欧拉幻矩阵的张量构造及9、27、81、243阶真例
Tensor constructions for Euler magic matrices and proper examples of orders 9, 27, 81 and 243
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中文总结 AI 辅助
本文提出一种张量构造，仅需因子满足正交条件，即可在三的幂次阶上生成真欧拉幻矩阵，并给出9、27、81、243阶的具体实例，部分结果已用Lean 4形式化验证。
中文摘要 AI 辅助
欧拉幻矩阵是满足 $MM^{\mathsf{T}}=\gamma I$ 以及两个对角平方和条件的整数矩阵 $M$；当其元素平方两两不同时，称为真欧拉幻矩阵。M{ü}ller 证明了除 $3$ 阶外，每个阶数都存在欧拉幻矩阵，且 $3$ 阶欧拉幻矩阵完全不存在，而真例的限制则严格得多。我们描述一种张量构造，其因子仅需满足正交方程 $AA^{\mathsf{T}}=\gamma I$：对于行指标群 $\mathbb{F}_3^k$ 上满足显式支撑条件的线性重索引 $L$，$k$ 个这样的 $3\times3$ 因子的重索引 Kronecker 积在 $3^k$ 阶上满足完整的欧拉幻条件。选择元素平方具有两两不同乘积的因子，我们得到 $9$、$27$、$81$ 和 $243$ 阶的真欧拉幻矩阵。因此，尽管 $3$ 阶不存在欧拉幻矩阵，该构造仍能在三的幂次上产生真例，而从因子到乘积的过渡正是欧拉条件被创造而非继承之处。我们给出 $L$ 的显式支撑条件，并证明它刻画了迫使每个半幻因子数组元组满足两个欧拉对角恒等式的线性重索引；在 $\mathbb{F}_p$ 上且仅有两个因子时，这样的重索引仅当 $p\le3$ 时存在。四个存在性结果在 Lean 4 中形式化，用于获得它们的两个构造实例也已形式化；$243$ 阶证书取自存档开发，未在撰写本文时重建，但其见证在此通过精确整数运算重现。$729$ 和 $2187$ 阶的进一步真例仅通过精确整数计算验证。
英文摘要
An Euler magic matrix is an integer matrix $M$ satisfying $MM^{\mathsf{T}}=γI$ together with two diagonal square-sum conditions; it is proper when its entry squares are pairwise distinct. M{ü}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 $AA^{\mathsf{T}}=γI$: 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.
