I suggest using the search function. This topic has been discussed in one form or another many times like everything else related to owning a Norton. You'll get more opinions with a search.
I've run synth and standard gear oil of all kinds in my P11 AMC gearbox. Makes little to no difference as long as it is gear oil. The gearbox does like to be treated like a baby, but mine has survived many shifts at 6000 RPM. I do change gearbox oil like I change engine oil though... Frequently
I've had my AMC gearbox apart 3 times, but not because it was broken. I just changed out the stock mainshaft for a Commando mainshaft to get proper engine sprocket and clutch sprocket alignment with an NEB clutch. All good in there. I have settled on cheap automotive 80-90wt hypoid oil in gallon jugs. Matter of fact what I put in there last night was from Walmart. lol Shifts the same and the bushes are not worn out because of using standard automotive gear oil. My best guess is the reported broken gearboxes would have broken no matter what oil was used in them.
There will undoubtably be disagreement on what my not so cool choice is, but I am not caring. I do me.