Divisor-bound conjecture for the symbolic generator

Let $\Delta_n\in\mathbb{Z}_{>0}$, and let $r\in\mathbb{Z}_{>0}$ be a divisor of $\Delta_n$. Consider the parametrized formulas for $A,B,C,D$ constructed in Sections 2.1 and 2.2, and require them to yield positive, non-zero integer values. Divisor-bound conjecture. It is necessary when $n=2$, and empirically observed when $n=3$, that

$$r<\sqrt[n]{\Delta_n}.$$

This inequality ensures that expressions such as $\frac{\Delta_n-r^2}{2r}$ are non-negative integers; for $n=3$, it improves the likelihood that $\sqrt{12\cdot\Delta_3\cdot r-3\cdot r^4}$ is a perfect square, enabling closed-form rational solutions. The statement is proved only as a necessary condition for $n=2$ in the supplied wording; its status for $n=3$ is empirical rather than established.