7 Fibonacci and Lucas Embeddings
A remarkable discovery: Fibonacci and Lucas numbers map exactly to GIFT constants.
7.1 Fibonacci Embedding
\(F_0 = 0, F_1 = 1, F_{n+2} = F_n + F_{n+1}\)
\(F_3 = 2 = p_2\) (Pontryagin class)
\(F_6 = 8 = \mathrm{rank}(E_8)\)
\(F_8 = 21 = b_2\)
\(F_{12} = 144 = (\mathrm{dim}(G_2) - p_2)^2 = 12^2\)
Complete embedding \(F_3\) through \(F_{12}\) in GIFT constants.
7.2 Lucas Embedding
\(L_0 = 2, L_1 = 1, L_{n+2} = L_n + L_{n+1}\)
\(L_4 = 7 = \mathrm{dim}(K_7)\)
\(L_5 = 11 = D_{\mathrm{bulk}}\) (M-theory dimension)
\(b_3= 77 = L_4 \times L_5 = 7 \times 11\)