Extensions of One-Sided Box Geometry and Pyramid Invariants to gd-Sets and qm-Spaces

What the math gives to ML

The paper develops an asymmetric, measure-aware notion of functional domination: a target family of observables is considered representable by a source family when, under a coupling, almost all mass lies on pairs where every target observable is uniformly approximated by some source observable. The transferable asset is the combination of one-sided approximation, trimmed high-probability support, and an explicit worst-case error, rather than ordinary symmetric feature matching. This suggests a robust asymmetric distillation or domain-adaptation loss in which teacher observables must be recoverable from student features on a learned high-mass coupling while a small fraction of mismatched examples is ignored. The construction is useful when the target contains information the source should preserve, but exact pointwise correspondence is unavailable or contaminated by outliers.