Improved Lower Bound for Frankl's Union-Closed Sets Conjecture

We verify an explicit inequality conjectured recently by Gilmer, thus proving that for any nonempty union-closed family $F\subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2}\approx 0.38$ fraction of the sets in $F$. One case, an explicit one-variable inequality, is checked by computer calculation.