The claim that the market is good at predicting future earnings.

This is conceptually very easy to test; get historical stock price and earnings data, compute P/E ratios at each snapshot, compute earnings growth across snapshots, and then look at the relationship between the two. Vanguard ran the numbers here (page 7), and two ways of calculating the P/E ratio were the strongest two factors. (As one would expect, 1-year returns were very difficult to predict at all, and they were mostly useful for 10-year returns.)