Prime multipliers of order ten: norm spectra, local charts, and selective lifting

Abstract

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.