FB(S³)R: A Compact S³ Spectral Consistency Framework for Ring-like Ultra-Large-Scale Structure at z ≈ 0.8

A. M. Lopez, R. G. Clowes "A Giant Ring on the sky" Recent observations at z ≈ 0. 8 reveal a striking configuration of ultra-large-scale structures within a single field: the Giant Arc (~1 Gpc), the Big Ring (~400 Mpc), and the Giant Ring (≲1 Gpc), the latter detected at >4σ statistical significance. Their nested, quasi-concentric arrangement exceeds the standard ΛCDM homogeneity scale (~370 Mpc) and is not generically reproduced as a non-random field in simulations such as FLAMINGO-10K under two-dimensional power spectrum analysis. These results raise a fundamental question: do such structures represent statistical anomalies, or do they reflect deeper geometric constraints? This work does not claim that the observations prove a specific global topology. Instead, it demonstrates that a compact, simply-connected spatial geometry S³ (Rₑff) provides a natural and mathematically consistent framework in which ring-like ultra-large-scale structures arise as structural consequences rather than statistical outliers. The central mechanism is spectral: on compact manifolds, the Laplacian spectrum is discrete, and its eigenfunctions are globally supported. As a result, correlations are governed by spectral structure rather than by local decay, fundamentally altering large-scale behaviour. Within the linearised free-mode regime, four key results are established: (i) Correlation functions built from globally supported eigenmodes cannot exhibit a purely exponential decay of the form e^ (−r/ξ₀) at all separations. (ii) On S³ (Rₑff), the two-point correlation function is bounded, oscillatory, and does not generically decay to zero at large geodesic distances. (iii) A spherical shell of constant phase on S³ (Rₑff) projects onto ring-like or arc-like structures when intersected with a narrow redshift slice, providing a direct geometric origin for observed uLSS patterns. (iv) A natural hierarchical organisation emerges: a Fibonacci stratification of eigenmodes generates preferred scalesrₙ = πRₑff / Fₙ, with asymptotic ratio rₙ / rₙ₊₁ → φ. A phenomenological identification is proposed in which the Giant Ring, Big Ring, and the characteristic clustering scale correspond to Fibonacci levels n = 3, 4, 5, yielding an effective spectral curvature scaleRₑff ≈ 475 Mpc within the z ≈ 0. 8 shell. Importantly, this scale is not the global curvature of the Universe and does not conflict with cosmological constraints such as Rglobal > 10 Gpc from Planck data. It instead characterises the local spectral structure of the observed field. Taken together, these results suggest a shift in perspective: ultra-large-scale structures need not arise solely from stochastic growth in an effectively infinite space. They may instead emerge as projections of globally organised spectral modes on a compact manifold. In this view, the observed rings are not anomalies, but signatures of underlying geometric order encoded in the topology and spectral properties of space.