AI Solved What Mathematicians Could Not

If your Highland Park or Lake Forest law firm still treats AI as a glorified autocomplete tool, stop. In May 2026, an OpenAI reasoning model disproved the Erdős unit distance conjecture, an 80-year-old open problem in mathematics that had stumped some of the most decorated experts in the field. A team that included a Fields Medalist reviewed the proof and endorsed it for publication in the Annals of Mathematics without reservation. The model was not trained specifically for mathematics. It was a general-purpose reasoning system.

If that does not change how you think about AI in your firm, nothing will.

The Problem That Stumped Experts for 80 Years

The Erdős unit distance problem, known to mathematicians as Erdős problem 90, asks how many pairs of points can be positioned exactly one unit apart when you arrange n points on a flat plane. Hungarian mathematician Paul Erdős proposed a conjecture about the maximum count in 1946. For eight decades, no mathematician could settle whether his guess was correct or find a configuration that beat it.

In May 2026, an OpenAI reasoning model found a counterexample. It constructed a configuration using high-dimensional grids projected back into two dimensions, combined with techniques from algebraic number theory, that produced far more unit-distance pairs than Erdős's original grid approach. The AI proved Erdős wrong. As Scientific American reported, Cambridge mathematician and Fields Medalist Timothy Gowers said he would recommend the paper for publication in a top mathematics journal "without any hesitation." University of Toronto's Daniel Litt called it "the first result produced autonomously by an AI that I find interesting in itself."

The important context, as TechCrunch noted, is that OpenAI had claimed a similar breakthrough in October 2025, only to have that claim fall apart under scrutiny. This time, the same mathematicians who debunked the earlier claim are endorsing the result. The day after OpenAI's announcement, Google DeepMind published results showing their system had solved nine additional open Erdős problems.

Why AI Succeeded Where Human Mathematicians Struggled

The AI's advantage was not brilliance in the Hollywood sense. It was breadth and stamina. Jacob Tsimerman, one of the mathematicians who reviewed the proof, noted that the approach the AI took was one humans had considered and set aside. Such proof strategies "consume much time and frequently don't work out," he said. Humans stop when the odds look bad. A reasoning model does not.

The model also pulled together techniques from multiple mathematical subfields into a single coherent strategy. As The Conversation explained in their analysis, the new proof uses tools from algebraic number theory to show there are arrangements of points that far exceed the square grid approach for unit-distance pairs. That cross-disciplinary synthesis is exactly what makes hard problems hard for human experts. Deep specialization in one area creates blind spots in adjacent fields. AI has access to the entire published literature and can hold all of it in context while following a proof strategy that a human would have abandoned as too tedious.

The proof was then cleaned up, extended, and verified by the human team. This is the realistic picture of AI in expert work right now: the model generates the core insight, humans provide the judgment and rigor to confirm and publish it. Neither alone got there in eighty years of trying.

The Wrong Lesson Most Law Firms Are Drawing

The response from professional service firms is predictable: "That is mathematics. My work involves judgment, nuance, and client relationships. AI cannot replicate that." There is a grain of truth there. A reasoning model still needs human oversight for anything that carries real legal or fiduciary weight. No responsible practitioner should remove final judgment from a qualified attorney.

But that framing misses the more consequential fact. The gap between where AI actually performs today and where most law firms believe it performs has grown very large. AI is past the autocomplete stage. It is past the research-assistant stage. It is now contributing novel ideas to problems that stumped leading academic experts for eighty years. If your mental model of the technology is "it makes things up and hallucinates," you are working from a picture that is several product generations out of date.

What AI can do right now for a Winnetka or Lake Forest law firm: draft motion templates and first-cut memos, review contracts for specific risk clauses, generate research summaries from case law databases, cross-reference regulatory updates against a client's existing agreements, and flag deadline risks in matter management systems. None of that requires solving a geometry conjecture. It requires the same properties that let AI solve the Erdős conjecture: broad knowledge, no fatigue, and a willingness to handle tedious work that consumes attorney time without adding billable value.

For a closer look at what proactive AI agents can do inside a North Shore law practice, see the guide on deploying AI agents for intake and matter management at Lake Forest law firms.

Your Competitive Risk Is Widening

Every quarter your firm treats AI as an experiment rather than a practice tool, firms with better adoption are billing more matters in the same time or reducing first-draft costs without cutting quality. The math breakthrough matters because it settles a debate that has kept many professionals on the sidelines: is this technology capable of real expert-level work, or is it still a sophisticated search tool?

The official OpenAI announcement of the Erdős result makes clear that the model used was a general-purpose reasoning system, not a math-specific tool. The reasoning that cracked an 80-year open problem in geometry is the same underlying capability available in the tools your attorneys already have access to today. The question is whether they are using them.

The risk for a Highland Park or Glencoe firm is not that AI replaces your attorneys tomorrow. It is that a competing firm uses AI to handle a much larger share of first-draft and research work in the same time, freeing attorneys for the judgment-heavy work that clients value and that generates the highest margin. That is a gap that compounds. By the time it shows up in revenue per attorney, it has already become structural.

Not sure where your firm sits on the adoption curve? Take the free AI readiness quiz to get a prioritized starting point built around your practice type.

One Step You Can Take This Week

You do not need to restructure your firm's technology stack to get started. The single most direct action this week is getting your attorneys using Claude or GPT-4o for one specific, repeated task. Case law research summaries work well as a first use case because the output is easy to verify and the time savings are clear. Contract clause review is another strong entry point.

The goal is not full automation. It is giving your attorneys a fast, knowledgeable thinking partner for the portions of their work that currently consume time without requiring attorney judgment. The Erdős result tells you the thinking partner is more capable than most people assumed. The only variable is whether your firm is putting it to work.

If you want to move faster than a self-serve experiment, the Bace Agency service packages for North Shore law firms include hands-on setup, specific use-case identification for your practice area, and a 90-day implementation plan.

The Erdős result is not a distant warning about some future version of AI. It is a data point about the version your attorneys already have access to right now. The only variable is whether your firm is using it.

Ready to find out where AI fits in your practice? Schedule a free 30-minute AI audit and we will map out the highest-value starting point for your firm.

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