Arithmetic Tent Dynamics of Signed Power-GCD Maps on Divisor Lattices: Necklace Cycles, Mean Counts, Exact Basin Trees, and Asymmetric Powers

For \(n\ge 1\), \(r\ge 2\), and \(\varepsilon\in\{-1,+1\}\), thispreprint studies the divisor-lattice map\[T^{(r)}_{n,\varepsilon}(d)=\gcd\!\left(n,\left|d^r+\varepsilon(n/d)^r\right|\right).\]The map is reduced primewise to\[F_{\alpha,r}(\beta)=\min\{\alpha,r\beta,r(\alpha-\beta)\},\]which proves sign independence and identifies each normalized coordinatewith a truncated tent map. The relation with Yao-Qiang Li's multi-branchgeneralized tent maps is stated explicitly: the maps coincide when \(r=2\),and for even \(r>2\) the nonzero periodic subsystem uses Li's two extremebranches. Nonzero local cycles on the GCD-induced finite grid are classifiedby primitive binary necklaces and an exact reduced-denominator divisibilitycondition.The denominator of a necklace gives its least realization exponent, anexact mean formula for local cycle counts, and, together with the\(r\)-power content of the grid multiplier, the exact depth, layer sizes,and total size of the cycle's basin tree. Every positive integer isrealized as an exact period. The paper also treats asymmetric powers\[\gcd\!\left(n,\left|d^a+\varepsilon(n/d)^b\right|\right)\]in the sharp clean range \(a,b\ge2\). It obtains a weighted-necklaceclassification and an explicit exact \(k\)-cycle for every \(k\). Version2.2 adds a proof that reduced denominators are invariant around everyweighted periodic orbit and states the exact off-cycle tree shape withoutthe earlier binary-tree ambiguity. The degree-one boundary is shown to bequalitatively different because sign dependence can survive below the gcdcap. Computational verification uses exact integer and rational arithmetic andchecks 52,311 cases. The computations support but are not used in theproofs.