Axiomatic Emergence AC; The Physical Origin of Mathematical Structures from Information and Computation

This article proposes and systematically expounds a research program based ontwo fundamental physical and computational principles—Information Conservation(A1) and Computability (A2). Its aim is to derive, starting from these two axioms,the core structures of modern mathematics and their realization in the physical universe. The program indicates that under the constraints of A1 and A2, discrete andself-referential dynamical systems spontaneously organize into hierarchical mathematical objects, including sets, categories, higher-order categories, toposes, spacetime geometry, and non-classical logics. A salient feature of this program is thateach stage of mathematical structure is bound to a specific, testable astronomicalor physical observational prediction, thereby placing the entire theoretical systemwithin an empirically testable framework. If all predictions are confirmed, it supports the proposition that ”mathematical structures are the inevitable emergenceof information conservation in a computable universe”; if any prediction is falsified,it only provides grounds for revising the formalization of axioms or the derivationpath, without undermining the established mathematical system.