A conjugate point at which there are many geodesics

Fix a unit $v\in T_p{S}_R^n$. By [F1], $c_p(v)=\pi R$ is finite and $\mathrm{Cut}(p)=\{-p\}$. By [F2] the cut point of $p$ along ${\gamma }_v$ is ${\gamma }_v(\pi R)$, so ${\gamma }_v(\pi R)\in \mathrm{Cut}(p)=\{-p\}$, and $d_g(p,{\gamma }_v(\pi R))=\pi R$. By [F3] the curve ${\gamma }_v$ is the maximal geodesic with ${\gamma }_v(0)=p$ and ${\dot{\gamma }}_v(0)=v$, and by [F4] it has unit speed, so $L_g({\gamma }_v{\mathrm{\mid }}_{[0,\pi R]})=\pi R=d_g(p,{\gamma }_v(\pi R))$ and the segment is a minimizing geodesic from $p$ to $-p$.

By [F6] choose a chart at $p$; its coordinate derivations ${\mathrm{\partial }}_1{\mathrm{\mid }}_p,\dots ,{\mathrm{\partial }}_n{\mathrm{\mid }}_p$ form a basis of $T_p{S}_R^n$. The first two of these form a linearly independent list, so Gram--Schmidt [F7] supplies orthonormal $e_1,e_2\in T_p{S}_R^n$ with $g_p(e_i,e_j)={\delta }_{ij}$.

By [F5], for every unit $v$ the point $-p={\gamma }_v(\pi R)$ is conjugate to $p$ along ${\gamma }_v$ with multiplicity $n-1$. Thus every minimizing meridian of step 1.1 ends at a conjugate point, and the conjugacy hypothesis of the refuted claim is met by each of them.

For $m\in \mathbb{N}$ define $u_m=\frac{e_1+m{e}_2}{\sqrt{1+m^2}}\in T_p{S}_R^n,$ a linear combination of tangent vectors. By [F8] the metric is bilinear and symmetric, so orthonormality in [F7] gives $g_p(e_1+m{e}_2,e_1+m{e}_2)=g_p(e_1,e_1)+2m{g}_p(e_1,e_2)+m^2{g}_p(e_2,e_2)=1+m^2,$ and then, using $(\sqrt{1+m^2}{)}^2=1+m^2$ and $1+m^2>0$ from [F9], $g_p(u_m,u_m)=\frac{1+m^2}{(\sqrt{1+m^2}{)}^2}=1.$ So every $u_m$ is a unit tangent vector at $p$.

Suppose $u_m=u_k$ with $m,k\in \mathbb{N}$. Since $e_1,e_2$ are orthonormal they are linearly independent, so the coefficients of a vector in their span are unique; comparing the coefficients of $e_1$ in $u_m=\frac{1}{\sqrt{1+m^2}}{e}_1+\frac{m}{\sqrt{1+m^2}}{e}_2$ and in the same expression with $k$ gives $\frac{1}{\sqrt{1+m^2}}=\frac{1}{\sqrt{1+k^2}},$ hence $\sqrt{1+m^2}=\sqrt{1+k^2}$ and, squaring and using [F9], $1+m^2=1+k^2$, that is $m^2=k^2$. Since $m,k\ge 0$ in $\mathbb{R}$ by [F9], the equivalence $a\le b⟺a^2\le b^2$ on nonnegative reals gives $m\le k$ and $k\le m$, so $m=k$. Therefore $m\mapsto u_m$ is injective.

For each $m\in \mathbb{N}$ put ${\gamma }_m:={\gamma }_{u_m}$ on $[0,\pi R]$; by [F3], ${\gamma }_m(0)=p$ and ${\dot{\gamma }}_m(0)=u_m$. If ${\gamma }_m={\gamma }_k$ as maps, then their derivatives at $t=0$ coincide, so $u_m=u_k$; by step 3.1 this forces $m=k$. Hence $m\mapsto {\gamma }_m$ is injective, and the meridians ${\gamma }_m$ are pairwise distinct.

By steps 1.1 and 2.1 every ${\gamma }_m$ is a minimizing geodesic from $p$ to $-p$ along which $-p$ is conjugate to $p$ with multiplicity $n-1$, and by step 4.1 the members of the family are pairwise distinct. The set $G$ of minimizing geodesic segments from $p$ to $-p$ is infinite: if $G$ were finite, then its subset $G^{\mathrm{\prime }}=\{{\gamma }_m:m\in \mathbb{N}\}$ would be finite by [F10], so $G^{\mathrm{\prime }}\approx \mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }$ with $\mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }\in \mathbb{N}$ by [F10]; but $m\mapsto {\gamma }_m$ is a bijection $\mathbb{N}\to G^{\mathrm{\prime }}$ by step 4.1, hence $\mathbb{N}\approx \mathrm{\mid }{G}^{\mathrm{\prime }}\mathrm{\mid }$, contradicting the statement of [F10] that $\mathbb{N}$ is not equinumerous with any natural number. Thus the conjugate point $-p$ is joined to $p$ by infinitely many distinct minimizing geodesics, and the claim in the Statement refuted is false: no meridian ${\gamma }_m$ is the unique minimizing geodesic from $p$ to the conjugate point $q=-p$.

Boundary and choice audit. The dimension hypothesis $n\ge 2$ is used exactly at steps 1.2 and 2.2 to obtain two orthonormal tangent directions and a one-parameter family of unit directions; dimensions $n=0$ (where $T_p{S}_R^0$ is the zero space, so no unit direction exists) and $n=1$ (where the antipode has only the two semicircles) are outside the quantified claim, and nothing is asserted for them. The zero vector is never used: all $u_m$, including $u_0=e_1$, are unit vectors by step 2.2, and the family is still injective at $m=0$ by step 3.1. The witness is nonempty because $p$ is supplied, and $p\mathrm{\ne }-p$ because $\mathrm{\mid }p\mathrm{\mid }=R>0$; the interval $[0,\pi R]$ is nondegenerate because $R>0$, and the meridians are nonconstant geodesics of positive length. Both endpoints of $[0,\pi R]$ are included and the derivative at $0$ is one-sided for the injectivity argument of step 4.1. Exactly the inherited ${\mathrm{A}\mathrm{C}}_{\omega }$ is assumed: it is spent only through the two sphere examples quoted in [F1] and [F5], while the orthonormal pair of step 1.2 comes from one finite basis list and the family of step 2.2 is given by an explicit formula, so no choice function on a family of directions is invoked. The refuted claim is a one-way uniqueness implication, so no converse case arises.