M104 air pump run after warm

[tan man]

1995 E320 with 224K
When I start the car cold, the air pump works for about 2 minutes, then cuts off for 2 minutes. But then it will go on again for 30 seconds, go off for 2 minutes, then on again for 30 seconds and finally shut off for good.
Additionally when I start the car cold, o for a 2 mile drive then pull to a stop, the air pump will come on for 30 seconds.
What/where is the brain that controls the air pump? I supposed that is what is faulty that I need to get at. Too bad this car has a serpentine belt, otherwise I would bypass the air pump altogether.
Unrelated or perhaps related, I've had a code 4 & 26 for years which is related to the upshift delay system. The green vacuum hose running to the trans is not holding vacuum and it is too difficult to get at the upshift cable mechanism so I plugged the system at the firewall. Codes came back So I assume the upshift circuit is also faulty because the firewall junction for vacuum tube is open/sucks even when car is warm/hot.
Any insight appreciated.