Cristi,
"What remains then" is indeed a legitimate question about my setup, which is quite minimalistic in its most general form. It is also worth asking if the causal metric hypothesis trivializes deep and subtle issues. My view is that one of the principle reasons manifold models have dominated physics is because they are so convenient mathematically; once you know about the continuum and the complex numbers their lure is almost irresistible. Hence, more primitive and messy approaches may have been neglected.
Coming from a math background and working mostly with algebraic schemes and complex manifolds, it is hard for me to believe that the physical world behaves in such a convenient way. Conceptual simplicity and mathematical convenience are very different! This essay and all the unpublished work associated with it represent my attempt to "think physically" rather than just mathematically; my focus here is the basic physical principles, and the associated math is not nearly as convenient as the math encountered in mainstream physics. In any case, I think approaches like this deserve more attention.
You seem to have some of the same philosophical motivations, refusing to reject singularities just because they are "mathematically ugly."
Take care,
Ben