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 (weight-class exclusions through the vertex dichotomy, via an annihilation lemma), the staircase transport with cleared polynomial certificates, and a machine campaign on the lone open degree pair (72,108) below 125, built directly on Guccione-Guccione-Horruitiner-Valqui (arXiv:2204.14178, Proposition 4.3). Stated within the certified scope: the campaign provides strong machine evidence toward the closure of (72,108), not yet a complete proof; the remaining step, a simultaneous-in-all-coefficients certificate, is stated precisely, together with an adversarial audit of the whole chain. Every certificate is verified identically over the rationals and persisted. All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).