Redmare, the only reason you don't see Phalaris in the pedigree is that you haven't looked back far enough. Both Northern Dancer and Native Dancer are intensely inbred to Phalaris, as are some other names you might not recognize. What you have to do is go to allbreedpedigree.com and take every name in the right-hand column of the 5-generation pedigree that displays and click on each and every one. All the names are hotlinks. When you click on one of them, it opens the five-generation pedigree of that horse. You can go on doing this in fact until you get back to Old Bald Peg (1680's).
What you are going to find in all probability is that 50% or more of those horses trace back to Phalaris, so that the total ancestry of the horse you own is heavily Phalaris.
That's how they breed racehorses these days: carelessly. To the point that it is now difficult to find even an old broodmare who is not so bred: so that it is impossible to produce a foal with no Phalaris or even with a low percentage of Phalaris. See my previous articles on this subject in EQUUS Magazine, vis., the one on Secretariat, the one on American Pharoah, and the one on Justify.
And yes, I am STILL having fits getting a tablet that is compatible with my system, so you'll have to continue to wait patiently for that. Cheers -- Dr. Deb