Only just finished after a sorting out an error that meant I could not get the second 9-digit number to work. Fortunately I guessed that the highlighted cells might show a particular characteristic, so added the squares of two sets of numbers in the grid and found the sum matched all but the last digit. I'd forgotten to revise the SW corner when I spotted the common feature of the entries about a third of the way through my solve.
I agree that it was a long tedious slog, though the grid is indeed very elegant.
Thanks to those who posted that useful link. I wish I'd visited this forum earlier because that link would have saved me a lot of calculations. However, it helped with the endgame and it'll be useful for double-checking everything.