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Exclusion of the Two Surviving Newton Configurations at the (72, 108) Jacobian Frontier

Preprint - not yet peer reviewed.

This release contains the author-prepared manuscript and a dependency-closed, SHA-256-bound exact proof archive.

The theorem proved in the manuscript

Over every algebraically closed field k of characteristic zero, there is no pair P,Q in k[x,y] with [P,Q]=x^2 having either exact Newton configuration listed 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)}

For the smaller configuration, a lossless Laurent reduction leads to a degree-21 Belyi map, a degree-35 exact edge field, and a terminal lift incompatibility. For the larger configuration, the layer-five obstruction is split into the exhaustive branches D != 0 and D = E = 0; exact factorwise subresultants and one fixed 11 x 11 Sylvester determinant close every branch.

Conditional degree consequence

Conditional on the explicitly cited and pinned external preprint reduction chain, the two exclusions imply

max(deg(P), deg(Q)) >= 125

for every hypothetical plane Jacobian counterexample in characteristic zero. The two exact Newton exclusions are the theorem proved in this manuscript; the global degree bound is a dependency-bearing corollary.

This work does not prove the plane Jacobian conjecture. It does not exclude degree 125 or larger.

Frozen proof boundary

  • Theorem-core commit: 7639822e66de9464e141e748088b62d73c388169
  • Theorem-core tag: larger-polygon-logical-proof-2026-08-09
  • Public preprint tag: public-preprint-v1-2026-08-10

The passed theorem-core proof code at the frozen tag is unchanged. Later commits add manuscript revisions, reproducibility hooks, packaging, licensing, and public-release documentation.

Exact environment

  • SageMath 10.9
  • Singular 4.4.1
  • Docker image: sagemath/sagemath@sha256:e068670ae5863b54b2550e72437ec637b0283acb0dc712c8584c124dbf44e667

Downloads and integrity check

Download all four assets into one directory:

raising_degree_bound_125.pdf
Reproducibility_Supplement.pdf
jacobian_125_supplement_flat.zip
SHA256SUMS.txt

On Linux/macOS, run:

sha256sum -c SHA256SUMS.txt

On PowerShell, compare each line of SHA256SUMS.txt with:

Get-FileHash -Algorithm SHA256 .\raising_degree_bound_125.pdf
Get-FileHash -Algorithm SHA256 .\Reproducibility_Supplement.pdf
Get-FileHash -Algorithm SHA256 .\jacobian_125_supplement_flat.zip

The outer ZIP is intentionally flat. Extract it, verify its internal SHA256SUMS.txt, unpack jacobian_125_dependency_closed_archive.tar.gz into a fresh directory, and run:

python tools/verify_supplement_manifest.py .

Expected result:

MATCH=790 MISMATCH=0 MISSING=0 UNLISTED=0 MALFORMED=0

Then follow REPRODUCE_LARGER_POLYGON.md. The theorem replay first verifies the smaller configuration, regenerates the larger source chain, verifies the exceptional chart, closes the D != 0 and D = E = 0 branches, and finally runs the logical coverage and conditional global-degree audits. Optional force-recompute commands for expensive resultants are documented separately and are not required to check the distributed finite identities.

AI-assisted workflow and responsibility

During preparation, Ziwei Guo used OpenAI Codex for mathematical exploration, orchestration of exact computer-algebra checks, LaTeX drafting, reproducibility packaging, and editorial revision. Codex is not an author and cannot assume responsibility for the work. Ziwei Guo accepts sole responsibility for the mathematical claims, the computer-assisted proof, and the released text. No numerical approximation or unverified generative output is used as a theorem premise.

Licensing and citation

  • Author-created code and verification scripts: MIT
  • Author-created certificates, data, manifests, and logs: CC0 1.0
  • Author-prepared preprint and LaTeX source: CC BY 4.0
  • Third-party literature snapshots: original copyrights retained; excluded from the preceding license grants

See LICENSES.md and CITATION.cff in the repository. Preferred citation:

Ziwei Guo, Exclusion of the Two Surviving Newton Configurations at the (72, 108) Jacobian Frontier, preprint, 2026. ORCID: 0009-0008-6271-8583. Frozen theorem-core commit 7639822e66de9464e141e748088b62d73c388169; public release public-preprint-v1-2026-08-10.