Computational Evidence Against Quadratic-Cubic Factorization for the Second Cuboid Quintic

Abstract: Let $Q_{p,q}(t)\in\mathbb{Z}[t]$ be Sharipov's even monic degree-$10$ second cuboid polynomial depending on coprime integers $p\neq q&gt;0$. Writing $Q_{p,q}(t)$ as a quintic in $t<sup>{2}$ produces an associated monic quintic polynomial. After the weighted normalization $r=p/q$ and $s=r<sup>{2}$ we obtain a one-parameter family $P_s(x)\in\mathbb{Q}[x]$ such that [ Q_{p,q}(t)=q^{20}\,P_s!\left(\frac{t^{2}}{q^{4}}\right)\qquad\text{with}\qquad s=\left(\frac{p}{q}\right)^{2}. ] Assuming a quadratic divisor $x<sup>{2}+ax+b$ with $a,b\in\mathbb{Q}$, we reduce divisibility of $P_s(x)$ to the vanishing of an explicit remainder [ R(x)=R_{1}(s,a,b)\,x+R_{0}(s,a,b). ] A key structural observation is that $R_1$ and $R_0$ are quadratic in $b$ and that, on the equation $R_1=0$, the second condition becomes linear in $b$. This yields a one-direction elimination to a plane obstruction curve $F(s,a)=0$ with $F\in\mathbb{Z}[s,a]$, without any lifting-back issues: when the linear coefficient is nonzero, the parameter $b$ is forced to be the rational value $b=C/L$. We isolate the degenerate locus $L=C=0$ and show it produces only $s=\pm 1$ (hence only $s=1$ in the cuboid domain $s&gt;0$). Let $\overline{C}\subset\mathbb{P}<sup>{2}$ be the projective closure of $F(s,a)=0$. Using Magma we perform a height-bounded search for rational points on $\overline{C}$. With bound $H=10<sup>{7}$, the search returns $8$ rational points, whose affine part has $s\in{-1,0,1}$. In particular, no affine rational point with $s&gt;0$ and $s\neq 1$ is found up to this bound. This provides strong computational evidence that for rational $s&gt;0$, $s\neq 1$, the quintic $P_s(x)$ admits no quadratic factor over $\mathbb{Q}$ (equivalently, no $2+3$ (quadratic-cubic) factorization over $\mathbb{Q}$), and yields a conditional exclusion assuming completeness of the rational-point enumeration on $\overline{C}$.