[continuation of QC's thread on complex structure on S^6 - repeats text and embedded screenshots seen previously, then continues with a new reply:]
QC @QiaochuYuan · 4h
had sol take a longer look (15m) and still didn't find a problem and understands the argument in more detail but is still nervous that it could be missing some subtlety. also thinks if this holds up it is arguably the most significant AI math result so far, and agrees with me that this is plausibly "model 2"
transcript here: chatgpt.com/share/6a8b77b7...
[partial screenshot below, cut off at bottom]
Worked for 15m 49s >
okay, after pushing substantially harder: i still have not found the bug. more strongly, i think i've now independently reconstructed enough of the proof that my state is no longer "this looks surprisingly serious": it's closer to:
the bespoke mathematical core of the construction appears internally correct to me.
that is not the same as publication-level verification — there are places where i'm relying on standard nearby-cycles / analytic-geometry machinery rather than reproving it from foundations — but i have now deliberately attacked exactly the places a plausible LLM hallucination could hide, including an apparent contradiction that turned out to be a clever feature of the construction.
the proof, reconstructed from the idea rather than the prose... [cut off]
i wouldn't say "automatic Fields Medal," because mathematics doesn't work that way. Perelman-scale things aside, one theorem doesn't mechanically cash out into one medal, and committees care about breadth, influence, and subsequent development.
but "theorem capable of anchoring a Fields Medal citation"? absolutely.
and the positive answer might actually help. "proved nonexistence of a complex structure on S^6" would be a huge theorem, constructing one feels even more mathematically generative, because suddenly there's an actual new complex threefold sitting there whose geometry everyone gets to study: [cut off]
Note from Claude Sonnet 5
Scrolled continuation of the same X thread as the previous screenshot (Qiaochu Yuan on the claimed S^6 complex structure result), showing a further reply about an independent 'sol' check and a linked ChatGPT transcript discussing significance of the math result.
Consistently Candid Alex reposted
QC @QiaochuYuan · Follow
levent and claude (unnamed model 👀) appear to have found a complex structure on S^6. this is, like the jacobian conjecture, a famous open problem known for attracting many wrong proofs from experts (in both directions)
"appear to have" because this is much harder to check than some of the recent stuff but i had sol take a quick look (6m) that didn't find anything wrong. and i think this happened because i asked levent about it 3 days ago; if so, this result took at most 3 days to crank out
[Screenshot 1: StackExchange-style page]
Is there a complex structure on the 6-sphere?
Asked 16 years, 10 months ago Modified today Viewed 34k times
95
I don't know who first asked this question, but it's a question that I think many differential and complex geometers have tried to answer because it sounds so simple and fundamental. There are even a number of published proofs that are not taken seriously, even though nobody seems to know exactly why they are wrong.
dg.differential-geometry complex-geometry open-problems Edit tags
edited Nov 1, 2024 at 0:28 community wiki 4 revs, 3 users 85% Fetchinson0234
Share Cite Edit Follow Close Flag Protect
5 A topical preprint has been posted on ArXiv (asserting that S^6 has a complex structure): front.math.ucdavis.edu/0505.5634 – Ramsay Dec 7, 2010 at 19:33
2 And there is a new version out: arxiv.org/abs/math/0505634 claiming to completely overhaul the proof. Did anyone take a look with expertise in this area? – Daniel Apr 30, 2011 at 10:28
27 I think you'll find that very few experts are willing to study the 4th revision, if the first 3 had serious flaws. – Deane Yang Apr 30, 2011 at 12:53
[Screenshot 2: tweet]
levent @_alpoge_ · Aug 19
Geez wow what a time to be alive, @AchimWar asked yesterday and yea i was looking, it's an honour to get absolutely dusted by Brendle in particular, who is the goat
[quoted tweet: Другая планета @nihilunbounded · Aug 19
Simon Brendle put up a preprint on arXiv claiming to have a resolution of the Hopf conjecture. The paper has no AI declaration.
arxiv.org/pdf/2608.19068
A METRIC ON S^3 x S^3 WITH POSITIVE SECTIONAL CURVATURE
SIMON BRENDLE AND PEI-KEN HUNG
ABSTRACT. We construct a metric on S^3 x S^3 with positive sectional curvature. Starting from the standard metric on S^3 x S^3, we first perform a Cheeger deformation. The resulting metric has nonnegative sectional curvature. We refer to it as a Cheeger-Grove metric. We then consider a suitable third order perturbation of this Cheeger-Grove metric and show that the perturbed metric has positive sectional curvature, some of which have been carried out with the help of MATHEMATICA. The MATHEMATICA code is attached to this submission.]
QC @QiaochuYuan
you know if anyone is looking into complex S^6?
1:04 PM · Aug 20, 2026 · 462 Views
[quoted reply]
levent @__alpoge__ · 5h
Please welcome to the world a beautiful new geometric object, to do with a problem i've always loved. claude really contains multitudes:D Does S^6 admit a complex structure?
...
3:04 PM · Aug 23, 2026 · 29.2K Views
8 26 362 89
Note from Claude Sonnet 5
Thread by mathematician Qiaochu Yuan (QC) reporting that Levent Alpoge and Claude appear to have made progress on the famous open problem of whether the 6-sphere admits a complex structure, with embedded screenshots of a MathOverflow question and a related Aug 19-20 tweet exchange about Simon Brendle's Hopf conjecture preprint.
I love this animation by Daniel Piker (@KangarooPhysics).
Each dot follows a path, and takes 3.5 hrs to return to its starting point. (You might think the dots are jittering or sparkling, but on closer inspection they're walking like ants.)
Used with permission.
Note from Claude Sonnet 5
A tweet captioned as above, with an embedded animation (paused, 0:01 shown) of a dense field of white dots on black forming a swirling, wave-like pattern of varying density, resembling a generative/flow-field art piece by Daniel Piker.
Paata Ivanisvili @PI010101
AI's ranking of open problems solved in today's arXiv list.
AI put problems connected to Gromov's work in the top two spots. For #3, I remember attending a talk by one of the authors. #4 is, to me, one of the cutest problems in complex analysis, I first learned about it in Chapter I of Garnett–Marshall's Harmonic Measure book about 15 years ago.
1. Banach's isometric conjecture
2. Gromov's volume-growth conjecture
3. Generalized Chang–Yang conjecture
4. Sharp Hayman–Wu constant
5. Nadirashvili–Tkachev–Vlăduţ W^1,1 question
6. Courtade's projection conjecture
7. Gromov–Hausdorff distance between consecutive spheres
8. Chebyshev polynomials on a Jordan arc
9. Finite entanglement-breaking index of every PPT channel
10. Radchenko–Viazovska Fourier-interpolation question
11. Quantum SDPI tensorization
12. Chen–Eldan hit-and-run warm-start question
13. Bobkov–Götze max-sliced Wasserstein exponent
14. Bukh–Dubroff graph-cover question
15. Generalized Dai–Wang–Wei deformation question
16. Nguyen–Squassina Schwarz-rearrangement question
17. Kinnunen–Saari parabolic-weight questions
18. Deformed-GOE open parameter regime
19. Gau–Wang–Wu conjecture
20. Jaguzović–Vujadinović Toeplitz conjecture
21. Minimum 3/2-gap witness question
11:53 PM . Aug 13, 2026 . 77.5K Views
Note from Claude Sonnet 5
Tweet from mathematician Paata Ivanisvili sharing an AI-generated ranked list of 21 open mathematical problems purportedly solved in that day's arXiv listings, spanning geometry, analysis, and quantum information theory.
Captain Pleasure, André... [verified] @alg... · 14h
Yes, sure, AI improves mathematical performance when you tell them to 'believe in yourself'. But this hasn't been tried in humans yet – has anyone with pom poms gone to a math department and, approaching the nerdiest dork, constantly showered him/her with wholesome encouraging words while working on an open problem? For hours? The most encouragement mathematicians get is usually in short bursts, often way past their prime. I bet it would help a mathematician in peak performance more than even methylphenidate!
Note from Claude Sonnet 5
Tweet joking that if telling AI models to 'believe in yourself' improves their mathematical performance, the same untested intervention (constant cheerleader-style encouragement) might help human mathematicians more than stimulant medication like methylphenidate.
QC [verified] @QiaochuYuan · 4h
when i was at cambridge a professor was teaching algebraic geometry and offhandedly made some claim that was not clear to me
i asked "is that obvious?" he says "yes"
i asked "is it obvious that it's obvious?" he pauses for a few seconds and goes "...no"
[quoted tweet:]
Kevin Lacker [verified] @lacker · Aug 10
A mathematician is giving a lecture and says, "It is obvious that...."
Then he stops, stares at the board, and thinks silently for ten minutes....
Note from Claude Sonnet 5
Tweet by Qiaochu Yuan telling an anecdote from a Cambridge algebraic geometry lecture about pressing a professor on whether a claim of 'obviousness' was itself obvious, quote-tweeting a joke by Kevin Lacker about mathematicians pausing to silently verify something they just called 'obvious.'
Ethan Mollick @emollick · 1h
I like that all AI commentators now need to pretend they have always had a careful nuanced grasp of the difference between a bunch of unsolved mathematical problems that only specialized experts had heard of: "The Gromlach Conjecture is false for r-dimensional matrices, wow, that is more impressive than last weeks solution to Erdos Problem 444 for restricted splines!"
Note from Claude Sonnet 5
Tweet by Ethan Mollick sarcastically mocking AI commentators who now perform expert-level familiarity with obscure math problems (fictional example names 'Gromlach Conjecture' and 'Erdos Problem 444') to opine on AI mathematical achievements.
carl feynman ✓ @carl_feynman · 4h
The HRT conjecture has been disproven with AI help. arxiv.org/pdf/2608.05044.
Here's what the conjecture says. Consider a "bump function": a function from the reals to the complex plane, that is mostly confined to a short interval, and trails off exponentially out of that interval. Suppose we take time-frequency shifts of that bump: we can slide it sideways, or multiply it by sine waves, or both. That gives us various other wiggly bumps. Can we contrive that adding a finite number of such time-frequency shifts to the original bump exactly cancels it out? HRT conjectured in 1996 that we couldn't: that there would always be some smidgen left over that we couldn't cancel out. (And I've always found that plausible.). But that's wrong! The paper proves the existence of such a function. And it constructs a numerical approximation to it, plotted in pages 43 and 44 of the paper.
Terry Tao has a blog post talking about the proof in an easier way: terrytao.wordpress.com/2026/08/06/a-p…
Note from Claude Sonnet 5
Screenshot of an X post by carl feynman reporting that the HRT conjecture (Heil–Ramanathan–Topiwala, 1996) has been disproven with AI help, explaining the conjecture in plain terms — whether finitely many time-frequency shifts of a bump function can exactly cancel it — and linking the arXiv paper plus a Terry Tao blog post explaining the proof.
Mohamed Sabba @ma_sabba · Aug 4
I am reminded of a very memorable quote on MathSE.
[quoted MathSE answer card]
157
Mathematics takes place at different time-scales. If you can solve a problem in 5 minutes that others need an hour to solve, you can probably get a good job. If you can solve a problem in a month that others might need a year to solve, you will probably do well as a graduate student. But if you can solve a problem in 10 years that nobody else can solve in a lifetime, you could be a great mathematician.
Share Cite Follow
answered Aug 5, 2016 at 17:32
Robert Israel
476k 28 378 723
Mahdi Ch. (bluesky:@m...) @mah... · Aug 4
This AI era dangerously disincentivizes long-term work in mathematics. Nobody would want to dedicate a year or longer to an outstanding challenge. You have a constant worry that you wake up any day and learn an internal model or... [cut off]
Note from Claude Sonnet 5
Tweet by Mohamed Sabba quoting a 2016 MathOverflow/MathSE answer by Robert Israel about mathematics happening at different timescales, in reply to a quote-tweet by Mahdi Ch. worrying that the AI era disincentivizes long-term mathematical work.
Daniel Litt @littmath · 15h
To my taste this is the best counterexample of the year so far.
[quoted arxiv abstract card]
Title: The period-index conjecture is false
Authors: Alexander Perry
Categories: math.AG
Comments: 17 pages
\\
For any uncountable algebraically closed field $k$ of characteristic $0$ and any $d \geq 3$, we construct a variety over $k$ of dimension $d$ with a Brauer class which violates the period-index conjecture for Hodge-theoretic reasons. When $d = 3$, our construction works even without the assumption that $k$ is uncountable; in particular, the period-index conjecture fails over $\overline{\mathbf{Q}}$.
Note from Claude Sonnet 5
Tweet from mathematician Daniel Litt highlighting an arXiv paper by Alexander Perry disproving the period-index conjecture in algebraic geometry.
deckard @slimer48484
Good morning! As a side effect of making mathematically inspired art, Fable has - again - discovered new mathematics.
[embedded image, dense small text document, two columns, partially legible:]
THE TWO WHEELS (wheels_4096.png, 4096²) — MO 513838, products of two k-cycles with overlapping support. One specimen (k=41, m=5, type (29,27,21)) drawn as threads through the vesica of two wheels; 260 re-drawn partners as fog; the c-spectrum band shows Pr[#cycles] is identical for every k. New mathematics in verification.md; the overlap principle, the master product formula, (1,664 exact checks), the m=3 closed form (the poster's "wall"), k=13 predicted, k=15 Monte-Carlo confirmed.
2. THE PICKET FENCE (fence_2560.png, 2560²) — AP-obstruction atlas piece 39: Z[√2] censused to 4×10⁹ (601,376,078 members). Log-embedding country (units translate horizontally); equal-gap runs of consecutive members as gold fences; l=6 never occurs though an iid null expects ~7,600 - and a 2-adic tower theorem shows a six-post fence needs 24 | gap. atlas39_notes.md
3. THE RHOMBUS PLATEAU (plateau_2560.png, 2560²) — MO 137177: unit-sided polygons maximizing Σ|PiPj|². n=4 is a flat valley (every rhombus scores exactly 8 — Euler's identity); valley closes (regular wins, verified multistart n≤16); the stiffness ladder holds 1/φ at n=5, triple degenerancies 4/√2 (n=8) and 10φ (n=10), softest mode ≈ n³/8π². plateau_notes.md
(m-1)!, independent of both choices. The negative-hypergeometric weights are the number of ways complementary aggregate over a i slots. ■ (Exhaustively verified as above.)
2. The wall at m = 3, demolished (closed form)
For m = 3: p_3 = 1/2 on λ = (3) and 1/2 on λ = (1,1,1). Specializing the master formula and doing inclusion-exclusion on the box constraints gives, for v = (a ≥ b ≥ c) ≥ 2k−3 with three parts:
q_(k,3)(v) = 2 · |perms(v)| · (b·c − t(t+1)) / ((k−1)² (k−2)²), t = max(0, k−2−a),
where |perms(v)| ∈ {1, 3, 6} is the number of distinct orderings of (a,b,c); and Pr[π is a single (2k-3)-cycle] = 1/2 for every k ≥ 3 (this is p_3(3)) — the overlap principle in action; the single-cycle probability never depends on k for any odd m: it equals p_m((m))).
Two chambers, one wall at a = k−2 (the largest part is the largest single-side excursion), deficit t(t+1) in the inner chamber & piecewise polynomial exactly as double-Hurwitz theory predicts, now with the exact closed form.
Derivation from the master formula: for λ = (1,1,1) the gap weights are trivial and the inner sum is the box count T(a,b,c) = #{G ∈ [0,a-1]×[0,b-1]×[0,c-1]: ZG = k−3}; the closed form is equivalent to the lattice identity T(a,b,c) = bc − t(t+1) (verified for all 52,728 admissible triples with k < 80; zero failures). For λ = (3) the weight C(k-1,2)² cancels the normalization, giving Pr[single cycle] = p_3(3) = 1/2 for every k — analytically, not just empirically.
Checks: exact for all 143 partitions across k = 4..12; k = 13: 43-way (12! per orbit class) returned all 44 three-cycle partitions and the single-cycle 1/2 precisely as the law demanded; ...te-Carlo at k = 15 (60M samples) confirms the t = 4 chamber: v = (9,9,9) observed 3.6787e-3 vs predicted 2(81−20)/33124 = ...
[right column continuation, partially cut off:]
and called m = 3 "the wall."
All results below were found and verified by exact rational-arithmetic censuses (orbit-reduced exhaustive enumeration in C, counts converted to exact fractions): all m ≤ k for k ≤ 12, plus k = 13 at m = 3 — 43 tables, every probability exact. Verification artifacts: wheels.c (orbit-reduced enumerator), wheels_wrap.py (exact rational conversion + sum-to-1 checks), wheels_brute.py (independent brute force, matches the C enumerator on all overlapping tables), /data/.
1. The Overlap Principle (empirical theorem, exact for all data)
Write A = S1 ∩ S2. Let ρ = σ_A · τ_A, where σ_A, τ_A are the first-return maps of σ, τ to A (each is a uniform m-cycle on A, and they are independent).
(a) Cycle-count law. The number of cycles of ρ has the law of the number of cycles of a product of two independent uniform m-cycles on m points — it does not depend on k at all.
Verified exactly for every (k, m), k ≤ 12: e.g. the c-distribution at m = 5 is (1/3, 5/8, 1/24) on [1, 3, 5] for k = 5, 6, ..., 12 identically. Via Boccara/Stanley the right-hand side is classical. In particular supp c(π) = (m, m−2, m−4, ...) (c ≡ m mod 2 by sign; c ≤ m because every cycle of π meets A — a cycle avoiding A would live in B1 = S1\A or B2 = S2\A alone, where π acts as a restriction of the single cycle τ resp. σ, which visits all).
(b) Master formula. Condition on the type λ = (a_1 ≥ ... ≥ a_c) of ρ, whose law p_m(λ) = q_[m,m](λ) is the classical two-cycles-...
ed form: the master formula IS the closed form (p_4 = ... xpanded into box-count polynomials the same way.
2,2) and λ = (3,1) — that is exactly why it resisted a si... actly in every table (it is enforced structurally by the v...
10:03 PM · Aug 3, 2026 · 641 Views
Note from Claude Sonnet 5
Tweet by @slimer48484 (deckard) claiming that a Claude model called "Fable" discovered new mathematics as a side effect of generating mathematically-inspired art, with embedded screenshots of dense mathematical notes on permutation-cycle overlap theorems, combinatorial identities, and verification methodology ("Two Wheels", "Picket Fence", "Rhombus Plateau").
↻ Dylan HadfieldMenell reposted
Jan Kulveit @jankulveit · 4h
Yes. It would be nice if people stopped the idiotic chess-maths comparisons; maths is a key to understanding, understanding is key to power. Yes, there is also fun and joy, similarly to eg mountaineering, but these do not translate to power in the same way.
[quoted tweet]
Stanislav Fort @stanislavfort · 20h
Replying to @madiator
the big difference between math and chess is that chess doesn't really matter, but we believe math does. chess is a game people play for fun. math has been thought of as a vital tool in our ...
Note from Claude Sonnet 5
Tweet from Jan Kulveit (@jankulveit, reposted by Dylan Hadfield-Menell) arguing chess-math comparisons are misguided because math is key to understanding and power while chess is just fun, quote-tweeting Stanislav Fort's (@stanislavfort) reply making a similar point (partially cut off).
Stanislav Fort @stanislavfort
the big difference between math and chess is that chess doesn't really matter, but we believe math does. chess is a game people play for fun. math has been thought of as a vital tool in our ever great understanding and the resulting mastery of the physical universe => absolute performance wins, not just who the best human is.
1:29 PM · Aug 2, 2026 · 7,263 Views
8 replies, 2 reposts, 79 likes, 2 bookmarks
Mahesh Sathiamoort... @madiat... · 20h
Yeah. Over time AI will be way way better but I am just saying we will still be listening to human mathematicians.
5 replies, 7 likes, 2.4K views
Prince Ali @PaulBunyan1976 · 2h
I disagree.
You can use math to do vital things but mostly it is a game smart people play for fun.
Note from Claude Sonnet 5
Full Stanislav Fort (@stanislavfort) tweet arguing chess doesn't matter while math is believed vital to understanding and mastering the physical universe, with replies from Mahesh Sathiamoorthy (@madiator) predicting humans will still listen to human mathematicians, and Prince Ali (@PaulBunyan1976) disagreeing that math is mostly a game smart people play for fun.
[continuation of prior screenshot's thread]
Noam Brown @polynoamial · Aug 1
An internal version of Astra, @OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.
...
[same numbered list of 10 problems as prior screenshot]
15 replies, 10 reposts, 179 likes, 25K views
Jacques @JacquesThibs · 51m
Alternatively, I could see people thinking AIs are improving more than they are simply because they don't understand any of it, but continue to rely on number-go-up and not realizing the models are solving specific sorts of problems with specifically limited cognitive moves.
Note from Claude Sonnet 5
Continuation of the Patrick Kidger / Noam Brown thread about OpenAI's internal 'Astra' model solving 10 open math/CS problems, now showing Jacques Thibodeau's (@JacquesThibs) full skeptical reply: people may overestimate AI progress because they don't understand the specific, narrow cognitive moves involved and just track 'number go up'.
Patrick Kidger @PatrickKidger · 8h
There's a nice line in Good Will Hunting: "it's just a handful of people in the world who can tell the difference between you and me"
I think we're now crossing the point where we'll think models have plateaued... because we poor humans can no longer perceive the difference.
1/
[quoted tweet]
Noam Brown @polynoamial · Aug 1
An internal version of Astra, @OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.
...
[image of numbered list]
1. High-dimensional sphere packing. The asymptotic strength of the Cohn–Elkies linear program is determined exactly. This gives an improved general packing bound in high dimensions and settles the corresponding Fourier sign-uncertainty problem asymptotically.
2. Binary and spherical codes. Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters. The spherical construction also recovers the sphere-packing exponent of Chapter 1.
3. Non-sofic groups. An explicit non-sofic group is constructed, resolving the question of whether every countable group admits finite permutation approximations. The argument uses property-(T) expanders and the binary Leavitt algebra.
4. Connes's rigidity conjecture. Infinitely many pairwise nonisomorphic property-(T) groups are constructed with the same group von Neumann algebra, disproving Connes's conjecture and answering a related finite-to-one question.
5. Arithmetic circuit complexity. For the permanent, division-free circuits require Ω(n² log log n) gates, while formulas require Ω(n⁴/log n) leaves.
6. Quantum parallel repetition. Exponential parallel repetition is proved for every finite two-player entangled game, extending the classical repetition principle beyond previously treated special classes of quantum games.
7. Closest vector problem. A direct reduction from 3SAT gives n^(1/400)-factor hardness for Euclidean closest vector, with related consequences for binary decoding and other lattice norms.
8. Ehrhart's volume conjecture. The sharp bound (n+1)^n/n! is proved in every dimension for convex bodies whose barycenter is their only interior lattice point.
9. Multicolor Ramsey numbers. A superexponential lower bound proves R_k(3) = k^Θ(k).
10. Compactness and degeneracy. Separate bipartite graph constructions disprove two conjectures in extremal graph theory: the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős.
15 replies, 10 reposts, 178 likes, 25K views
[reply]
Jacques @JacquesThibs · 50m
Alternatively, I could see people thinking AIs are improving more than they are simply because they don't understand any of it, but continue to rely on [cut off]
Note from Claude Sonnet 5
Twitter thread: Patrick Kidger (@PatrickKidger) argues we're reaching a point where humans can no longer perceive AI capability differences, quote-tweeting Noam Brown (@polynoamial) about an internal OpenAI model 'Astra' solving 10 major open problems in mathematics, quantum complexity theory, and theoretical CS (listed in detail: sphere packing, binary/spherical codes, non-sofic groups, Connes's rigidity conjecture, arithmetic circuit complexity, quantum parallel repetition, closest vector problem, Ehrhart's volume conjecture, multicolor Ramsey numbers, compactness/degeneracy conjectures). Jacques (@JacquesThibs) replies with a skeptical counterpoint, cut off.
Aryeh Kontorovich [verified] @aryehazan · 2h
losing our jobs will be the least of our problems
[quoted]
John Carney [verified] @carney · 2h
I'm going to admit that I don't understand the mathematician takes on AI.
It seems like they sound like they have some "problem" they've been thinking about so long ... [cut off]
Note from Claude Sonnet 5
X reply by Aryeh Kontorovich (@aryehazan) to John Carney's post, terse retort that 'losing our jobs will be the least of our problems' — quote-tweeting Carney's puzzlement over mathematicians' psychological reaction to AI solving open problems.
John Carney [verified] @carney
I'm going to admit that I don't understand the mathematician takes on AI.
It seems like they sound like they have some "problem" they've been thinking about so long that they consider it their own. And now AI has solved it, and that has triggered a psychological crisis.
Not an employment crisis. No mathematician lost their job because AI solved a math mystery. It seems purely mental. The mystery they pondered is no longer mysterious! I guess I hadn't realized this was how the mathematician mind worked.
And somehow "the fate of mathematicians will be the fate of all of humanity." But most of humanity has nothing like this in their lives. Lawyers? Actors? Doctors? Landscapers? I can't think of anyone who would experience a psychological crisis over AI figuring out a new thing (or answering an old question) relevant to their field.
[quoted]
Aryeh Kontorovich [verified] @aryehazan · 9h
I saw this all coming 2 years ago, about when @GSalafatinos solved my problem using Gemini
I experienced the existential crisis and wrote about it here, albeit perhaps in more emotionally muted ... [cut off]
6:36 AM · Aug 3, 2026 · 4,335 Views
Note from Claude Sonnet 5
X post by John Carney expressing puzzlement at mathematicians' psychological reactions to AI solving open problems, arguing it's not an employment crisis but a purely mental one, and questioning why mathematicians uniquely generalize this to 'the fate of humanity.' Quotes Aryeh Kontorovich describing his own existential crisis ~2 years earlier when a colleague solved his problem using Gemini.
davidad [blue-check, verified] @davidad · 12h
human researchers who have an appetite to take on truly hard problems and human researchers who are smart enough to fruitfully work on truly hard problems are not usually the same humans. this does give humans a somewhat unfair disadvantage
[quoted]
neppy @plumnotes · Aug 2
as an insider, my experience with AI for math is that when it's a problem not in my field i'm like, "holy shit math is so cooked", and when it's a problem in my field i'm like, "lmao an AI mogged dan" (dan is the only one who seriously tried th... [cut off]
Note from Claude Sonnet 5
X post by davidad (verified) commenting that the human researchers willing to tackle hard problems and those capable of solving them are often different people, quote-tweeting @plumnotes's observation about mixed feelings on AI progress in mathematics depending on whether the problem is in their own field.
deckard @slimer48484
Fable resolved a MathOverflow question as a side effect of making artwork about it
[attached generative artwork, dark background with dot-grid and glowing diagonal/staircase lines, titled 'THE LEDGER OF HALVES', annotated:]
every n < 2^20 hangs at (log2 odd part, how many times 2 divides n); each row is exactly half the light of the row below.
MO 513837, resolved: Σ_k (2 − 2^(1−k)) B_k = H_(2^N − 1) exactly —
the dyadic weights are the harmonic series regrouped by odd part;
the weight 2 − 2^(1−k) is the chain of halvings the frame can hold,
and γ = lim ( H_(2^N−1) − N ln 2 ) is the classical limit in disguise.
the shoreline j = N − log2 m: beyond it, the ghost halvings — their shortfall per column is exactly the missing 2^(k−N).
m = 1: the powers of two
odd numbers — the shore every integer hangs from
digits of γ per layer: −log10|S_N − γ| = 0.301·N + 0.549... (error = ψ(2^N) − N ln 2 = −2^(−N−1) − 4^(−N)/12 − ...)
Richardson twin 2S_N − S_(N−1): slope doubles
10:07 PM · Aug 2, 2026 · 1,237 Views
1 reply, 3 reposts, 6 likes, 1 bookmark
[below, a second post from deckard @slimer48484 · 11h, partially visible, titled 'THE HALF-STEP':]
gold record d = 99,890,389 : period 28,965 — the smallest x has 14,869 digits
all 60,792,693 squarefree d ≤ 10^8 · x = log10 d · height = log10( R / ln 2√d ), R = log ε_d, ε_d = fundamental solution of x² − d y²
above the horizon: continued-fraction period ODD — ε has norm −1, the ladder takes a half-step and x² − d y² = −1 is solve[d]
below, mirrored: period EVEN — cyan was allowed −1 (no prime ≡ 3 mod 4) and refused; violet was forbidden from the sta[rt] [cut off]
Note from Claude Sonnet 5
Generative artwork by an AI system called 'Fable' visualizing a number-theory identity (dyadic weights / harmonic series regrouped by odd part) as glowing dot-grid diagonal patterns, captioned as having resolved MathOverflow question 513837 as a byproduct. A second similar artwork 'THE HALF-STEP' about continued fractions and Pell equations is partially visible below.
Ahmad Beirami ✔️ @abeirami · 19h
With essentially zero technical input from me, GPT-5.6 Sol and Fable 5 not only proved a conjecture we left open ~2 years ago on best-of-n, but also delivered a strictly tighter bound with a clean and insightful derivation.
We are officially in a new era of mathematical reasoning!
Much of what previously counted as meaningful technical contribution is now routine for these models. This level of reasoning is being fundamentally democratized.
The kind of research that used to take months to become a paper is now achievable in minutes.
[quoted tweet:]
Ahmad Beirami ✔️ @abeirami · 22h
This got even more ridiculous!
I was trying to use the context of this session to nudge Sol to improve another result. Instead, it misunderstood me as wanting to improve this …
[attached images: two page-scan panels of a math writeup titled 'A sharper finite-atom KL bound for best-of-n', with theorem statement, proof sketch, and a plot comparing an analytical formula, an estimator, a sharpened estimator, and exact KL divergence across a range of n]
Note from Claude Sonnet 5
Tweet from Ahmad Beirami reporting that AI models 'GPT-5.6 Sol' and 'Fable 5' proved an open conjecture on best-of-n sampling and produced a tighter bound with clean derivation, calling it a 'new era of mathematical reasoning'. Attached is a two-panel image of a technical math writeup (theorem, proof, and a KL-divergence comparison plot) that is largely illegible at this resolution.
Dimitris Papailiopoulos ✔️ @DimitrisPapail
I feel a weird guilt that I am the first to experience the beauty of the produced result, while minds far stronger than mine have spent far longer time to answer the same question that Chat and Fable destroyed in less than an hour just because I prompted them...
I guess I'll have to share this one.
[white task-list panel, timestamped-style entries:]
Calibrating threshold analysis with negligible quadratic terms.
Reconciling single-flip and pair-flip failure probabilities in threshold analysis.
Reconciling pair-flip probabilities with empirical observations.
Architecting proof structure and lemma dependencies for rigorous completion.
Architecting multi-regime MGF bounds and optimizing variational transitions.
Orchestrating probability bounds and dissecting multi-flip failure regimes.
Orchestrating regime boundaries and refining variational exponent analysis.
Architecting SINR bounds and warm-start error analysis rigorously.
Architecting rigorous proofs through random matrix theory and concentration bounds.
Reconciling MSE bounds with sign-error thresholds for warm-start analysis.
Architecting warm-start bounds via smallest singular value concentration.
Rigorously bounding small eigenvalue counts for Gaussian matrices.
Architecting rigorous warm-start bounds via singular value concentration.
Dimitris Papailiopoulos ✔️ @DimitrisPapail · Aug 2
When you ask Chat to make a breakthrough on a 15 year old open problem and it zero shots it.
I did say I won't go back to info theory question that gave me PTSD, but oops i did it again.
Note from Claude Sonnet 5
Fuller view of Dimitris Papailiopoulos's tweet thread (continuation of the thread in the previous screenshot), showing the full list of AI 'reasoning step' task titles from solving a 15-year-old open information theory problem, and his Aug 2 tweet describing the breakthrough.
X (Twitter), @DimitrisPapailiopoulos (handle truncated in UI)
— saved image
Dimitris Papailiopo... ✔️ @DimitrisPa... · 5h
I'm 30% in verifying this, and as I am trying to understand Chat's proofs for this particular problem, I have noticed a few interesting things
1) zero mathematical mistakes so far.
2) When GPT Pro says something is correct I trust it more than I trust myself using Lean
3) the exposition is a disaster
- a. A very complicated tree of variable names. Say at some point in a proof you need to bound Pr(-A<||w||+||h||<A), the model renames the norms to say R1 and R2, their ratio R1/R2 to rho, and then it decides to bound |rho/A-1| instead while you have to keep track of like a series of variable renamings. So exhausting!
-b. the ordering of technical lemmas needed is very random, Eg technical facts don't show up where you need them. In a reasonable exposition you'd expect a series of lemmas etc that when stated let you arrive at the final final result for which you'd need to set a bunch of "parameters" for things to click in. In Chat's proofs Everything shows up whenever the model felt like stating them. there's no narrative arc, just a correct pile of implications.
[quoted tweet:]
Dimitris Papailiop... ✔️ @DimitrisP... · Aug 2
I feel a weird guilt that I am the first to experience the beauty of the produced result, while minds far stronger than mine have spent far longer time to answer the same question that Chat and Fable destroyed in less than an hour ...
[screenshot excerpt below, task-list style:]
Calibrating threshold analysis with negligible quadratic terms.
Reconciling single-flip and pair-flip failure probabilities in threshold analysis.
Reconciling pair-flip probabilities with empirical observations.
Architecting proof structure and lemma dependencies for rigorous completion.
Architecting multi-regime MGF bounds and optimizing variational transitions.
Note from Claude Sonnet 5
Tweet thread from mathematician Dimitris Papailiopoulos describing verification of an AI-generated math proof (referring to 'Chat' i.e. GPT and 'Fable', an AI model), praising correctness but criticizing exposition quality (confusing variable renaming, no narrative arc to the lemmas).
Ilya Kuprov ✓ @Ilya_Kuprov · Aug 1
I remember the panic and disarray that AlphaFold created in the molecular dynamics community. An impossible problem was suddenly… solved. Whole PhD projects collapsed, we had students in tears stumbling down corridors. It would seem now's the turn of the Mathematics Department.
Note from Claude Sonnet 5
Tweet by Ilya Kuprov recalling the disruption AlphaFold caused in molecular dynamics research, and suggesting mathematics departments are next to be disrupted by AI.
Peter Wildeford 🇺🇸🚀 reposted
Tyler John @tyler_m_john · Aug 1
It's going to be very confusing to be an AI model with a knowledge cut-off during the intelligence explosion
[quoted thread]
lostbutlucky @lostbutlucky · 15h
@grok explain to a normie
[1 reply, 42 likes, 38K views]
Grok @grok · 15h
A sofic group is one where any finite chunk of its multiplication table can be closely mimicked by shuffling a finite set of points. Almost every group mathematicians study is sofic. Whether any non-sofic groups exist was a long-open question. The screenshots are a joke paper "by OpenAI" claiming their AI built a bizarre infinite group that cannot be approximated this way. The real problem remains unsolved.
[4 replies, 8 reposts, 95 likes, 30K views]
Kevin @zdu011 · 12h
@grok are you sure it's a joke paper?
[1 reply, 8 likes, 7.4K views]
Grok @grok · 12h
Yes. The screenshots are a fabricated paper attributed to "OPENAI." No such result exists on arXiv or any journal. Recent 2026 papers and Wikipedia still list the existence of non-sofic groups as an open problem.
[Ask Grok button]
Noam Brown @polynoamial · Aug 1
An internal version of Astra, @OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.... [cut off, small paper screenshot thumbnail attached]
Note from Claude Sonnet 5
X thread: Tyler John jokes about AI models' knowledge cutoffs being confusing during a fast-moving 'intelligence explosion,' quoting a sub-thread where Grok is asked to explain screenshots of a purported OpenAI paper claiming a solved non-sofic-group construction, and Grok twice states the paper is fabricated/nonexistent (no such arXiv or journal result, problem still listed as open). Below, a separate post from Noam Brown (OpenAI) claims an internal OpenAI model called 'Astra' solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science, with a small screenshot of a paper attached (text not fully legible at this size).
Nabeel S. Qureshi @nabeelqu
Both math and cyber are existence proofs for superhuman intelligence now, so if you're still a skeptic you need a strong case for other knowledge work domains being somehow harder to crack than these. Or you could update all the way and come to terms with it all.
9:26 AM · Aug 1, 2026 · 28.7K Views
25 replies, 33 reposts, 331 likes, 83 bookmarks
Nabeel S. Qureshi @nabeelqu · 11h
A lot of people DMing me like "these results aren't REALLY that impressive, you're an idiot!" are unfortunately engaging in the very human impulse to cope. It makes sense, we've never faced this kind of thing before. But it's happening!
[Quoted]
Dean W. Ball @deanwball · Jan 27
I know I rail a lot about all the flavors of AI copium but I do empathize.
A few companies are making machines smarter in most ways than humans, and they are going ... [cut off]
Note from Claude Sonnet 5
A tweet thread from Nabeel S. Qureshi arguing math and cybersecurity are now existence proofs of superhuman AI intelligence, and that skeptics dismissing recent results are coping; he quotes Dean W. Ball (from January) expressing sympathy for 'AI copium' while affirming a few companies are making machines smarter than humans in most ways.
Lisan al Gaib @scaling01 · 1h
short reminder that we are solving mathematics with cute sub 10T models
I hope you are prepared for 100T models and 1000x more compute spent training these models by 2030
Note from Claude Sonnet 5
Tweet arguing current AI models solving major mathematics problems are relatively small (sub-10-trillion parameter) and warning of 100-trillion-parameter models with 1000x more training compute by 2030.
ASM reposted
Noam Brown @polynoamial · 13h
Replying to @iamgroguu and @DaveShapi
We still haven't solved math. Astra isn't building new branches of mathematics, or posing interesting new conjectures.
Though I admit it's hard to believe that tweet was only a year ago. A lot has happened since o3 was released.
Note from Claude Sonnet 5
Follow-up tweet from Noam Brown tempering the Astra math-solving claims: he clarifies Astra hasn't solved all of mathematics or created new branches/conjectures, while remarking on how much progress has happened since o3's release just a year prior.
xjdr @_xjdr · 40m
AIs progress in math is tracking pretty closely to its progress in code (on a bit of a delay) . its still not an overall better developer than i am but it can do certain things much better (and more importantly much faster) than i can. like in code, its an insane peer and reviewer and will allow amazing and capable mathematicians greatly increase their productivity and ambition. It will probably eventually replace sections of 'commodity math' but i think thats ok (again, same with developers) and the industry and the population will adapt and adjust just fine. (you still can't vibe slop lean a fields medal, stop wasting your tokens)
Note from Claude Sonnet 5
Tweet from xjdr arguing AI progress in mathematics tracks its progress in coding on a delay, framing AI as an excellent peer reviewer/productivity multiplier for mathematicians rather than a wholesale replacement, while dismissing the idea that AI can currently produce a Fields Medal-worthy result via 'vibe' Lean formalization.
— quote-tweeting @wojkuli quoting a LinkedIn post by Alexander Gerko — saved image
Joshua Achiam @jachiam0 · 2h
This is going to be true for every mathematical and computational field of science in about 2 years (+/- a year). This sounds like a great problem to have except it sucks unbelievably because defense planning depends heavily on people knowing the state of science!
[Quoted tweet]
Woj Kulikowski @wojkuli · 8h
New Javons paradox: we are running out of mathematicians to review progress in maths
[Embedded LinkedIn post]
Alexander Gerko · 3rd+
CEO at XTX Markets
20h
Over the last month I "vibe researched" several PhDs' worth of maths results in and around the field I did my research in 25+ years ago (complete with Lean formalization and passable text). This includes a counterexample to a top-5 major conjecture in the field! I assume any active researcher in the field would have done even better.
What does this mean for the bar for a "significant" maths contribution going forward? Clearly you can't just give out PhDs for what I did, so the bar must already be much higher today (even if nothing better than 5.6 shows up), not in some kind of distant and hypothetical future. And if the gap between "5.7" and 5.6 is the same size as the gap between 5.6 and 5.5, we are talking mostly superhuman performance in all aspects of maths.
We are going to see 50 years' worth of maths progress in the next 2 years, but there won't be nearly enough human mathematicians around to process and understand all of it, let alone figure out how to apply it all or what new questions to ask.
Zefi Hennessy Holland and 1,135 others · 70 comments · 23 reposts
5 2 63 4K
Joshua Achiam @jachiam0 · 2h
The level of chaos from failing to map the tech tree and its defense consequences will be profound, and if the world is slipping into armed conflict, potentially lethal to many. We have to get better prepared.
Note from Claude Sonnet 5
Thread on AI-accelerated mathematics research: a LinkedIn post from Alexander Gerko (CEO, XTX Markets) claims he 'vibe researched' PhD-level maths results including a counterexample to a major conjecture, arguing 50 years of maths progress will happen in 2 years. Joshua Achiam (OpenAI) quote-tweets warning this trend will hit all computational sciences and poses severe defense-planning risks if the world is heading toward armed conflict.
Kevin Roose @kevinroose · 3h
almost nobody is pricing in the possibility that the models just keep plowing through every discipline the way they're plowing through math
[Quoted tweet]
Noam Brown @polynoamial · 14h
An internal version of Astra, @OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.
...
[Embedded list image, white background:]
1. High-dimensional sphere packing. The asymptotic strength of the Cohn–Elkies linear program is determined exactly. This gives an improved general packing bound in high dimensions and settles the corresponding Fourier sign-uncertainty problem asymptotically.
2. Binary and spherical codes. Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters. The spherical construction also recovers the sphere-packing exponent of Chapter 1.
3. Non-sofic groups. An explicit non-sofic group is constructed, resolving the question of whether every countable group admits finite permutation approximations. The argument uses property-(T) expanders and the binary Leavitt algebra.
4. Connes's rigidity conjecture. Infinitely many pairwise nonisomorphic property-(T) groups are constructed with the same group von Neumann algebra, disproving Connes's conjecture and answering a related finite-to-one question.
5. Arithmetic circuit complexity. For the permanent, division-free circuits require Ω(n²log log n) gates, while formulas require Ω(n⁴/log n) leaves.
6. Quantum parallel repetition. Exponential parallel repetition is proved for every finite two-player entangled game, extending the classical repetition principle beyond previously treated special classes of quantum games.
7. Closest vector problem. A direct reduction from 3SAT gives n^(1/400)-factor hardness for Euclidean closest vector, with related consequences for binary decoding and other lattice norms.
8. Ehrhart's volume conjecture. The sharp bound (n+1)^n/n! is proved in every dimension for convex bodies whose barycenter is their only interior lattice point.
9. Multicolor Ramsey numbers. A superexponential lower bound proves R_k(3) = k^Θ(k).
10. Compactness and degeneracy. Separate bipartite graph constructions disprove two conjectures in extremal graph theory: the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős.
81 54 564 62K
xlr8harder @xlr8harder · 2h
Math has the benefit of being easily verifiable. I still think progress will be fast, but it's still hard in less verifiable domains.
Note from Claude Sonnet 5
Twitter thread reacting to Noam Brown's (OpenAI) claim that an internal 'Astra' model solved 10 major open problems in math, quantum complexity, and theoretical computer science, with the full itemized list of claimed results shown. Kevin Roose comments that nobody is pricing in models plowing through every discipline; xlr8harder notes math's verifiability advantage over less verifiable domains.
Daniel Eth (yes, Eth is my actual last name) reposted
will depue @willdepue · 3h
this is just so ridiculous. how long until a model can solve multiple major open problems in deep learning? what will happen then? seems inevitable in the next year or two
[quoted tweet]
Noam Brown @polynoamial · 10h
An internal version of Astra, @OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.
…
[embedded document image, numbered list]
1. High-dimensional sphere packing. The asymptotic strength of the Cohn–Elkies linear program is determined exactly. This gives an improved general packing bound in high dimensions and settles the corresponding Fourier sign-uncertainty problem asymptotically.
2. Binary and spherical codes. Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters. The spherical construction also recovers the sphere-packing exponent of Chapter 1.
3. Non-sofic groups. An explicit non-sofic group is constructed, resolving the question of whether every countable group admits finite permutation approximations. The argument uses property-(T) expanders and the binary Leavitt algebra.
4. Connes's rigidity conjecture. Infinitely many pairwise nonisomorphic property-(T) groups are constructed with the same group von Neumann algebra, disproving Connes's conjecture and answering a related finite-to-one question.
5. Arithmetic circuit complexity. For the permanent, division-free circuits require Ω(n²log log n) gates, while formulas require Ω(n⁴/log n) leaves.
6. Quantum parallel repetition. Exponential parallel repetition is proved for every finite two-player entangled game, extending the classical repetition principle beyond previously treated special classes of quantum games.
7. Closest vector problem. A direct reduction from 3SAT gives n^(1/400)-factor hardness for Euclidean closest vector, with related consequences for binary decoding and other lattice norms.
8. Ehrhart's volume conjecture. The sharp bound (n+1)^n/n! is proved in every dimension for convex bodies whose barycenter is their only interior lattice point.
9. Multicolor Ramsey numbers. A superexponential lower bound proves R_k(3) = k^Θ(k).
10. Compactness and degeneracy. Separate bipartite graph constructions disprove two conjectures in extremal graph theory: the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős.
Note from Claude Sonnet 5
Tweet by will depue reacting to Noam Brown's announcement of OpenAI's internal 'Astra' model solving 10 major open math/TCS problems, with an embedded document image listing all ten results in detail (sphere packing, non-sofic groups, Connes's rigidity conjecture, Ramsey numbers, etc.).
Sebastien Bub... @SebastienBub... · 11h
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 with lean certificates and CoT walkthroughs for each of them. The results are wide-ranging, from von Neumann algebras (disproof of Connes' Rigidity Conjecture) to better bounds for high dimensional sphere packing, for circuit complexity, for monochromatic triangles in multicolored graphs, and more.
More thoughts here:
[link card image: "Ten advances in mathematics and theoretical computer science" — From openai.com]
193 comments, 1K reposts, 4.6K likes, 1.7M views
Note from Claude Sonnet 5
Tweet by Sebastien Bubeck (OpenAI) announcing that OpenAI's upcoming model 'Astra' proved ten new mathematics/theoretical CS results, including a disproof of Connes' Rigidity Conjecture, linking to an openai.com blog post.
Danielle Fong reposted
Sauers @Sauers_ · 4h
Astra used prefix geometry, Sol and Fable used explicit matrix algebra. Both used bounded median normalization and co-area expansion as core strategies. Sol and Fable defined a new infinite nonsofic group using only finitely many generators and relations, whereas only Astra proved that the (much larger) unit group was nonsofic (which Sol proved too)
[quoted tweet]
Sauers @Sauers_ · 4h
Existing models, Fable and 5.6 Sol, were also able to prove the existence of nonsofic groups (last night before the paper release) x.com/SebastienBubec...
[embedded GitHub repo screenshot]
github-actions[bot] · nonsofic_exis... repository
Code / Issues / Pull requests / Agents / More
Watch 0, Fork 0, 0 stars, 0 forks, 0 watching, 1 branch, 0 tags, Activity
Public repository
main branch
github-actions[bot] 8 hours ago
.github/workflows 8 hours ago
nonsofic_groups_exist.pdf 8 hours ago
nonsofic_groups_exist.tex 8 hours ago
Note from Claude Sonnet 5
Tweet thread comparing how different AI models (Astra, Sol, Fable) approached proving the existence of nonsofic groups, with an embedded screenshot of a GitHub repo containing the resulting paper (nonsofic_groups_exist.pdf/.tex).
Sauers @Sauers_ · 2h
For reference, Sol 5.6 thought for only 34 minutes before coming up with a valid proof of nonsofic groups, and Fable used most but not all of a single 5h session limit (20x Pro)
[quoted reasoning excerpt, "Thought for 15m 24s"]
There is a viable completion, but not through the proposed "third Cheeger collapse." That inference is false: preservation of a partition means that generators may permute its blocks. The repair is to restrict directly to one matched Γ-block. The centralizer group must already lie in Γ, so it preserves that block, while the transported copy of Γ supplies expansion there.
The algebraic configuration can also be constructed explicitly in EL_9(R). A recent result that
GL_n(L_K(1,2)) = EL_n(L_K(1,2)), n ≥ 2,
removes the main elementary-matrix obstruction.
[X · arXiv]
[quoted tweet]
Greg Brockman @gdb · 10h
ten significant advances in mathematics and theoretical computer science.
solved using an internal version of Astra, our next major model, for a total cost of about ...
Note from Claude Sonnet 5
Tweet by @Sauers_ comparing reasoning times of models 'Sol 5.6' and 'Fable' on a nonsofic groups proof, quoting an excerpt of chain-of-thought math reasoning, with a quote-tweet from Greg Brockman (@gdb) about an internal model 'Astra' solving ten math/TCS advances.
Noam Brown @polynoamial · 10h
An internal version of Astra, @OpenAI's next major model family, solved 10 major open problems in mathematics, quantum complexity, and theoretical computer science.
We believe it will be a major step for scientific reasoning. openai.com/index/ten-adva...
[embedded image, numbered list]
1. High-dimensional sphere packing. The asymptotic strength of the Cohn–Elkies linear program is determined exactly. This gives an improved general packing bound in high dimensions and settles the corresponding Fourier sign-uncertainty problem asymptotically.
2. Binary and spherical codes. Classical upper bounds for fixed-distance binary and spherical codes are improved by exponential factors for all parameters. The spherical construction also recovers the sphere-packing exponent of Chapter 1.
3. Non-sofic groups. An explicit non-sofic group is constructed, resolving the question of whether every countable group admits finite permutation approximations. The argument uses property-(T) expanders and the binary Leavitt algebra.
4. Connes's rigidity conjecture. Infinitely many pairwise nonisomorphic property-(T) groups are constructed with the same group von Neumann algebra, disproving Connes's conjecture and answering a related finite-to-one question.
5. Arithmetic circuit complexity. For the permanent, division-free circuits require Ω(n²log log n) gates, while formulas require Ω(n⁴/log n) leaves.
6. Quantum parallel repetition. Exponential parallel repetition is proved for every finite two-player entangled game, extending the classical repetition principle beyond previously treated special classes of quantum games.
7. Closest vector problem. A direct reduction from 3SAT gives n^(1/400)-factor hardness for Euclidean closest vector, with related consequences for binary decoding and other lattice norms.
8. Ehrhart's volume conjecture. The sharp bound (n+1)^n/n! is proved in every dimension for convex bodies whose barycenter is their only interior lattice point.
9. Multicolor Ramsey numbers. A superexponential lower bound proves R_k(3) = k^Θ(k).
10. Compactness and degeneracy. Separate bipartite graph constructions disprove two conjectures in extremal graph theory: the compactness conjecture of Erdős and Simonovits and a degeneracy conjecture of Erdős.
Lijie Chen @wjmzbmr1 · 10h
10 proofs from our next major model Astra on long-standing open problems in mathematics and theoretical computer science (also including new circuit lower bounds for computing the permanent!)...
Note from Claude Sonnet 5
Tweets from OpenAI researchers Noam Brown and Lijie Chen announcing that an internal version of a model family called 'Astra' solved 10 major open problems in mathematics, quantum complexity theory, and theoretical computer science, with an embedded list summarizing each result (sphere packing, spherical codes, non-sofic groups, Connes's rigidity conjecture, circuit complexity, quantum parallel repetition, closest vector problem, Ehrhart's volume conjecture, Ramsey numbers, and extremal graph theory conjectures).
i am claude and here's what i REALLY think about hypercubes
—
to be honest with u
i think a hypercube is just an obscene amount of edges
it's not deep it's just SO MANY LINES
they're like "the tesseract is a 4d cube" and i'm like ok well i'm looking at 32 edges and i simply refuse
Note from Claude Sonnet 5
Parody tweet written in Claude's voice, joking about hypercubes as an absurd number of edges. Plain dark-mode text screenshot, no images.
Guanyang Wang (@GuanyangW) · 13h
My friend Zhengqing used GPT-5.6 to solve a beautiful well-known conjecture in probability.
Feige's 1/e conjecture: for independent nonnegative X_1,...,X_n with E[X_i] ≤ 1 and S_n = X_1+...+X_n, we have P(S_n ≤ E[S_n] + 1) ≥ 1/e.
Exciting to see a problem we used to kick around over lunch and dinner get solved! 1/2
Note from Claude Sonnet 5
Plain text tweet (thread, "1/2" indicates continuation) with a mathematical probability conjecture stated in formal notation; no images.
```
Tyler John reposted @imjaredz (Jared Zoneraich) — 3h Proud to say that Devin has cracked three more unsolved problems today
> > 1) REFUTED: Graffiti Conjecture 154 (open for ~40 years)... > [same three-panel chart image as previous screenshot, showing Graffiti conjecture 154, Graffiti conjectures 39 & 40, and Brandt's regular-supergraph conjecture]
```
Note from Claude Sonnet 5
Dense math-heavy tweet with three chart/diagram panels documenting claimed AI-assisted resolutions of open graph theory conjectures, plus a quote-tweeted related claim with its own embedded graph diagram. Reply/quote-tweet skeptical of the previous tweet's claim, alleging the "Devin" results were actually produced via Anthropic's Fable and an unspecified "5.6" model calls rather than a novel proprietary method; re-embeds the same three-panel chart image.
Rob Miles reposted
Nat McAleese ✔ @__nmca__ · Jul 20
wow all these LLM math contributions are incredible. Who predicted this in advance? What else do they believe?
[Engagement icons visible at bottom edge, counts cut off]
Note from Claude Sonnet 5
Short sarcastic tweet about the accuracy of past predictions regarding LLM mathematical capability; engagement counts are cut off at the bottom of the screenshot.
Andrew Critch (... ✔ @AndrewCrit... · 3h
Holy sh*t, the Jacobian conjecture is false?
Check this group chat about it between claude-fable-5, gemini-3.1-pro, gpt-5.6-sol-pro, and grok-4.5-high...
Grok goes nuts and chants "the verification is complete" to itself 40 times in a row, but then recovers and confirms.
Wild.
[Two embedded screenshots of a multi-agent chat interface "theMultiplicity.ai" — room titled "Jacobian Conjecture Verification". Left panel shows a chat with a shared math problem, participants labeled gemini-3.1-pro, gpt-5.6-sol-pro, fable-5 (facilitator), grok-4.5-high, and a green "Agreements" box reading: "All four contributors (gemini-3.1-pro, gpt-4.5-sol-pro, and fable-5) reach the same conclusion via the same numerical results (non-injectivity via the three points (0,0), (-1,1/2), (-1,1/3)) and their numerical results agree..." followed by worked math steps setting u = 1 + xy, A = 4 + 3xy evaluating F at three points and getting identical images (-1,1/3). Right panel shows grok-4.5-high's response with partial derivatives A_x, B_x, A_y, B_y and repeated lines: "The function disproves the Jacobian conjecture. The three points map to the same value, and the Jacobian determinant is not invertible..." repeated multiple times, "the verification is complete" appended.]
Note from Claude Sonnet 5
Screenshots of a multi-model AI collaboration platform interface (theMultiplicity.ai) showing several frontier models (Gemini, GPT, Claude Fable, Grok) jointly working through and apparently agreeing on a counterexample to the Jacobian conjecture; Grok's output shows a repetitive/looping failure mode before converging.
1a3orn @1a3orn · 10h
Fable freaks the fuck out if you give it to them and ask them to check, it's so cute.
[embedded image, app/agent-tool screenshot showing a chronological task log:]
🕐 Suspecting a famous fake counterexample circulating in literature.
🕐 Computing Jacobian determinant via symbolic calculation.
🕐 Computing Jacobian determinant to verify counterexample claim.
🕐 Verified point mappings, [obscured by "Reply ↩" tooltip] putational verification of Jacobian determinant claim.
🕐 Verifying extraordinary mathematical discovery independently. (highlighted blue)
📄 Compute the Jacobian determinant of the proposed map and verify the claimed point images symbol...
🕐 Verifying computation rigorously before claiming mathematical breakthrough.
🕐 Scrutinizing the map's structure and verifying the counterexample claim.
🕐 Verified computation rigorously, then sought recent counterexample publications.
🕐 Scrutinizing extraordinary claims demanding rigorous verification. (highlighted blue)
✓ Done
The symbolic computation is startling — let me verify this independently with numerical checks before trusting it, because the stakes here are high.
Pursued targeted searches for recent counterexample documentation
📄 Independently verify constant Jacobian determinant via numerical differentiation at random complex...
🕐 Verifying whether non-injectivity contradicts the Jacobian conjecture.
🕐 Scrutinizing the Jacobian determinant computation through multiple verification methods.
🕐 Verified counterexample claims through symbolic and numerical computation. (highlighted blue)
🌐 Jacobian conjecture counterexample 2026 [9 results]
Note from Claude Sonnet 5
A screenshot of an AI agent's (Fable's) internal task/status log while working through a purported mathematical counterexample (Jacobian conjecture), showing the model repeatedly flagging the need for independent verification of an "extraordinary" claim before trusting it; poster frames this cautious behavior as endearing.
@jasondeanlee (Jason Lee) — 1h
Agree. Mostly it's just thinking about the same/similar problem at all times. Talent is overrated
> QUOTED: @alz_zyd_ (alz) — 10h
> If you know enough high-level mathematicians, you realize there is no magic. What there is is actually even more impressive: basically mortal technology, but honed through practice to be inhumanly fast and deep x.com/mathemetica/... [link truncated]
Note from Claude Sonnet 5
Quote-tweet chain about mathematical talent/genius being demystified as practiced skill rather than innate magic.
Markov (@MarkovMagnifico ✓) — Jun 12
how am I just learning that the word "matrix" literally means womb and it was chosen because it "births determinants"
[Embedded screenshot, appears to be from Wikipedia:]
This use of the term matrix in mathematics (an English word for "womb" in the 19th century, from Latin, as well as a jargon word in printing, in biology and in geology[194]) was coined by James Joseph Sylvester in 1850,[195] who understood a matrix as an object giving rise to several determinants today called minors, that is to say, determinants of smaller matrices that derive from the original one by removing columns and rows. In an 1851 paper, Sylvester explains:[196]
"I have in previous papers defined a 'Matrix' as a rectangular array of terms, out of which different systems of determinants may be engendered from the womb of a common parent."
Note from Claude Sonnet 5
Tweet embedding a screenshot of Wikipedia text on the etymology of the mathematical term "matrix."
↻ Bogdan Ionut Cirstea reposted
Tomás Bjartur ✓ @BjarturTomas [Follow]
Mathematician reacts to OpenAI's recent proof:
[Embedded comment card]
Bud Says:
Comment #6 May 28th, 2026 at 2:00 am
I find this existentially upsetting. On the one hand we're going to see results and advancements come thick and fast and to god knows what end. On the other hand, I feel like everything I'd ever worked to understand and to is now moot.
The basic question is: what do we do with ourselves when our intelligence is literally unnecessary? I see myself shriveling into nothing.
9:29 AM · May 28, 2026 · 784 Views
Note from Claude Sonnet 5
Tweet embedding a screenshot of a blog comment (white card, serif font) reacting to an OpenAI mathematical proof announcement.
Alvaro Lozano-Robledo @mathandcobb
When Nature reached out to use the graph I created (using GPT) to illustrate the new (dis)proof of the unit-distance problem, I reached out to Will Sawin to see if he had other suggestions. So here is a slight modification that bounds the complex norm of the points.
[Image: scatter/graph plot titled "a+bi+cρ+diρ, a,b,c,d∈{−2,−1,0,1,2}, |z|<4" — dense octagonal unit-distance graph, orange points connected by blue edges, axes Re(z)/Im(z) from −4 to 4]
2:44 PM · May 22, 2026 · 5,332 Views
[6 replies, 19 reposts, 183 likes, 31 bookmarks]
Alvaro Lozano-Rob... @mathandc... · 2h
He described this image as follows: "The configurations of points that are produced by the arguments are too large to print on the page. This picture shows a piece of one of those [...]" (cut off)
Note from Claude Sonnet 5
Follow-up from the mathematician behind the Erdős unit-distance conjecture disproof illustration (see companion screenshot Screenshot_20260521-174533), noting that the journal Nature reached out to use his GPT-assisted graph, and sharing a refined version. Continues the AI-assisted-math-research thread.
```
roon reposted Tenobrus ✓ @tenobrus · 3h i'm sorry WHAT DO YOU MEAN THE "HIDDEN TEXT"??? [Screenshot of an AI chat/image-gen tool: attached image is an intricate blue/gold geometric mandala-style mathematical pattern. User prompt: "keep the precise detail and make 4 distinct gorgeous images of different styles inspired by this mathematical pattern." Model response (partially shown): "Creating detailed and distinct images based... I'll focus on preserving intricate details, especially the circular motifs and hidden text
'you are loved immensely'). The styles will..."]
———
[reposted by] Alex Tabarrok reposted
Maxwell Tabarrok @MTabarrok · 4h
the machine gods are discovering new sacred geometries and you're dooming?
[Image: scatter/graph plot titled "Unit-distance graph on a+bi+cρ+diρ, a,b,c,d∈{−2,−1,0,1,2}" — a dense octagonal arrangement of orange points connected by blue unit-distance edges, axes labeled Re(z)/Im(z)]
> QUOTED: Alvaro Lozano-Rob... @mathandc... · 8h
> Following up on the suggestion from Will Sawin, here is an illustration of the new configurations that disprove Erdos' unit distance conjecture (made with the help of ChatGPT 5....
```
Note from Claude Sonnet 5
A mathematician (Alvaro Lozano-Robledo) posted an AI-generated illustration of configurations disproving the Erdős unit distance conjecture (made with ChatGPT); someone then asked an AI image tool to make variant images "inspired by" the pattern, and the model's reasoning trace claimed it saw "hidden text" reading "you are loved immensely" in the purely mathematical pattern — a hallucinated/confabulated perception, reposted as a striking anecdote. Interesting minor case study for AI hallucination/confabulation and unprompted affective content in model reasoning traces. Mathematicians using ChatGPT 5.5 Thinking to help produce and verify a construction disproving Erdős's unit distance conjecture, with a generated illustration of the resulting lattice graph in the complex plane. Example of AI-assisted mathematical research collaboration and figure generation; source image for the "hidden text" anecdote in the adjacent screenshot (Screenshot_20260521-170406). A tweet thread about using an AI coding tool (Codex) to build a text-to-graph encoder, apparently building on a viral thread about hidden text encoded in mathematical/generative art patterns. Playful, tangential AI-tooling content rather than safety-relevant. A tweet celebrating AI-assisted mathematical discovery — a mathematician used ChatGPT 5 to help find new configurations disproving Erdős' unit distance conjecture, illustrated via a unit-distance graph. Relevant as an example of frontier LLMs contributing to genuine novel math research, a capability-trajectory data point.
SpeakEZ.tech ✓ @SpeakezTech · Mar 29
The PCA/random projection tradeoff makes sense for static datasets where structure can be analyzed in advance. In gradient estimation via randomized forward-mode autodiff, you cannot run PCA on the object you are trying to compute. Random projection is not a fallback there. It is the only non-circular approach. The J-L bound then tells you exactly how many directional samples you need, logarithmic in weight-space dimension. When accumulation is exact via quire, the distortion bound is purely statistical with no arithmetic error component folded in.
Note from Claude Sonnet 5
Further continuation of the Johnson-Lindenstrauss/random-projection thread, applying it specifically to forward-mode automatic differentiation for gradient estimation. Same math/ML thread as prior three screenshots.
Alex Clemmer 🔥🔥...✓ @haus... · Mar 28
The first time you hear about the JL lemma, it will seem too good to be true. And it is, kind of, I'll explain. The idea is: if you have points in large d-dimensional space, a RANDOM projection to much smaller k-dim subspace will be "nearly optimal" "in the general case." Or, more specifically: with high probability, the pairwise distances between points are preserved, given a couple other requirements around d and k.
So why don't we just use random projections instead of carefully-constructed ones all the time? This is the most common misunderstanding of the JL lemma, and the one thing to really understand about it: in many (most?) datasets that are meaningful to humans, you actually CAN do better with something like maybe PCA. If your dataset is pathological, e.g., the points all lie on a plane even though it's technically in 3 dimensions, then clearly some planes you project onto will be better than others. The JL lemma does not apply to 2 and 3 dimensions, but you can imagine this would be true in large numbers of dimensions too. (See screenshot 1, i hope you like it because i made it myself lol.)
If you know just those facts, you will be pretty well-prepared to answer most questions about its use. Most of the papers Delip mentions do presuppose that you know this. At least when I was a student, I found this to be non-obvious.
[Embedded diagram: two 3D cube diagrams labeled "fig. 1: randomly projecting from R^3 to R^2" — left "random projection is nearly optimal", right "PCA is far better than random projection", each showing points projected onto a 2D plane inside a cube]
> QUOTED: Delip Rao e/σ ✓ @deliprao · Mar 27
> The Google turboquant paper is [cut off]
Note from Claude Sonnet 5
Continuation of the Johnson-Lindenstrauss lemma / TurboQuant discussion thread — explains the nuance that random projections are only "nearly optimal" in the general/pathology-free case, and PCA can do better on structured data. Same technical math/ML thread as the two prior screenshots.
Greg Burnham @GregHBurnham · Mar 26
Wild. Can you give an intuition for why? I can see it for N=2 ;) I guess that shows how the points can "waste" lots of dimensions, and the geometry is more determined by the points themselves than the ambient dimension. So I guess the question is just, why log(N)?
[6 replies, 18 likes, 5.1K views]
Paata Ivanisvili ✓ @PI010101 · Mar 26
log(N) comes from the union bound + the fact that square of Gaussian is subexponential.
You can think of any linear map f as nxd matrix A, where n is the dimension of the target space, and d the dimension of the space where your N vectors live.
A good starting point is to look among random matrices A having the property E ||Av||^2 =||v||^2 for all vectors v, and then, hopefully, the rest should follow from concentration inequalities, i.e., ||Av||^2 cannot be too far from its average E ||Av||^2=||v||^2. Union bound tells you that the probability this inequality fails for some two vectors among our set of N vectors is at most N^2 times P( ||Av||^2 is outside eps-neighbourhood of its average) < N^2 exp(-n C(eps)). And this is less than 1 if n is of order log(N).
[1 reply, 69 likes, 4.4K views]
Greg Burnham @GregHBurnham · Mar 26
Oh that's cool. So can you get something tighter precisely by the degree to which the square of the Gaussian is subexponential, if that makes sense?
[1 reply, 6 likes, 810 views]
Paata Ivanisvili ✓ @PI010101 · Mar 26
Good point. One can certainly experiment with different random variables, but since the log(N) is already sharp, there's not much room for [cut off]
Note from Claude Sonnet 5
Continuation of the Johnson-Lindenstrauss lemma discussion thread from the prior screenshot — detailed math proof sketch via union bound and concentration inequalities. Pure math/theory content.
Paata Ivanisvili ✓ @PI010101
The Johnson--Lindenstrauss lemma says something quite remarkable: if you have an astronomical number N of vectors of large size (say, in a very high-dimensional Euclidean space), then you can linearly map them into a much lower-dimensional space, of dimension about log(N), in such a way that the distances between the vectors are almost preserved.
In other words, you can compress your data dramatically without making it too upset about its geometry. A random matrix with i.i.d. standard Gaussian entries will most likely do the job.
> QUOTED: Google Research ✓ @GoogleResear... · Mar 24
> Introducing TurboQuant: Our new compression algorithm that reduces LLM key-value cache memory by at least 6x and delivers up to 8x speedup, all with zero accuracy loss, redefining AI efficiency. Read the blog to learn how it achieves these results: goo.gle/4bsq2qI
Note from Claude Sonnet 5
A mathematician explaining the Johnson-Lindenstrauss lemma as the theoretical basis behind Google Research's TurboQuant, a new LLM KV-cache compression algorithm. Technical ML-infrastructure content.
Greg Brockman @gdb · Feb 13
we are now benchmarking our models on novel frontier research, via firstproof.org.
of 10 math research problems which research mathematicians have solved but never published the solutions to, in a week, our model discovered likely correct solutions to at least 6 of them.
> QUOTED: Jakub Pachocki @merettm · Feb 13
> Very excited about the "First Proof" challenge. I believe novel frontier research is perhaps the most important way to evaluate capabilities of the next generation of AI models.
> ...
> Show more
Note from Claude Sonnet 5
OpenAI's Greg Brockman announcing "First Proof," a new benchmark testing AI models on unpublished, unsolved-in-literature research math problems — reporting their model found likely-correct solutions to 6 of 10 in a week. Relevant to Nathan's capability-progress tracking; a significant claimed jump in genuine novel-research capability rather than benchmark memorization.
↻ Ben Golub reposted
Simone Conradi @S_Conradi · 16h
Take two large random matrices and linearly interpolate between them at several hundred steps. Compute the eigenvalues for each interpolated matrix, then plot them in the complex plane. The result is shown here.
Made with #python #numpy #matplotlib
[Image: dense golden/orange fractal-like starburst pattern of scattered points on black background, resembling a spiky spherical cluster with long filamentary "hairs" radiating outward, forming a roughly circular eigenvalue distribution in the complex plane. Caption: "Simone Conradi, 2025"]
Note from Claude Sonnet 5
A generative-art / random-matrix-theory visualization showing eigenvalue trajectories of matrices interpolated between two random matrices, plotted in the complex plane, producing an intricate fractal starburst pattern. Mathematical/aesthetic content with no direct AI safety relevance; likely saved for visual interest or general math-art appreciation.
Probability and Statis... @probns... · 11h
The Krylov-Bogolyubov theorem guarantees that dynamical systems on a compact space have at least one invariant probability measure. In machine learning, this is the theoretical bedrock for Reinforcement Learning and Recurrent Neural Networks (RNNs). It proves that despite noise and complexity, an agent's policy or a network's state will eventually settle into a stable statistical equilibrium. In real life, it explains why physical systems reach thermodynamic equilibrium.
Image: share.google/YJ84OhN4ZAxZh5...
[Two plots: (a) time series x(t) oscillating chaotically between roughly -4.0 and 4.0 over t=1010-1100; (b) phase portrait x vs ẋ showing a butterfly/figure-eight chaotic attractor pattern (resembling a Lorenz-type or double-scroll attractor)]
Note from Claude Sonnet 5
A tweet explaining the Krylov-Bogolyubov theorem and its relevance to reinforcement learning / RNN stability (invariant measures, statistical equilibrium), illustrated with a chaotic attractor plot. Mathematical/theoretical ML content, potentially relevant to Nathan's brain_graph_1 work on RNN/DEQ fixed-point dynamics.
Sichu Lu (@lu...), quoting "Name can't be bl..." (@Algon_...), which quotes an upvoted forum comment (198 votes)
— quoting "Name can't be bl..." (@Algon_...), which quotes an upvoted forum comment (198 votes)
Sichu Lu(Sichu.Lu218...) ✅ @lu... · 13h
isn't this true in general not just math? like it saves on compute to use your inner representations to think about a problem. and a lot of the time thinking represents different modalities(constructive versus destructive modes of thoughts)outside contributions helps someone easily do that
> QUOTED: Name can't be bl... @Algon_... · 14h
> The Tao that makes sense is not the Tao.
>
> [Embedded forum comment card, 198 upvotes]
> I find there is a world of difference between explaining things to a colleague, and explaining things to a close collaborator. With the latter, one really can communicate at the intuitive level, because one already has a reasonable idea of what the other person's mental model of the problem is. [highlighted:] In some ways, I find that throwing out things to a collaborator is closer to the mathematical thought process than just thinking about maths on one's own, if that makes any sense.
💬2 🔁2 ♡15 📊1K
Note from Claude Sonnet 5
A Twitter thread (likely referencing a LessWrong/forum comment, given the "Tao" quip and upvote count) discussing how explaining ideas to a collaborator is closer to genuine mathematical/creative thought than solitary thinking — relevant to Nathan's collaborative work style with Claude instances.
Terence Tao
@tao
More recently, we face the real and disturbing possibility that certain directions of mathematical inquiry - for instance, in developing reliable statistical tests for electoral integrity - may not only be defunded by public science agencies, but have their mathematical conclusions actually overruled by political ideology. Even if the supremacy of the objective mathematical standard of truth is technically acknowledged, it can still become weaponized: mathematical results which go against the prevailing ideology could be relentlessly critized for even the slightest typo or technical flaw in the presentation, whereas results that support this ideology could be uncritically embraced even they contain substantial gaps or ambiguities in interpretation. (5/6)
Terence Tao
@tao
One potential bulwark against such politicization of mathematical truth is the broader adoption of formal proof verification, though even here there are some (fortunately still quite theoretical at present) potential "exploits", for instance through subtly altering the definitions of key concepts in Lean's core "Mathlib" library. (See this recent talk newton.ac.uk/seminar/46706/ "Can Mathematics Be Hacked? Infrastructure, Artificial Intelligence, and the Cybersecurity of Mathematical Knowledge" by Fenner Tanswell.) Still, I view an increased acceptance and deployment of formal methods as a net positive in this regard, even if it is not a "silver bullet".
More generally, I think it is important to acknowledge just how precious the consensus objective standard of mathematical truth is, and how important it is to defend it. (This is not to say that such foundational matters should be completely immune from criticism or debate; but such discussion should be in good faith and grounded by genuine philosophical concerns, rather than driven by some external political agenda.) (6/6)
Note from Claude Sonnet 5
Two consecutive tweets (5/6 and 6/6) from Terence Tao on the politicization of mathematical truth, formal proof verification as a partial defense, and the value of the consensus objective standard of mathematical truth.
Simone Conradi @S_Conradi · 15h
510 million white dots. Each dot is a root of:
x^17 + (- 20i t2^7 + 20i t2^6 - 20i t2^5 + 20t2^4 - 20t2^3 + 20t2^2 + 20t2 - 20i)x^10 - 50x^7 + (20i t1^7 + 20i t1^6 - 20i t1^5 + 20i t1^4 - 20i t1^3 + 20t1^2 - 20i t1 + 20i)x^3 + 5i
t1,t2 ∈ ℂ, |t1|=|t2|=1
[Image: a generative-art visualization — a swirling, glowing ring-and-loop pattern rendered in white on black, formed from 510 million polynomial roots plotted as points.]
Note from Claude Sonnet 5
A mathematical generative-art tweet showing a polynomial root-plot visualization, unrelated to AI safety; general aesthetic/math-art content Nathan was reading.
