27 Extended Observables
Approximately 50 physical observables derived from topology with zero free parameters. Mean deviation: \(0.24\% \).
27.1 Weak Mixing Angle (14 Expressions)
\(\sin ^2\theta _W := 3/13\).
Primary: \(\sin ^2\theta _W = N_{\mathrm{gen}} / (\mathrm{dim}(G_2) - 1) = 3/13\). Deviation: \(0.20\% \) from experiment.
27.2 PMNS Neutrino Mixing
\(\sin ^2\theta _{12} := 4/13\) (solar mixing angle).
\(\sin ^2\theta _{23} := 6/11 = 6/D_{\mathrm{bulk}}\) (atmospheric mixing angle).
\(\sin ^2\theta _{13} := 11/496 = D_{\mathrm{bulk}}/ (2 \times \mathrm{dim}(E_8))\) (reactor angle).
27.3 Quark Mass Ratios
\(m_s / m_d := 20\).
\(m_b / m_t := 1/42 = 1/(2b_2)\) — the structural invariant 42 appears!
\(m_c / m_s := 246/21 = (\mathrm{dim}(E_8) - p_2) / b_2\).
\(m_u / m_d := 79/168\).
27.4 Boson Mass Ratios
\(m_H / m_W := 81/52\).
\(m_H / m_t := 8/11 = \mathrm{rank}(E_8) / D_{\mathrm{bulk}}\).
\(m_W / m_Z := 37/42\).
27.5 CKM Quark Mixing
\(\sin ^2\theta _{12}^{\mathrm{CKM}} := 56/248 = \mathrm{dim}(\mathrm{fund.}\; E_7) / \mathrm{dim}(E_8)\).
\(A := 83/99 = 83/H^*\).
\(\sin ^2\theta _{23}^{\mathrm{CKM}} := 7/168 = \mathrm{dim}(K_7) / (8 \times b_2)\).
27.6 Cosmological Parameters
\(\Omega _{\mathrm{DM}} / \Omega _b := 43/8 = (1 + 2b_2) / \mathrm{rank}(E_8)\).
\(h := 167/248 = 167/\mathrm{dim}(E_8)\). Deviation: \(0.09\% \) from Planck 2018.
\(\sigma _8 := 17/21 = 17/b_2\).
\(Y_p := 15/61 = 15/\kappa _T^{-1}\).
27.7 Universality
The structural invariant \(2b_2= 42\) appears in both particle physics (\(m_b/m_t = 1/42\)) and cosmology (\(\Omega _{\mathrm{DM}}/\Omega _b = 43/8\)).
\(\mathrm{rank}(E_8)/D_{\mathrm{bulk}}= 8/11\) appears in both \(m_H/m_t\) and \(\sin ^2\theta _{23}^{\mathrm{PMNS}}\).