The distance from p is smooth on m minus p

For every open $U\subseteq {\mathbb{R}}^m$, $q^{-1}(q(U))={\bigcup }_{\lambda \in \mathrm{\Lambda }}(U+\lambda )$, so [F1] makes $q$ open. For distinct classes $q(x)\mathrm{\ne }q(y)$, [F6] attains each ${\delta }_i={\min }_{k\in \mathbb{Z}}\mathrm{\mid }{y}_i-x_i+k{L}_i\mathrm{\mid }$ and $\delta =({\sum }_i{\delta }_i^2{)}^{1\mathrm{/}2}>0$; therefore $q(B(x,r))$ and $q(B(y,r))$ are disjoint open neighborhoods when $0<r<\delta \mathrm{/}3$. Since $q$ is open, the images of a countable rational-ball basis of ${\mathbb{R}}^m$ give a countable basis of $Q$. Choose $2\epsilon <{\min }_i{L}_i$; $U_x={\prod }_i(x_i-\epsilon ,x_i+\epsilon )$ has pairwise disjoint lattice translates and $q{\mathrm{\mid }}_{U_x}$ is a homeomorphism onto the open set $q(U_x)$. These charts have translation overlap maps, so $Q$ is a smooth boundaryless manifold with the descended Euclidean metric. Finally, $q^{-1}(q(U_x))={\bigcup }_{\lambda \in \mathrm{\Lambda }}(U_x+\lambda )$ is a disjoint union of sheets, each mapped homeomorphically onto $q(U_x)$, and the metric is preserved on each sheet; thus $q$ is a covering and a local isometry.

The paths $s\mapsto q((1-s)x+sy)$ make $Q$ path-connected and [F5] makes it connected. At $p_0=q(0)$ identify $T_{p_0}Q$ with ${\mathbb{R}}^m$. For each $w$, ${\gamma }_w(t)=q(tw)$ exists for all real $t$; on every periodic chart its coordinates are affine and metric coefficients constant, so [F7] gives $\mathrm{\Gamma }=0$ and [F8] makes it a geodesic. Uniqueness [F9] and the exponential definition [F10] give ${\exp }_{p_0}(w)=q(w)$ for every $w$, so ${\mathcal{E}}_{p_0}=T_{p_0}Q$. This nonempty connected boundaryless model is metrically complete by [A1, F11].

Let $\alpha :[0,1]\to Q$ be any piecewise-$C^1$ path from $q(x)$ to $q(y)$ and lift it from $x$ by [F15]; its lift is piecewise $C^1$ after a finite subdivision into covering charts and ends at $y+\lambda$ for some $\lambda \in \mathrm{\Lambda }$. The local isometry in step 1.1 preserves speed on each subinterval, so [F14] gives $L_g(\alpha )=L_{{\mathbb{R}}^m}(\tilde{\alpha })$. Put $z=\tilde{\alpha }(1)-x$. If $z\mathrm{\ne }0$, set $u=z\mathrm{/}\mathrm{\mid }z\mathrm{\mid }$ and $H(t)=⟨u,\tilde{\alpha }(t)⟩$; on every smooth piece [F17] gives $H^{\mathrm{\prime }}=⟨u,{\tilde{\alpha }}^{\mathrm{\prime }}⟩\le \mathrm{\mid }{\tilde{\alpha }}^{\mathrm{\prime }}\mathrm{\mid }$. Both integrands are continuous and integrable by [F16], so [F18], [F19] and [F20] give $\mathrm{\mid }z\mathrm{\mid }=H(1)-H(0)={\sum }_j{\int }_{I_j}{H}^{\mathrm{\prime }}\le {\sum }_j{\int }_{I_j}\mathrm{\mid }{\tilde{\alpha }}^{\mathrm{\prime }}\mathrm{\mid }=L_g(\alpha )$. For $z=0$ the same lower bound follows from nonnegative speed. Hence each path has length at least $\mathrm{\mid }y-x+\lambda \mathrm{\mid }$ for its endpoint lift.

For every $\lambda \in \mathrm{\Lambda }$, the projection of the straight segment from $x$ to $y+\lambda$ has constant speed and length $\mathrm{\mid }y-x+\lambda \mathrm{\mid }$ by [F13] and [F19]; step 3.1 gives the lower bound for every competing path. The floor calculation in step 1.1 attains the nearest representative in each coordinate. Taking the infimum over paths in [F12] yields $d_g(q(x),q(y))={\min }_{\lambda \in \mathrm{\Lambda }}\mathrm{\mid }y-x+\lambda \mathrm{\mid }=\sqrt{{\sum }_{i=1}^m({\min }_{k\in \mathbb{Z}}\mathrm{\mid }{y}_i-x_i+k{L}_i\mathrm{\mid }{)}^2}$.

For $m=1$, $L_1=L>0$, and $p=q(0)$, step 4.1 gives $d_g(p,q(x))=\min (x,L-x)$ for $0\le x\le L$. The unit geodesic ray $\gamma (t)=q(t)$ minimizes for $0\le t\le L\mathrm{/}2$, and for every $t>L\mathrm{/}2$ the nearest representative has absolute value at most $L\mathrm{/}2<t$. Thus [F21] gives $c_p(\dot{\gamma }(0))=L\mathrm{/}2$, [F22] makes $q(L\mathrm{/}2)$ a cut point, the lifts $L\mathrm{/}2$ and $-L\mathrm{/}2$ give distinct minimizing paths to it, and $q(L\mathrm{/}2)\mathrm{\ne }p$.

In the smooth coordinate $s\mapsto q(L\mathrm{/}2+s)$ around the antipode, $r_p(q(L\mathrm{/}2+s))=L\mathrm{/}2+s$ for $s<0$ and $r_p(q(L\mathrm{/}2+s))=L\mathrm{/}2-s$ for $s>0$. The left derivative is $1$ and the right derivative is $-1$, so $r_p$ is not differentiable at the cut point $q(L\mathrm{/}2)\mathrm{\ne }p$.

For $m=2$ with periods $a,b>0$ and $p=q(0)$, the distance formula gives the closed Dirichlet cell $D=[-a\mathrm{/}2,a\mathrm{/}2]\times [-b\mathrm{/}2,b\mathrm{/}2]$: its interior has one nearest lift, the relative interior of each face has two tied lifts, and each corner has four. For any unit vector $u$, its radial geodesic stays minimizing through the first exit time $t_0=\min \{a\mathrm{/}(2\mathrm{\mid }{u}_1\mathrm{\mid }),b\mathrm{/}(2\mathrm{\mid }{u}_2\mathrm{\mid }):u_i\mathrm{\ne }0\}$; every $t>t_0$ lies outside $D$ in some coordinate, and shifting that coordinate by its period strictly shortens the lift. Hence $c_p(u)=t_0$, the set $\{ru:\mathrm{\mid }u\mathrm{\mid }=1,0\le r<c_p(u)\}$ is $\mathrm{int}D$, and the cut locus is $q(\mathrm{\partial }D)$: every ray's first exit lies in $\mathrm{\partial }D$, and every point of $\mathrm{\partial }D$ is the first exit on its radial ray. The empty manifold has no base point and a zero-dimensional manifold has no unit direction; neither affects this existential counterexample. The one-dimensional circle witness has $L>0$, includes its finite cut endpoint, and loses minimization strictly at every later time. Exactly ${\mathrm{A}\mathrm{C}}_{\omega }$ is used through the geodesic/exponential and Hopf--Rinow cut-time interfaces [A1, F9, F10, F11, F21, F22]; the quotient, path-lifting, floor and derivative calculations make no further selection and use no full AC. The item states no biconditional. [A1, F9, F10, F11, F21, F22, step 4.1, step 5.1, step 6.1] QED