Arithmetic Periodicity of Gonality under Regular Subdivision

For each fixed finite connected loopless multigraph G and degree d, we give an effective finite union of finitely generated additive submonoids describing all positive k for which the regular k-subdivision has divisorial gonality at most d. Consequently the gonality sequence eventually depends only on gcd(k,M) for a computable M. We also prove that every tricycle in the family of van Dobben de Bruyn, Smit and van der Wegen has gonality five for even k and six for odd k; midpoint necessity yields the all-parameter formula. Their rational rescaling method, tricycle construction, degree-five reduced-divisor restriction and known one-to-two subdivision drop are explicitly credited, together with prior integer-programming certificates. The paper includes complete written proofs and portable exact finite checks. It does not claim an efficient algorithm, small period bound, resolution of odd-fold drops on other graphs, or exhaustive closure of the broad Egres question AMR-029-0040. AI-assisted self-audited preprint; no independent review, formal verification or absolute-priority certification is claimed.