Thanks for that !

Same type of TE and same era, also a major brand, so that means my TE was probably trying to solve the same problem.
So that means it's probably not a two stage power ramp up, but an overcurrent protection.
It starting to make sense now.
I just replaced all the x13 "small" tubes, one at a time. TE never buzzed, and all tubes light up.
But as soon as I plug either or both of the big 6080 tubes, it buzzes.
So that makes sense. These two big tubes means big power hence the ones an overload protection system would want to monitor / protect.
I had a look at the manual for the HP 212A, PSU department. They have a couple 6AS7, similar to the 6080, and their relay seems related to these tubes.
So my relay must be not to switch the full on power, but rather like HP to protect the x2 6080 tubes if they are overloaded, hence why it only buzzes when these tubes are plugged in, and doesn't care about any of the other tubes.
So I guess this means my relay is kaput, as it fails to cut power (either to the tube, or the entire instrument, depending how it's implemented), and rather it's the fuse blowing.
So I need to check that as well once the root cause of the problem is fixed.
So this also means I must have an actual overload issue, since the fuse blows, rather than a "false alarm", the relay being triggered when it should not.
Also... I found out in my Tek tube spreadsheet, that I have a few of those 6080 in some of my scopes. There is one in each of my seven

type 317.
So I picked one... was actually marked 6AS7 not 6080, but well.... tried it anyway, looked the same. Result = it still buzzes.
Then in doubt, I tried pulling the 6080 from my only known good type 317... the one I restored years ago. It's an actual 6080 not an 6AS7, so confidence was higher.
It still buzzes.
So we are making progress !
1) Buzzer and relay most likely an over current protection circuitry
2) Mlost likely protecting whatever those two 6080 tubes are feeding.
3) The two 6080 are not at fault.
4) Buzzer most likely like HP, trigger at the same time as the relay
5) Relay fails to protect the instrument since the fuse blows instead. So eitehr defective, or its circuitry.
6) Relay secondary issue but not the root cause.
So either the rails fed by those 6080s are shorted, or the circuitry driving the grids of those tubes, is faulty and grounds the grids, making them conduct fully.
However it still kinda intriguing that just when it buzzes, and before the fuse blows... all power rails are up, which would mean none are shorted, and still intriguing that all of them read 50% of what they should be...
So maybe we have a third issue, maybe we do have a two stage power up sequence, and it's faulty.
This is repair is not an easy one... zero schematic, 15 tubes just for a PSU... I will keep at it fora few days but I am not too hopeful, looks like more than I can chew
