Axiomatic Emergence D; The Physical Origin of Higher-Order Categories

This paper is the third part of the research program ”Deriving MathematicalStructures from the Axioms of Information Conservation and Computability.” Weprove that in a self-referential discrete quantum dynamical system satisfying Information Conservation (A1) and Computability (A2), to achieve self-consistencyin the multi-scale description of the system’s evolution, its mathematical framework necessarily extends from an ordinary category to a higher-order category(∞-category) and topological ∞-stack. The core mechanism is that the noncommutativity of the system’s ”coarse-graining” operation sequences naturally givesrise to 2-morphisms describing the relationships between different ”computationalpaths.” Furthermore, the self-referentiality of the system, which implies a decisionlimit isomorphic to the halting problem, guarantees that this tower of morphismsmust extend to infinite higher orders (i.e., forming an ∞-category). The InformationConservation axiom constrains all higher-order morphisms to be reversible ”entropypreserving homotopies.” To coordinate theoretical descriptions from different localobservational perspectives, these local ∞-categories need to satisfy global consistency conditions, thereby naturally organizing into a topological ∞-stack. Thegeometric structure of this ∞-stack encodes topological information of the physicaltheory space and leads to a specific, falsifiable cosmological prediction: there shouldexist a sinusoidal oscillatory modulation with an amplitude of approximately 0.008and a frequency around 10−4 Hz in the primordial gravitational wave power spectrum produced during the early universe’s inflationary epoch. This prediction canbe tested by future space-based gravitational wave detectors such as LISA.