Let \( z\in \mathbb {H}{:}{=}\{z\in \mathbb {C}: \operatorname {Im}(z)>0\}\) , \(\Lambda {:}{=}\sqrt{\frac{1}{\operatorname {Im}(z)}}\Big ({\mathbb {Z}}\oplus z{\mathbb {Z}}\Big )\) and \( \theta (\alpha , z){:}{=}\theta (\alpha , \Lambda )=\sum _{(m,n)\in \mathbb {Z}^2} e^{-\pi \alpha \frac{ |mz+n|^2}{\operatorname {Im}(z) }}. \) We give a complete characterization of \(\mathop {\textrm{maxima}}\limits _{z\in \mathbb {H}}\frac{\partial }{\partial \alpha }\theta (\alpha ,z)\) for all \(\alpha >0\) . Precisely, we prove that there exist two thresholds \(0<\alpha _a<\alpha _b<1\) such that \(\begin{aligned} \begin{aligned} \mathop {\textrm{maxima}}\limits _{z\in \mathbb {H}}\frac{\partial }{\partial \alpha }\theta (\alpha ,z)= {\left\{ \begin{array}{ll} \;e^{i{\pi }/{3}}, & \hbox {if}\;\; \alpha \in (\alpha _b,\infty ),\\ \;i\;\hbox {or}\;e^{i{\pi }/{3}}, & \hbox {if}\;\; \alpha =\alpha _b,\\ \;i,\; & \hbox {if}\;\; \alpha \in [\alpha _a,\alpha _b),\\ \;iy_\alpha ,\; y_\alpha >1,\;\;& \hbox {if}\;\; \alpha \in (0,\alpha _a). \end{array}\right. } \end{aligned} \end{aligned}\) The location of the maximizer does not remain fixed at the hexagonal point. Instead, as \(\alpha \) decreases from \(\infty \) , the maximizer starts at \(e^{i\pi /3}\) , then at \(\alpha = \alpha _b\) undergoes a transition to the square lattice point i, where it remains for \(\alpha \in [\alpha _a, \alpha _b)\) . Finally, for \(0<\alpha < \alpha _a\) , the maximizer moves continuously along the imaginary axis to purely imaginary points \(iy_\alpha \) with \(y_\alpha > 1\) . This reveals a new and intricate maximization pattern, in contrast to the classical results for the single theta function (Montgomery [33]) and for differences of theta functions (our prior work [28]), in which the hexagonal lattice \(e^{i\pi /3}\) prevails as the optimizer. As a consequence, we establish that a class of modular invariant functions admits minimizers that are exactly the hexagonal-square lattices without passing through any intermediate rhombic lattices. This is the first rigorous result of its kind and provides a positive answer to an open problem posed by Conti and Zanzotto [15].
