Two-dimensional Jacobian research laboratory

This repository is a reproducible, claim-tracked investigation of the plane Jacobian conjecture. It does not claim a solution. The baseline date is 2026-07-23.

Current theorem and public preprint

The current manuscript, Exclusion of the Two Surviving Newton Configurations at the (72, 108) Jacobian Frontier, proves over every algebraically closed field of characteristic zero that no pair P,Q in k[x,y] with [P,Q]=x^2 has either of the two exact Newton polygon pairs in Proposition 4.3 of the pinned Guccione--Guccione--Horruitiner--Valqui preprint:

L: Newt(P)=conv{(0,0),(1,0),(8,14),(8,16),(0,8)}
   Newt(Q)=conv{(0,0),(2,1),(12,21),(12,24),(0,12)}

S: Newt(P)=conv{(0,0),(1,0),(8,14),(8,16)}
   Newt(Q)=conv{(0,0),(2,1),(12,21),(12,24)}

Conditional on the explicitly pinned external reduction chain, these two exclusions imply max(deg(P),deg(Q)) >= 125 for every hypothetical characteristic-zero plane Jacobian counterexample. The exclusions themselves are the theorem proved here; the degree bound is a dependency-bearing corollary. The work does not prove the plane Jacobian conjecture and does not exclude degree 125 or larger. The manuscript is a preprint and has not yet been peer reviewed.

The theorem-core proof code is frozen at commit 7639822e66de9464e141e748088b62d73c388169 and tag larger-polygon-logical-proof-2026-08-09. Later documentation, packaging, licensing, and reproducibility changes do not modify the passed proof code at that tag. The public preprint release is public-preprint-v1-2026-08-10.

The first run independently verifies the announced three-dimensional map, studies its weighted mechanism, and tests three direct descents. Exact outputs live in results/; research status and evidentiary labels live in STATUS.md and CLAIMS_LEDGER.md.

The second run reproduces the complete hyperbolic G_m-equivariant plane no-go theorem and derives a quotient-Jacobian weight formula explaining why the announced three-dimensional quotient has vanishing exponent two.

The third run asks whether a hypothetical minimal plane counterexample can be reduced to that equivariant class. It proves a positive-weight degeneration dichotomy, exhibits exact preservation counterexamples, and isolates a marked infinity filtration as the remaining target. The decision is INSUFFICIENT_EVIDENCE, not a claim about the truth of the plane conjecture.

The fourth run audits Nguyen's original 1999 and 2004 papers and corrects the target: the nonzero Delta=2 Wronskian is separative/polar data, not the dicritical endpoint that retains a nonproper target curve. It proves that the full Z^2_lex flag-Rees algebra is non-Noetherian, establishes a sharp dimension/domain trilemma for the natural two-pole models, proves a sufficient marked-boundary preservation theorem, and disproves the broad log-equivariant endpoint by the exact map (x^2*y,y). The cycle decision remains INSUFFICIENT_EVIDENCE; no two-pole Keller class is claimed excluded.

The fifth run starts the five-branch breakthrough program. It classifies exactness and lattice torsion for general pointed saturated rank-two cone monoids, separates ordinary from logarithmic boundary ramification, proves the exact n+2g finite-end ramification formula for irreducible exactly-two- pole Keller fibers, and identifies the precise surviving degree-108 family (8,28),(3,2). Exact countermodels block the form-equality-only bridge, and a Groebner certificate excludes only the vertex-only skeleton of the viable Newton polygons. The live decision is CONTINUE_PROGRAM; no breakthrough gate is claimed.

The sixth run audits the live 2026 (72,108) machine-certificate program and the original Druzkowski 1991 branch theorem. It obtains a fully coordinate-invariant, but known, lower bound of three generic-fiber branches at infinity for every hypothetical nonautomorphic Keller map. It also upgrades the Newton computation from a vertex skeleton to both complete coefficient spaces: explicit augmented minors exclude nonempty Zariski-open loci, with an independent second specialization and prime. Possible survivors are now confined to determinantal exceptional hypersurfaces; those hypersurfaces are not yet excluded, so the decision remains CONTINUE_PROGRAM and no original breakthrough gate is claimed.

The seventh run makes the closed exceptional problem structural. For the smaller surviving polygon, the exact Laurent change z=x*y^2,t=1/y converts the complete 25/47 coefficient system into five univariate differential identities. Its bottom identity forces a rigid degree-21 Belyi passport (2^10,1),(3^7),(17,1^4). An exact, independently reconstructed transitive permutation triple shows that edge is genuinely viable and has monodromy A_21; the remaining Gate-5 problem is the finite lift through four equations, not the original undifferentiated 72-variable system.

The eighth run closes that finite problem. The passport has exactly five dessin classes. After exact normalization, all 35 scaled edge representatives form one degree-35 number field, whose invariant quotient is an irreducible quintic. Exact layer-by-layer linear algebra over that field forces B=E=0 and then G'=0, contradicting the required G_12 vertex. A second Sage implementation changes column orders and kernel bases and obtains the direct unit ideal (1) in K[q,r,s]. Thus the smaller 25/47 Proposition 4.3 Newton pair is completely excluded: MAJOR_BREAKTHROUGH, Gate 5. The larger 61/125 polygon and the full (72,108) degree pair remain open.

The ninth run closes that remaining larger configuration without using the historical unbounded Macaulay/CRT reconstruction as a theorem premise. The layer-five obstruction splits into the exhaustive charts D != 0 and D = E = 0; exact factorwise subresultants and a fixed 11 x 11 Sylvester determinant close all branches. Exact source-chain reconstruction binds the terminal certificates to all 302 Laurent-Jacobian coefficient equations. Together with the eighth run, this proves the two exact Newton exclusions stated above. The global degree-125 consequence remains explicitly conditional on the pinned external reduction chain.

Reproduce the first run

cd jacobian-2d-research
python src\verify_3d_counterexample.py
python src\independent_verifier.py
python experiments\phase_b_first_three.py
python src\equivariant_no_go.py
python src\independent_equivariant_verifier.py
python src\quotient_jacobian_formula.py
python src\independent_quotient_verifier.py
$env:PYTHONPATH = "experiments"
python -m graded_reduction.preservation_counterexamples
python -m graded_reduction.independent_verifier
python -m flagged_infinity.primary_verifier
python -m flagged_infinity.independent_verifier
python -m breakthrough_program.primary_verifier
python -m breakthrough_program.independent_verifier
docker run --rm -v "${PWD}:/work" -w /work sagemath/sagemath@sha256:e068670ae5863b54b2550e72437ec637b0283acb0dc712c8584c124dbf44e667 `
  sage experiments/breakthrough_program/belyi_lift_over_number_field.sage
docker run --rm -v "${PWD}:/work" -w /work sagemath/sagemath@sha256:e068670ae5863b54b2550e72437ec637b0283acb0dc712c8584c124dbf44e667 `
  sage experiments/breakthrough_program/independent_belyi_lift_verifier.sage
python src\hash_results.py
python -m pytest -q

The primary verifier uses SymPy. The first independent verifier implements sparse polynomial arithmetic over fractions.Fraction; the flagged-infinity independent verifier imports none of its primary experiment modules. The Gate-5 certificates were run in the official SageMath 10.9 container with Singular 4.4.1. Lean was not needed for the finite decision; see ENVIRONMENT.md.

Claim discipline

Every research claim uses exactly one status:

• VERIFIED_THEOREM
• VERIFIED_EXACT_COMPUTATION
• REPRODUCED_KNOWN_RESULT
• COMPUTATIONAL_EVIDENCE
• CONJECTURAL
• INCOMPLETE_ARGUMENT
• DISPROVED_LEMMA
• FAILED_EXPERIMENT
• SOURCE_REQUIRED
• MAJOR_BREAKTHROUGH

External results are not promoted from an abstract or secondary page to a fully checked theorem. Finite-field screens are filters only.

Licensing and citation

Author-created software is licensed under the MIT License. Author-created data, exact certificates, and logs are dedicated under CC0 1.0. The preprint and its author-created LaTeX source are licensed under CC BY 4.0. Third-party literature snapshots retain their original copyrights and are excluded from these grants. See LICENSE, LICENSES.md, LICENSE-DATA.md, and LICENSE-PREPRINT.md for the exact scope.

Citation metadata is provided in CITATION.cff. The preferred citation is the manuscript by Ziwei Guo, ORCID 0009-0008-6271-8583, together with the frozen theorem-core commit and the public release tag.