22 pages, 7 figures. Exploratory manuscript. We introduce and study the family of arithmetic transformations parameterized by a prime 𝑘 and integers (𝑖,𝑗), generalizing the Collatz/Syracuse iteration. Two special regimes are analyzed rigorously by induction (global divergence for 𝑘=𝑖=𝑗; kernel 2-cycle for k=i,j=0). The remaining results (stability condition, cycle structure, division density, residue distribution, resistance, stopping time, coalescence, footprint) are conjectural and supported by numerical experiments on n≤10^7 ,k≤47. Python code available at: https://github.com/aniskh/sufes