Hives from deformed GUE minor processes

We construct random hives from deformed GUE minor processes. Starting from two independent diagonally deformed GUE matrices $$X=\sqrt{n}(wG+uD),\qquad Y=\sqrt{n}(w'G'+u'D'),$$ where $D,D'$ are diagonal and have GUE spectra, we use their minor processes to form a double hive and then apply the octahedron recurrence. Under the matching condition $$\frac{u}{w^2}=\frac{u'}{(w')^2},$$ we prove that the resulting hive law is close, in relative entropy, to a GUE hive law. More precisely, if $$a^2=w^2+u^2,\qquad b^2=(w')^2+(u')^2,$$ then the produced hive density $q_n$ satisfies $$D_{\mathrm{KL}}\!\left( q_n\, \middle\|\, \operatorname{Density}\bigl(H_n(a\sqrt n,b\sqrt n,c_{**}\sqrt n)\bigr) \right) = O(n\log n).$$ The third scale $c_{**}$ is determined by a limiting tetrahedral optimization problem; equivalently, writing $\delta=u+u'$, $$\delta^2 = \frac{ 2c_{**}^4(c_{**}^2-a^2-b^2) }{ (c_{**}^2-a^2+b^2)(c_{**}^2+a^2-b^2) }.$$ Thus the construction realizes GUE hive laws, up to subleading relative entropy, throughout the right-angled and obtuse regime. The appendix records two explicit surface-tension approximations and numerical comparisons which motivated the construction.