Is the universe infinite? Orthodox cosmology answers ambiguously: the observable universe is finite (bounded by the particle horizon), but the “full” universe may be infinite, depending on the spatial curvature. The ΛCDM model with Ω_k = 0 (spatially flat) admits both finite and infinite topologies — the theory is silent on global structure.

Category τ answers unambiguously: the universe is finite, structured, and closed. This chapter proves this claim through a chain of four theorems, each building on the last. The Finite Motif Theorem (§) shows that only finitely many distinct topological motifs can exist in τ³. The Saturation Radius Theorem (§) shows that beyond a characteristic scale R_sat, all structure is repeated. The Absorbing Pattern Theorem (§) shows that the large-scale pattern converges: long-range structure is eventually determined by the local pattern. The Global Finiteness Corollary (§) assembles these results into the definitive statement: the universe admits no infinite hierarchy of nested structures, no infinite plain of new motifs, and no fractal abyss of self-similar complexity. Everything that can exist, already exists within a finite portion of τ³.