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 0.23 completes the weighted three-parameter T_B restriction. On d=0, quotienting the structural kernel [P,P]=0 and combining determinants proportional to a^107 and b^95 with the exact origin rank gap closes the complete quotient boundary. On d not equal to zero, exact component ideals close G. A row basis selected independently on the finite L and Q residual schemes is shared; its 23-term determinant descends as A^87 R(A^3,B). A direct unit ideal closes L, while an exact B^36 zero-set split, quadratic reduction, and unit univariate gcds close Q. This proves a complete cover of the three-parameter T_B restriction, not 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).