What question did this study set out to answer?

The aim is to create a unified reference connecting cosmological topology and exceptional algebraic structures.

April 12, 2026Open Access

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

Key Points

The aim is to create a unified reference connecting cosmological topology and exceptional algebraic structures.
Constructed through five mathematical constructions: icosian ring, resolution of Kleinian singularity, McKay correspondence, Clifford algebra, and ADE classification.
Established a dictionary with twelve entries, synthesizing known results in related mathematical fields.
Identified gaps and open questions within the existing theories.
Ten entries verified through known theorems in invariant and representation theories.
One entry identified a structural gap concerning the lack of an E₇ intermediate.
One entry remained open, addressing eigenmode multiplicities and E₈ branching coefficients.

Abstract

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 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 eva@m0ss.io

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

Key Points

Abstract

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