An Explicit GCD Evaluation for Three-Direction Torus Arrangements

We give an explicit gcd evaluation for three families of closed curves on the two-dimensional torus. The vertical coordinates form the union of two common-origin cyclic grids of orders N and R, while the other families are level sets of x+y and x-y on grids of orders Y and K. A finite-group calculation gives the triple intersections, and a cellularity proof justifies the passage from incidences to geometric faces. With d=gcd(Y,K), L=lcm(Y,K), and g=gcd(N,R), the count is F=(N+R-g)(Y+K)+2YK-d[gcd(N,2L)+gcd(R,2L)-gcd(g,2L)]. For R=12,Y=27,K=60, this becomes F(N)=87N+4248-84gcd(N,12)-3gcd(N,1080), giving 6264 at N=27. The note is an explicit specialization of established toric-arrangement and arithmetic-Tutte theory. It also identifies its triple-point count as a specialization of the author's earlier cyclic-grid exposition. Companion source and author-supplied reports of 284 passing finite checks are provided; the proof does not depend on extrapolating those checks. The author supplied the motivating idea, initial hypotheses and research basis, and pursued the work with AI assistance. Assistance from ChatGPT (OpenAI) and Claude (Anthropic), including calculations, proof arguments, exposition, literature comparison and code across the workflow, is disclosed. Internal mathematical inspection was AI-assisted. Independent human peer review and completed personal verification of every proof step are not claimed. The article, its manuscript source and identified separate documentation are licensed under Creative Commons Attribution 4.0 International (CC BY 4.0). The original supplementary software—hv27_torus_counts.py, run_hv27_checks.py and RUN_HV27_WINDOWS.bat—is licensed under MIT. Validation reports are covered by CC BY 4.0 only to the extent of rights held by the author; no exclusive rights are asserted in mathematical facts or unprotected data. See LICENSING_SCOPE.txt and the accompanying notices for exact file-level scopes. The accompanying files are the author-review preparation v08 of the preprint and its selected public supplement. The supplement contains the manuscript source, license notices, software, explanatory documentation and supplied validation reports. It excludes internal preparation records.