AI 中文总结
本文通过计算机辅助证明70阶与143阶傅里叶矩阵的所有主子式非零，利用提升定理简化问题并完成有限域计算验证，附带自包含标准库验证器。
AI 中文摘要
我们给出计算机辅助证明：70×70阶与143×143阶傅里叶矩阵的每个主子式均非零。Caragea、Lee、Malikiosis与Pfander的提升定理将这两个论断分别简化为特征7下10阶傅里叶矩阵、特征13下11阶傅里叶矩阵的所有主子式非零的问题。我们在有限域𝔽₇⁴与𝔽₁₃¹⁰中实现本原根，通过精确的无除法运算计算全部2¹⁰与2¹¹个主子行列式，结果均无零值。该提升定理实际得出更强结论：70阶矩阵的每个10阶主子式、143阶矩阵的每个11阶主子式均非零。本文附带有限域计算的自包含标准库验证器。
英文摘要
We give computer-assisted proofs that every principal minor of each of the \(70\times70\) and \(143\times143\) Fourier matrices is nonzero. A lifting theorem of Caragea, Lee, Malikiosis, and Pfander reduces the two assertions to the nonvanishing of all principal minors of the Fourier matrix of order \(10\) in characteristic \(7\), and of order \(11\) in characteristic \(13\), respectively. We realize primitive roots in \(\mathbb F_{7^4}\) and \(\mathbb F_{13^{10}}\) and evaluate all \(2^{10}\) and \(2^{11}\) principal determinants by exact, division-free arithmetic. None vanishes. The lifting theorem in fact yields the stronger conclusions that every \(10\)-principal minor of the order-\(70\) matrix and every \(11\)-principal minor of the order-\(143\) matrix is nonzero. Self-contained standard-library verifiers for the finite-field calculations accompany the paper.
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