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April 12, 20260 citationsOpen Access

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

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MEMoss Eva

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

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Cite This Study

Moss Eva (2026) studied this question.

synapsesocial.com/papers/69db38274fe01fead37c64d1https://doi.org/10.5281/zenodo.19490217
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