Remainder Structures

This paper introduces Remainder Structures, a framework that treats modular reduction as a primitive mathematical operation rather than merely a derived arithmetic procedure. The framework is built upon a selectively chosen system of axioms governing a binary remainder operation. By examining different combinations of these axioms, the paper investigates how familiar modular arithmetic can be characterized and how alternative remainder systems can be constructed. From these structures, the paper develops the concept of Residue Clusters: transformation monoids generated by compositions of reduction maps. Their algebraic properties are studied, including idempotence, noncommutativity, maximal subgroups, and structural relationships with the underlying remainder systems. The paper further introduces Reduction-Compatible Semirings, extending the investigation to algebraic systems in which addition and multiplication interact compatibly with modular reductions. Several structural results are established, alongside finite computational investigations and independently cross-checked implementations. The computational results are explicitly distinguished from general mathematical proofs, and their limitations are discussed. Finally, the paper presents open problems concerning the classification, asymptotic behavior, reconstruction, and infinite extensions of residue clusters and related algebraic structures. The aim of this work is to explore modular reduction as an independent algebraic primitive and to investigate the mathematical structures that emerge from it.