Is there not a situation where the total P1 time can exceed the sector time? For instance, if the pilot flying is regaining currency, under the supervision of a current and qualified pilot (who is therefore 'in command'), then can the pilot flying not claim P1 (actually P1/S), and the supervising pilot also claim P1?
Straight off the top of my head, with a mixture of JAA/FAA (and other) experience. So it might be a bit mixed up and I stand ready to be corrected.