The directed Θ₆-graph partitions the plane around each point into six 60° cones and connects points based on proximity within each cone. The spanning ratio — the worst-case ratio between the shortest path in the graph and the straight-line distance — measures how well this sparse construction preserves distances. It was known to be between 4 and 7. Bonichon & Bose (arXiv:2603.09048) prove it's exactly 5.
This is the first tight bound for any directed Θ_k-graph. Previous work gave bounds with gaps of 2 or more. Closing the gap completely requires qualitatively different techniques: the lower bound must construct a point configuration that forces a path exactly 5 times the direct distance, while the upper bound must prove no configuration can do worse.
The lower bound uses a converging series: a sequence of point placements, each adding a tiny detour, that converges to ratio 5 in the limit. No finite configuration achieves exactly 5 — the bound is a supremum, approached but never reached. The upper bound uses linear programming: for any candidate path between two points, the LP certifies that no sequence of edges in the Θ₆-graph can exceed ratio 5.
The two proof techniques are from different mathematical traditions. The converging series is analytic — a limit argument from real analysis. The linear program is algebraic — a finite-dimensional optimization. They meet at 5.
For computational geometry, tight spanning ratios determine whether a graph is suitable for routing. A ratio of 5 means the longest detour through the Θ₆-graph is at most 5 times the direct distance. This is usable for wireless sensor networks, where energy-efficient routing through sparse graphs is the design constraint. The exact answer — not “between 4 and 7” but “exactly 5” — turns an engineering guideline into a guarantee.