Universality of Layered Programmable Decompositions by Fourier Absorption

arXiv:2510.19397v3 Announce Type: replace Abstract: Decomposing arbitrary unitary transformations into physically realizable operations is fundamental to quantum information, quantum control, and linear optics. Layered programmable architectures implement such transformations by alternating tunable phase masks with fixed mode mixers. However, establishing their universality at fixed depth remains challenging: we show that neither a sufficient parameter count nor successful synthesis of sampled targets guarantees global coverage. We prove that an architecture is universal exactly when every reachable transformation can absorb a diagnostic input Fourier multiport by reprogramming its phase masks. The associated Fourier absorption residual (FAR) requires no external target ensemble, vanishes exactly for universal architectures, and bounds worst-target synthesis error. For four-mode Fourier architectures, a permutation obstruction rules out universality with five masks, even under small unitary mixer perturbations, whereas a constructive factorization establishes universality with seven. Beyond exact universality, comparison with a universal reference of the same depth yields certified uniform approximation bounds in terms of unitary mixer errors. Finally, numerical simulations using a geodesic phase optimizer illustrate the distinction between Fourier absorption and direct target synthesis.