28 Mass Gap Ratio and Universal Law
The bare spectral gap \(\lambda _1 = \mathrm{dim}(G_2) / H^*= 14/99\) and the universal spectral law \(\lambda _1 \times H^*= \mathrm{dim}(G_2)\).
28.1 Spectral Theory Foundations
Abstract compact Riemannian manifold with dimension and volume.
The Laplace–Beltrami operator \(\Delta \) on a compact Riemannian manifold.
\(\Delta _{\mathrm{gap}} := \inf \sigma (\Delta ) \setminus \{ 0\} \) (smallest positive eigenvalue of \(\Delta \)).
\(\Delta _{\mathrm{gap}} {\gt} 0\) for compact manifolds.
28.2 Bare Mass Gap Ratio: 14/99
\(\lambda _{\mathrm{num}} := \mathrm{dim}(G_2) = 14\).
\(\lambda _{\mathrm{den}} := H^*= 99\).
\(\lambda _1 := \mathrm{dim}(G_2) / H^*= 14/99\).
\(\gcd (14, 99) = 1\): the fraction \(14/99\) is irreducible.
The ratio \(14/99\) arises as \(\mathrm{dim}(\mathrm{Hol}) / (b_2+ b_3+ 1)\) — holonomy divided by cohomological complexity.
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28.3 Universal Spectral Law
\(\lambda _1 \times H^*= \mathrm{dim}(G_2) = 14\): the universal spectral identity.
\(\gcd (14, 99) = 1\).
\(\Delta _{\mathrm{phys}} = (14/99) \times 200\; \mathrm{MeV} \approx 28.3\; \mathrm{MeV}\).
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