Saturated Aperiodic Packings with Finite Stability

We give explicit one-parameter families of saturated unit-ball packings in \(\mathbb{R}^2\) and \(\mathbb{R}^3\) with trivial translational stabilizer and exact square or cubic grid contact graphs. Every finite subfamily is locally isolated when its complement is fixed. For rational parameters, all centers and the planar equilibrium stresses are rational. The planar proof uses a positive reciprocal stress and a finite-support rigidity argument; a layered extension gives the three-dimensional result. Nevertheless, varying the parameter produces a nontrivial global motion continuous in the uniform displacement norm and preserving every contact. The result concerns finite stability, not rigidity against arbitrary motions of infinitely many balls. It is a scoped construction related to Musin's aperiodic-jamming question, not an unrestricted or strict-jamming solution. This unrefereed, AI-assisted preprint makes no absolute priority claim.