math.NT — Number Theory

An explicit upper-bound certificate for the essential minimum of the Zhang–Zagier height

Contributed by Ricardo Pinho

We give an exact certificate proving $C_{82}<0.2534449$ for the essential minimum of $h(\alpha)+h(1-\alpha)$, where $h$ is the absolute logarithmic Weil height. We use Doche's limit-point method, building on Grinsztajn's factor library and, in particular, Épi's unequal-multiplicity construction. We retain Épi's denominator factors, add Doche's $P_7$, and add 25 numerator factors found by LP-dual-weighted evaluation lattices with LLL. Joint linear-program re-optimisation is followed by exact rational rounding. Two separately written ball-arithmetic checkers certify the bound; a further adversarial computational audit accepts the certificate.