Tensor constructions for Euler magic matrices and proper examples of orders 9, 27, 81 and 243

Abstract

An Euler magic matrix is an integer matrix $M$ satisfying $MM^{\mathsf{T}}=\gamma I$ together with two diagonal square-sum conditions; it is proper when its entry squares are pairwise distinct. M{\"u}ller proved that Euler magic matrices exist in every order other than $3$, and that no Euler magic matrix of order $3$ exists at all, while proper examples are considerably more restrictive. We describe a tensor construction whose factors are only required to satisfy the orthogonality equation $AA^{\mathsf{T}}=\gamma I$: for a linear reindexing $L$ of the row index group $\mathbb{F}_3^k$ obeying an explicit support condition, the reindexed Kronecker product of $k$ such $3\times3$ factors satisfies the full Euler magic conditions in order $3^k$. Choosing factors whose entry squares have pairwise distinct products, we obtain proper Euler magic matrices of orders $9$, $27$, $81$ and $243$. The construction therefore produces proper examples in powers of three even though order three admits no Euler magic matrix, and the passage from the factors to the product is exactly where the Euler conditions are created rather than inherited. We give an explicit support condition on $L$ and show that it characterises the linear reindexings forcing the two Euler diagonal identities for every tuple of semi-magic factor arrays; over $\mathbb{F}_p$ with two factors, such a reindexing exists only when $p\le3$. The four existence results are formalised in Lean 4, as are the two instances of the construction used to obtain them; the order-$243$ certificate is taken from the archived development and was not rebuilt in preparing this paper, although its witness was reproduced here by exact integer arithmetic. Further proper examples of orders $729$ and $2187$ are verified by exact integer computation only.