lvl 6 awaken gem?

No, only uncorrupted.

There is an odds chart here

https://www.poewiki.net/wiki/Doryani%27s_Institute

Awesome, explains it well. I think that first list of outcomes would be useful on the Vaal orb page as well. They have some of that information, but it is more detailed here.

Checks out, with this discarding method it is correct that it is in fact double.

On first roll, 1/8 chance of success, with 6/8 chance of miss with a second roll, 1/8 chance of miss with chance of level being removed.

On second roll, where level chance was not discarded, 1/6 chance of success.

There by giving the total probability of +1 upgrade from Doryani institute to be:

1/8+6/8*1/6 = 1/8+1/8 = 1/4

Thanks Piousqd, I have looked for this explanation for some time and never seen it before, perhaps it has been updated or I was looking in the wrong place.

^yep exactly.

The fact that the double corrupt cannot choose the same main outcome is the key here.