The Icosahedral Spectrum: A Survey of the Scalar Laplace Spectrum of S³/2I and the Numerical Semigroup ⟨6,10,15⟩

This is a survey. The scalar Laplace spectrum of the Poincaré homology sphere S³/2I is classically established (Ikeda 1980, 1995; Bär 1996 for the Dirac companion), and its explicit eigenmodes have been constructed across the cosmic-topology literature — Weeks (2006), Lachièze-Rey & Caillerie (2005), Kramer (2005) via Klein's icosahedral invariant, and via the Hopf map Lachièze-Rey & Weeks (2008) and Dowker (2009). The organising fact — that the surviving harmonic degrees are exactly those expressible as sums of the vertex, face, and edge counts 12, 20, 30 — is likewise established: Weeks (2006, §10) states it outright, names the first gap, performs the halving to k̂ = k/2 = a·6 + b·10 + c·15, records a closed-form complement criterion, and derives the killing of every odd degree from the antipodal map. What this survey does is collect these results in a common icosahedral and numerical-semigroup language, and draw one connection across fields. The surviving set is the submonoid S = ⟨12, 20, 30⟩ = 2·⟨6, 10, 15⟩, and ⟨6, 10, 15⟩ is a numerical semigroup in the strict sense — Frobenius number 29, genus 15, fifteen gaps — so the killed even degrees are exactly twice those gaps — fifteen of them, the largest 58 = 2·29 — while every odd k dies separately, to the central element −I. The spectral gap is λ₁ = 168 at k = 12, the value Paper 113 relies on, and the surviving set lies within the even / integer-spin / real sector, selected there by A₅-invariance, the odd / spinorial sector belonging to the Dirac operator. Alongside this spectral reading the survey sets a second, already-known one: the same numerical semigroup ⟨6, 10, 15⟩ is the weight system of the E8 Brieskorn singularity x² + y³ + z⁵ whose link is S³/2I — a neighbourhood developed in its own literature — the invariant degrees halved, both readings descending from the (2, 3, 5) icosahedral data through Klein's correspondence. The survey's act is the juxtaposition, not the singularity side, which it records rather than claims. All statements are checked against direct character computation for all k ≤ 120.