Global Periodic-Point Statistics for Higher-Order Sum-GCD Dynamics: Euler Products and a Negative Secondary Term

We study the global arithmetic statistics of periodic points arising from thehigher-order sum-GCD divisor-lattice dynamics\[d\longmapsto \gcd\!\left(n,\left|d^r+(n/d)^r\right|\right),\qquad r\ge 2.\]For a fixed iterate length \(m\), we prove a self-contained word--gridcorrespondence for the local prime-power system, obtaining a multiplicativefixed-point counting function \(A_{r,m}(n)\). More generally, a finitedenominator spectrum with a unique minimum is shown to satisfy an abstractEuler-product and summatory transfer principle. Applied to the GCD dynamics,this gives the explicit factorization\[\sum_{n\ge1}\frac{A_{r,m}(n)}{n^s}=\zeta(s)\frac{\zeta((r+1)s)}{\zeta((r+2)s)}L_{r,m}(s),\]where \(L_{r,m}\) is absolutely convergent in\(\operatorname{Re}s>1/(r+2)\). The isolated factor at the unique smallestnonzero symbolic denominator yields the two-term summatory asymptotic\[\sum_{n\le X}A_{r,m}(n)=\mathfrak C_{r,m}X+\mathfrak D_{r,m}X^{1/(r+1)}+O_{r,m,\varepsilon}\!\left(X^{1/(r+2)+\varepsilon}\right),\]with explicit Euler-product constants and\(\mathfrak D_{r,m}<0\). Thus a local symbolic denominator gap produces aglobal, signed secondary term. We also derive exact-period and orbit-countingidentities by Möbius inversion, explicitly allowing cancellation of thesecondary coefficient, and extend the mechanism to asymmetric branch slopes,where the secondary exponent is controlled by the right branch. Version 1.2adds a self-contained asymmetric word--grid proof and a weighted-denominatorinvariance argument. The special case \(m=1\) is identified with the classical count of unitarydivisors that are perfect \((r+1)\)-st powers; the novelty lies in theall-iterate symbolic Euler factors, the universal negative secondary term, andthe asymmetric extension. Exact computations accompany the paper asverification and reproducibility evidence but are not used as proofs.