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\[d \!(n,|d^r+(n/d)^r|), r≥ 2.\]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 \(Ar,m(n)\). 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\[∑n≥1{Ar,m(n)}{n^s}=ζ(s)ζ((r+1)s)/ζ((r+2)s)Lr,m(s),\]where \(Lr,m\) is absolutely convergent in\(Res>1/(r+2)\). The isolated factor at the unique smallestnonzero symbolic denominator yields the two-term summatory asymptotic\[∑n≤ XAr,m(n)= Cr,mX+ Dr,mX1/(r+1)+Or,m,ε\!(X1/(r+2)+ε),\]with explicit Euler-product constants and\( Dr,m<0\). 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.