Finite capture and the closure of roots of restricted polynomials

We study how a countable algebraic root set passes to a fractal connectedness locus. Let $D_n=\{-n+1,-n+2,\dots ,n-1\}$, and let $R_n$ be the set of roots of monic polynomials whose non-leading coefficients lie in $D_n$. We study $\overline{R_n}\setminus \bar{\mathbb{D}}$. Outside the closed unit disk this set equals a connectedness locus $M_n$ for a collinear affine iterated function system, or equivalently the zero set of reciprocal power series $1+{\sum }_{k\ge 1}{d}_k{c}^{-k}$ with $d_k\in D_n$. For non-real parameters in the lens $X_n=\{c\in \mathbb{C}\setminus \bar{\mathbb{D}}:\mathrm{\mid }c\pm 1\mathrm{\mid }<\sqrt{2n}\}$ we construct a canonical trap and enclosure for the associated difference attractor and use them to define finite-capture sets ${\mathrm{\Theta }}_k(n)$ for the marked point $2c$. Our main result is the uniform inclusion $\overline{{\mathrm{\Theta }}_k(n)}\cap (X_n\setminus \mathbb{R})\subset {\mathrm{\Theta }}_{k+2}(n)$ for every $k\ge 0$. Consequently, $(M_n\cap X_n)\setminus \mathbb{R}$ is exactly the closure of the finite-capture locus. The paper combines explicit trap geometry with certified inverse search. Moreover, $M_n\setminus \mathbb{R}\subset X_n$ for every $n\ge 20$, and this is sharp for $2\le n\le 19$. Thus, for $n\ge 20$, the non-real part of $\overline{R_n}\setminus \bar{\mathbb{D}}$ is exactly the closure of the finite-capture locus.