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| Format: | Recurso digital |
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Zenodo
2026
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| Online Access: | https://doi.org/10.5281/zenodo.19258780 |
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Table of Contents:
- <p>We compute the spectrum of the Laplacian on the Poincaré homology sphere S³/2I and prove that the surviving eigenvalues have a simple characterisation: the harmonic degree k contributes a non-trivial eigenspace if and only if k lies in the numerical semigroup generated by {12, 20, 30} — the vertex count, face count, and edge count of the regular icosahedron. The proof combines Klein's classification of the invariant ring ℂ[x,y]^{2I} (generated by invariants of degrees 12, 20, 30 with one relation at degree 60) with the Molien series P(t) = (1 − t⁶⁰)/((1 − t¹²)(1 − t²⁰)(1 − t³⁰)), verified independently by direct character computation for all k ≤ 120 with zero mismatches. The filter is finite: exactly 15 even modes are killed (the Frobenius number is 58), after which every even harmonic survives. The result demonstrates that the discrete spectrum of S³/2I is built from the combinatorial skeleton of the icosahedron: the integers that emerge from this manifold are not arbitrary but are structured by V = 12, F = 20, E = 30.</p>