24 Poincaré Duality and the GIFT Spectrum
Poincaré duality doubles the GIFT spectrum: the full Betti sequence \((1, 0, 21, 77, 77, 21, 0, 1)\) of \(K_7\) sums to \(198 = 2 \times H^*\). The central structural identity \(H^*= 1 + 2 \times \mathrm{dim}(K_7)^2\) shows the effective cohomological dimension is determined by the manifold dimension alone.
24.1 Full Betti Sequence and Total Betti Number
The total Betti number \(\beta _{\mathrm{total}} = \sum _{k=0}^{7} b_k = 198\).
\(\beta _{\mathrm{total}} = 2 \times H^*= 198\).
\(198 = 2 \times N_{\mathrm{gen}}^2 \times D_{\mathrm{bulk}} = 2 \times 9 \times 11\).
24.2 Structural Identity: \(H^*= 1 + 2 \mathrm{dim}(K_7)^2\)
\(b_2+ b_3= 2 \times \mathrm{dim}(K_7)^2 = 98\).
\(H^*= 1 + 2 \times \mathrm{dim}(K_7)^2 = 99\).
24.3 Holonomy Embedding Chain \(G_2{\lt} \mathrm{SO}(7) {\lt} \mathrm{GL}(7)\)
\(\mathrm{dim}(\mathrm{SO}(7)) = \binom {7}{2} = 21\).
The second Betti number equals the dimension of \(\mathrm{SO}(7)\).
\(\mathrm{dim}(\mathrm{GL}(7)) - \mathrm{dim}(G_2) = \binom {7}{3} = 35\): the number of independent 3-forms on \(K_7\).
\(\mathrm{dim}(G_2) + \mathrm{dim}(K_7) + 28 = \mathrm{dim}(\mathrm{GL}(7))\), i.e., \(14 + 7 + 28 = 49\).
24.4 \(G_2\) Torsion Decomposition
\(1 + \mathrm{dim}(K_7) + \mathrm{dim}(G_2) + \mathrm{dim}(J_3(\mathbb {O})) = \mathrm{dim}(K_7)^2 = 49\).
\(1 + \mathrm{dim}(K_7) + \mathrm{dim}(J_3(\mathbb {O})) = \binom {7}{3} = 35\).
24.5 SU(3) Branching Rule
\(\mathrm{dim}(G_2) = \mathrm{dim}(\mathrm{SU}(3)) + 2 N_{\mathrm{gen}} = 8 + 6 = 14\).
24.6 Betti–Torsion Bridge
\(b_2+ b_3= 2 \times (1 + \mathrm{dim}(K_7) + \mathrm{dim}(G_2) + \mathrm{dim}(J_3(\mathbb {O})))\).
24.7 Master Certificate
All 12 conjuncts of the Poincaré duality certificate are proven.