Geometric Constraints and Combinatorial Complexity in the Toroidal <em>N</em>-Queens Problem: Part II

The toroidal N-Queens problem imposes modular constraints on queen placements, modeled as a simplicial complex XN where edges encode conflict-free pairs and simplices represent consistent configurations. We prove solutions exist if and only if gcd(N, 6) = 1, leveraging modular arithmetic and toroidal symmetries. Topological obstructions, analyzed via cohomology, limit global solutions for composite N, while elliptic curve embeddings reveal geometric structure. The solution space TN grows exponentially (|T5| = 10, |T7| = 28), challenging enumeration for N &gt; 200 due to the torus&rsquo;s periodic constraints. An energy function &epsilon;(&pi;) identifies hyperstable solutions as isolated minima, offering insights into combinatorial complexity and high-dimensional discrete optimization.