Pairs of commuting isometries via new core operator

What the math gives to ML

The paper introduces a nonstandard operator-valued certificate for a commuting pair of isometries: a specific inclusion-exclusion combination of range projections and second iterates, required to be positive semidefinite. This is more informative than separately enforcing orthogonality or commutation because it measures how the ranges of two directional operators overlap across one and two steps. A plausible neural transfer is a two-axis recurrent or state-space module whose transition operators are approximately isometric and commuting, with the paper's core operator used as a stability and geometry regularizer. The available extraction does not expose the full Hardy-space model, so the implementable proposal below uses only the explicit core-operator formula and tests whether its positivity improves long-horizon sequence behavior.