Deterministic Integer-Valued Information Routing on Four-Dimensional Discrete Lattices

This paper develops a discrete mathematical framework fordeterministic information routing on structured four-dimensionalinteger lattices. The framework represents computational states onvertices of $\mathbb{Z}^{4}$ and defines local interaction throughthe eight axial neighbors obtained under an $L_{1}$-metricneighborhood of radius one. A discrete routing operator is introducedusing the quotient--remainder decomposition $n=8q+r$, where aninteger quantity is distributed among the eight neighboring states byassigning $q$ units to each destination and one additional unit to adeterministic subset of $r$ destinations. The resulting operator isformulated and analyzed as a discrete state-transition rule. Inparticular, the paper establishes its conservation of the routedinteger quantity, bounded distributional imbalance, and deterministicbehavior under identical initial conditions. The framework furtherdefines source-state evolution and provides a mathematical basis forextending local routing to hierarchical and multi-scale discrete statespaces. The work focuses on the mathematical formulation andproperties of the routing system rather than on a physicalinterpretation of the underlying lattice. Computationalimplementations are considered only as verification mechanisms forthe derived mathematical properties. The results provide a formalfoundation for subsequent investigation of discrete multi-scalecomputation, invariant-preserving information transport, andspecialized computational architectures.