A Formal Exposition of the Recursive Cosmos: Axioms, Structure, and Foundational Theorems

We propose a new theoretical framework, the Recursive Cosmos, founded on the principle that physical reality is an infinitely nested hierarchy of self-contained "Systems." The central postulate of this theory is the RG-Physical Correspondence Principle: the physical process that transforms the laws of nature between hierarchical levels is mathematically identical to the flow of the Renormalization Group. Using the languages of Category Theory and the Functional Renormalization Group, we construct a mathematically rigorous and self-consistent model. This framework offers a unified perspective on a wide range of fundamental puzzles, proposing that phenomena such as the Hierarchy Problem, Dark Matter, Dark Energy, and the Black Hole Information Paradox are not independent problems, but interconnected consequences of our universe's position within the cosmic hierarchy. The theory's predictive power is enforced by a "Principle of Hierarchical Consistency," which demands that a single set of universal parameters simultaneously accounts for multiple, unrelated observables. This leads to a web of sharp, falsifiable predictions, including specific relations for the Hubble tension, the muon g-2 anomaly, and the scaling of black hole jet power. We present the Recursive Cosmos not as a final theory, but as the foundation for a new, testable research program, offering a novel, relational worldview and a clear path toward experimental verification or falsification.