From Discrete Rigidity to Fractal Residuals: Constraint Persistence Across Continuum Limits

We study a minimal constrained orbit system exhibiting complete rigidity in a finite discrete setting.We show that this rigidity collapses entirely under an isometric continuum limit, leading to full reachability. We then introduce a contractive continuum model, in which a nontrivial residual set emerges as the attractor of an iterated function system.The Hausdorff dimension of this residual set is computed explicitly using standard IFS theory and verified numerically via box-counting simulations. The work clarifies how global constraints may disappear or persist under different continuum limits.