Analytic local resolution of Medvedev's Morse index conjecture for the critical hyperbolic catenoid in $\mathbb{H}^3$

Let $\Sigma_a\subset B^3(r(a))\subset\mathbb{H}^3$ ($a>1/2$) be the critical hyperbolic catenoid of the Mori family, a free boundary minimal surface in the geodesic ball. The Medvedev conjecture [9] states ind$(\Sigma_a)=4$ for all $a>1/2$. We study its strong form: ind$(\Sigma_a)=4$ and nul$(\Sigma_a)=2$. The nullity condition nul$(\Sigma_a)=2$ combines the mode-$|k|=1$ result $\text{nul}_R(\Sigma_a)|_{|k|=1}=2$ of [11, Cor. 4.4] with vanishing kernel in modes $|k|=0,|k|\ge2$; the latter, not in [11], is established here for $a\in(1/2,1/2+\delta_0)$. The main result is the analytic local resolution of the strong Medvedev conjecture: $\exists\delta_0>0$ s.t. ind$(\Sigma_a)=4$, nul$(\Sigma_a)=2$ for all $a\in(1/2,1/2+\delta_0)$. This follows from the expansion $H(a):=\sinh r(a)/K(a)=\sigma_*\cosh\sigma_*+C_0(a-\frac12)+O((a-\frac12)^2)$ as $a\to(1/2)^+$, with $C_0=\frac{\sigma_*\cosh\sigma_*(\sinh^2\sigma_*-1)(3\sinh^2\sigma_*-2)}{12\sinh^2\sigma_*}$, where $\sigma_*>0$ the unique positive root of $\sigma=\coth\sigma$, and $C_0>0$ by $\sigma_*>\log(1+\sqrt2)$. The proof proceeds via three reductions: $(i)$ the Medvedev conjecture is equivalent to $\mu_0^{\mathrm{even}}(2)>0$ $(E)$ and $\mu_2(0)>0$ with non-degeneracy in mode $0$ $(F)$; $(ii)$ $\mu_2(0)>0$ reduces, via a Sturm shooting-count argument, to $\phi_a>0$ of the parametric Jacobi field on the principal branch; $(iii)$ $\phi_a>0$ reduces, under $\sinh r(a)>2K(a)$ $(G)$, to $H'(a)>0$ via a constant Wronskian and Sturm separation. Auxiliary results: a Picone identity (base $f_*$) closing unconditionally the odd radial sector for $|k|\ge2$; a second Picone identity (base $B$) proving $(E)$ unconditionally on $(1/2,1]$ and, via Hardy estimates, on $(1/2,A_*]$ ($A_*>1$); analytic closure of $(G)$ on $(1/2,1]$ via strict concavity of a transcendental function; an alternative proof of ind$(\Sigma_a)\ge4$ via Lorentz ambient coordinates.