The XOR problem has long been cited as proof that neural networks require nonlinear activation functions to learn nonlinearly separable decision boundaries. We argue that this interpretation is reversed. XOR is not evidence that nonlinearity is required; it is evidence that learning is fundamentally topological. We demonstrate that XOR can be solved by a purely piecewise-linear model, provided that the input space is partitioned into local patches and those patches are reconnected by a learned transition rule.No nonlinear activation is applied to the value branches. The softmax gate acts solely as a transition rule defining the topological partition of the input domain. In this view, learning is the act of cutting and gluing regions of input space—a process that induces a nontrivial holonomy similar to a monopole-like singularity. Because topological quantities are invariant under continuous deformation, all solutions of XOR—whether through ReLU, sigmoid, tanh, high-dimensional lifting, or discrete gating—are merely different coordinate representations of the same topological object.Continuity, differentiability, and analytic activation functions are not the essence of learning; the topology of how input regions are divided and re-attached is.