How to measure contract complexity

The duo kept their program running in the background for more than a decade. During that time, a couple of computers from their scattered group were exposed to overheating and even fire. “There was one that actually sent sparks out,” Brittenham said. “That was kind of fun.” (He added that those machines were “honorably retired.”)

Then, in the fall of 2024, a research paper on A Failed attempt to use machine learning To refute the collective conjecture he brought to the attention of Brittenham and Hermeler. They may have thought that machine learning wasn’t the best approach to this particular problem: If there was a counterexample to the addition intuition, it would be a “needle in a haystack,” Hermiller said. “That’s not exactly what things like machine learning are about. It’s about trying to find patterns in things.”

But that reinforced the suspicion the couple already had, that maybe their carefully polished gym nets could find the needle.

The tie that connects

Brittenham and Hermeler realized that they could take advantage of the uncomplicated sequences they had discovered to look for possible counterexamples to the addition conjecture.

Imagine again that you have two nodes whose undeciphered numbers are 2 and 3, and you are trying to decode their connected sum. After changing the jumper once, you get a new node. If the collective guess is to be believed, the number of the original node should be 5, and the number of this new node should be 4.

But what if the untwisted number of this new node is already known to be 3? This means that the original knot can be untied in just four steps, which takes the guesswork out of it.

“We’ve got this middle knot,” Brittenham said. “What can we learn from them?”

He and Hermeler already had the perfect tool for the occasion in their laptop suite: the database they had spent the past decade developing, with its upper limits on uncomplicated numbers of thousands of nodes.

Mathematicians began adding pairs of nodes and working through uncomplicated sequences of their connected groups. They focused on continuous sums whose uncomplicated numbers were rounded only in the broadest sense, with a large gap between their highest and lowest possible values. But that still leaves them with a huge list of nodes to solve, “certainly in the tens of millions, perhaps hundreds of millions,” Brittenham said.

For several months, their computer program applied cross-changes to these nodes and compared the resulting nodes with those in their database. One day in late spring, Brittenham scanned the program’s output files, as he did most days, to see if anything interesting had turned up. To his great surprise, there was a line of text: “CONNECT SUM BROKEN.” It was a message that he and Hermeler had encoded in the software, but they never expected to actually see it.