Divisor-overlap geometry on the integers: exact structural results and a negative benchmark for interconnect topology

This work develops and analyzes a metric on the positive integers derived from normalized common-divisor structure. Let τ(n) denote the number of positive divisors of n. The construction defines a similarity kernel from the divisor count of (n,m), normalized by the divisor counts of n and m. The associated distance is shown to admit an exact Euclidean realization: every integer can be represented by a normalized divisor-incidence vector, with the kernel equal to the Euclidean inner product of the corresponding vectors. The resulting distance is therefore a genuine Euclidean metric on the positive integers. The work establishes several exact arithmetic properties of this geometry, including monotonicity with respect to common-divisor structure, conditional multiplicative self-similarity under the necessary coprimality condition, and an exact upper bound relating similarity to the relative divisor counts of two integers. The equality case of this bound recovers the divisibility relation, connecting the construction directly to the classical divisor graph. The metric is then converted into finite threshold graphs and examined as a possible mathematical model for network interconnect topology. The principal benchmark uses the integers 2,…,129 and a threshold of t=1.01, producing a graph with 128 vertices and 1,485 edges. Its average shortest-path length, diameter, maximum degree, and normalized betweenness are compared with reference graph constructions, including random graphs with matched vertex and edge counts. Additional experiments at N=64, 128, and 256, using 20 independent matched random-graph realizations at each scale, show that the tested divisor-overlap construction exhibits substantially greater concentration of shortest-path traffic than the corresponding random reference. The observed disadvantage increases across the tested range. The engineering conclusion is deliberately negative: the tested divisor-overlap topology does not demonstrate an advantage over a random topology with the same wiring budget on the principal network metrics examined. This result is treated as a mathematical and methodological finding rather than as a failure of the underlying construction. The principal contribution is therefore the mathematical framework itself: a concrete Euclidean geometry of divisor structure and a reproducible family of arithmetic similarity graphs connecting number theory, metric geometry, and graph topology. The work does not make claims concerning quantum computation, quantum entanglement, cryogenic systems, physical hardware, or experimentally realized physical phenomena. All constructions and benchmarks are classical mathematical and computational objects. The paper also identifies several directions for further study, including threshold spectra, degree distributions, spectral properties, arithmetic community structure, alternative randomized null models, and the asymptotic scaling of the resulting graphs. Author: Nadia SahraouiORCID: 0009-0002-7049-9794DOI: 10.5281/zenodo.21905803