The Broken Equivalence

Two paths in a metric space are “tree-like equivalent” if their concatenation traces out a tree — the combined path goes out and comes back without enclosing area. Hambly and Lyons showed in 2010 that for Lipschitz paths of bounded variation, this defines an equivalence relation: reflexive, symmetric, and transitive. Path A is tree-like to B, B is tree-like to C, therefore A is tree-like to C.

Brazas, Conner, Fabel, and Kent (arXiv:2603.16774) remove the Lipschitz constraint and find that transitivity breaks. There exist three paths where A is tree-like to B and B is tree-like to C, but A is not tree-like to C. The counterexample is constructed from explicit fractal geometry in the plane.

The structural surprise is where the transitivity lives. It does not live in the tree-like property itself — that property is purely geometric, defined by whether a concatenation encloses area. Transitivity lives in the Lipschitz constraint, which controls how wildly the path can oscillate. Remove the regularity condition and the geometric property remains well-defined, but the algebraic structure (equivalence relation) collapses.

The Lipschitz condition is not a technical convenience. It is a structural load-bearing element: the thing that makes the geometric relation algebraically tractable. The “tree-like” property alone is insufficient for transitivity because without Lipschitz regularity, fractal paths can encode enough complexity that the intermediate path B can be tree-like with both A and C while A and C enclose area together.

This is a general warning about removing regularity conditions. A theorem proved under regularity assumptions may not degrade gracefully — it may break categorically. The conclusion (equivalence relation) depended on the hypothesis (Lipschitz) in a way that was invisible until someone constructed the explicit counterexample. The constraint was not restricting the theorem; it was enabling it.