Structure-Preserving Coefficient–Operator–Mellin Transport Theorem

Structure-Preserving Coefficient–Operator–Mellin Transport Theorem One Composition Law, Three Working Realizations This theorem establishes an explicit, invertible transport among three classical realizations of multiplicative arithmetic: arithmetic coefficients, scale-transfer operators, and Dirichlet/Mellin symbols. Its central correspondence is: arithmetic coefficients under Dirichlet convolution↔ scale-transfer operators under composition↔ Dirichlet/Mellin symbols under multiplication The point is operational: Move the identity. Keep the law. An arithmetic relation can be moved into the representation where its relevant operation becomes easiest to see or manipulate, worked there, and transported back while preserving the compositional structure established by the theorem. The Three Realizations Coefficient space Arithmetic information is encoded by coefficient sequences, with Dirichlet convolution as the native composition law. Operator space The same information is represented by scale-transfer operators, where Dirichlet convolution becomes ordinary operator composition. Mellin/Dirichlet space Mellin monomials diagonalize the scale-transfer action, converting operator composition into multiplication of the corresponding Dirichlet symbols. These are not three unrelated descriptions. They are three working realizations of the same algebraic mechanism. What the Theorem Makes Possible The theorem turns representation choice into a usable mathematical move. It allows a researcher to ask: Where is this identity easiest to work with? Can an arithmetic convolution be converted into operator composition? Can an operator relation be diagonalized into scalar multiplication? Can a result obtained in one realization be transported faithfully into the others? Which features survive transport automatically, and which require additional structure? The contribution is therefore not merely that the three classical ingredients exist. It is their assembly into a single explicit transport instrument with a stated preservation boundary. Preservation Boundary The theorem preserves the relevant algebraic and compositional structure. It does not assert that every metric, geometric, spectral, topological, or analytic property automatically transfers between realizations. Such properties require their own hypotheses and arguments. That distinction is deliberate: transport what the theorem actually preserves; do not smuggle additional structure across the boundary. Relation to Relation-Primary Arithmetic This theorem is AP-001 of the larger Relation-Primary Arithmetic (RPA) research architecture. RPA generalizes the same research motion: transport → compare → measure loss → lift → cross-constrain → exclude AP-001 is the cleanest exact example of that architecture in action: the same compositional law made native in three different mathematical habitats. Parent architecture:Relation-Primary Arithmetic: Moving Problems Across Representations — A 133-Component Architecture Toward the Riemann HypothesisDOI: 10.5281/zenodo.22731871 Summary The Structure-Preserving Coefficient–Operator–Mellin Transport Theorem provides a faithful three-way transport among Dirichlet convolution, scale-transfer composition, and Dirichlet/Mellin multiplication. It is designed to be used as a mathematical switching device: move a problem into the representation where the needed operation becomes native, work there, and return without losing the law that connects the three realizations. Citation: Woffinden, A. (2026). Structure-Preserving Coefficient–Operator–Mellin Transport Theorem, v1.0. Zenodo. DOI: 10.5281/zenodo.22731873License: © 2026 Aaron Woffinden. Released under CC BY-NC 4.0.