6 Physical Relations
6.1 Weinberg Angle
The weak mixing angle \(\theta _W\) is one of the most precisely measured parameters in the Standard Model. GIFT derives an exact prediction.
The numerator is \(b_2= 21\).
The denominator is \(b_3+ \mathrm{dim}(G_2) = 77 + 14 = 91\).
Cross-multiplication: \(21 \times 13 = 273 = 3 \times 91\).
\(\frac{3}{13} = 0.230769\ldots \) vs experimental \(0.23122 \pm 0.00004\) (deviation: 0.19%).
6.2 Koide Formula
The Koide formula relates the masses of charged leptons. It remained unexplained for 43 years until GIFT derived it from topology.
The numerator is \(\mathrm{dim}(G_2) = 14\).
The denominator is \(b_2= 21\).
Cross-multiplication: \(14 \times 3 = 42 = 21 \times 2\).
The Koide formula \(Q = 2/3\) was discovered empirically in 1981 and remained unexplained for 43 years. GIFT derives it in two lines from topology.
6.3 Fine Structure Constant
\(\alpha ^{-1}_{\mathrm{alg}} = \frac{\mathrm{dim}(E_8) + \mathrm{rank}(E_8)}{2} = \frac{248 + 8}{2} = 128\)
\(\alpha ^{-1}_{\mathrm{bulk}} = \frac{H^*}{D_{\mathrm{bulk}}} = \frac{99}{11} = 9\)
\(\alpha ^{-1}_{\mathrm{base}} = 128 + 9 = 137\)
With torsion correction:
(experimental: \(137.035999...\), deviation: 0.002%)
6.4 Strong Coupling
\(\mathrm{dim}(G_2) - p_2 = 14 - 2 = 12\)
\(\alpha _s = \frac{\sqrt{2}}{12}\), where \(12 = \mathrm{dim}(G_2) - p_2\)
6.5 Lepton Mass Ratios
\(m_\mu /m_e\) base: \(\mathrm{dim}(J_3(\mathbb {O})) = 27\) (exceptional Jordan algebra)
\(m_\mu /m_e \approx 27^\phi \) where \(\phi = (1 + \sqrt{5})/2\) is the golden ratio.
\(3477 = 3 \times 19 \times 61 = N_{\mathrm{gen}} \times p_8 \times \kappa _T^{-1}\)
6.6 Higgs Quartic
\(\lambda _H^2\) numerator: \(\mathrm{dim}(G_2) + 3 = 17\)
6.7 Cosmological Parameters
The spectral index \(n_s = \zeta (11)/\zeta (5)\) uses:
\(11 = D_{\mathrm{bulk}}\) (M-theory dimension)
\(5 = \) Weyl factor
\(\Omega _{DE} = \ln (2) \times \frac{98}{99} = \ln (2) \times \frac{H^*- 1}{H^*}\)