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 0.18 retains the source-validated original-pair gates and transport audit, then advances the constructible certificate-strata method from a first chart transition to a complete exact cover of the declared two-coefficient GGHV slice. On the normalized residual curve, two primitive maximal minors are F(z)=21-96z-1024z^2 and G(z)=(8z+1)^14, with z=u^9. Persisted integer polynomials satisfy A(z)F(z)+B(z)G(z)=17^14, proving the two minors generate the unit ideal over Q[u,u^-1]. A connected mod-9 row/column grading extends to every nonconstant direction in the persisted lower family, with weight law w_(p,q)=q-p+1 mod 9. This closes one exact slice and supplies a scalable graded lift; it does not close the full (72,108) family or decide the planar conjecture. All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).