20 G\(_2\) Metric Properties
Properties of the numerical \(G_2\) metric on \(K_7\), computed via physics-informed neural networks (PINNs) and compressed to a single \(7 \times 7\) symmetric positive-definite matrix.
20.1 Non-Flatness of \(K_7\) (Bieberbach Bound)
For compact flat \(n\)-manifolds (Bieberbach, 1911), the third Betti number satisfies \(b_3\leq \binom {n}{3}\), with equality realized by the \(n\)-torus. For \(n = 7\): \(\binom {7}{3} = 35\).
\(\binom {7}{3} = 35\).
\(b_3(K_7) = 77 {\gt} 35 = \binom {7}{3}\), so \(K_7\) does not admit a flat Riemannian metric.
\(b_3- \binom {7}{3} = 77 - 35 = 42 = 2b_2\).
20.2 Spectral Degeneracy Pattern \([1, 10, 9, 30]\)
The first four eigenvalue multiplicities of the Laplacian on \(K_7\) are observed at \(5.8\sigma \) significance.
The first four eigenvalue multiplicities: \([m_1, m_2, m_3, m_4] = [1, 10, 9, 30]\).
\(m_2 = 10 = b_2(\mathrm{CI}(2,2,2))\), the second Betti number of the complete intersection building block.
\(m_4 = 30 = 13 + 12 + 4 + 1\) (K3 + fiber + neck + twist modes).
\(m_1 + m_2 + m_3 + m_4 = 50 = b_3- \mathrm{dim}(J_3(\mathbb {O}))\).
20.3 SPD\(_7\) Metric Parametrization
The space of \(7 \times 7\) symmetric positive-definite matrices has \(\mathrm{dim}(\mathrm{SPD}_7) = 7 \cdot 8/2 = 28\) independent entries.
\(\mathrm{dim}(K_7) \cdot (\mathrm{dim}(K_7) + 1) / 2 = 28\).
\(28 = 2 \times \mathrm{dim}(G_2) = 2 \times 14\): the metric degrees of freedom are twice the holonomy dimension.
\(1{,}070{,}471 / 28 = 38{,}231\): compression from PINN parameters to the analytical metric matrix.
20.4 Triple Derivation of \(\det (g) = 65/32\)
Three algebraically independent paths yield the same determinant.
\(\mathrm{Weyl} \times (\mathrm{rank}(E_8) + \mathrm{Weyl}) = 5 \times 13 = 65\) and \(2^{\mathrm{Weyl}} = 2^5 = 32\).
\(p_2 \times (b_2+ \mathrm{dim}(G_2) - N_{\mathrm{gen}}) + 1 = 2 \times 32 + 1 = 65\) and \(b_2+ \mathrm{dim}(G_2) - N_{\mathrm{gen}} = 21 + 14 - 3 = 32\).
\(H^*- b_2- \alpha _{\mathrm{sum}} = 99 - 21 - 13 = 65\).
All three paths yield \(\det (g) = 65/32\).
\(\gcd (65, 32) = 1\): the fraction \(65/32\) is irreducible.
20.5 \(\kappa _T^{-1} = 61\) Structural Decomposition
\(\kappa _T^{-1} = 61 = \mathrm{dim}(F_4) + N_{\mathrm{gen}}^2 = 52 + 9\).
\(61\) is prime.
\((\kappa _T^{-1} + 1) \times (b_3- b_2) + \mathrm{Weyl} = 62 \times 56 + 5 = 3477 = m_\tau /m_e\).
20.6 Master Certificate
All 12 conjuncts of the \(G_2\) metric master certificate are proven: non-flatness, spectral degeneracies, SPD\(_7\) parametrization, \(\det (g) = 65/32\) triple derivation, and \(\kappa _T\) decomposition.