21 TCS Piecewise Metric Structure
On the TCS manifold \(K_7= M_1 \cup M_2\), the analytical metric is piecewise-constant:
where \(J\) is an involutory orthogonal matrix from the automorphism group of the Fano plane \(\mathrm{PG}(2,2)\).
21.1 Building Block Asymmetry
\(M_1\) is the quintic in \(\mathbb {CP}^4\) with \(b_2 = 11\), \(b_3 = 40\). \(M_2\) is the complete intersection \(\mathrm{CI}(2,2,2)\) in \(\mathbb {CP}^6\) with \(b_2 = 10\), \(b_3 = 37\).
\(b_3(M_1) - b_3(M_2) = 40 - 37 = 3 = N_{\mathrm{gen}}\).
\(b_2(M_1) - b_2(M_2) = 11 - 10 = 1\).
21.2 Effective Degrees of Freedom per Block
\(H^*(M_i) := 1 + b_2(M_i) + b_3(M_i)\) for each building block.
\(H^*(M_1) = 1 + 11 + 40 = 52 = \mathrm{dim}(F_4)\), where \(F_4\) is the automorphism group of the exceptional Jordan algebra \(J_3(\mathbb {O})\).
\(H^*(M_2) = 1 + 10 + 37 = 48 = h(G_2) \times \mathrm{rank}(E_8) = 6 \times 8\), where \(h(G_2) = 6\) is the Coxeter number of \(G_2\).
\(H^*(M_1) + H^*(M_2) = 52 + 48 = 100 = H^*+ 1\). The extra \(1\) accounts for the double-counted \(b_0\) from each connected block.
21.3 Matrix Space Decomposition
The \(7 \times 7\) matrix space decomposes as \(49 = 28 + 21 = 2 \cdot \mathrm{dim}(G_2) + b_2\). The symmetric part (28 = metric DOF) equals \(2 \times \mathrm{dim}(G_2)\); the antisymmetric part (21 = torsion 2-forms) equals \(b_2\).
21.4 Fano Plane Automorphism Group
The Fano automorphism group has order \(168 = 8 \times 21\).
21.5 Fano Incidence Arithmetic
From the Fano plane: 7 points \(\times \) 3 lines per point \(= 21 = b_2\).
21.6 Kovalev Involution Eigenspace
The Kovalev twist \(J\) splits \(\mathbb {R}^7\) into \(V_+ \oplus V_-\) with \(\mathrm{dim}(V_+) = N_{\mathrm{gen}} + 1 = 4\) and \(\mathrm{dim}(V_-) = N_{\mathrm{gen}} = 3\).
The number of Kovalev-type involutions (with 4 fixed directions) equals the dimension of \(\Lambda ^3(\mathbb {R}^7)\): both equal 35.
\(C(7,3) + 14 = 35 + 14 = 49 = 7^2\).
21.7 Master Certificate
All 10 conjuncts of the TCS piecewise metric master certificate are proven.