The Deligne–Drinfeld conjecture

The explanation

Take two letters, x and y. Combine them with a bracket: [a,b] means ab minus ba. With three letters you can build [x,[x,y]], which expands to xxy − 2xyx + yxx. You can also build [y,[x,y]]. Every other three-letter bracket expression is a mix of these two. The number of letters is called the weight. Counting gets large fast. Weight 11 allows 186 independent bracket expressions. Weight 29 allows 18,512,790.

Now demand that an expression pass three short tests. The first says swapping x and y flips its sign. The second adds three versions with x and y replaced by other mixes, and asks for zero. The third, the pentagon, balances five versions written in symbols for four braid strands. Expressions that pass all three are the solutions. The theorem pins down how many survive. In weight 3 one survives. In weight 11 two survive. In weight 29 only 120 survive out of 18,512,790.

Where do those counts come from? The prediction is a building recipe. Start with one basic piece in each odd weight: 3, 5, 7 and upward. Combine pieces with a bracket, and their weights add. Weight 8 has one combination, the bracket of the 3-piece with the 5-piece. Weight 11 has two: the 11-piece alone, and the 3-piece bracketed twice with the 5-piece. Counting all such combinations gives 1, 2 and 120 above. The prediction says the solutions are exactly these combinations, with no hidden coincidences among them.

The solutions form the Grothendieck–Teichmüller Lie algebra. It records the tiny symmetries of the rules for regrouping and crossing braid strands. Solutions combine under a special operation, the Ihara bracket. The Deligne–Drinfeld conjecture predicts that this structure is free on one generator in each odd weight from 3. Free means the generators obey no relations beyond the bracket's own rules. Generator here means one of the basic pieces.

The prediction has three parts: the basic pieces exist, they combine freely, and nothing else solves the equations. The first two were known. Drinfeld's 1990 paper credits Ihara with the odd-weight pieces. Brown's 2012 work gave a free family of solutions. In 2026 Willwacher showed that suitable families of pieces are always free. The obstacle was the third part. Nobody could rule out extra solutions in every weight. Computations by Naef and Willwacher settled the count only up to weight 29.

The paper's main estimate is an upper bound on the number of solutions in every weightSource 1. It matches the recipe's count. The key step studies the pentagon in characteristic twoSource 2. That is arithmetic where 1 + 1 equals 0. There, the equation simplifies enough to count what can survive. Separately, the paper builds a solution in each odd weightSource 3. Each has a nonzero coefficient on the word x…xy, which has one y. This relies on a convergent integral being nonzero, not on any unproved fact about its valueSource 4. Together the two halves give existence, freeness and the exact count at onceSource 5.

So the theorem forbids any surprise solutionSource 6. In weight 30 the recipe gives 151 solutions among 35,790,267 bracket expressions. No computer had checked that weight, and the theorem says 151 is exact. It also shows that the Ihara bracket of two solutions is again a solutionSource 7. The basic pieces it uses are one valid choice among many; the paper says they are not canonicalSource 8. The paper also claims a consequence for Kontsevich's graph complex, an algebraic structure built from drawings of graphsSource 9. That step relies on Willwacher's earlier theorem. It describes only one layer of that structure, the layer called degree zero, and no other.

How sure is this? A computer proof checker called Lean has checked a formal version of the main theorem. It covers the rational solutions of the three equations with the Ihara bracket. It also covers the version for infinite sums by weight. The regularized-transport proposition and the graph-complex consequences are outside that check. No specialist review has been published yet.

What it doesn't show. This does not give a canonical choice of the basic pieces; the paper says its generators are not canonical.