The GPS system knows where the aircraft WAS...
The navigation message consists of 30-second frames 1,500 bits long, divided into five 6-second subframes of ten 30-bit words each. Each subframe has the
GPS time in 6-second increments.
GPS signal is about 6 seconds long from each sat... you need at least 4 sats, and for most systems, with redundancy and error trapping you need 6.
Meanwhile, the ac is moving at 400kts ...
The system correlates all of this into where the ac WAS.
Due to all of the latencies involved, the Kalman filters estimate where the aircraft is.
GPS satellites also talk to each other and to their controlling ground station and pass information about the health of the system and the time, synchronized by the ground station.
No, the GPS signal does not include integrity. Integrity has more to do with the atmospheric conditions of the broadcast when receiving. If a SAT is off, it is shut down until repaired.
WASS does broadcasts and GBAS broadcasts do include integrity corrections.
Within the FMC, not the signal, there are integrity factors for accuracy of location...especially for waypoints associated with a procedure. There is the HIL right there on the FMS.