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Pump Runs w/Ignition Switch Off; "...The relay probably has the points welded..." by Seabronc
Or
Pump Runs w/Ignition Switch Off; "...If it stays running with the truck off, then most likely you have a bad fuel pump relay; or EEC Relay hanging closed; If the EEC relay goes bad, it could be sending the signal to the fuel pump relay to turn the fuel pumps on; One easy test for this is: Without the key in ignition, bump the starter over @ the starter relay using a screwdriver. If it is the EEC relay, the truck will start without the key in the ignition..." MIESK5 NOTE; have parking brake on and someone in driver seat with foot on brake and transmission in park or neutral.
from Ford EVTM; The Control Module (PCM) runs the pump{s} for one second when it receives an ignition- on signal. It also runs the pumps as long as it receives a PIP signal from the Hall-effect devices, it continues pump operation even after the key is released from START. If the PIP signals fall below 120RPM, the control module cuts off the signal to the fuel pump relay or the integrated relay control module. The pump will also run when the terminals of the fuel pump test connector are jumped. the Control module signals the pump when it receives a CRANK signal, and when the Control module gets PIP signals that the engine is running. the pump does not run if the PIP indicates the engine is not running even with ignition ON {except for that first one second}.