I think it may be possible to automate finding what I would call proof game kernels, which is just the subsequence of pawn moves (including promotions) and pawn

I think it may be possible to automate finding what I would call proof game kernels, which is just the subsequence of pawn moves (including promotions) and pawn captures. One could further simplify the search by abstracting away from the exact location of pawns, but recording only their color and order within each file. Many of the 381 illegal positions can be proven illegal just by absence of such proof kernels. Proof game kernels can also guide construction of full proof games.