  In this paper, we introduce a novel variational framework rooted in algebraic
geometry for the analysis of the Hardy $Z$-function. Our primary contribution
lies in the definition and exploration of $\Delta_n(\overline{a})$, a newly
devised discriminant that measures the realness of consecutive zeros of $Z(t)$.
Our investigation into $\Delta_n(\overline{a})$ and its properties yields a
wealth of compelling insights into the zeros of $Z(t)$, including the corrected
Gram's law, the second-order approximation of $\Delta_n(\overline{a})$, and the
discovery of the G-B-G repulsion relation. Collectively, these results provide
compelling evidence supporting a new plausibility argument for the Riemann
hypothesis.
