arXiv2026
Disordered stealthy hyperuniform materials combine liquid-like statistical isotropy with crystal-like homogeneity, suppressed density fluctuations at large length scales, bounded holes, and an isotropic structure factor that vanishes for a finite range of wavevectors. This combination yields unusual physical properties, including optical transparency, effective delocalization, ultrafast spreadability, optimal conductivity, and complete isotropic photonic bandgaps. Gyromorphs, point patterns whose structure factor includes rings of Bragg-like peaks arranged with discrete $G$-fold rotational symmetry, were recently introduced as counterexamples: disordered media that can somehow achieve the same physical properties, in some cases with higher performance, without stealthiness or hyperuniformity. In this paper, we resolve the puzzle of how gyromorphs fit consistently with the stealthy hyperuniform studies. We first show that gyromorphs are actually hyperuniform and, in the large-$G$ limit where they become nearly isotropic, belong to the weakest form of hyperuniformity, known as Class III. Thus, gyromorphs should have comparatively degraded physical properties compared to stealthy hyperuniform media, which belong to the strongest form of hyperuniformity, known as Class I. We verify this expectation using the rigorous spectral Green's matrix method for the calculation of the density of states (DOS) and Purcell factors in large arrays of electric dipoles. We find that gyromorphs display size-dependent pseudogaps richly populated by localized states rather than smooth band gaps like those found for highly stealthy hyperuniform materials or in deterministic structures such as Vogel spiral and triangular lattices. Furthermore, we predict similar disorder-induced degradation relative to stealthy hyperuniformity with regard to transparency, spreadability and diffusion properties.