Sharp onset indices for all recorded empirical examples, upgrading them from observations to theorems: Weintraub's g_{k+25}(3114)=729·g_k(3114) holds for all k≥5 and fails at k=4 (the record says k≥6); Steinerberger's five r=9 starts have onsets 37–39 with common first certificate point 8892342; 385 has onset 13 (certificate 251505), 1570 onset 28 (18755712), 1702 onset 15 (218700), and the r=14 outliers 3393/6175/6969 onsets 2/7/2. Cambie's 738, 148646, 4325798 carry certificates at k=0 (recorded as k≥1).