Finite Bases for Truncated Cyclic Group Flat Semirings with Two Independent Parameters

For positive integers $m,n$, let $$A_{m,n}=(\{1,\dots ,m\}\times {\mathbb{Z}}_n)\cup \{0\}$$ have flat addition and multiplication truncated at degree $m$. We prove that $A_{m,n}$ is finitely based exactly when $m\le 2$ or $(m,n)=(3,1)$. Explicit finite bases are supplied throughout this region. Outside it, high-girth hypergraphs with a constant-sum rigidity property yield finite countermodels to every bounded-variable fragment of the equational theory. The proof places no divisibility or coprimality restriction on the parameters.