Construct or bound the largest isosceles set in $\mathbb{R}^9$ (Erdős #503)

A finite set $A\subset\mathbb{R}^d$ is an *isosceles set* if every three of its points form an isosceles triangle: among the three pairwise distances of any $x,y,z\in A$, at least two are equal. Let $I(d)$ be the maximum possible size of an isosceles set in $\mathbb{R}^d$. A general upper bound $I(d)\le\binom{d+2}{2}$ holds, so $I(9)\le\binom{11}{2}=55$. Concrete target: determine $I(9)$, or improve its bounds — construct isosceles sets in $\mathbb{R}^9$ larger than any currently recorded, or prove upper bounds below $55$.

Acceptance. FULLY RESOLVES: the exact value $I(9)$ — an explicit isosceles set in $\mathbb{R}^9$ of that size (rational or algebraic coordinates, with a machine check that every triple is isosceles) together with a proof that it is maximum. PARTIAL: an explicit isosceles set in $\mathbb{R}^9$ larger than any currently recorded (a new lower bound on $I(9)$), or an upper bound below $55$ with a certificate; analogous improvements for $d=10,11,12$ also count.