The duo kept their show running in the background for over a decade. During this time, several computers from their destroyed collection overheated and even caught fire. “There was one that actually sent sparks,” Brittenham said. “It was a lot of fun.” (These machines, he added, “retired with honor.”)
Then, in the fall of 2024, an article was published about failed attempt to use machine learning the refutation of the additivity hypothesis attracted the attention of Brittenham and Hermiller. Perhaps, in their view, machine learning is not the best approach to this particular problem: If there were a counterexample to the additivity hypothesis, it would be a “needle in a haystack,” Hermiller said. “That’s not really what machine learning is about. It’s about finding patterns in things.”
But it reinforced a suspicion the pair already had – that maybe their more carefully crafted sneaker might find the needle.
A tie that binds
Brittenham and Hermiller realized that they could employ the discovered untangled sequences to search for potential counterexamples to the additivity hypothesis.
Imagine again that you have two nodes whose unresolved numbers are 2 and 3, and you are trying to untangle their sum of connections. After one intersection change, you will receive a fresh junction. If the additivity assumptions are to be believed, the number of solutions to the original node should be 5 and the value of the fresh node should be 4.
But what if we already know that the number of unresolved nodes of this fresh node is 3? This means that the original knot can be untied in just four steps, which undermines the assumption.
“We have these middle nodes,” Brittenham said. “What can we learn from them?”
He and Hermiller already had the perfect tool for the occasion, humming away on their laptops: the database they had been working on for the previous decade, with its upper limits at an infinite number of thousands of nodes.
Mathematicians began adding pairs of nodes and working on unsolved sequences of their connection sums. They focused on call totals, whose unresolved numbers were only approximate in the loosest sense, with a immense difference between their highest and lowest possible values. But that still left them with a huge list of knots to work through – “definitely tens of millions, probably hundreds of millions,” Brittenham said.
Their computer program spent months applying crossover changes to these nodes and comparing the resulting nodes with nodes in the database. One day in delayed spring, Brittenham checked the program’s output files, as he did most of the time, to see if anything engaging had turned up. Much to his surprise, a line of text appeared: “CONNECT SUM BROKEN.” It was a message he and Hermiller encoded in the program, but they never expected to see it.