A planar program for the two-variable Jacobian conjecture: a theorem ladder, staircase transport, and machine certificates at the (72,108) frontier

Paper B of a three-paper research record. The two-variable Jacobian conjecture remains open after the 2026 counterexample in dimension 3. This paper develops the planar program: an unconditional theorem ladder, the staircase transport with cleared polynomial certificates, and an exact-arithmetic campaign on the lone open degree pair (72,108) below 125, built directly on Guccione-Guccione-Horruitiner-Valqui (arXiv:2204.14178, Proposition 4.3). Version 0.16 retains the source-validated original-pair gates from v0.15: seven possible northeastern vertices for a nonzero inner polynomial, and deg_x Res_y(P,Q)=21 with an 84+24 major/minor root partition. It also adds the exact transport audit. Because the reduction swaps the selected eliminant, localizes x, and inverts a boundary divisor, absolute resultant degree 21 is not a direct equation on the 51 final Laurent coefficients without a boundary-divisor ledger. This closes an invalid compute route without excluding (72,108). The reduced-system campaign remains strong evidence toward closure, not a proof; a simultaneous-in-all-coefficients certificate is still required. All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).