  This is the sixth in a series of articles devoted to showing that a typical
covering map of large degree to a fixed, regular graph has its new adjacency
eigenvalues within the bound conjectured by Alon for random regular graphs.
  In this article we show that if the fixed graph is regular Ramanujan, then
the {\em algebraic power} of the model of random covering graphs is $+\infty$.
This implies a number of interesting results, such as (1) one obtains the upper
and lower bounds---matching to within a multiplicative constant---for the
probability that a random covering map has some new adjacency eigenvalue
outside the Alon bound, and (2) with probability smaller than any negative
power of the degree of the covering map, some new eigenvalue fails to be within
the Alon bound without the covering map containing one of finitely many
"tangles" as a subgraph (and this tangle containment event has low
probability).
