Counting intersections of rational cyclic grids: an application of Smith normal form

Choose two points from equally spaced grids on a circle. How many choices put both their sum and their difference on specified target grids? A small example leads from fractions to integer congruences. We then use classical Smith normal form to obtain the exact count, carefully allowing the two input grids to have different sizes. Inclusion–exclusion handles a union of grids. The article is an exposition and application of established counting methods, not a claim of a new general theorem. Author contribution: The author reports that he proposed the motivating hypothesis and provided logical and scientific direction for this exposition, with AI assistance. This contribution statement makes no originality or priority claim for the classical mathematics used here. AI assistance and review status: ChatGPT (OpenAI) provided the main AI assistance with generative drafting, editing and reorganization, mathematical inspection, reference checks, and examples. Codex assisted the present manuscript preparation. The author reports limited early-stage assistance from Claude (Anthropic); its specific contribution to this note and the exact historical model versions are not established by the retained records. This assistance was not limited to language correction. The author reports having read the manuscript. Mathematical inspection was AI-assisted; completed personal verification of the mathematical proof is not claimed. No independent human peer review or proof-assistant kernel verification is claimed. Funding: The author reports no external funding, financial assistance, or institutional or company support. This is not a peer-reviewed journal publication.