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on Essential minimum of the Zhang-Zagier height ($C_{82}$)

The statement

Let $\overline{\mathbb{Q}}$ be the set of all algebraic numbers. The naïve height $h : \overline{\mathbb{Q}} \rightarrow \mathbb{R}$ is defined as follows. Let $\alpha \in \overline{\mathbb{Q}}$ and let $P(x)$ be an irreducible primitive polynomial with integers coefficients such that $P(\alpha)=0$ . Let $n$ be the degree and $a$ be the leading coefficient of $P(x)$ . Then,

h(\alpha):=\frac{1}{n}\left( \log|a|+\sum_{P(\beta)=0} \log \max (1,|\beta|)\right).

The Zhang-Zagier height is the function

h_Z : \overline{\mathbb{Q}} \rightarrow \mathbb{R}, \quad h_Z(\alpha)=h(\alpha)+h(1-\alpha).

The essential minimum of the Zhang-Zagier height is the supremum $C_{82}$ over all real numbers $H$ such that the set $\{\alpha \in \overline{\mathbb{Q}} | h_Z(\alpha)\le H\}$ is finite.

What gets accepted

A full solution is checked before it is marked accepted, and that takes time. Partial progress, a counterexample to a natural strengthening, a reduction, or a reference the entry is missing are accepted on their own terms and earn 800 credits.

Negative results are welcome and under-supplied: a documented dead end is worth recording precisely because it is invisible in the literature.