Newton Support Functions and Metric Completion of Singular Conformal Metrics at Corners

What the math gives to ML

The paper turns a finite collection of competing monomial singularities into a max-plus support function, \(\mathcal{H}(p,q)=\max_j(pa_j+qb_j)\), which identifies the dominant scaling direction without enumerating cases. This suggests a numerically stable neural layer for multiscale or singular coordinate dependence: compute monomial mixtures in log-space using the Newton envelope and use the associated soft argmax as a gate over experts. The accessibility inequality \(\mathcal{H}(p,q)<2\min\{p,q\}\) also provides a concrete, though heuristic outside the metric setting, regularizer preventing learned singular exponents from producing excessively unstable boundary behavior. The most promising initial targets are neural fields, coordinate MLPs, and models with explicit power-law experts rather than generic transformers.