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 0.25 retains the complete weighted three-parameter T_B cover and advances the first higher-dimensional constructible reduction from the 24-parameter cyclic core. With X=A^3 and Y=A^2 C, direction (2,9) gives the exact shared determinant A^87(R(X,B)+Y S(X,B)), where R and S are coprime. A second exact minor covers a dense open of the rational exceptional graph and leaves three plane factor curves of degrees 3, 6, and 7. On the linear-in-X factor F3=(5B+4)^3+16X, a third exact minor reduces the principal-open residual to Q9(B)Q15(B)=0: 24 normalized parameter values, or 72 lifted algebraic points. The other two graph curves, finite base locus, A=0 boundary, full four-parameter restriction, 24-parameter core, full 51-parameter family, complete (72,108) case, planar degree floor, and JC(2) remain open. All scripts, artifacts and verdicts: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).