Johnson optimality and uniqueness for distance ratio $1:\sqrt{2}$
We prove that a two-distance set in $\mathbb R^d$ with distance ratio $1:\sqrt{2}$ has at most $d(d+1)/2$ points for every $d\ge9$. Equality in full affine dimension characterizes the Johnson construction; for $d\ge10$, the full-dimensional condition is automatic. The proof separates configurations according to the connectivity of the short-distance graph. The connected case uses root lattices and intersection theorems, while the disconnected case uses an orthogonal decomposition into component direction spaces and the span of their center differences. We also prove a relative bound for Euclidean two-distance sets and show that equality forces sphericity. Together with the Larman--Rogers--Seidel theorem, these results give maximum cardinality $d(d+1)/2$ for $9\le d\le21$ and $275$ for $d=22$, without exhaustive enumeration.