Robin nullity in mode $|k|=1$ and asymptotic radius of the critical hyperbolic catenoid

For each parameter $a>1/2$, the critical hyperbolic catenoid $\Sigma_a$ is a rotationally symmetric, free boundary minimal annulus in a geodesic ball $B^3(r(a))\subset\mathbb{H}^3$, in the family of Mori, do Carmo--Dajczer, and Medvedev. We establish three analytic results about $\Sigma_a$. (I) Robin nullity and index in mode $|k|=1$. The Robin nullity of the Jacobi operator $L_{\Sigma_a}=\Delta_g+(|II|^2-2)$ in angular Fourier mode $|k|=1$ equals $2$, with kernel spanned by the Killing--Jacobi fields associated to the rotations $L_{12},L_{13}\in\mathfrak{so}(3,1)$ that fix the geodesic axis of $\Sigma_a$ and send $\partial B^3(r(a))$ to itself. The radial profile admits the closed form $f_*(s)=\partial_s\Phi_a^0(s,0)=\frac{d}{ds}[A(s)\cosh\varphi(s)]=\sinh r(s)\cdot r'(s)$, where $r(s)$ is the geodesic distance from $p_0=(1,0,0,0)$. By Sturm--Liouville theory, the Robin Morse index of $\Sigma_a$ in mode $|k|=1$ also equals $2$, refining the lower bound $\mathrm{ind}(\Sigma_a)\geq 4$ of Medvedev. (II) Asymptotic radius. The boundary radius satisfies $r(a)=\tfrac{3}{2}\log a+d_\infty+o(1)$ as $a\to\infty$, with $d_\infty=\log[\sqrt{2}\,\Gamma(1/4)^2/\pi^{3/2}]=\log[2\sqrt{2\pi}/\Gamma(3/4)^2]$. The closed form for $d_\infty$ follows from a Beta-function evaluation of $I_\infty=\int_0^{\infty}\cosh(2t)^{-3/2}\,dt$. (III) Degenerate limit. As $a\to(1/2)^+$, $r(a)=c_*\sqrt{a-1/2}\,(1+o(1))$ with $c_*=\sigma_*\cosh\sigma_*$, where $\sigma_*$ is the unique positive fixed point of $\sigma=\coth\sigma$. The proof of (I) follows the mode-by-mode strategy of Devyver for the Euclidean critical catenoid, with $\mathfrak{so}(3,1)$ replacing $\mathfrak{so}(3)$, supplemented by the closed-form identification $f_*=\partial_s\Phi^0$ specific to the hyperbolic ambient. The proof of (II) is a Laplace-type asymptotic analysis of the implicit free boundary condition.