The case of a Rook on a circular board is similar to a Silver General on a rectangular board: there are virtually no forced wins, and the few that exist are very short. If one would consider such piece 'major' for that reason, then 'major' basically becomes a useless concept. The only reason for distinguishing majors from minors in chess theory is to imply the prospects of winning the game. That there could be an extremely small number of exceptions (which are even less likely to occur in practice than their number suggests) is not very important. Checkmating a bare King is not the only concern inchess (variants). One also wants to recognize more complex elementary wins, e.g. 2 minors vs bare King, or a strong major vs a much weaker defender, such as Queen vs Rook. The latter is generally won, but there are some draws, where the Rook can deliver a perpetual, and you cannot escape it without losing the Queen.
So a more practical approach would be to abandon absolutism, and allow a small number of exceptions (say a few percent of the positions). The Rook on a circular / cylinder board then is a minor, the Woody Rook on 8x8 a major. Just a slightly imperfect one.
For cases that are really intermediate, such as the Evil Wolf, (or in fact the orthodox Pawn), we must admit that they are neither major nor minor, and that the winning prospects cannot be judged by material balance alone.
The case of a Rook on a circular board is similar to a Silver General on a rectangular board: there are virtually no forced wins, and the few that exist are very short. If one would consider such piece 'major' for that reason, then 'major' basically becomes a useless concept. The only reason for distinguishing majors from minors in chess theory is to imply the prospects of winning the game. That there could be an extremely small number of exceptions (which are even less likely to occur in practice than their number suggests) is not very important. Checkmating a bare King is not the only concern inchess (variants). One also wants to recognize more complex elementary wins, e.g. 2 minors vs bare King, or a strong major vs a much weaker defender, such as Queen vs Rook. The latter is generally won, but there are some draws, where the Rook can deliver a perpetual, and you cannot escape it without losing the Queen.
So a more practical approach would be to abandon absolutism, and allow a small number of exceptions (say a few percent of the positions). The Rook on a circular / cylinder board then is a minor, the Woody Rook on 8x8 a major. Just a slightly imperfect one.
For cases that are really intermediate, such as the Evil Wolf, (or in fact the orthodox Pawn), we must admit that they are neither major nor minor, and that the winning prospects cannot be judged by material balance alone.