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 0.17 retains the source-validated original-pair gates and the v0.16 transport audit, then corrects the certificate-cover formulation. Principal-open localized lifts collapse to one global polynomial covector; the genuinely stronger method is a constructible rank stratification with new syzygies after closed specialization. The method is applied to the actual GGHV matrix. On the first two interacting coefficients, the selected augmented minor factors exactly as (st-8)^6(2^15 s^9-(st-8)^7)/2^39. An alternative minor covers the component st=8, leaving a rational residual curve, and a third minor is proved nonzero at one point of that curve. This is the first exact chart transition on the reduced coefficient family, not a closure of (72,108). The complete two-parameter curve, the other coefficients, and the planar conjecture remain open. All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).