22 Selection Principle and Yang–Mills
Selection of the canonical neck length \(L^* = \sqrt{\kappa H^*}\) with \(\kappa = \pi ^2/14\), Cheeger inequality, and Kaluza–Klein spectral bridge.
22.1 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).
\(L^* := \sqrt{\kappa H^*} = \sqrt{99\pi ^2/14}\).
\(7 {\lt} L^* {\lt} 9\).
\(\lambda _1 = \pi ^2 / (L^*)^2\); conditional on the selection principle, this equals \(\mathrm{dim}(G_2) / H^*= 14/99\) (bare topological ratio). The analytical formula gives \(\lambda _1 = 6\pi ^2/475 \approx 0.12467\).
The spectral gap equals holonomy dimension divided by cohomological complexity.
Complete selection principle certificate.
22.2 Cheeger–Buser Inequality
\(h(M) := \inf \frac{\mathrm{Area}(\Sigma )}{\min (\mathrm{Vol}(A), \mathrm{Vol}(B))}\) over all separating hypersurfaces \(\Sigma \).
Cheeger bound: if the bare ratio \(14/99\) is used as lower bound, then \(\lambda _1 \geq h^2/4 = (14/99)^2/4 = 49/9801\).
The bare ratio satisfies \(14/99 {\gt} (14/99)^2/4 = 49/9801\) (Cheeger-consistent).
Cheeger–Buser bounds satisfied.
22.3 KK Spectral Bridge and Yang–Mills
Via the KK spectral bridge (Axiom KK_EFT), the bare algebraic ratio gives \(\Delta _{\mathrm{KK}} = (14/99) \times \Lambda _{\mathrm{QCD}}\).
\(\Delta \approx 28.28\; \mathrm{MeV}\) (with \(\Lambda _{\mathrm{QCD}} = 200\; \mathrm{MeV}\)).
The KK spectral bridge provides a candidate connection to the Clay Millennium Prize Yang–Mills mass gap problem, via \(\lambda _1 {\gt} 0\) on \(K_7\) (one axiom: KK_EFT).
The algebraic ratio \(\mathrm{dim}(G_2) / H^*\) has purely topological origin.
Complete KK spectral bridge certificate.