Near-Parseval orbit frames for irreducible unitary representations: from mixing and expansion

What the math gives to ML

The paper gives a constructive route to representing many vectors using one seed vector and a family of unitary transformations while keeping the resulting orbit close to a Parseval frame. The transferable asset is the combination of near-isometric frame bounds, sequential selection of weakly correlated orbit elements, and quotienting transformations that differ only by a phase. This can become a compact associative-memory or attention-key generator in which one learned seed replaces a large bank of independently learned slots. A finite-dimensional implementation can test the theorem's structural idea using empirical Gram-spectrum control even when the exact representation-theoretic decay assumptions are unavailable.