Generalized Multidimensional N-Queens on Toroidal Grids, Part III: Non-Linear Polynomials and Topological Insights

In this third installment of our series on the \textit{N}-Queens problem, we extend the classical puzzle to \( D \)-dimensional toroidal grids under the \emph{Unrestricted Directional Attack Model} (URDAM), where queens threaten along every nonzero vector in \( \{-1,0,1\}^D \setminus \{\mathbf{0}\} \). We model hyperdiagonal interactions via a conflict hypergraph equipped with an energy-minimization principle, and eliminate residual threats through non-linear polynomial mappings of degree \( \geq 3 \). Our main theoretical result establishes that a conflict-free arrangement exists if and only if: \( \gcd\!\left(N,\; \prod_{\substack{p \leq D \\ p \text{ prime}}} p\right) = 1. \) We develop constructive algorithms based on Latin hypercubes and orthomorphisms, and interpret solution spaces through toric geometry. Numerical experiments for dimensions \( 3 \leq D \leq 5 \) confirm both correctness and computational feasibility.