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

This article proposes and systematically expounds a research program based on
two 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 uni
verse. The program indicates that under the constraints of A1 and A2, discrete and
self-referential dynamical systems spontaneously organize into hierarchical mathe
matical objects, including sets, categories, higher-order categories, toposes, space
time geometry, and non-classical logics. A salient feature of this program is that
each stage of mathematical structure is bound to a specific, testable astronomical
or physical observational prediction, thereby placing the entire theoretical system
within an empirically testable framework. If all predictions are confirmed, it sup
ports the proposition that ”mathematical structures are the inevitable emergence
of information conservation in a computable universe”; if any prediction is falsified,
it only provides grounds for revising the formalization of axioms or the derivation
path, without undermining the established mathematical system.