Normal logic—like "if A is like B and B is like C"—actually falls apart once you start tracing paths on a fractal.
arXiv · March 18, 2026 · 2603.16774
The Takeaway
Researchers studying 'tree-like' paths—routes that retrace their own steps—found that this property isn't transitive. Using a complex fractal construction, they proved you can have two paths that are 'essentially the same,' but that relationship fails to carry over to a third, shattering a basic assumption of how we categorize similarity.
From the abstract
The notions of tree-like loop and Lipschitz tree-like loop were introduced by Hambly and Lyons in their 2010 Annals of Mathematics paper. They showed that the Lipschitz tree-like property determines an equivalence relation on the set of paths of bounded variation in a given metric space and then asked if this notion could be extended to paths without the Lipschitz requirement. We show that after eliminating the Lipschitz requirement, the resulting relation is no longer transitive and thus is not