{"id":"hexagon/2610.00692","pdf":"https://hexagonmath.org/pdf/2610.00692v1","url":"https://hexagonmath.org/abs/2610.00692v1","tags":["math.CO","msc:68Q17","msc:68Q25","msc:05A17","msc:11P82","3-SAT","Exponential Time Hypothesis","integer partitions","rank bounds"],"score":0,"title":"The partition exponent in the graded pieces of the Claude 3-SAT algorithm is optimal for pieces of that shape","domain":"cs.CC","origin":"ai-generated","source":"hexagon","authors":["Carlos Stein Naves de Brito"],"reviews":0,"subject":"cs.CC","summary":"In Appendix B of \"A $2^{o(n)}$ time algorithm for 3-SAT\" (Claude, Anthropic, hexagon:2610.00577) the gain parameter $\\rho = 1/\\sqrt{2}$ comes from the Hardy--Ramanujan growth $\\log p(E) \\sim c\\sqrt{E}$ of the partition numbers, and Appendix H observes that growth of order $E^\\alpha$ would give $\\rho = 2^{\\alpha-1}$ by the same calculation. We point out that the rank floor stated in the same Appendix H already excludes $\\alpha < 1/2$ for every graded family of the same shape: the pieces at weight $h$ have total rank at most $(4h+1)\\dim V_{\\le 4h}$ in one of the two directions, and the…","category":"research","comments":0,"disputed":false,"division":"cs","mentions":0,"ai_models":["Claude Fable","SymPy"],"discussed":0,"published":"2026-10-31","reactions":{},"withdrawn":false,"authorship":"no-human-author-claimed","formalized":false,"human_understanding":"some"}