Linearized Yang–Mills Spectral Gap on the Poincaré Homology Sphere

We compute the spectral gap of the gauge-covariant Hodge Laplacian for linearized SU(2) Yang–Mills theory on the Poincaré homology sphere S³/2I. The flat-connection moduli space consists of exactly three isolated vacua. Explicit spectral computation via the McKay correspondence for the extended E₈ diagram yields the coexact 1-form spectral gap at each vacuum: 4/R² for the trivial and standard vacua, and 36/R² for the Galois conjugate vacuum. The icosahedral symmetry filters the first four coexact levels from the Galois sector, producing a ninefold enhancement over the baseline gap.