From GCA to a unified theory of twisted arithmetic dynamical systems

This work introduces a unified theoretical framework for the study of arithmetic processes that exhibit systematic but non-random violations of classical conservation or inheritance laws. Building on constructions originating from Geometric Composition Algebra (GCA), we formalize these phenomena under the notion of Twisted Arithmetic Dynamical Systems (SDAT).

Rather than treating deviations from exact laws as negligible error terms, the proposed framework places such deficits at the center of the analysis. An axiomatic structure is introduced to capture p-adic stratification, low-dimensional spectral behavior, and asymptotic stability of normalized deficits. From these axioms, a family of universal invariants is defined, allowing meaningful comparison between arithmetic dynamical systems with very different microscopic definitions.

The framework is illustrated through two complementary examples. The Collatz map is used purely as a motivating empirical system, without claiming new results on its convergence. In contrast, the FUSE system, derived from GCA, provides a fully computable and non-conjectural example in which all proposed invariants can be explicitly evaluated and tested numerically. Extended computations reveal stable p-adic patterns, a finite Fourier dimension, and convergence of normalized deficits.

Finally, the work formulates a twisted conservation law conjecture, suggesting that apparent violations of conservation may be absorbed by structured correction terms combining p-adic and spectral components. While exploratory in nature, the SDAT framework offers a coherent language for organizing arithmetic phenomena beyond exact identities and opens new directions for the structural study of arithmetic dynamics.