(as soon as bit 63 is set, the number would be in 'overflow state', but at the same time the lower 63 bits would preserve the wrap-around value in case that's useful).
On the other hand, compilers are already free to check the CPU carry flag bit and trap on it. Those flag bits are just not exposed to programming languages (which IMHO is a serious 'dissonance' between assembly and high level languages), also CPU flags can't be carried around with their associated result value and go 'viral' like NaNs (but whether that's even useful instead of trapping immediately is another question).
Also, look around you and realize that non-64-bit microprocessors outnumber 64-bit machines by orders of magnitude. There is more to the world than your laptop and smartphone. (Just within 1m of myself at this very moment I see 4 devices with a 32 bit microprocessor in them, and I'm sitting in a coffee shop reading a book. Well, and browsing HN now and then.)