十阶素乘数：范数谱、局部图与选择性提升
Prime multipliers of order ten: norm spectra, local charts, and selective lifting
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中文总结 AI 辅助
该论文针对复合基10证明了近乎完整的素乘数定理，确定了十阶傅里叶矩阵主子式的范数谱，解决了多数范数例外情形，讨论了推广到一般无平方因子阶的困难。
中文摘要 AI 辅助
记P(N)为“阶N傅里叶矩阵的每个主子式均非零”的断言。此前，无平方因子的主子式猜想仅对少数均匀的小乘数族成立，包括多个含两个素因子的族；更广泛的多因子结果是非均匀的，仅存在孤立的精确验证。我们针对复合基10证明了近乎完整的素乘数定理：对所有素数p∉{2,5,11}，P(10p)成立。证明始于十阶傅里叶矩阵主子式的完全分圆范数谱，其有理素支撑为{2,3,5,11,31}。普通有限特征提升解决了范数安全的乘数；一个已知的小乘数定理处理了一个范数例外情形，另一个则通过保留每个载体退化的素理想图得以解决，这得到了一个标志选择性提升引理和一个活动秩范数预算。该图分析还分离了剩余无平方因子例外处的局限：活动载体可覆盖每个局部图，故一阶论证终止。我们讨论了这一障碍、固定基提升的非均匀性，以及推广到一般无平方因子阶的困难。所有有限计算均为精确的，且已被独立确认。
英文摘要
Write P(N) for the assertion that every principal minor of the Fourier matrix of order N is nonzero. The square-free principal-minor conjecture was previously known for a few uniform small-multiplier families, including several families with two prime factors; broader higher-factor results were non-uniform, apart from isolated exact verifications. We prove a near-complete prime-multiplier theorem for the composite base 10: P(10p) holds for every prime p \notin {2,5,11}. The proof begins with the complete cyclotomic norm spectrum of the principal minors of the order-ten Fourier matrix. Its rational-prime support is {2,3,5,11,31}. Ordinary finite-characteristic lifting settles the norm-safe multipliers. A known small-multiplier theorem handles one norm-exceptional case; another is settled by retaining the prime-ideal chart in which each carrier degenerates. This yields a flag-selective lifting lemma and an active-rank norm budget. The chart analysis also isolates the limitation at the remaining square-free exception: active carriers can cover every local chart, so the first-order argument stops. We discuss this obstruction, the non-uniformity of fixed-base lifting, and the difficulties in passing to general square-free orders. All finite calculations are exact and have been independently confirmed.
