19 Dimensional Hierarchy
The electroweak–Planck hierarchy \(M_{\mathrm{EW}}/M_{\mathrm{Pl}} \approx 10^{-17}\) is derived from \(K_7\) topology via:
19.1 Cohomological Suppression
The cohomological exponent \(H^*/ \mathrm{rank}(E_8) = 99/8\).
\(S_{\mathrm{cohom}} := e^{-H^*/\mathrm{rank}(E_8)} = e^{-99/8}\).
\(10^{-6} {\lt} e^{-99/8} {\lt} 10^{-5}\).
\(S_{\mathrm{Jordan}} := \varphi ^{-54} = (\varphi ^{-2})^{27}\), where \(27 = \mathrm{dim}(J_3(\mathbb {O}))\) and \(\varphi ^{-2} \approx 0.382\).
\(\varphi ^{-54} {\lt} 10^{-10}\).
\(R_{\mathrm{hier}} := S_{\mathrm{cohom}} \times S_{\mathrm{Jordan}} = e^{-99/8} \times \varphi ^{-54}\).
\(R_{\mathrm{hier}} {\lt} 10^{-15}\). The hierarchy problem is solved by topology.
\(-39 {\lt} \ln (R_{\mathrm{hier}}) {\lt} -38\), matching \(M_{\mathrm{EW}}/M_{\mathrm{Pl}} \approx 10^{-17}\).
19.2 Vacuum Structure
\(N_{\mathrm{vacua}} := b_2= 21\) associative 3-cycles on \(K_7\).
\(N_{\mathrm{vacua}} = b_2= 21\).
\(\mathrm{dim}(\mathcal{M}) = b_3= 77\).
\(b_3= 40 + 37\) (quintic and CI blocks).
Complete vacuum structure derived from topology.
19.3 \(E_8\to E_6\) Symmetry Cascade
\(\mathrm{dim}(\mathrm{fund.}\; E_6) = 27 = \mathrm{dim}(J_3(\mathbb {O}))\).
\(248 = 78 + 8 + 2 \times 27 \times 3 = \mathrm{dim}(E_6) + \mathrm{dim}(\mathrm{SU}(3)) + 162\).
19.4 Absolute Lepton Masses
\(m_\tau / m_e = (b_3- b_2)(\kappa _T^{-1} + 1) + W = 56 \times 62 + 5 = 3477\).
\(3477 = 3472 + 5\) where \(3472 = 56 \times 62\) (Betti \(\times \) kappa) and \(5 = W\) (Weyl factor).
\(3477 = 3 \times 19 \times 61\) (all prime factors are GIFT-expressible).
\(206 {\lt} 27^\varphi {\lt} 208\) (Jordan algebra base with golden exponent).
\(y_\tau = 1/98 = 1/(H^*- 1)\) (tau Yukawa from cohomology).
All absolute mass formulas verified.