Neural network realization of binary refinement iterates via a two-chart atlas selector

What the math gives to ML

The paper supplies an exact, fixed-width ReLU realization of binary refinement cascades whose apparent digit-selection discontinuities are removed by a two-chart atlas on the circle. The transferable asset is not refinement itself, but a general construction for implementing piecewise-affine state machines with discrete regime changes without multiplying a continuously varying state by a hard selector. This suggests a weight-shared multiscale neural module: repeated refinement cells can represent exponentially many linear regions with depth linear in the number of refinement steps, while retaining exact continuity and a small state dimension. The most credible first use is as a parameter-efficient CPWL or implicit-signal backbone, initialized from a refinement mask and then optionally fine-tuned.