Dear Arved Huebler,
thanks for your comment. I think what HoTT does, is to facilitate a formal system which in its inner core states that without consistency there is no existence possible. If one assumes HoTT to be consistent itself, this leads to the impression that HoTT must capture and represent the fundamental truth. But not all consistent schemes do necessarily meet reality. Maybe HoTT does answer this question in the positive and says that all mathematically consistent formal systems are to be generated at some point (in time) and therefore indeed meet reality.
I think HoTT does state a triviality, namely that inconsistencies do not pave the path from the abstract to the concrete (especially physical reality). This path should symbolize the well known deductive principle in logics. Now, HoTT makes a huge leap by saying that if something suffices consistency, it must exist (somewhere, somehow). If one does understand HoTT as merely stating that consistency is necessary for existence, but not sufficient, i would wonder what the needed sufficient additional properties are to make something physically existent.
By assuming an entity of 'non-existence', and by stating that only consistent structures can come into existence, this 'non-existence' must be considered as existent - because within HoTT, it has the feature of consistency. Otherwise it could never produce the whole chain of events that led to our universe. Stated differently: the entity of 'non-existence' has a single property, namely the potential to produce something. This is a somewhat trivial logical conclusion based on our factual existence. Therefore one must understand your entity of 'non-existence' as existent. How can one and the same thing be existent and non-existent at the same instant? I really think that we cannot logically consistent assume such a non-existent entity being the first unmoved mover. And because this is logically inconsistent to me, i think, it must fail, although i nonetheless do not exclude that mathematics was generated by some other causes than itself.