The partition exponent in the graded pieces of the Claude 3-SAT algorithm is optimal for pieces of that shape
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…
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