29 Selection Principle and Yang–Mills
Selection of the canonical neck length \(L^* = \sqrt{\kappa H^*}\) with \(\kappa = \pi ^2/14\), Cheeger inequality, and connection to Clay Millennium Prize.
29.1 Selection Constant
\(\kappa := \pi ^2 / \mathrm{dim}(G_2) = \pi ^2/14\).
\(9/14 {\lt} \kappa {\lt} 10/14\), i.e., \(0.643 {\lt} \kappa {\lt} 0.714\).
\(b_2(M_1) + b_2(M_2) = b_2\) and \(b_3(M_1) + b_3(M_2) = b_3\) (Mayer–Vietoris).
\(L^* := \sqrt{\kappa H^*} = \sqrt{99\pi ^2/14}\).
\(7 {\lt} L^* {\lt} 9\).
\(\lambda _1 = \pi ^2 / (L^*)^2 = \mathrm{dim}(G_2) / H^*= 14/99\).
The spectral gap equals holonomy dimension divided by cohomological complexity.
Complete selection principle certificate.
29.2 Cheeger–Buser Inequality
\(h(M) := \inf \frac{\mathrm{Area}(\Sigma )}{\min (\mathrm{Vol}(A), \mathrm{Vol}(B))}\) over all separating hypersurfaces \(\Sigma \).
\(h(K_7) \geq 14/99\).
\(\lambda _1 \geq h(K_7)^2/4 = (14/99)^2/4 = 49/9801\).
Cheeger–Buser bounds satisfied.
29.3 Yang–Mills Mass Gap
GIFT predicts a mass gap \(\Delta = (14/99) \times \Lambda _{\mathrm{QCD}}\).
\(\Delta \approx 28.28\; \mathrm{MeV}\) (with \(\Lambda _{\mathrm{QCD}} = 200\; \mathrm{MeV}\)).
GIFT provides a candidate solution for the Clay Millennium Prize Yang–Mills existence and mass gap problem.
The mass gap has purely topological origin: \(\mathrm{dim}(G_2) / H^*\).
Complete Yang–Mills connection certificate.
29.4 Connes Bridge
The set \(\{ 2, 3, 5, 7, 11, 13\} \) from Connes’ Weil positivity approach.
\(|\{ 2,3,5,7,11,13\} | = 6 = \) Coxeter number of \(G_2\).
The largest Connes prime is \(13 = \mathrm{dim}(G_2) - 1\).
\(2 \times 3 \times 5 = 30 = \) Coxeter number of \(E_8\).
Complete Connes bridge certificate.