So, this happened.

First, the Courtade-Kumar conjecture.

This one has been open since 2013. Take a bunch of bits and pass them through independent noise. The question is which Boolean function preserves the most information about the original bits. The conjecture says the best answer is almost stupidly simple. Just read one bit. These are called dictator functions. Courtade and Kumar conjectured that no more complicated Boolean function can do better.

Now we have multiple claimed proofs. One is computer-assisted. Vu Khac Ky and Tuan Tran also posted an independent proof using a different approach. Two different proofs appearing at about the same time doesn’t automatically settle anything, but it certainly makes this more interesting. Both are new, and both now get handed over to everyone else to check.

Computer-assisted proof on arXiv

Ky and Tran proof on arXiv

Then there is the Kahn-Saks conjecture.

This is a problem about partially ordered sets and their linear extensions. A partial order might tell us that $a$ has to come before $b$ and $c$ has to come before $d$ without specifying where everything else goes. A linear extension fills in the rest and gives us a complete ordering that still obeys those restrictions. The Kahn-Saks conjecture deals with how balanced those possible orderings can be.

Max Aires has posted a proof showing that as the width of the partial order becomes sufficiently large, the best balancing probability gets arbitrarily close to 1/2. For now, I would still call this a claimed proof. The paper is new and people need time to go through it.

Proof on arXiv

The third result involves AI. Because apparently we can’t go a week anymore without one of these.

This is a major open problem in percolation theory (something I did not know existed). Imagine an infinite grid where every edge is independently turned on with some probability $p$. If $p$ is small, you don’t get an infinite connected cluster. Make $p$ large enough and you do.

Somewhere between those two behaviors is the critical probability $p_c$.The question is what happens exactly at $p_c$. For nearest-neighbor Bernoulli bond percolation on $ℤᵈ$, mathematicians have been trying to prove that the probability of an infinite cluster at the critical point is zero for every $d ≥ 2$. The missing dimensions were 3 through 10.

An Anthropic project now claims to have finished them. The result comes from proving a conjecture of Gady Kozma and Asaf Nitzan using a new gluing inequality. The proof was produced with AI assistance and formally checked in Lean.

The Lean part makes this especially interesting, but it doesn’t mean everybody packs up and goes home. Lean verifies the formal statement it was given. Mathematicians still need to make sure the formalization says exactly what everyone thinks it says and understand the proof itself. So I’m being a little careful with the word “solved.” If it holds up, though, the remaining dimensions of a major percolation problem are gone.

Anthropic formal mathematics project

More from Dogmathic

For more math news, explanations, proofs, examples, and commentary, check out Dogmathic Math Notes and the latest Dogmathic videos. You can also grab free PDFs and LaTeX resources or sign up for occasional email updates when something worth sharing gets published.

Blogmathic | Videos | Free PDFs | Email Updates | Contact