Global Invertibility of Function in H2(C+)

We introduce a structural framework for Hardy-space expansions, termed Atomic Riesz Expansion, in which analytic functions are decomposed into a geometric map, a spectral modifier, and a symmetry preserver. This layered architecture produces analytic atoms that simultaneously preserve zero configurations, encode functional symmetries, and admit stably invertible synthesis with Riesz-type bounds. Conceptually, the framework shifts the focus from norm-based decompositions to structural representation: local geometric information, spectral stability, and analytic modulation are disentangled and recombined to yield a globally faithful expansion. Through controlled Gram–spectral interactions, local data upgrades to global analytic control, providing a unified perspective on stability, approximation, and operator-theoretic reconstruction. Beyond Hardy spaces, this paradigm offers a general mechanism for reversible, structure-preserving analytic representation, with potential applications wherever invertibility, zero-geometry, and analytic symmetry must coexist.