Complete Classification of Signed Higher-Order Sum-GCD Dynamics on Divisor Lattices: Tent Maps, Binary Necklaces, and Every Finite Period

We study the divisor-lattice self-map\[T⁽ʳ⁾n,ε(d)=\!(n,|d^r+ε(n/d)^r|), r≥2, ε∈\{-1,+1\}.\]Writing \(n=∏_p pα_p\) and\(d=∏_p pβ_p\), we prove the exact coordinate formula\[v_p(T⁽ʳ⁾n,ε(d))=min\{α_p,rβ_p,r(α_p-β_p)\}.\]Consequently, the positive and negative signs define exactly the samemap for every \(r≥2\), and all prime coordinates evolve independently.After normalization, each coordinate is a truncated tent map. Weclassify all periodic orbits by primitive binary necklaces togetherwith a reduced-denominator divisibility condition, derive exact localand global periodic-point counts, classify all fixed points, andreconstruct the full functional graph by inverse branches. Mostnotably, every positive integer occurs as an exact period: for every\(k≥1\), every \(r≥2\), and every prime \(p\), the choice\(n=pr^k+1\), \(d=p^r\) yields the exact cycle\(p^r↦ pr^2↦⋯↦ pr^k↦ p^r\).The upload includes the paper and an exact-arithmetic Pythonverification script.