computer controller sounds like a good bet right now, especially if it is a "timed" thing. all the doors are vacuum controlled and something is telling them to open and close. dont sound like you are loosing vacuum, sounds like it is being turned off.

 

Here's an excerpt from the service CD: 2002 F150 Climate Control.pdf

Go to the section starting on page 7 for a schematic view of how the airflow is controlled. There is no single "mode" door. There is a temperature blend door that is electrically controlled (item #5). There are three airflow control doors, all vacuum operated, that control inlet source and outlet direction. The vacuum is switched by electrically-operated solenoids within the EATC. ALL four of these doors are controlled by the EATC module. The EATC also controls the blower speed via the blower speed control module.

On page 11 is the EATC self-test that you can execute from the front panel. It may give you additional clues. I'd suggest trying it when the fault occurs.

I'd lean very strongly towards condemning the EATC control head (computer) based on your reports of the repeatability and specific timing of the appearance of the fault. I'd also keep in mind that the EATC does communicate with the PCM so there might be some interaction there.

Steve