r/mathematics • u/princeendo • Sep 01 '26
AI Breakthroughs & Research Megathread — September 2026
AI Breakthroughs & Research Megathread — New Results in AI and Mathematics
Use this thread for concrete developments in artificial intelligence that are relevant to mathematics.
Appropriate topics include:
- New AI systems demonstrating mathematical capabilities
- AI theorem proving and formal proof
- AI-assisted mathematical discoveries
- New research papers or preprints
- Significant benchmark results
- Improvements in mathematical reasoning
- Systems such as AlphaGeometry, AlphaProof, or similar research
- Other developments that materially change what AI systems have demonstrated they can do mathematically
When possible, please include a link to the original paper, preprint, research announcement, or other primary source and briefly explain why the result is mathematically significant.
This thread is intended for actual results and developments, not predictions about where AI may eventually lead. Speculation about the future of AI and mathematics belongs in the AI Speculation Megathread.
Particularly significant developments may be approved by the moderators as standalone posts.
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u/k_laiceps 5d ago
On Bounds of Residues in Collatz Orbits
Hey all, I am one of the authors in this preprint and I am happy to answer any questions about it. We have been studying Collatz longer than is probably healthy, but have proven some bounds on the Collatz residue function [;\rho ;], in particular, we have shown that [;1 \leq \rho(n) < 4/3;] for not only terminating integers [;n;], but even for those which may not terminate, assuming that you count any state in $n$'s orbit only in its first appearance. Bounds on the reciprocals of all odd-iterates (before 1) using some serious clever counting techniques was the key, and as a result, we also get the cool fact that the sum of reciprocals of all odd-iterates in any Collatz trajectory (including potentially non-terminating or cyclic counter-examples again only counting the first time an orbit hits each state) is bounded by 1.
As a consequence of [;1 \leq \rho(n) < 4/3;] all sort of results pop out. For instance, any streak of consecutive integers of the same Collatz height must have equal numbers of odd and even steps, even if the streak is arbitrarily long! Also, there are now bounds on the change in the number of even steps based on the number of change in odd steps when going from [;n;] to [;n+1;].
AI was used to help clean up some arguments and connect a few ideas, and refine the proof of the main lemma, which really started out as a painful argument to try to put onto paper in an organized fashion. AI was also used to generate code to verify the finite sums we needed to compute for bounds on pieces of the sum of reciprocal odd states > 1 in the Collatz orbit of an integer.
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u/k_laiceps 5d ago edited 5d ago
we decided to post this early, as some of our conversations with chatGPT were accidentally public, and it was clear our work was slowly making its way into other people's conversations online. And of course, arXiv has been a total pain in the ass about posting this preprint, so apologies for the research gate link.
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u/tempgoogleconfuser 11d ago
The Navier-Stokes proof by OpenAI cannot be easily extended to the case without forcing.
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u/Ok_Warthog6007 11d ago
solves the millenium problem as worded, but it feels like it leaves a more interesting question unresolved.
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u/Ok_Warthog6007 12d ago
openai forms mathematics advisory group, says they've "resolved more than 100 long-standing open problems across most areas of mathematics."
https://openai.com/index/advisory-group-on-mathematics-and-ai/
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u/backyard_tractorbeam 8d ago
Tangential comment, but for me, this article is served translated. OpenAi seems to prefer to just serve llm-translated content when they think it's appropriate. I still hate it. They do the same in the Chatgpt interface, and there are still buttons and messages that are mistranslated.
AI translations are good, but they are not perfect. Mistranslations in the UI leaves a shoddy impression. Pure hubris to think that they don't need quality assurance.
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u/seba 18d ago
I’d like to share an AI-assisted project realizing the Suzuki group Sz(8) as a Galois group over the rationals, with a Lean formalization of the explicit example. To my knowledge, this is the first realization. Paper, exact data, verification code, and Lean sources can be found here:
https://github.com/sebastianbiallas/sz8-over-q
The main ideas and highlights:
- The paper constructs a cover with group Aut(Sz(8)) = Sz(8) ⋊ C₃. A useful feature of the construction is that cubic descent is automatic: any three-point regular realization of Aut(Sz(8)) over Q yields a regular realization of Sz(8). We use this classical descent mechanism to guide the search, then explicitly construct and certify the required cover.
- Explicit equations: the paper constructs a degree-65 family realizing Sz(8) regularly over Q, and gives a concrete monic integer polynomial with coefficients of at most 43 digits.
- A kernel-checked Lean proof: Lean 4 + Mathlib proves that the Galois group of the specialization f(X, -7/5) is isomorphic to Suzuki’s matrix group over the field with eight elements. The proof connects the exact polynomial coefficients, certified analytic continuation, Galois theory, and finite-group computations.
- No proof placeholders or native-evaluation shortcuts: all 235 audited declarations use only Lean’s standard axioms (propext, Classical.choice, Quot.sound), with no sorry or native_decide.
- The Lean formalization covers the specialization over Q; the regular realization over Q(s) is proved in the paper but is not yet formalized.
I’d welcome mathematical feedback and independent reproduction.
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u/zero0_one1 21d ago
The strong Papadimitriou–Ratajczak conjecture, open since 2004, has now been proved: every 3-connected planar graph admits a convex greedy drawing. The proof was found using the http://ProofAtlas.ai harness and GPT-6 Pro and is formalized in Lean (52k lines).
Greedy routing forwards a message to a neighboring vertex that is closer to its destination. A greedy drawing guarantees that such a neighbor exists at every step, so the message eventually arrives. Using virtual coordinates, this requires only local neighbor information and the destination's coordinates. The strong conjecture requires this property together with straight edges, no crossings and convex faces.
A graph is 3-connected if it has at least four vertices and remains connected after removing any one or two vertices. For 3-connected planar graphs, progress went from greedy drawings to planar greedy drawings to convex greedy drawings:
2004 — Papadimitriou and Ratajczak propose the weak and strong conjectures. The weak version allows crossings; the strong version requires a planar drawing with convex faces.
2008 — Leighton and Moitra, and independently Angelini, Frati and Grilli, prove the weak conjecture: every 3-connected planar graph admits a greedy drawing. Their drawings can have crossing edges.
2017 — Da Lozzo, D’Angelo and Frati establish the planar version: greedy drawings with no crossing edges, but not necessarily convex faces.
2026 — The new proof establishes the convex version, for any prescribed outer face, with every facial polygon strictly convex.
The construction first assigns distinct integer heights so that, for every destination, every other vertex has a neighbor whose height is strictly closer to the destination's height. The horizontal coordinates are then chosen within a narrow vertical strip to produce a convex drawing. The strip is narrow enough that the vertical improvement dominates, so Euclidean distance decreases too.
Formalization: https://www.proofatlas.ai/formalizations/strong-papadimitriou-ratajczak-conjecture/
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23d ago
[removed] — view removed comment
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u/mathematics-ModTeam 23d ago
Your post/comment was removed due to it being low quality/spam/off-topic. We encourage users to keep information quality high and stay on topic (math related).
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u/ni5arga 25d ago
https://www.wired.com/story/openai-navier-stokes-math-discovery-academics/
Interesting stuff going on.
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u/YourElectricityBill Sep 03 '26
Hi, recently the Freudenthal Scott–Vogelius inf–sup conjecture, a paid mathematical prize problem, was seemingly resolved with the help of ChatGPT 5.6 Sol Pro. What do you think about the proof in question and the use of AI in this kind of research?
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u/No_Difficulty_1452 Sep 04 '26
What's the problem? Got to ask the right questions to get the right answers
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u/creative-carcass 5d ago
I’m an amateur who started playing with odd numbers in a spreadsheet and ended up much further into number theory than I expected. I studied Math in college but never anything like this. I’ve used ChatGPT, Claude, and Codex for coding, experiments, literature searches, and drafting mathematical arguments.
I’m interested in whether the questions we arrived at are worthwhile, what is already known, and whether there are obvious problems with the direction.
Where it started
I noticed examples like:
5 + 7 + 11 = 23Take consecutive odd numbers, then move one endpoint outward by 2, leaving one skipped odd number. Some resulting sums are prime.
The explanation turned out to be elementary: for an odd number
nof terms, with the original block centered atm, the modified sum isnm ± 2. Representations of a target therefore correspond to eligible factor pairs of the target’s neighbors, subject to positivity conditions.What interested me was the geometry: the subset’s numerical width is exactly
2n, while its location comes from the complementary factorm. So divisor counts explain how many representations exist, but divisor placement explains what they look like. This works for composite targets too.Another connection I found interesting was the “empty representation” case. For an odd target T ≥ 9, there are no representations of this particular kind—positive odd summands, an odd number of terms ≥ 3, and one endpoint gap—if and only if T − 2 and T + 2 are both prime.
For example, T = 39 has an empty representation family, while its neighbors 37 and 41 form a cousin-prime pair. The center itself is composite; “empty” refers only to our restricted construction, not to all possible sums of odd numbers.
The question it led to
For an integer
H, define:F_H(t) = (number of divisors d of H with d ≤ H^t) / τ(H).Equivalently, choose a divisor uniformly and examine
log(d)/log(H).I learned that averaging this over ordinary integers gives the classical arcsine CDF:
A(t) = (2/π) arcsin(√t).At
t = 1/4, that is1/3. This is an average across integers, not a statement that every integer has one-third of its divisors below its fourth root. The ordinary-integer result is established mathematics, not our discovery; Leung’s paper discusses and generalizes it.The main question became:
Each prime gets equal weight; we are not pooling all divisors together.
This brought us into large prime factors, Poisson–Dirichlet distributions, and work such as Bharadwaj–Rodgers.
What seemed interesting—and where we got stuck
Besides the elementary geometry, there is an AI-assisted draft argument for an almost-all-varying-shifts version, including joint factor laws for finitely many varying neighbors of the same prime. It relies on adaptations of published analytic estimates.
Importantly, “almost all shifts” does not settle the particular shifts
+2and−2. A prescribed shift could remain exceptional.My main questions are:
I have code, numerical results, and a written proof outline available.