Can one theoretically prove that these moves are marginally better? I heard that for the first move, 1. e4, 1. d4, 1. c4 and 1. Nf3 are equally good and none of them is better than others, even marginally. However, after 1. e4, both 1... e5 and 1... c5 are marginally better than 1... e6 and 1... c6; but none of 1... e5 and 1... c5 is better than the other. I am wondering if there is a way to theoretically prove this, either using an engine, or using statistics, or using opening theories.
So my question is: Can the following statement be proven?

After 1. e4, both 1... e5 and 1... c5 are marginally better than 1... e6 and 1... c6; but none of 1... e5 and 1... c5 is better than the other.
You can't really &quot;prove&quot; it. But, presumably, players want to play the best move, so which move is more often played may be an indication of which move the players think is best.
I looked at two databases with master games, and the most popular moves after 1.e4 in each were c5, e5, e6, and c6, in that order. This supports the first half of your statement (since e6 and c6 are played less often than e5 and c5) but not the second half (since c5 is played more than e5.)