Abstract
In 1986 Bapat and Sunder asked whether per(𝐴) is the largest eigenvalue of F𝐴 = [𝑎𝑖,𝑗 per(𝐴(𝑖, 𝑗))] for every positive semidefinite Hermitian 𝐴, where 𝐴(𝑖, 𝑗) deletes row 𝑖 and column 𝑗. Drury disproved this over the complex field in 2018 and conjectured that the property holds for real positive semidefinite matrices. We disprove Drury's conjecture with a rank-4, 32 × 32 integer Gram matrix and an integer vector whose Rayleigh quotient exceeds per(𝐴); verification uses integer arithmetic. Together, the two counterexamples resolve the Bapat–Sunder eigenvalue conjecture negatively over both fields. Counterexamples exist of every order 𝑛 ≥ 32. As an application to quantum optics, the violation yields a 36-mode orthogonal interferometer and a direction of relative arrival-time delays among 32 photons with identical-variance Gaussian wave packets along which every sufficiently small nonzero delay increases the probability that all photons are detected within four designated output modes.