Axiomatic Emergence E; The Physical Origin of Toposes

This paper systematically explores the origin of the mathematical frameworkrequired to describe the physical world, based on the two physical axioms of ”Information Conservation” and ”Computability”. The study proves that in a universemodel satisfying these two axioms, all possible local ”causal slices” and the physicaltheories defined on them, when satisfying global consistency conditions, naturallyconstitute an ∞-topos. The internal logic of this topos is proven to be intuitionistic, stemming from the locality and finiteness of physical observation. Its core realnumber object can only be defined by computable Cauchy sequences, i.e., ”computable real numbers”. This mathematical construction is not a purely theoreticalassumption; it leads to a precise, testable cosmological prediction: there exists acharacteristic positive offset of approximately 4.3 microkelvin in the quadrupoleamplitude of the cosmic microwave background (CMB) temperature fluctuations.Next-generation CMB experiments (e.g., CMB-S4) will be able to test this prediction, thereby providing empirical adjudication for the entire research paradigm of”physical principles driving the selection of mathematical structures”.