Planar Keller Maps after the Counterexample: Machine-Certified Newton-Polygon Exclusions, the Staircase Transport, and the Recalibrated Frontier

Abstract

The July 2026 counterexample settled the Jacobian conjecture in dimensions N ≥ 3 and left JC(2) as the open case. This paper develops an exact-arithmetic program on the planar case. The Keller condition J(P, Q) = 1 is linear in Q for fixed P; the resulting machine (eliminate, parametrize the consistency variety, complete, invert) closes the columns min deg ≤ 2 uniformly (one closed inverse for the whole quasi-triangular class), forces perfect-power tops at bidegrees (2, n) and (3, n), and inverts every consistent stratum it has met. Four unconditional exclusion theorems follow from a weight-class calculus: a two-term P = x + a x^u y^v is never a Keller component at any partner degree; neither is any lower-weight perturbation of it; no Keller component carries a Newton top edge anchored at the linear vertex x with interior direction; and if x is a vertex of the Newton polygon then P = x + f(y) exactly (the vertex dichotomy). The residual open core is the transport of the Keller constant along the staircase of Newton vertices swallowing x: we show the completion window is block-triangular under the edge grading, exhibit the classwise transport chain as an executable elimination whose obstruction always localizes at the constant's class, and prove the corresponding fifth exclusion: for primitive tops the certificate-tower induction closes it unconditionally at all partner degrees (cleared base certificates with monomial pairings such as -13860 a^7, plus a resonance-reduction lemma showing window growth can never repair the constant's row). A recalibration against the primary sources shows that our composite-gcd certificates replicate literature-covered territory (the verified floor is gcd ≤ 8, primes, 2p, then Heitmann's B ≥ 16 and the Guccione-Guccione-Valqui dichotomy B = 16 or B > 20), and that the open frontier is the B = 16 normal form together with the single surviving degree pair (72, 108) below 125. For that open pair, an exact instantiation of the primary sources gives two necessary conditions on the original pair: seven possible northeastern vertices for a nonzero inner polynomial, and deg_x Res_y(P, Q) = 21 with an 84 + 24 major/minor root partition; neither condition alone excludes the pair. A constructible determinantal-strata calculation closes the first actual three-coefficient GGHV slice exactly. On the residual curve, two normalized maximal minors reduce, with z = u^9, to 21 - 96z - 1024z^2 and (8z + 1)^14; an explicit integer Bezout identity has constant 17^14. After adding ε_(0,7), the only two-chart residual is a degree-12 fiber over z = -1/8. A third exact minor has coprime degree-13 fiber determinant, certified by a second integer Bezout identity. A connected mod-9 row/column grading extends to every nonconstant direction in the 26-parameter lower family. This is a strict slice result and does not exclude the full (72, 108) family. At the level of the full family, after removal of the structural constant column and with the complete 302-row pool (the forced-only construction has 289 rows), one selected augmented minor compresses to a 36-column block depending on 24 of the 51 parameters. Two connectivity-generating triples factor exactly; one graph-active direction cancels identically, while the other determinant becomes weighted homogeneous with weights (7, 3, 9) after shifting the forced coefficient. On the d = 0 boundary, the tautological kernel [P, P] = 0 can be quotiented exactly. One quotient determinant factors into b^32, one five-term weighted factor, and five binomials; a second is a nonzero scalar times a^107. Together with the exact origin rank gap, these charts close the complete quotient boundary. On d ≠ 0, a first alternative chart reduces each selected component G, L, Q to a finite intersection. A third exact determinant factors as X^30 LQR, for X = A^3, and exact component ideals close G. A row basis selected independently on the finite L and Q residuals is in fact shared. Its 23-term determinant descends as A^87 R(A^3, B); exact unit-ideal certificates close both residuals. Together with the quotient boundary this closes the complete three-parameter T_B restriction. This is still a strict slice result, not closure of the 24-parameter core, the full family, (72, 108), or JC(2). Restoring direction (2, 9) gives the first higher-dimensional recursive atlas. The shared determinant is A^87 (R(X, B) + Y S(X, B)), X = A^3, Y = A^2 C, with gcd(R, S) = 1. A second exact minor covers a dense open of this rational graph; its residual factors into three plane curves F_3 F_6 F_7 = 0. On the linear-in-X component F_3 = (5B + 4)^3 + 16X, a third exact minor reduces the principal-open residual to Q_9(B) Q_15(B) = 0: 24 normalized values, or 72 lifted algebraic points. On the quadratic-in-X component F_6, the same exact section has a nonzero linear quotient class U(B)X + V(B). Its degree-74 function-field norm has five irreducible factors of degrees 2, 3, 6, 18, 30. Exact same-point arithmetic places the first three on X = 0 or R = S = 0; only the degree-18 and degree-30 factors survive on AS ≠ 0. Thus F_6 also becomes finite: 48 normalized values, or 144 lifted algebraic points. A distinct exact section on F_7 has a degree-58 norm. Its B^16 factor lies on A = 0, and its degree-12 factor lies on R = S = 0; the surviving irreducible degrees 3, 9, 18 give 30 normalized values and 90 lifts. Exact CRT comparison then identifies the degree-9 and degree-18 blocks with the same geometric points already projected from F_3 and F_6, reducing the seven factor appearances to five squarefree blocks of total degree 75. The h_7 section covers the unique degree-15 and degree-30 blocks. Two further exact maximal-minor sections generate the unit ideal on the remaining F_7 blocks of degrees 3, 9, 18. Hence the accumulated atlas closes the entire AS ≠ 0 rational graph. Exact factorwise subresultants then identify the principal-open base locus as a reduced 90-dimensional algebra with field-block degrees 3, 6, 12, 69. Existing sections cover the degree-12 and degree-69 blocks; one new exact section completes degrees 3 and 6. Bezout identities, independent resultants, multiplication-matrix norms, and a direct determinant control certify the finite atlas uniformly in Y. Thus the complete A ≠ 0 sector of the four-parameter restriction is covered. Returning to the original 302-by-125 augmented matrix before invariant division, the direct A = 0 boundary admits two exact minors independent of C. Their divisors are (5B + 4)^3 (25B^2 - 20B + 16)^3 and B^95 (109375B^6 - 110592)(21875B^6 - 4800B^3 - 24576), up to nonzero scalars. An explicit Bezout identity makes their gcd one. Combined with the d = 0 quotient cover, this closes the complete four-coefficient restriction. Adjoining the transverse coefficient at (2, 8), EXP-132 closes the entire direct A = 0, d = 1 boundary of that five-coefficient restriction. Three normalized maximal minors generate the unit ideal in Q[B, C, T]: after one exact row combination, two depend only on B and are coprime, with an explicit characteristic-zero Bezout certificate [MV/D]. The inherited EXP-123 and EXP-124 bases vanish identically on this specialization, so the certificate requires a new SCC-guided basis rather than a parameter substitution into the four-coefficient atlas. EXP-135 then proves that the EXP-124 section is unchanged by (2, 8) on its complete normalized (A, B, C) ambient chart. The transverse core has ranks 7, 3, 0 through its first three powers. Separate determinant degree bounds (25, 24, 6, 7), a complete 136,500-control tensor grid over 30 primes, and a CRT modulus exceeding the explicit characteristic-zero coefficient bound certify det(H + T K_(2,8)) = det(H) [MV/D]. Hence the F_3 F_6 F_7 divisor of the four-coefficient restriction is retained for this section. The other residual sections and the transverse d = 0 quotient remain open, so the complete five-coefficient restriction and every larger target remain open. A new multiplication formula for x^m-anchored edges shows the B = 16 leading form cannot produce the Keller constant on its top edge for m ≥ 2: that case, too, is a staircase transport. All computations are exact and reproducible; every claim is labeled machine-verified [MV], derived [D], or conjectural [C]. Source code, data and computational records: https://github.com/fsantibanezleal/CAOS_RESEARCH (problems/algebraic-geometry/jacobian-conjecture).