A polyhedral formula for $n\times2\times2$ Kronecker coefficients via cluster algebras

We study the triple-invariant algebra $$\Bbbk[\Bbbk^3\otimes\Bbbk^2\otimes\Bbbk^2]^{U_3\times U_2\times U_2}.$$ A quotient slice and the induced logarithmic top form determine a signed Markov chart, realized as the fiber $\zeta=-1$ of an ordinary cluster family. We prove $$\mathscr U_{\mathrm{gen}}=\mathcal M_u[u_\Delta],$$ where $\mathcal M_u$ is a specialized middle cluster algebra and $u_\Delta$ is the discriminant of weight $(220;22;22)$. Its theta cone has a sixteen-element Hilbert basis. Pairing its positive- and negative-degree generators reduces each triple-weight space to a single discriminant level determined by the weight. Counting the resulting two-dimensional slice gives an explicit nonnegative finite-sum formula. Determinant reduction extends the formula to all $n\times2\times2$ Kronecker coefficients.