The Ehrhart series of magic squares of order seven

Let $\mathrm{IMS}_n(m)$ denote the number of $n\times n$ nonnegative integer matrices whose row sums, column sums, and two main diagonal sums are all equal to $m$. We determine the Ehrhart series $F_7(q)=\sum_{m\ge 0}\mathrm{IMS}_7(m)q^m$ as a reduced rational function. The denominator has degree $373$ and cyclotomic factors of order at most $15$; the numerator is a palindromic polynomial of degree $366$ with nonnegative integer coefficients.
Using the SimpCone decomposition, the associated polytope is represented as a sum of $166$ million signed simplicial cones. The LRQC evaluator computes their generating functions over finite fields; a typical cone requires only one or two quotient characters, and the cost per character is nearly linear in the truncation degree $T$. An explicit common denominator together with Ehrhart reciprocity reduces the rational reconstruction to the prefix up to $T=1256$, while an explicit counting bound supplies the coefficient bounds needed for deterministic lifting from the prime fields to $\mathbb Z$. This prefix is independently computed for the whole cone family in eight prime fields. Exact Chinese remaindering lifts the verified residues to equality over $\mathbb Z$, the finite-prefix criterion proves the rational identity, and exact polynomial gcds prove that the displayed denominator is reduced.