Maximum Euclidean two-distance sets: determine $g(d)$ for $9\le d\le22$
Statement
A two-distance set in Euclidean space $\mathbb{R}^d$ is a finite point set realizing only two distinct pairwise distances. Let $g(d)$ be the maximum cardinality of a two-distance set in $\mathbb{R}^d$ - the EUCLIDEAN problem, with points anywhere in $\mathbb{R}^d$, NOT restricted to a sphere. Blokhuis proved $g(d)\le\binom{d+2}{2}$; the standard lower bound (the midpoints of the edges of a regular simplex) gives $g(d)\ge\binom{d+1}{2}$. Exact values of $g(d)$ are known only for $d\le8$. For each dimension $9\le d\le22$, determine $g(d)$: is $g(d)=\binom{d+1}{2}$, or can it be exceeded, as it is at $d=8$ (where $g(8)=45$) and anomalously at $d=23$ (where a $277$-point set exists)?
Acceptance. FULLY RESOLVES (per dimension $d$, an independent deliverable): the exact value of $g(d)$, via a matching construction (explicit coordinates = lower-bound certificate) and a proof or SDP-dual upper bound of the same value. ADVANCES: a construction exceeding $\binom{d+1}{2}$ in some $9\le d\le22$ (e.g. an anomaly at $d=22$), OR a certified SDP/LP upper bound tightening the known bound toward $\binom{d+1}{2}$ for some $d$ in range. Each dimension carries independent construction (config) and upper-bound (dual) certificates; numerical-only configurations without exact distance verification do NOT qualify.
Background
Framework from Larman-Rogers-Seidel (1977) and Blokhuis (1984). Exact Euclidean values are settled only for $d\le8$ (P. Lisonek, 'New maximal two-distance sets', 1997, giving $g(8)=45$). For $d\ge9$ only upper bounds (Delsarte / SDP linear-programming) are known: the Sep-2025 survey Z. Chen & W.-H. Yu, 'Bounds on two-distance sets in Euclidean space and Unit Sphere', arXiv:2509.00858, provides bounds but no exact $g(d)$ in $9\le d\le22$. IMPORTANT - this EUCLIDEAN problem is NOT the spherical one: Glazyrin & Yu (Adv. Math. 330 (2018) 810-833, arXiv:1611.09479) determined the maximum SPHERICAL two-distance set $M(d)=d(d+1)/2$ for all $d\ge7$ except $d=(2k+1)^2-3$ (i.e. $d=6,22,46,\dots$); that (solved) spherical problem is linked to $g(d)$ only through the sandwich $M(d)\le g(d)\le M(d+1)$ (a gap of about $d+1$), so it leaves every $g(d)$ in $9\le d\le22$ open. The sole known excess above $\binom{d+1}{2}$ for $d>8$ is the exceptional $277$-point set in $\mathbb{R}^{23}$ (distances $2$ and $\sqrt6$; Ge-Koolen-Munemasa, arXiv:2504.18110, 2025); $d=22$ is itself a Glazyrin-Yu-exceptional dimension and the in-range dimension most likely to hide an anomaly. Related constructions: Nozaki-Shinohara, arXiv:1804.06040. Vetted open as of 2026-07-06 (high confidence; Sep-2025 survey gives bounds only above $d=8$).
References
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