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\[T^{(r)}_{n,\varepsilon}(d)=\gcd\!\left(n,\left|d^r+\varepsilon(n/d)^r\right|\right),\qquad r\ge2,\quad \varepsilon\in\{-1,+1\}.\]Writing \(n=\prod_p p^{\alpha_p}\) and\(d=\prod_p p^{\beta_p}\), we prove the exact coordinate formula\[v_p(T^{(r)}_{n,\varepsilon}(d))=\min\{\alpha_p,r\beta_p,r(\alpha_p-\beta_p)\}.\]Consequently, the positive and negative signs define exactly the samemap for every \(r\ge2\), 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\(k\ge1\), every \(r\ge2\), and every prime \(p\), the choice\(n=p^{r^k+1}\), \(d=p^r\) yields the exact cycle\(p^r\mapsto p^{r^2}\mapsto\cdots\mapsto p^{r^k}\mapsto p^r\).The upload includes the paper and an exact-arithmetic Pythonverification script.