Attack log — Smale 16 / plane Jacobian conjecture

Chronological attempts, newest last. A finite search changes the record only when its scope and certificate are both complete.

2026-08-27 — research before search

• Replayed the classical reductions and the current finite-degree literature. Moh's old boundary is 100; Nguyen gives 104; Guccione–Guccione–Horruitiner– Valqui prove that below 125 only $(72,108)$, up to transposition, survives.
• Chased a forum lead to Helali's immutable Git/Zenodo archive instead of trusting the forum number. A clean exact replay passed. Direct comparison with Theorem 2.1 and Proposition 4.3 confirmed the degree implication, both polygon pairs, the Laurent coordinate change, and the normalization.
• Kept the 2026 three-variable announcement separate. It is not community consensus, and neither accepting nor rejecting it changes the plane problem.

2026-08-27 — ten imagined end-states

The certificate was imagined first in each case; “die” names the earliest check that would make the route unfit for a bound.

1. Exclude the $(72,108)$ pair exactly. Certificate: unit-ideal identities for both Proposition 4.3 systems plus an independently checked bridge to Theorem 2.1. It might work because the systems are finite; it dies if a polygon, branch, or normalization was omitted.
2. Classify all homogeneous plane Keller perturbations. Certificate: a coefficient theorem reducing $F=I+H_d$ to a shear and a sparse exact inverse checker. It might work because the two homogeneous pieces of the determinant have different degrees; it dies as a record attack because this family is classical and contains no counterexample.
3. Obstruct the tangent-sweep mechanism in the plane. Certificate: a formal determinant identity for every polynomial parametrized curve, checked independently on generic coefficient arrays. It might work because a raw two-dimensional sweep has no spare coordinate; it dies as a full obstruction if a twisted or non-tangent construction evades the template.
4. Re-enumerate every sub-125 degree pair from numerical semigroup data. Certificate: a second program reproducing Theorem 2.1's unique exception. It might catch a missing degree pair; it dies if the paper's geometric admissibility tests cannot be reconstructed without importing its proof.
5. Make the 89 MB hard identity compact. Certificate: a deterministic modular seed plus rational reconstruction yielding the same exact multiplier vector. It might make review easier; it dies if reconstruction still needs essentially the full 1,925-coordinate solution.
6. Classify cubic plane Keller maps coefficient by coefficient. Certificate: a Gröbner basis and inverse for every component. It is finite; it dies as a dent because the degree lies far below the published record.
7. Classify sparse Keller maps at maximum degree 125. Certificate: a support-complete list and unit ideals for every non-automorphic support. It might reach the new frontier; it dies unless the support restriction follows from a published reduction.
8. Bound the Bass–Connell–Wright inverse tree expansion. Certificate: a vanishing theorem for all rooted-tree terms beyond an explicit index. It would prove polynomial invertibility in a reduced class; it dies when nilpotence cancels only sums of trees rather than individual trees.
9. Turn one Magnus inner-polygon conjecture into a finite obstruction. Certificate: exact inequalities eliminating every lattice vertex outside Lee–Li's proved region. It might narrow approximate roots; it dies while the remaining conjectural step is still equivalent to the original problem.
10. Formalize the degree-125 bridge in Lean. Certificate: a theorem from the two unit-ideal hypotheses to the degree inequality. It might remove prose ambiguity; it dies as the first move because importing the 89 MB algebraic-number-field certificate is the hard part.

2026-08-27 — selection and pruning

• Picked 1: it is the only route that can immediately improve the documented finite record, and an exact certificate already exists to be replayed.
• Picked 2: it gives an all-degree positive control family and a small exact inverse certificate, useful for testing every local algebra convention.
• Picked 3: it directly probes the new higher-dimensional mechanism while making a universal, finite, checkable statement in two variables.
• Pruned 4 and 5 because they duplicate the expensive parts of the two sources without strengthening the implication. Pruned 6 because degree 3 is already classical. Pruned 7 because sparsity is not justified. Pruned 8 and 9 at their still-conjectural steps. Deferred 10 until the external exact identity has a practical formal representation.

2026-08-27 — q1: exceptional pair, executed backwards

• Imagined certificate: two exact unit-ideal identities; an exhaustive split for Case 1; a hash-bound artifact; and a small verifier connecting its variables to Proposition 4.3 and Theorem 2.1.

• First contradiction check: compared the source polygons rather than the archive's prose. They agree vertex for vertex. Independent lattice enumeration gives $(61,125)$ points in Case 1 and $(25,47)$ in Case 2.

• Checked the coordinate change $t=xy^2$, $z=y^{-1}$. Python and Rust both derive $[t,z]=-1$, $x^2=t^2z^4$, and the same five coefficient identities. They also give determinant 14 for the three-vertex normalization matrix, so no forbidden division by a variable coefficient is hidden there.

• Bound the large certificate to commit c530fe44e5f53b17840110931803e7c7c5a24cde and archive SHA-256 232204bdb598cc2ea0368e154c8573e18bbfdc69fa631c8878de4b884b38bb18. Eight central internal hashes also matched.

• Ran the entire archive in a fresh Python 3.14 environment. All serialized certificates passed. The separate gmpy2 evaluation checked 13,410 number-field products, equivalent to 335,250 scalar products. Both hard branch identities and the branch involution passed.

• Result: dent. The exact computation makes both Proposition 4.3 systems inconsistent. Combined with Theorem 2.1, every hypothetical plane counterexample satisfies

  $$\max\{\deg P,\deg Q\}\ge125.$$

  This is a finite degree improvement, not a proof of the plane Jacobian conjecture. As with any computer-assisted result built on a published reduction, specialist review of that reduction remains welcome.

2026-08-27 — q2: homogeneous perturbations

• Started with the inverse certificate. For a linear form $l=ax+by$, the maps

  $$F=(x+cb l^d,\ y-ca l^d), \qquad F^{-1}=(x-cb l^d,\ y+ca l^d)$$

  preserve $l$, which makes both compositions and the determinant immediate.

• Worked backwards from a general homogeneous perturbation $I+H_d$. Homogeneous degree separation forces both the divergence and $\det JH_d$ to vanish. The first makes $H_d$ Hamiltonian; the second gives a binary form with zero Hessian. The binary Hessian lemma makes that form a power of one linear form, recovering exactly the displayed family.

• The exact sparse verifier passed at degrees 2, 3, 7, and 125. The last case is a useful high-degree sign and overflow check, but the classification is classical and gives no finite-record improvement. Dropped as a bound route.

2026-08-27 — q3: raw tangent sweeps

• Imagined obstruction: the determinant of every raw tangent sweep in the plane has a visible parameter factor.

• For $S(w,\gamma)=K(w)+\gamma K'(w)$, its Jacobian columns are $K'+\gamma K''$ and $K'$. Exact expansion gives

  $$\det JS=\gamma\det(K'',K').$$

• A formal-coefficient implementation verified the identity for generic polynomial curves of every degree from 1 through 12. The formula itself is degree-free. Thus a raw two-dimensional tangent sweep has zero or nonconstant determinant and cannot be Keller.

• This is a checkable obstruction to one template, not to all possible plane constructions. Extra twists and parameters are outside its scope. No bound changed; the broader search is residue.