Conjugate points and multiplicity are invariant under affine reparametrization

Define $\tilde{\gamma }=\gamma \circ \varphi$. Since $\varphi$ is affine, [F4] makes $\tilde{\gamma }$ a geodesic; by [F3], composing the smooth local-frame coefficients of a field $J$ with the smooth map $\varphi$ gives a smooth field $J\circ \varphi$ along it. The intervals are nondegenerate by $a<b$ and $c<d$.

Locally write $J(t)=e(\gamma (t))j(t)$ and $B(t)={\omega }_{\gamma (t)}(\dot{\gamma }(t))$; then $\dot{\tilde{\gamma }}=\lambda \dot{\gamma }\circ \varphi$ and $\tilde{B}(s)={\omega }_{\gamma (\varphi (s))}(\dot{\tilde{\gamma }}(s))=\lambda B(\varphi (s))$. The coefficient column becomes $j\circ \varphi$, so the chain rule [F7] and frame formula [F5] give $D_s(J\circ \varphi )=\lambda (D_{t}J)\circ \varphi$ and, applying it again, $D_s^2(J\circ \varphi )={\lambda }^2(D_t^{2}J)\circ \varphi$. These local identities are intrinsic and hold at included endpoints with one-sided derivatives.

The velocity scales by $\dot{\tilde{\gamma }}=\lambda \dot{\gamma }\circ \varphi$; trilinearity of curvature [F6] gives $R(J\circ \varphi ,\dot{\tilde{\gamma }})\dot{\tilde{\gamma }}={\lambda }^2(R(J,\dot{\gamma })\dot{\gamma })\circ \varphi$. Hence the full Jacobi operator of $J\circ \varphi$ equals ${\lambda }^2(D_t^{2}J+R(J,\dot{\gamma })\dot{\gamma })\circ \varphi$. By [F1], a Jacobi field pulls back to a Jacobi field; applying the same calculation to ${\varphi }^{-1}$, whose slope is $1\mathrm{/}\lambda$, proves the converse.

Since $\varphi$ bijects the endpoints, $J$ vanishes at $a,b$ exactly when $J\circ \varphi$ vanishes at $c,d$, even when $\lambda <0$ exchanges their order. Steps 1.1, 1.2, and 2.1 show $P_{\varphi }$ maps the endpoint-vanishing Jacobi space to the other one with inverse pullback by ${\varphi }^{-1}$; composition preserves every real linear combination, so [F8] makes $P_{\varphi }$ linear and [F9] makes it a linear isomorphism.

Both spaces are finite-dimensional by [F2], so their isomorphism gives equal dimensions by [F10]. Since $P_{\varphi }$ and its inverse preserve nonzero fields, each space contains a nonzero Jacobi field exactly when the other does; [F2] then gives both directions of conjugacy and equality of multiplicities. If $\dim M=0$, [F11] and [F12] make every tangent fiber zero, so both endpoint spaces are zero; if $\gamma$ is constant in any dimension, [F2] gives the same conclusion. No choice principle is needed because ${\varphi }^{-1}$ is explicit. [F2, F10, F11, F12, step 3.1, given] $\mathrm{\square }$