  An $N=\infty$ equivalence among quenched $U(N)$ models on finite lattice tori
of $V$ sites is proven to all orders in planar perturbation theory by putting
circulant lattice momenta together with group indices on 't Hooft's double
lines. Known estimates for the number of order $N^2$ diagrams, $N\gg V$, and
the simultaneous presence of UV and IR cutoffs, suggest a positive radius of
convergence for the planar perturbative expansion before the limit $N=\infty$
is taken.
