Computational Resolution of Hadamard Product Factorization for 4 × 4 Matrices

We computationally resolve an open problem concerning the expressibility of 4 × 4 full-rank matrices as Hadamard products of two rank-2 matrices. Through exhaustive search over mathbbF2, we identify 5,304 counterexamples among the 20,160 full-rank binary matrices (26.3%). We verify that these counterexamples remain valid over mathbbZ through sign enumeration and provide strong numerical evidence for their validity over 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 × 4 rank-2 matrices). This emergent low-dimensional structure suggests deep algebraic constraints governing Hadamard factorizability.