= Intersection test for nonisometric finite covers
Let two finite locally isometric covers carry uniquely length-identified primitive closed <geodesics> $\alpha_i,\beta_i$. If their geometric intersection numbers differ, the covering surfaces are not <isometric>, because an <isometry> preserves both lengths and intersections. For base simple curves intersecting once, lift components correspond to cycles of their <monodromy permutations>. Their intersection number is the number of sheets common to the two cycles. This distinguishes covers even when every individual permutation has the same cycle type in both actions, as in <Gassmann equivalence>.
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