**Version 2 (January 14, 2026) ** Extended computational results: • NEW: m (7, 3) ≥ 27, m (8, 3) ≥ 41, m (9, 3) ≥ 61 (lower bounds) • NEW: Theory-computation feedback loop section (1000× algorithmic speedup) • Updated all tables and analysis through n=9 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) ≤ dᵢ ≤ 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 |Fₘax|/NS (n) from 0. 84 to 0. 42, point to specific directions for asymptotic improvement.
Cody Shane Mitchell (Wed,) studied this question.