FB(S³)R: Dark-Sector Residuals in Compact S³ FLRW Cosmology

This work develops a geometrically minimal framework for analysing dark-sector residuals within a compact positively curved FLRW cosmology on the three-sphere S³. Instead of introducing phenomenological exotic fields, the paper investigates whether part of the observed large-scale cosmological residual structure may emerge from the spectral properties of compact spatial geometry itself. The central idea is structurally simple: on a compact manifold the Laplacian spectrum becomes discrete. As a consequence, infrared structure appears geometrically through admissible global modes rather than through arbitrary cutoffs. Cosmological dynamics are therefore studied through compact spectral decompositions governed by eigenmodes of the S³ Laplacian: Δψₙ = −λₙψₙ where the eigenvalue hierarchy λₙ determines the organisation of long-wavelength cosmological structure. Within compact FLRW geometry, residual large-scale contributions arise from the finite mode structure associated with positive curvature and global topological closure. A key result of the work is that compact topology modifies infrared behaviour without violating local relativistic causality or standard Einstein dynamics. Locally, spacetime remains compatible with conventional FLRW evolution, while globally the compact geometry constrains admissible spectral organisation. In this perspective, dark-sector residuals are interpreted not as additional unconstrained matter sectors, but as consequences of global geometric mode structure on S³. The framework further explores how compact geometry influences spectral degeneracies, finite-volume mode stability, and large-scale residual organisation. Rather than postulating new particles, the work studies the hierarchy:geometry → spectrum → cosmological residuals as a mathematically controlled mechanism linking topology and cosmological observables. The paper is presented as a compact-topology consistency analysis and as part of a broader programme investigating whether global geometric structure may contribute nontrivially to dark-sector phenomenology in cosmology.