Geometric Insertion Optimizer for TSPLIB

This paper introduces the Geometric Insertion Optimizer (GIO), a deterministic heuristic approach to the Traveling Salesman Problem (TSP). Unlike stochastic metaheuristics, GIO maps 2D spatial distributions into a 1-dimensional insertion sequence via a Fibonacci-inspired Archimedean spiral projection. The algorithm executes a structured parameter sweep across a torsion constant (Φ) and a geometric tension threshold (Ratio), initializing the topological expansion from every node in the dataset. GIO performs route construction using continuous floating point geometric analysis and angular penalization to mitigate spatial traps, while discrete distance evaluation is used for final validation. Empirical results on TSPLIB instances show that GIO consistently achieves optimal or best-known solutions for small instances (up to 100 nodes in the experiments), and maintains competitive near-optimal gaps (typically sub-1%) for medium sized instances. Results suggest that solution quality is strongly influenced by the initial geometric ordering induced by spatial projection, offering a deterministic alternative to stochastic construction heuristics