Discrete Mode Structure and Observational Signatures of a Three‑Torus Universe

Cosmology on a spatial three‑torus exhibits a discrete Laplacian eigenmode spectrum with a strict infrared cutoff, modified density of states, and characteristic degeneracy patterns. These features imprint distinctive signatures on the CMB temperature and polarization anisotropies, the matter power spectrum, and the statistics of large‑scale structure.This work develops the full perturbation formalism for scalar and tensor modes on , derives the resulting power spectra, and identifies observational discriminants relevant to current and forthcoming surveys. The three‑torus provides a conservative, falsifiable extension of standard CDM, capable of explaining or constraining several large‑scale anomalies through its discrete mode structure.The manuscript also discusses conceptual implications for cosmic variance, quantum cosmology, and the emergence of time in compact universes.