The symmetric 2-tensors decompose as conformal trace (\(1\)) plus traceless part (\(27 = \mathrm{dim}(J_3(\mathbb {O}))\)): \(28 = 1 + 27\).

The 2-forms decompose as standard (\(7\)) plus adjoint (\(14\)): \(21 = \mathrm{dim}(K_7) + \mathrm{dim}(G_2)\).

The endomorphism algebra uses all four fundamental \(G_2\) representations: \(49 = 1 + 7 + 14 + 27 = \mathrm{dim}(K_7)^2\).

The 3-forms decompose as \(C(7,3) = 35 = 1 + \mathrm{dim}(K_7) + \mathrm{dim}(J_3(\mathbb {O}))\). The singlet is the associative 3-form \(\varphi _0\); the \(27\) is the same representation as in \(\mathrm{Sym}^2_0(V_7)\).

The metric has \(28 - 27 - 1 = 0\) free parameters:

1. \(\mathrm{dim}(\mathrm{SPD}_7) = 28\) total metric DOF

2. \(G_2\) holonomy kills \(\mathrm{dim}(J_3(\mathbb {O})) = 27\) traceless directions

3. \(\det (g) = 65/32\) fixes the remaining conformal modulus

For the isotropic metric \(g = c^2 \cdot I_7\): \(\det (g) = c^{2 \times \mathrm{dim}(K_7)} = c^{\mathrm{dim}(G_2)}\). The holonomy dimension determines the conformal equation.

\(27 = 3^3\): the traceless symmetric space dimension equals the cube of the generation number.

The moduli gap \(77 - 21 = 56 = \mathrm{dim}(\mathrm{fund.}\; E_7) = \mathrm{rank}(E_8) \times \mathrm{dim}(K_7) = 8 \times 7\).

All 9 conjuncts of the conformal rigidity certificate are proven.