a16zInside OpenAI’s Breakthroughs in Mathematical Reasoning
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A Renaissance In Reachable Results 0:00
The conversation opens with two OpenAI researchers, Mark Sellke and Mehtaab Swani, both former practicing mathematicians, describing why AI-assisted math feels like it has entered a new era. A human mathematician often has an idea, tries it for hours or weeks, and gives up if it does not pan out quickly, since the risk-reward calculation favors abandoning long shots. A model like GPT does not weigh its time the same way. If a human suggests a direction, it will simply pursue it, however long that takes. This difference in how effort gets rationed, rather than any single new capability, is offered as the reason so many previously untouched problems are suddenly being cracked.
How Two Mathematicians Got Pulled In 1:00
Sellke says he joined OpenAI shortly after seeing the model win an IMO gold medal last summer and wanted to understand how it was done. Swani followed after Sellke gave him a GPT-5 account in the fall, and he quickly became convinced the models were worth taking seriously. Swani's own turning point came from Erdos's collected open problems in combinatorics, a list that is notoriously hard to search because the literature is scattered. He plugged one unresolved-looking problem into GPT-5 and within five minutes it found a reference resolving a question he and friends had spent hours on without knowing if it was even still open. That experience repeated itself roughly ten more times as they explored further.
Search Skill Versus Real Reasoning 4:30
Beyond just knowing the literature well enough to make connections, the researchers point to a second strength: once the model has an idea, it executes the finicky details, like checking whether epsilon is smaller than delta, far more reliably than a human, who tends to get lost in that bookkeeping. The unit distance problem is used as an example, where the underlying approach traces back to Erdos-style ideas, but pushing it through to completion took enormous, meticulous effort. A human might abandon that path because the odds don't seem to justify the time; the model, told to pursue it, simply keeps going.
Making The Right Bets 7:00
Sellke describes mathematics as often a matter of gambling against a problem, deciding whether a risky-seeming approach is worth trying. In the unit distance conjecture case, the model combined its knowledge with what looks like good taste, correctly betting on approaches that many serious mathematicians had also tried, which is part of what made the result striking. Unlike a human whose brain can get anchored to a failed approach, the model can be redirected without that residue, more easily abandoning unproductive paths and updating on failure the way a fresh, unbiased collaborator would.
Where This Reasoning Ability Comes From 13:00
Asked how this mathematical reasoning emerges given that math papers and textbooks are poor teaching material, since they hide the struggle behind polished proofs, Sellke and Swani note this isn't a lean-based or auto-formalization-driven approach. It reflects OpenAI's broader push to train general-purpose reasoning models across many domains, so behaviors like backtracking and restarting an approach are not math-specific tricks but general features of sustained reasoning that show up wherever a model reasons hard enough. They add that the summarized chains of thought were released partly so people could see that the model isn't guessing wildly but reasoning in a way that resembles reading a colleague's raw notes.
The Sphere Packing Problem 16:30
The two turn to a favorite among the ten recently released problems: sphere packing, the question of how densely unit spheres can fill space in various dimensions. In one dimension it's trivial, in two dimensions the answer is the familiar hexagonal lattice arrangement proven only in the 1960s, and in three dimensions it's the orange-stacking arrangement proven by Hales in the 2000s with a proof running several hundred pages of messy linear programming and geometry. The only other known exact dimensions are 8 and 24, solved in 2017 using two exceptionally dense, highly symmetric structures called the E8 and Leech lattices.
How Little Is Known, And Astra's Bound 20:00
Otherwise almost nothing is known precisely; researchers only have a crude lower bound of 2 to the minus d and know the true packing density must decay exponentially in high dimensions, since spheres increasingly can't touch. For decades the best known bound came from a two-page 1970s paper by Russian mathematicians giving an oddly specific exponential rate. The discussion sets up Astra's contribution, a new bound built on the linear-programming approach originated by Cohn and Elkies and advanced by Fields Medalist Maryna Viazovska, who constructed special functions in dimensions 8 and 24 to nail the exact answer there, work that earned her the Fields Medal.
A Mysterious Bound Explained 26:30
The discussion returns to a linear programming bound in high dimensions, a problem where a numeric paper by Cohn and others had guessed the right answer without knowing why it was true. The model constructs a function that achieves this bound and then proves no better function exists, turning a guess into an equality. One of the speakers mentions having worked on this exact problem for six months as a graduate student and making zero progress, so seeing the model's short, elegant complex-analysis proof was a pleasant surprise. He describes the proof as the kind of good mathematics where, once you see it, you wonder why nobody found it sooner.
Spherical Codes and Cube Codes 29:30
The second problem concerns spherical codes, which are essentially sphere packings drawn on the surface of another sphere rather than in flat space, and binary codes, the same idea placed on the corners of a cube. Binary codes are explained through error correction: if you send a string of bits and some get corrupted, you want a dictionary of code words spaced far enough apart, measured by Hamming distance, that a single error can still be traced back to the original word. As the spheres in a spherical code shrink, the curvature stops mattering and the problem starts to look just like ordinary sphere packing, so the two problems turn out to be closely related.
Representation Theory Improves the Bounds 34:04
For the sphere and cube cases, the model's improvement leaned on representation theory, using the heavy symmetry both shapes have, rather than the complex analysis used for the full-space packing problem. Pushed far enough, the representation-theory formulas for spherical codes reproduce the earlier full-space sphere-packing result when the small spheres shrink toward a point, showing the two results are genuinely connected rather than coincidental. Notably, this was the one case where the researchers interacted with the model step by step, first asking it to improve the code bound, then asking it to push further, which led it to discover the deeper connection.
Judgment, Task Orientation, and Taste 36:31
The conversation turns to whether the model shows judgment or just follows instructions literally. One researcher notes models tend to be task-oriented: told to improve a bound by a certain factor, it does exactly that and stops, even if further progress is possible, until asked again. They describe taste in a utilitarian way, as whatever lets you solve problems faster through better judggment, and note that solving harder problems already implies better taste, since it requires choosing among a vast space of possible paths. One speaker suggests it might even work to separate one system responsible for taste from another responsible for long, grinding work on a hard problem, similar to a supervisor and an underling.
What Sofic Groups Are 44:30
The topic shifts to sofic groups. A group is a set of elements with a multiplication operation that is associative, has inverses, and has an identity element, and it is the standard way mathematicians formalize symmetry, whether from numbers, matrices, or rotations. A group can be finite, like the four rotations of a square, or countably infinite, like the integers. A group is called sofic if it can, in some precise sense, be approximated by finite groups, the way the integers are approximated by integers taken modulo n, which look locally the same as the full integers until you step back far enough to see the whole circle. It had been unknown whether every group is sofic, and Astra proved that a non-sofic group exists.
Why the Question Mattered 48:00
Sofic groups matter because many nice properties, including a property called surjunctive, can be proven for finite groups and then carried over by approximation, so mathematicians had hoped every group would turn out to be sofic. This connects to a related and stronger conjecture in probability, the Aldous-Lyons conjecture, which claims that any infinite, unimodular random graph can be approximated by large finite graphs, again illustrated with the integers pictured as a line graph approximated by large circular graphs. The Aldous-Lyons conjecture had already been disproved two years earlier through an enormously complex 250-page argument built on quantum complexity theory that few people could fully follow. By contrast, the direct proof that a non-sofic group exists is only about fifteen pages, stays entirely within ordinary group theory, and builds on existing results by mathematicians named Cohn and Ta, without needing any of the quantum complexity machinery.
A Delicate Combinatorics Argument 55:01
You hear the team describe one proof as a very delicate combinatorics argument, one where the model had to find the right statement, identify the relevant pieces from the existing literature, and execute the logic correctly. The hard part of the problem was that it resisted approximation by any finite structure. When they did a postmortem, they found a concrete combinatorial obstruction that previous papers had implicitly grappled with but couldn't rule out. Astra found that adding one extra algebraic fact eliminated the problematic case, essentially finishing what earlier authors had already been circling. The speakers note that a year ago they expected AI proofs to be sprawling, thousand-page outputs that no human could parse, but instead these proofs have turned out short and elegant, to the point where only humans currently produce the very long, 200-page proofs.
Mathematicians Building on AI Results 57:00
Follow-up work has already begun from researchers who built on this proof, which the team sees as exactly the outcome they hoped for, since they don't want to be writing all the follow-up papers themselves. They discuss how the math community is gradually accepting AI's role, though attribution and co-authorship remain unsettled questions. One speaker points out that if models can produce sophisticated mathematics, they can also help people understand it, describing how feeding a paper into a model and asking for the proof strategy is often faster than reading the introduction directly. This suggests AI both accelerates the creation of mathematics and eases the burden of absorbing it.
How the Field Might Reorganize 1:00:02
Historically, proving a hard result was so difficult that understanding, internalizing, and explaining it came along for free, since the prover was forced to master it. Now that proving is less of a bottleneck, the speakers expect the value to shift toward people who can organize and communicate mathematical knowledge to others, a role that was previously implicit but may become an explicit, rewarded part of the field. They note math has a very high ceiling of difficulty, so even fast AI progress may leave problems like P versus NP unsolved, potentially pushing the field toward big mysteries rather than routine ones. They close by expecting broader access: people outside professional mathematics will understand more, and fields needing math expertise won't always need to find a world expert, which could speed up applied mathematics generally.
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