A degenerate free boundary minimal annulus and non-uniqueness in spherical caps beyond the hemisphere

Let $\{\Sigma_a\}$, $a\in(0,1/2)$, be de Oliveira's family of embedded free boundary minimal annuli of revolution in geodesic balls $B(R(a))\subset\mathbb{S}^3$, $R(a)>\pi/2$. We prove that $R$ is real-analytic, tends to $\pi/2$ at both ends, and therefore folds: it has an interior maximum $R_*>\pi/2$ and is not injective. Hence each $B(\rho)$ with $\pi/2<\rho<R_*$ contains at least two, and only finitely many, mutually non-congruent annuli of the family. At every critical point of $R$ the annulus is degenerate modulo ball-preserving isometries: its Jacobi--Robin kernel contains a rotationally invariant, reflection-even field not induced by any Killing field of $\mathbb{S}^3$ preserving the ball; its nullity is at least three. These degenerate annuli form a nonempty discrete set; off it, the rotationally invariant even nullity vanishes. Those sitting at a maximizer of $R$, in the largest cap $B(R_*)$, are called degenerate annuli of maximal cap radius; each such annulus is also a strict local area maximizer in the family, since area and $R$ have the same critical points. Thus the hypothesis that all Jacobi fields of an embedded free boundary minimal annulus in a spherical cap are Killing-induced, used in the Naff--Zhu continuity approach to uniqueness, fails for some radius $R>\pi/2$. The exact identity $\dim K_0^{ev}(\Sigma_a)=\mathbf{1}_{\{R'=0\}}(a)$ detects the degeneration; it follows from the symmetry-free relation $\partial_\eta\varphi_a-\operatorname{ct}_\kappa(r(a))\varphi_a=r'(a)A_a(\eta,\eta)$, a Robin defect identity, in all space forms and dimensions.