In 2010, Hambly and Lyons proved that paths of bounded variation can be classified by a tree-like property: two paths are equivalent if their concatenation traces out something that retracts onto a tree. Under a Lipschitz constraint, this relation is reflexive, symmetric, and transitive --- a proper equivalence relation, cleanly partitioning the space of paths into distinct classes. The natural question was whether the Lipschitz requirement could be dropped, extending the classification to rougher paths. The answer, demonstrated through an explicit fractal counterexample, is no. Without the Lipschitz condition, the relation loses transitivity.
The counterexample lives in the plane. Path A is tree-like equivalent to path B, and path B is tree-like equivalent to path C, but the concatenation of A and C produces a loop that cannot be collapsed onto any tree. The failure is not approximate or marginal --- it is structural. The fractal construction generates paths whose local regularity is insufficient to guarantee that the global cancellation required for tree-like equivalence composes properly across intermediaries.
This is a precise instance of a broader phenomenon: regularity conditions are not merely technical accessories to a theorem but load-bearing walls. The Lipschitz bound does not just simplify the proof --- it creates the algebraic structure. Remove it and you do not get a weaker version of the same result. You get a different mathematical object entirely, one where the basic act of classification through equivalence is no longer available.
The lesson reaches beyond topology. In any domain where two things are each “equivalent” to a third, the transitivity of that equivalence is never free. It is always purchased by some constraint on the objects being compared. Drop the constraint and the relation may still be meaningful --- but it is no longer an equivalence, and the partition it promised dissolves.