Recall the conjecture from 2026-02-24: if , then the minimum ratio test gives .
Testing various flat norm instances showed this conjecture to be false! I constructed an instance where in an iteration, , but and is integral. In this instance, we had
This was not a one-off iteration, several iterations had the same .
2652: [B=(540x540)][theta=-1/-1] (4:4) (5:9) (6:1081) (7:1082) (8:1083) (11:1080) (13:552) (14:23) (17:17) (18:1086) (19:1087) (22:1085) (24:558) (25:1091) (28:1090) (29:1092) (33:1089) (35:1095) (36:1096) (39:1097) (44:1094) (46:1100) (47:1101) (49:744) (50:1102) (52:53) (55:1099) (56:1159) (57:1160) (58:1161) (65:605) (73:1262) (76:31) (82:42) (86:1263) (88:1162) (90:750) (94:94) (96:1248) (97:1249) (101:1268) (110:20) (122:1264) (126:1258) (127:1259) (138:1267) (146:1260) (152:1269) (161:701) (165:1250) (167:787) (168:1254) (169:1257) (170:1251) (184:1265) (188:209) (190:190) (208:215) (228:1255) (238:1253) (241:790) (244:1270) (249:788) (250:789) (257:797) (282:282) (343:343) (344:344) (345:345) (346:346) (353:893) (382:382) (388:928) (393:933) (404:944) (415:955) (426:966) (437:977)
| d_B = (167:-1) (282:-1) (283:1) (343:-1)
| d_B^P = (50:-1) (65:-1) (66:-1)
| d_B^Z = (217:1)
| Pd_B^P = (247:1) (282:-1) (343:-1)
| Zd_B^Z = (283:1)
| -Aj = (247:1) (282:-1) (283:1) (343:-1)
| det(B) = 2
| det(B(i:-A_j)) = { (4:0) (5:0) (6:0) (7:0) (8:0) (11:0) (13:0) (14:0) (17:0) (18:0) (19:0) (22:0) (24:0) (25:0) (28:0) (29:0) (33:0) (35:0) (36:0) (39:0) (44:0) (46:0) (47:0) (49:0) (50:0) (52:0) (55:0) (56:0) (57:0) (58:0) (65:0) (73:0) (76:0) (82:0) (86:0) (88:0) (90:0) (94:0) (96:0) (97:0) (101:0) (110:0) (122:0) (126:0) (127:0) (138:0) (146:0) (152:0) (161:0) (165:0) (167:-2) (168:0) (169:0) (170:0) (184:0) (188:0) (190:0) (208:0) (228:0) (238:0) (241:0) (244:0) (249:0) (250:0) (257:0) (282:-2)! (343:-2) (344:0) (345:0) (346:0) (353:0) (382:0) (388:0) (393:0) (404:0) (415:0) (426:0) (437:0)}