A nonzero pi class on a torus has a primitive embedded pi root

Choose a $C^2$ diffeomorphism of $L$ with the standard torus by [F5], and use its induced identification ${\pi }_1(L)={\mathbb{Z}}^2$. Write the nonzero class as $\alpha =m\beta$ with $m\ge 1$ and $\beta =(p,q)$ primitive, $\gcd (\mathrm{\mid }p\mathrm{\mid },\mathrm{\mid }q\mathrm{\mid })=1$. The smooth straight loop $t\mapsto t(p,q)$ modulo ${\mathbb{Z}}^2$ is embedded: if two parameter values in $[0,1)$ had the same projection, their difference times $(p,q)$ would be integral, and Bezout's identity would make that difference integral, hence zero. Transfer this loop through the inverse $C^2$ diffeomorphism to obtain an actual embedded regular $C^2$ loop $g$ on $L$. A path to the basepoint supplies the based primitive class. The given $\alpha$ class equals $m\beta$, so a compact based homotopy and fixed-flow fence transport in [F1] identify the original $\alpha$-fence loops with the $m$-fold loops of the $g$-fence. No smoothness of a topological cell map is used.

Let $h$ be the increasing one-sided holonomy of $g$. Since $\alpha$ has identity one-sided holonomy (its loops are null on the chosen side by the $\mathrm{\Pi }$ property), $h^m(t)=t$ for every sufficiently small positive $t$. An increasing map has no nontrivial finite orbit: if $h(t)>t$ its successive iterates strictly increase and if $h(t)<t$ they strictly decrease, so $h(t)=t$ and the short displaced loops $g_t$ close. Their $m$-fold loops are null by the transported compact homotopy and the $\mathrm{\Pi }$ property of $\alpha$; oriented surface fundamental groups are torsion-free, so $[g_t{]}^m=1$ implies $[g_t]=1$. Hence $\beta$ is a genuine nonzero embedded $\mathrm{\Pi }$ cycle on the same original leaf and the chosen side.

Use the one fixed transverse flow to construct the fence $F(u,t)={\mathrm{\Phi }}_{\tau (u,t)}g(u)$ with ${\tau }_t>0$. Short compact-leafwise separation for the compact loop $g(S^1)$ gives injectivity of the full annulus: if $F(u,t)=F(v,s)$, flow uniqueness gives ${\mathrm{\Phi }}_{\tau (u,t)-\tau (v,s)}g(u)=g(v)$, so the separation forces equality of the times and of $g(u),g(v)$; embeddedness of $g$ gives $u=v$ and strict positivity of ${\tau }_t$ gives $t=s$. Compact-to-Hausdorff then makes the fence an embedding, every $g_t$ is an embedded null curve in its actual leaf, and the surface-Jordan disk lemma of [F1] applied in the leaf, rather than only in its universal cover, gives an actual embedded disk bounded by $g_t$; the spherical-leaf alternative is excluded by the in-pair spherical stability item, ensuring the selected disk is unique. This supplies the embedded Reeb construction input, not a deduction of ambient cap embedding from universal-cover caps, and only finitely many fences and homotopies are used, hence only the standing countable choice from [F4].