I thought for a moment you might have had a point on a technicality there Cosmo. However I can't find any specific reference to "distress" in the context of PIC emergency authority.
For PIC to take provisions under emergency authority, there has to be imminent danger - would you not agree?
5.3.1.1 Distress and urgency traffic shall comprise all radiotelephony messages relative to the distress and urgency conditions respectively. Distress and urgency conditions are defined as:
a) Distress: a condition of being threatened by serious and/or imminent danger and of requiring immediate assistance.
b) Urgency: a condition concerning the safety of an aircraft or other vehicle, or of some person on board or within sight, but which does not require immediate assistance.
Mayday = imminent danger = emergency = PIC can do whatever he deems necessary.
E.g. if not deviating from the cleared SID, single engine, it would cause a crash into a mountain.
Or
E.g. if not descending immediately, following a rapid decompression, it would mean the occupants onboard the aircraft would die from oxygen deprivation.