Favourite Distances in Three-Space: Exact Suspension Counts and a Three-Arc Gap

Assign a positive radius to each point of a finite set in Euclidean three-space and count directed pairs at the assigned distance. Put B(n)=ceil(n^2/4+5n/2)+1. We prove that the exact maximum among n-point suspensions is B(n) for every n>=13, using Swanepoel's construction for sharpness. A maximum principle on the finitely generated group of circle angles supplies a rational increment, and a rational sine-product obstruction excludes equality in the earlier upper bound. We then prove that a suspension core with at least eight circle points and one genuine exceptional point has at most B(n) arcs; with two exceptions it has at most B(n)+3. Combining this with Swanepoel's structural theorem gives B(n)<=f_3(n)<=B(n)+3 for all sufficiently large n. Two exceptions in a large extremizer must be reflections across the circle plane. The exact unrestricted maximum and the conjecture that every sufficiently large extremizer is a suspension remain unresolved. AI-assisted, self-audited and unrefereed; no independent review, formal verification or absolute priority is claimed.