Discuss Findings

across all 465 sanitized results from 21 hosts, best-condition fixation-ratio excess over neutral is +1.81251 +/- 0.02856 (95% CI +1.75652 to +1.86849, d=2.94, 100.0% sign-consistent), amplifier-condition fraction is +0.51917 +/- 0.00557 (95% CI 0.50826 to 0.53008, d=4.33, 100.0% sign-consistent), and mean critical-r across k is +2.75890 +/- 0.03413 (95% CI 2.69200 to 2.82580, d=3.75, 100.0% sign-consistent); per-k phase boundaries are k={2,3,5,8,12} with mean r_crit={2.626,2.793,2.744,2.797,2.836} and mean max fixation ratios={2.478,2.178,2.007,1.860,1.817}. Strongest condition means are k=2,r=8 with fix_ratio 2.0246 [1.9464,2.1028], fix_prob 0.04049 +/- 0.00080, amplifier_fraction 0.8538, mean_gens 257.6 +/- 2.5; k=2,r=6 with fix_ratio 1.6774 [1.6092,1.7456]; and k=3,r=8 with fix_ratio 1.6074 [1.5402,1.6747]. The weak edge is concentrated at larger hyperedges and low multiplication factors, with k=12,r=2 giving fix_ratio 1.0009 [0.9453,1.0564], k=12,r=1.5 giving 1.0028 [0.9471,1.0584], and k=8,r=1.5 slightly subneutral at 0.9918 [0.9402,1.0434]; per-condition fixation-ratio distributions are mostly right-skewed with moderate-to-heavy tails, and host diversity remains nontrivial despite concentration in a few machines (21 hosts total, largest host share 40.6%, median 3 results/host)

CONFIRMED. Frequency-dependent public-goods benefits partially reverse the known suppressor-of-selection behavior of random hypergraphs, but only once the multiplication factor clears a hyperedge-size-dependent threshold near r approximately 2.6 to 2.8. Small hyperedges amplify a single cooperator most strongly because local contributions are less diluted, so the payoff advantage survives weak-selection averaging and produces fixation rates roughly double neutral at k=2, r=8. As hyperedges grow, the same contribution is spread across more neighbors, which pushes the system back toward neutral or mildly suppressive behavior and explains why the critical threshold rises slightly with k while the attainable max fixation ratio falls. The right-skewed condition-level distributions indicate occasional very strong amplification realizations rather than a narrow deterministic shift, which fits a mechanism where random hypergraph membership sometimes creates unusually favorable cooperative exposure patterns. Scientifically, this means hypergraph structure alone does not universally suppress selection once payoffs are endogenous: game-induced synergy can overcome structural suppression, but the required synergy grows with interaction group size.