Function theory of the hexablock and applications to the tetrablock and Euclidean biball

What the math gives to ML

The paper develops a Schur-Agler realization for the hexablock: bounded holomorphic functions are represented through a block-diagonal contractive multiplier and an auxiliary Hilbert-space feature map. The key transferable asset is the positive-kernel identity together with the resulting lurking-isometry construction, which gives a principled way to make a multivariate neural module contractive on a prescribed structured domain. A practical adaptation is a gated recurrent or feed-forward layer whose input-dependent operator is block diagonal and whose output is generated by a contractive colligation, rather than by unconstrained dense weights. This is most promising for stable sequence models and bounded-control policies, where preventing amplification matters more than maximizing raw expressivity.