7 Joyce Existence Theorem
Joyce’s perturbation theorem proves \(K_7\) admits torsion-free \(G_2\) structure.
7.1 PINN Verification
\(\varepsilon _0 = 0.0288\) (scaled: 288)
\(\| T(\varphi _0)\| = 0.00141\) (scaled: 141)
\(\| T(\varphi _0)\| {\lt} \varepsilon _0\) (20\(\times \) safety margin)
7.2 Existence
\(\exists \varphi : K_7\to \Omega ^3\), torsion-free \(G_2\) structure.
All conditions verified: topological (\(b_2= 21\), \(b_3= 77\)), analytic (contraction mapping), existence.