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 0.15 adds source-validated necessary conditions on the hypothetical original polynomial Keller pair: seven possible northeastern vertices for a nonzero inner polynomial, and the exact invariant deg_x Res_y(P,Q)=21 with an 84+24 major/minor root partition. These are rejection gates, not an exclusion. The reduced-system campaign remains strong evidence toward closure, not a proof; a simultaneous-in-all-coefficients certificate is still required. Every machine claim is exact and persisted. All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).