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 and an exact-arithmetic campaign on the lone open degree pair (72,108) below 125, built on Guccione-Guccione-Horruitiner-Valqui (arXiv:2204.14178, Proposition 4.3). Version 0.24 retains the complete weighted three-parameter T_B cover and adds the first higher-dimensional constructible reduction from the 24-parameter cyclic core. An exact audit refutes a free lift of the shared residual-selected basis but isolates direction (2,9). With X=A^3 and Y=A^2 C, its exact selected determinant is A^87(R(X,B)+Y S(X,B)), where R and S have 23 and 18 terms and are coprime. On A nonzero, the selected exceptional locus is the rational graph Y=-R/S plus the finite base locus V(R,S). This is not a four-parameter cover and does not close the 24-parameter core, the full 51-parameter family, the complete (72,108) case, the planar degree floor, or JC(2). All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).