SDRIS: Limit Cycles in p-adic Topologies via Cayley-Dickson Algebras and H₄ Symmetries

We introduce within the Static-Dynamic Recursive Information Space (SDRIS) a background independent framework based on a discrete p-adic relational topology whose dynamics are coupled to the Cayley-Dickson construction. We analyze how the dimensional ascent through hypercomplex algebras leads to a progressive loss of algebraic structure, culminating in the emergence of zero-divisors. This algebraic mechanism induces a degradation of global connectivity within the underlying graph. When high-dimensional configurations are projected onto four-dimensional structures, the system is constrained by the non-crystallographic H₄ Coxeter group. The resulting geometry exhibits frustration, characterized by a positive angular deficit in icosahedral substructures. We discuss how this frustration can be interpreted, within a Regge-calculus framework, as a source of discrete curvature. By embedding the discrete constraints into a variational setting, we identify conditions under which the system may close into a dynamical cycle: at the terminal algebraic threshold, topological isolation favors a discrete resolution that returns the system to lower-dimensional configurations. While the precise form of this closure mechanism requires further specification, the framework provides a mathematically consistent setting in which geometric frustration and potentially cyclic behavior emerge from abstract algebraic rules alone. Data Availability Statement: The text and original figures of this manuscript are distributed under the terms of the Creative Commons Attribution-ShareAlike 4. 0 International License, with the exception of the source code reproduced in the appendix below. The included appendix code, along with the hosted Python repository, trained model weights, and associated simulation datasets, are permanently archived in the Zenodo repository under the copyleft European Union Public Licence (EUPL v1. 2).