A classic math inequality guarantees endless number systems with the right properties, and that supply mattered for a recent OpenAI geometry result.

This explores how the Golod–Shafarevich criterion guarantees an unlimited supply of number fields with the right properties, and why that supply mattered for a recent AI-assisted result in geometry. First, a limit: the collection covers this criterion only as one ingredient in a specific construction. It does not have a step-by-step account of the theorem itself, so the next few sentences fill in standard background rather than summarize the corpus. A number field is an extension of the rational numbers, and each one has a class group that measures how badly unique factorization fails in it. The Golod–Shafarevich inequality says this: if that group has many independent generators compared with the relations between them, you can never stop building bigger fields on top of the base field. That stack of fields is called a class field tower, and the inequality forces it to be infinite. Every level of the tower is a field of higher degree, but its discriminant (a measure of how complicated the field's arithmetic is), scaled by degree, stays bounded. That gives you infinitely many fields that keep growing in size while staying controlled.

That endless, controlled growth is what the corpus says made the difference. In OpenAI's counterexample to Erdős's unit distance conjecture, the classical tools were towers and number-field methods. The new move was letting the degree of the field go to infinity. With a fixed prime that splits in these fields, the class number and discriminant stay small, and that is enough to build point sets in the plane with more unit-distance pairs than Erdős expected What made OpenAI's unit distance counterexample succeed?. A follow-up construction used lattices built from CM fields, a special family of number fields, together with Golod–Shafarevich arguments. It turned 'more than linear' into an explicit exponent: more than n^1.014 unit-distance pairs among n points How many unit distances can points in a plane have?. So the criterion does not supply the answer. It supplies the raw material, a guaranteed infinite family, that the geometric argument then exploits.

The surprise is where these notes sit in the collection. They are not mainly about number theory. They belong to a larger debate about what it means when AI helps produce mathematics. Erdős problems became an AI testing ground because they are easy to state and vary widely in difficulty. One account reports that machine checking of a proof and human understanding of it have become separate outcomes, with understanding falling behind Why did Erdős problems become a popular AI testing ground?. Tao argues that an AI system's opacity matters less if its output is paired with a reliable checker, such as a proof assistant or a numerical method Can opaque machine learning models help prove new mathematics?. The Leiden Declaration takes the opposite emphasis: proofs exist to convey understanding as well as certainty, so human authors must keep sole responsibility for correctness Can AI-generated proofs ever replace human mathematical understanding?.

The Golod–Shafarevich case is a useful test for that debate. The most important ingredient was a decades-old theorem, used in a way that working number theorists can read and check. That makes it a counterexample to the worry that AI-assisted proofs must be unreadable. Formalizing it would still be hard, though. One note argues that even a single theorem needs a whole connected web of definitions and lemmas, and that statement-by-statement approaches only succeed by quietly leaning on existing libraries Can autoformalization work on individual statements alone?. Class field towers are exactly the kind of deep, layered theory that sits underneath a headline result.