A Spectral Construction of Natural-Number Arithmetic

We construct natural-number arithmetic on an ordered family of correlation modes and retain the expressions by which its elements are generated. Minimal unbiased compositional change determines a rotation plane and a Gaussian law. Local information and total normalized correlation determine a finite window; preservation of constants selects its Neumann closure within the separated homogeneous local boundary class. Modulation and composition of the oriented modes define an ordered Peano semiring with unique factorization. The modes retain quadratic energy and canonical scale transport, while winding degree certifies the arithmetic. Ordered expressions have finite evaluation fibres. Equal root-channel weights and independent subhistory assembly determine a positive history law whose Gaussian marginal is the original mode mass. Conditional readout separates value information from internal history variation by exact variance and entropy identities. Using the same weights, parent–child assembly defines a nonnegative self-adjoint trace-class operator with explicit Gaussian cutoff bounds. Its contraction evolution couples value coefficients to history differences with zero initial readout: an exact matrix element at value four proves this coupling on the complete history space. The coupling persists under small changes of the relevant conditional weights. Eliminating the history differences gives an exact integral equation for the value response. Lean formalizations accompany the arithmetic construction, history-source recursion and assembly results.Reading and research materialsSSRN preprintArticle and supplementary materialsJinku Guo · Northwestern Polytechnical University · ORCID 0009-0000-6600-6171.This metadata record provides a research summary and links to the work. Full texts, software and supplementary files are available at the linked repositories under their stated licences.