I agree with all your claims here in the literal sense, but suppose there's a minimum length, then it would seem to be at least theoretically possible to use di

I agree with all your claims here in the literal sense, but suppose there's a minimum length, then it would seem to be at least theoretically possible to use discrete mathematics to formulate an approximation to the "real number formulation of physics"? The fact that we don't have already a full system using discrete maths doesn't mean it is impossible, because our current system is based on a long tradition of belief in real numbers, and assuming physical space is continuous. I'd argue (admittedly unhelpfully) that unless we have actually tried to formulate physics using discrete mathematics and found a barrier that we prove unequivocally that it is impossible to overcome, we can't claim that physics must be formulated using real numbers/continuous math. There's a difference between "we don't know how to do this" vs "we know we can't do this".