(0) There could be a Siegel zero , like an "indentation" in the Riemann zeta function. The prime numbers need to all collaborate to "shift" the zeta function a little in order to "buckle" the zeta function at the Siegel zero, so there's only really enough room in the primes to accomodate one Siegel zero at most.
(1) Ultrafinitism  comes in many flavors, including a flavor where there is a single largest prime number . In this ultrafinite physical world, the largest prime determines how large any object can be before folding back in on itself. Since this would be a true statement about natural numbers, it would also be a sort of conspiracy amongst the primes, with only one instance.
Situations like this article remind me of both simultaneously: If there's any sort of conspiracy amongst the primes to forge a Siegel zero, it would have to be the same conspiracy which produces an ultrafinite prime candidate; the primes don't have bandwidth for two conspiracies.
This is why I don't like ultrafinitism: It insists on mixing levels, like an Escher print.
The world of pure mathematics isn't the physical world, for all the same reasons Santa Claus isn't real, and trying to bind pure math to some (theoretical!) properties of the physical world is just as wrong-headed as insisting that nobody can ever make songs about Santa because we know nobody can deliver toys that quickly.
The spacing is quasicrystalline.
The interesting thing about quasi-crystals, are that they can be modeled as higher dimensional crystals projected on a lower dimension....
(quasicrystals are super stable, which is why we find them in meteorites, we have no known way of manufacturing bulk quasicrystals).