{"ID":2888551,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2508.14901","arxiv_id":"2508.14901","title":"Computational Resolution of Hadamard Product Factorization for $4 \\times 4$ Matrices","abstract":"We computationally resolve an open problem concerning the expressibility of $4 \\times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\\mathbb{F}_2$, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3\\%). We verify that these counterexamples remain valid over $\\mathbb{Z}$ through sign enumeration and provide strong numerical evidence for their validity over $\\mathbb{R}$. Remarkably, our analysis reveals that matrix density (number of ones) is highly predictive of expressibility, achieving 95.7\\% classification accuracy. Using modern machine learning techniques, we discover that expressible matrices lie on an approximately 10-dimensional variety within the 16-dimensional ambient space, despite the naive parameter count of 24 (12 parameters each for two $4 \\times 4$ rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability.","short_abstract":"We computationally resolve an open problem concerning the expressibility of $4 \\times 4$ full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over $\\mathbb{F}_2$, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3\\%). We verify that these counterexamp...","url_abs":"https://arxiv.org/abs/2508.14901","url_pdf":"https://arxiv.org/pdf/2508.14901v1","authors":"[\"Igor Rivin\"]","published":"2025-07-31T21:00:28Z","proceeding":"math.RA","tasks":"[\"math.RA\",\"cs.LG\",\"math.AG\"]","methods":"[]","has_code":false}