  We eliminate 38 infinite families of possible principal graphs as part of the
classification of subfactors up to index 5. A number-theoretic result of
Calegari-Morrison-Snyder, generalizing Asaeda-Yasuda, reduces each infinite
family to a finite number of cases. We provide algorithms for computing the
effective constants that are required for this result, and we obtain 28
possible principal graphs. The Ostrik d-number test and an algebraic integer
test reduce this list to 7 graphs in the index range (4,5) which actually occur
as principal graphs.
