I tiakina i:
| Kaituhi matua: | |
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| Hōputu: | Recurso digital |
| Reo: | |
| I whakaputaina: |
Zenodo
2026
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| Ngā marau: | |
| Urunga tuihono: | https://doi.org/10.5281/zenodo.19258780 |
| Ngā Tūtohu: |
Tāpirihia he Tūtohu
Kāore He Tūtohu, Me noho koe te mea tuatahi ki te tūtohu i tēnei pūkete!
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| _version_ | 1866901626787201024 |
|---|---|
| author | Keeble, Clifford |
| author_facet | Keeble, Clifford |
| 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> |
| format | Recurso digital |
| id | zenodo_https___doi_org_10_5281_zenodo_19258780 |
| institution | Zenodo |
| language | |
| publishDate | 2026 |
| publisher | Zenodo |
| record_format | zenodo |
| spellingShingle | The Icosahedral Spectrum: Eigenvalues of S³/2I as Combinations of Vertices, Faces, and Edges Keeble, Clifford Poincaré homology sphere, binary icosahedral group, Laplacian spectrum, Molien series, Klein invariants, numerical semigroup, spectral gap, icosahedron, S³/2I, Bootstrap Universe Programme <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> |
| title | The Icosahedral Spectrum: Eigenvalues of S³/2I as Combinations of Vertices, Faces, and Edges |
| topic | Poincaré homology sphere, binary icosahedral group, Laplacian spectrum, Molien series, Klein invariants, numerical semigroup, spectral gap, icosahedron, S³/2I, Bootstrap Universe Programme |
| url | https://doi.org/10.5281/zenodo.19258780 |