秩至多为四的矩阵的定量Chollet不等式：谱界与九阶紧框架构造
Quantitative Chollet Inequalities for Matrices of Rank at Most Four: Spectral bounds and an order-nine tight-frame construction
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中文总结 AI 辅助
本文通过计算机辅助证明，为秩至多四的矩阵建立了两个Chollet猜想的定量版本，给出了显式小于1的谱界，并构造了九阶紧框架，推进了永久式不等式的研究。
中文摘要 AI 辅助
Chollet的永久式猜想询问，对于复Hermitian半正定矩阵，是否成立per(A∘B) ≤ per(A)per(B)，其中∘表示逐元素（Hadamard）乘积。我们提出了两个受限形式的计算机辅助证明，其显式常数严格小于1。对于每个整数n≥10以及每个秩至多为四的此类矩阵A，当分母非零时，自共轭比率per(A∘conjugate(A))/per(A)^2被999991742359/10^12所界定。对于九阶情形，我们为四个非零特征值均等于9/4的相关矩阵获得了界限999815240367/10^12。第一个论证结合了复球面积分、谱子空间倾斜、投影界、精确有限覆盖以及解析无穷尾。第二个论证使用了九个Gram向量之间的二次关系、十维二次型空间上的正算子以及严格有界的熵。所有决定性的有限计算均使用有理算术和完整的闭域证书。不受限制的猜想，包括一般的非紧九阶秩四矩阵，不在这些结果的范围之内。
英文摘要
Chollet's permanent conjecture asks whether per(A o B) <= per(A) per(B), where o denotes the entrywise (Hadamard) product, for complex Hermitian positive semidefinite matrices. We present computer-assisted proofs of two restricted forms with explicit constants strictly smaller than one. For every integer n >= 10 and every such matrix A of rank at most four, the self-conjugate ratio per(A o conjugate(A)) / per(A)^2 is bounded by 999991742359 / 10^12 when the denominator is nonzero. For order nine, we obtain the bound 999815240367 / 10^12 for correlation matrices whose four nonzero eigenvalues all equal 9/4. The first argument combines complex-sphere integrals, spectral subspace tilts, projection bounds, exact finite covers, and an analytic infinite tail. The second uses a quadratic relation among nine Gram vectors, a positive operator on the ten-dimensional space of quadratic forms, and rigorously bounded entropy. All decisive finite calculations use rational arithmetic and full closed-domain certificates. The unrestricted conjecture, including general non-tight order-nine rank-four matrices, is outside these results.
