AI Math Papers, Explained

Beginner-friendly explainers for the AI-generated research papers in openai/math.

In autumn 2026 OpenAI released a collection of 722 mathematical manuscripts in 372 result families, produced by an internal model working on open research problems. They are written for specialists. This repository explains them to beginners in each field.

Every explainer answers the same questions:

1. What does the paper actually say? The result in plain language.
2. What was the problem? The question, with enough background to follow it.
3. What's the history? Who tried before, and what they achieved.
4. Why does it matter? Significance and consequences.
5. What's the main idea? How the proof works, at several zoom levels.
6. Whose ideas does it build on? The people behind the tools.
7. What doesn't it prove? Caveats, scope and verification status.

Each explainer comes with visual material generated with Google NotebookLM: slide decks, infographics, mind maps, reports and a short audio overview. All of it is checked against the paper, with known errors listed in each paper's errata.

Explained so far

Field	Paper	Explainer
Algebraic geometry	The rational Hodge conjecture for CM abelian varieties (family 032). On every complex abelian variety with complex multiplication, every rational Hodge class comes from algebraic cycles. This is a major case of the Hodge conjecture, a Millennium Prize problem	Read it
Number theory	The exact Birch–Swinnerton-Dyer formula from low Selmer corank (family 002). For every elliptic curve over ℚ whose Selmer group has corank 0 or 1 at some prime, the full BSD formula holds, including finiteness of Ш. BSD is a Millennium Prize problem; this proves it for that class of curves, not all of them	Read it
Number theory	Goldfeld's conjecture for quadratic twists (family 006). For the quadratic twists of any elliptic curve over ℚ, analytic ranks 0 and 1 each occur half the time, and the average analytic rank tends to 1/2	Read it
Number theory	The Quasi-Riemann Hypothesis: a zero-free half-plane Re(s) > 7/8 (family 003). The Riemann zeta function and every Dirichlet $L$-function have no zeros with real part above 7/8	Read it
Operator algebras	An isomorphism of the free group factors (family 287). The von Neumann algebras of the free groups on 2 and 3 generators are isomorphic, so all free group factors, including $L(\mathbb{F}_\infty)$, are the same	Read it
Operator algebras	Kadison's similarity problem (family 288). Every bounded unital homomorphism from a C*-algebra into the operators on a Hilbert space becomes a *-homomorphism after a change of inner product	Read it
Topology	The Hilbert–Smith conjecture in every finite dimension (family 304). Every locally compact group acting faithfully on a connected finite-dimensional manifold is a Lie group, which completes the last open part of Hilbert's fifth problem	Read it
Algebra	A counterexample to Kaplansky's zero-divisor conjecture (family 196). A finitely presented torsion-free group whose group algebra over $\mathbb{F}_2$ has zero divisors, so Kaplansky's conjecture fails in characteristic 2	Read it
Group theory	Thompson's group F is nonamenable (family 248). The group of dyadic piecewise-linear homeomorphisms of [0, 1] admits no invariant mean, settling a question open since the 1970s	Read it
Combinatorics	Quasipolynomial bounds for arithmetic progressions (family 159). Erdős's conjecture is true: every set of positive integers whose reciprocals sum to infinity contains arithmetic progressions of every length	Read it
Number theory	Catalan's constant is irrational (family 005). $G = 1 - 1/9 + 1/25 - 1/49 + \cdots$ is not a fraction, settling a question that has been open since Euler's time	Read it
Number theory	The irrationality exponent of π is 2 (family 017). π cannot be approximated by fractions much better than a typical number can; as a consequence, the Flint Hills series $\sum 1/(n^3 \sin^2 n)$ converges	Read it
Number theory / logic	Hilbert's tenth problem over the rational numbers (family 004). No algorithm can decide whether a polynomial equation with integer coefficients has a rational solution	Read it
Theoretical computer science	Exact derandomization of logarithmic space: L = RL = BPL (family 103). Any problem a randomized algorithm solves with logarithmic memory can also be solved deterministically with logarithmic memory	Read it
Theoretical computer science	The Unique Games Theorem (family 102). Khot's Unique Games Conjecture is true: for unique games over a large enough alphabet, distinguishing almost-satisfiable instances from barely-satisfiable ones is NP-hard	Read it
Theoretical computer science	An upper bound of 9/4 for the matrix multiplication exponent (family 107). Over the complex numbers, $n \times n$ matrices can be multiplied with $O(n^{9/4+\varepsilon})$ arithmetic operations for every $\varepsilon > 0$	Read it
Combinatorial geometry	A nine-dimensional counterexample to Borsuk's conjecture (family 156). A compact set in 9-dimensional space cannot be split into 10 pieces of smaller diameter, so Borsuk's conjecture fails in every dimension from 9 up (the previous record was 63)	Read it
Combinatorial geometry	The Euclidean plane is not five-colorable (family 158). However you color the plane with five colors, some two points at distance exactly 1 get the same color, so the chromatic number of the plane is 6 or 7	Read it
Harmonic analysis	Kakeya in three and four dimensions (family 074). The Kakeya maximal conjecture holds in 3D, and every set in 4D containing a unit segment in every direction has full Hausdorff dimension 4	Read it
Convex geometry	The symmetric Mahler conjecture (family 087). Every origin-symmetric convex body $K$ in $n$ dimensions has volume product $\lvert K\rvert\ \lvert K^\circ\rvert \ge 4^n/n!$, with equality exactly for Hanner polytopes	Read it
Dynamical systems	Hilbert's sixteenth problem: uniform bounds for limit cycles (family 143). A planar polynomial vector field of degree $d$ has at most $B(d)$ limit cycles, a bound depending only on the degree	Read it

The full list of all 722 papers and their status is in CATALOG.md.

Repository layout

papers/
  <preprint-slug>/              # one directory per openai/math paper, same name as openai/math/preprints/<slug>/
    README.md                   # the explainer
    assets/
      README.md                 # inventory, notebook sources, errata
      figures/                  # hand-made diagrams
      notebooklm/               # NotebookLM outputs: slides, infographics, reports, mind maps, audio
CATALOG.md                      # every paper, grouped by discipline and family (generated)
PIPELINE.md                     # how an explainer is produced, step by step
scripts/
  build_catalog.py              # regenerates CATALOG.md from openai/math
  mindmap_to_markdown.py        # renders NotebookLM mind-map JSON as Markdown

Directory names match openai/math/preprints/ exactly, so every explainer maps to exactly one manuscript.

A note on trust

These are explainers, not peer review. The underlying papers were produced by an AI model, and openai/math itself notes that unformalized results may contain errors. Each explainer records what has been formally verified (for example in Lean) and what hasn't. NotebookLM outputs are AI-generated as well, so every paper's assets/README.md lists the mistakes found in them.

Contributing

Pick a paper marked ⬜ in CATALOG.md and follow PIPELINE.md. Corrections to existing explainers are very welcome. Open an issue or a pull request.

License

The explanations, figures and generated assets in this repository are licensed under CC BY-SA 4.0. The papers belong to openai/math, which is released under the Apache 2.0 license; this repository links to them rather than copying them.