A planar program for the two-variable Jacobian conjecture: a theorem ladder, staircase transport, and machine certificates at the (72,108) frontier

Abstract

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 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 0.32 retains the complete declared four-coefficient restriction and the direct transverse boundary theorem from v0.31, then proves an ambient transverse determinant identity for the accepted EXP-124 section. For its normalized 33-by-33 core H(A,B,C) and direction K_(2,8), exact ranks, separate degree bounds (25,24,6,7), 136,500 complete tensor-grid controls over 30 primes, and a CRT modulus exceeding an explicit characteristic-zero coefficient bound prove det(H+T K_(2,8))=det(H) in QQ[A,B,C,T]. Thus the old F3 F6 F7 divisor is retained for this section after adjoining (2,8). The other residual sections and transverse d=0 quotient remain open, so the complete five-coefficient 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).