On the Irreducibility of the Cuboid Polynomial $P_{a,u}(t)$

In this paper we consider the even monic degree-8 cuboid polynomial $P_{a,u}(t)$ with coprime integers $a\neq u>0$. We prove irreducibility over $\mathbb{Z}$ by excluding all degree-8 splittings. First, any putative $4{+}4$ factorization is shown to force a specific Diophantine constraint that has no integer solutions, via a short $2$- and $3$-adic analysis. Second, we exclude every $2{+}6$ factorization using an exact divisor criterion together with a discriminant obstruction. Finally, after ruling out $2{+}6$, the patterns $2{+}2{+}4$, $2{+}2{+}2{+}2$, and $3{+}3{+}2$ regroup trivially to $2{+}6$ and are therefore impossible. Consequently, $P_{a,u}(t)$ admits no nontrivial factorization in $\mathbb{Z}[t]$.