21 Mass Gap Ratio and Universal Law
The algebraic ratio \(\mathrm{dim}(G_2) / H^*= 14/99\) is a topological invariant (holonomy dimension divided by cohomological complexity). The analytical mass gap is \(\lambda _1 = 6\pi ^2/475 \approx 0.12467\), giving \(\lambda _1 \cdot H^*\approx 12.34\) (not \(14\) or \(13\)).
21.1 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.
21.2 Bare Algebraic Ratio: 14/99
\(\lambda _{\mathrm{num}} := \mathrm{dim}(G_2) = 14\).
\(\lambda _{\mathrm{den}} := H^*= 99\).
Bare algebraic ratio: \(\mathrm{dim}(G_2) / H^*= 14/99\). This is a topological invariant, not the spectral eigenvalue (the analytical mass gap is \(\lambda _1 = 6\pi ^2/475 \approx 0.12467\)).
\(\gcd (14, 99) = 1\): 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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21.3 Universal Spectral Law
\(14 = \mathrm{dim}(G_2)\) and \(99 = H^*\): the bare ratio components are topological. The analytical spectral product \(\lambda _1 \cdot H^*\approx 12.34\) (from \(\lambda _1 = 6\pi ^2/475\)).
\(\gcd (14, 99) = 1\).
Using the bare ratio: \(\Delta = (14/99) \times \Lambda _{\mathrm{QCD}}\) with \(\Lambda _{\mathrm{QCD}} = 200\; \mathrm{MeV}\) gives \(\approx 28.3\; \mathrm{MeV}\). The analytical mass gap \(\lambda _1 = 6\pi ^2/475\) gives \(\approx 24.9\; \mathrm{MeV}\).
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