A Polyhedral Formula for $n\times 2\times 2$ Kronecker Coefficients via Cluster Algebras

Let $$\mathbb{k}[{\mathbb{k}}^3\otimes {\mathbb{k}}^2\otimes {\mathbb{k}}^2{]}^{U_3\times U_2\times U_2}\mathrm{.}$$ We construct an ordinary cluster family with Markov principal part. At $\zeta =-1$, the intersection of its initial Laurent ring with the three adjacent Laurent rings equals $\mathcal{U}$, and $$\mathcal{U}={\mathcal{M}}_u[u_{\mathrm{\Delta }}],$$ where ${\mathcal{M}}_u$ is the middle algebra. Its theta functions are indexed by a cone with a sixteen-element Hilbert basis. Multiplication by $u_{\mathrm{\Delta }}$ pairs the Hilbert generators of mutable degrees $1$ and $-1$ and reduces each triple-weight space to the slice $\mathrm{\ell }={\mathrm{\ell }}_0$. Counting the lattice points in this slice gives a finite sum with nonnegative summands. Determinant reduction extends the formula to all $n\times 2\times 2$ Kronecker coefficients.