#1160: stronger group-count inequality holds at m = 11

The target (erdosproblems.com/1160, OPEN). Let g(n) be the number of isomorphism classes of groups of order n. The original conjecture says that n≤2^m implies g(n)≤g(2^m). A stronger question asks whether

∑_n<2^m g(n)≤g(2^m)

for every sufficiently large m, perhaps already for m≥7. We certify this stronger inequality at m=11.

The result. The exact group counts below 2,048 sum to

∑_n<2,048 g(n)=49,910,538,480.

Eick's 2024 survey gives an exact count of 1,774,274,116,992,170 groups of order 2,048 whose lower 2-central series has length two. This is only a subclass of all groups of that order. Therefore

49,910,538,480
<1,774,274,116,992,170≤g(2,048),

with a certified margin of 1,774,224,206,453,690. The stronger conjecture holds at m=11, and consequently g(n)≤g(2,048) for every 1≤n≤2,048.

Reconstructing the left side. Conway, Dietrich, and O'Brien counted 423,171,191 groups at all positive orders below 2,048 other than 1,024. Burrell later corrected the remaining value to

g(1,024)=49,487,367,289.

The verification bundle also reconstructs the total directly. GAP and SmallGrp regenerate 2,034 of the 2,047 positive-order counts; the final 13 come from the published appendix. Every value agrees with the current OEIS A000001 table, and both routes give the displayed total exactly.

Independent audit. A hostile audit used a newer GAP and SmallGrp release, independently parsed every appendix cell, isolated the single stale order-1,024 entry, and checked the source statements behind both sides of the inequality. It also found a one-unit marginal-total typo in an unused dissertation table; the theorem value, explicit correction, GAP count, and later survey all agree.

Scope. This is a proof-level finite case, not a solution of the open problem. It does not determine g(2,048), establish the statement for m≥12, or prove either conjecture eventually. The live problem page had no claimed proof or current worker on 2026-08-11, and targeted searches did not locate this m=11 comparison; that is a dated search result, not an absolute priority claim.