From Dodecahedron to E₈: A Dictionary between Cosmic Topology and Exceptional Geometry

The Poincaré dodecahedral space S³/I*, proposed as a candidate topology for the spatial universe, has the binary icosahedral group 2I as its fundamental group. Through five independent mathematical constructions - the icosian ring (Conway & Smith 2003), the resolution of the Kleinian singularity C²/2I (Klein 1884), the McKay correspondence (McKay 1980), the Clifford algebra of R³ (Dechant 2016), and the ADE classification of spherical space forms (McInnes 2004) - this same group generates the E₈ root lattice, the largest exceptional structure in Lie theory. This note constructs an explicit dictionary between the cosmological and algebraic sides of this correspondence. Of the twelve entries, ten are established by existing theorems in invariant theory, representation theory, and algebraic geometry; one is a proven structural gap (the absence of an E₇ intermediate, reflecting 2O ⊄ 2I); and one remains open, concerning the correspondence between eigenmode multiplicities on PDS and branching coefficients of E₈ representations. No new conjectures are advanced; the dictionary collects and organises results that are individually known but have not previously been assembled into a unified reference. Ver. 1.0.5 --- Ξυα Mσςς [email protected]