The corner parameter enumerates the symmetric order-three magic squares

A bijection onto an interval.

Let $S_{n}(t)$ denote the number of symmetric magic squares of order $n$ and line sum $t$, and let $\mathrm{symmParamSet}(e)=\{0,1,\dots,2e\}$ be the admissible corner parameters of the symmetric family, with $\mathrm{symmParamCount}(e)=2e+1$. The theorem states

That is, sending a symmetric magic square of order three and line sum $3e$ to its top-left corner $M_{00}$ is a bijection onto $\{0,\dots,2e\}$.

Proof. The classification MagicSquares.symmetric_magic_three_classify shows that such a square is $\mathrm{symmMagic3}(e,a)$ with $a=M_{00}$, whose $(0,1)$ entry is $2e-a$; the row-$0$ identity $a+(2e-a)+e=3e$ is then satisfiable in $\mathbb{N}$ only for $a\le 2e$, so the map lands in $\mathrm{symmParamSet}(e)$. It is injective because $a$ determines the whole square, and surjective because for every $a\le 2e$ the array $\mathrm{symmMagic3}(e,a)$ is symmetric, is magic of line sum $3e$, and has all entries at most $2e$, hence may be read over the ambient type $\mathrm{Fin}(3e+1)$ of the counting function.

Context. This is the order-three counterpart of the bijection magic_three_param_bij used in Mission I: there, MacMahon's two parameters $(a,c)$ enumerate the magic squares of line sum $3e$; here the extra symmetry condition cuts the parameter set down from the $\ell_{1}$ ball $\{(a,c):|a-e|+|c-e|\le e\}$ to its diagonal slice $c=e$, an interval of $2e+1$ points. Composing the bijection with the cardinality of an interval gives the closed form $S_{3}(3e)=2e+1$, and it also isolates why the count is linear rather than quadratic in $e$.