Corners and parity
As on the 3x3, with the 5x5's parities.
Corners and parity
Corners are solved as on the 3x3, from UFR; commutators that turn only outer faces work unchanged on the bigger cubes. Each kind has its own parity. On the 4x4 the corners' parity changes with each quarter turn of an outer layer and the wings' with each quarter turn of an inner one, so the two are independent: an odd number of corner targets ends with a corner swap, and an odd number of wing targets with a swap of two wings, which the 4x4's OLL parity algorithm does on its own.
2R' U2 2L F2 2L' F2 2R2 U2 2R U2 2R' U2 F2 2R2 F2
A swap of two wings and nothing else visible: the pure OLL parity algorithm. (22 HTM)
Checked by the engine: 2R' U2 2L F2 2L' F2 2R2 U2 2R U2 2R' U2 F2 2R2 F2 → 2 edges moved, 0 corners moved, 0 centers moved
On the 5x5 the midges move with the corners under every outer turn, so with the centres in place corner and midge parity come together, as on a 3x3, and the 3x3's parity algorithm handles both; the wings have their own parity, handled as on the 4x4.