We present a comprehensive computational and theoretical analysis of the Erdős-Szemerédi sunflower problem. We compute exact values m(n,3) = 2, 4, 6, 9, 13, 20 for n = 1,...,6—a sequence absent from the literature for the power-set formulation. Our central results establish proven slack in the Naslund-Sawin bound at multiple values: 41.7% at n=3 and 4.5% at n=6. We prove that each local "blocking" tensor has slice rank exactly 2, while the Naslund-Sawin proof implicitly uses factor 3—identifying the precise source of overcounting. We prove a Strong Balance Theorem: in any maximum sunflower-free family, element frequencies satisfy m(n,3) - m(n-1,3) ≤ d_i ≤ m(n-1,3) - 1, implying frequencies lie in approximately [0.33, 0.67]. We establish that admissible monomials satisfy a degree triangle inequality, constraining the polynomial structure. These structural insights, combined with the observed monotonic decay of the ratio |F_max|/NS(n) from 0.84 to 0.42, point to specific directions for asymptotic improvement.