claim about machine reasoning at the frontier of number theory. The other is a claim about what children need before they ever meet it.
OpenAI has published a paper, "Improved Short Gaps Between Primes," whose abstract states plainly that the proof is due to GPT-6 Astra — and it improves the standing record on one of mathematics' most famous open problems. The result is that the gap between consecutive primes keeps returning to at most 186 infinitely often, down from the 246 that the Polymath 8b project established a decade ago. The paper derives DHL[40,2] — every admissible set of forty integer shifts has infinitely many translates containing at least two primes — by combining equidistribution estimates from Polymath 8a and from Julia Stadlmann's more recent work with new factorization conditions that let the multidimensional Selberg sieve run over a larger support. The authors note, in a line that deserves to be widely quoted, that Stadlmann has independently and concurrently proved a bound of 240.
The evidence OpenAI is offering is unusually concrete for a lab announcement, and that is the interesting part. The companion repository ships a Lean 4 formalization of the main theorem, along with a Python certificate that recomputes the numerical bounds from scratch. The catch is right there in the name: the formalization is conditional. It rests on three explicit axioms that remain unproved inside Lean — two finite-field exponential-sum estimates traceable to Deligne's work, plus 104 outer and 45 inner numerical integral bounds plus three cap bounds. The project's own metadata is candid about this, listing the permitted axioms and marking the review as self-assessed. What has been verified is that the chain of reasoning from those assumptions to the result is correct, not that the assumptions hold.
That distinction is the story. Record-chasing in analytic number theory has always been a numerical game as much as a conceptual one — Zhang's original 70 million fell to 4,680, then 600, then 246 through better sieve weights and more computing. A model that can propose the right enlarged support and then have the arithmetic checked is doing something genuinely new, and the honest framing is not "AI proved a theorem" but "the part we can machine-check, we did." Independent verification by number theorists is the next domino, and it has not fallen yet.
New York City will impose a one-year moratorium on student-facing generative AI from 2-K through eighth grade, affecting nearly 600,000 students — roughly two-thirds of the largest school district in the United States. Mayor Zohran Mamdani and Schools Chancellor Kamar Samuels announced the policy Wednesday, days before the September 10 start of the school year. It covers all software using student-facing generative AI, including companion chatbots offering psychological support. High schoolers get a narrower regime: twice-yearly critical thinking modules and limited pilots. Teachers keep AI for lesson planning and operational tasks but are barred from using it to grade. City Hall says 38 existing education department programs are being cut or altered because of their AI use.
Enforcement is the obvious hole. Mamdani himself acknowledged the city may not know every AI tool in its buildings, since schools buy software outside citywide contracts, and the United Federation of Teachers said the policy "leaves many questions unanswered" — particularly about how the department will demand AI safeguards from the vendors it buys from. A moratorium that regulates the screen but not the supply chain is a statement of values more than an operating plan. Still, this is a reversal from a March framework that let students use AI for research and creative projects, and it lands in the same week that Washington is fighting over whether AI needs a regulator at all.
Separately, Microsoft AI shipped MAI-Transcribe-2, a speech recognition model it says beats Gemini 3.5 Transcribe and GPT-Transcribe on accuracy and speed, priced at ten cents per audio hour through the end of the year. That is roughly a 72% cut from the 36 cents an hour Microsoft charged for the first model in this line five months ago; language coverage has gone from 25 to 60 in the same span, and features rivals sell separately — speaker diarization, word-level timestamps, keyword biasing, code switching — are bundled into the base rate. This is the modality where Microsoft's strategy of building its own models to swap out of OpenAI's has moved fastest, and the price is the argument.
What to watch: whether a number theorist not employed by OpenAI confirms the 186 bound, and whether New York's taskforce report in April turns a moratorium into an actual framework.
If a model can produce a proof that Lean verifies — even a conditional one — does "the proof is due to GPT-6 Astra" read as authorship to you, or as something closer to instrumentation? Tell us in the comments.
Sources: Improved Short Gaps Between Primes (OpenAI) · PrimeGaps186 (GitHub) · Hacker News discussion · The Guardian — NYC AI ban · CBS New York — NYC AI ban · Microsoft AI — MAI-Transcribe-2 · VentureBeat — MAI-Transcribe-2
