Beyond the certificate: geometry, dynamics, and arithmetic of a three-dimensional Keller counterexample

Public repository: https://github.com/shadybrook/jacobian-counterexample-research

This repository develops a reproducible structural study of the polynomial map announced by Levent Alpoge on 20 July 2026. It begins with an independent exact certificate:

[ \det DF=-2, ]

and three distinct rational points have the same image. Under the standard definition, this disproves the Jacobian Conjecture in dimension three. The main purpose of the repository, however, is to understand the mathematics the example opens up: its fibers, Galois closure, monodromy, behavior at infinity, inverse-Jacobian dynamics, equivariant quotient, and obstruction to descending the construction to the affine plane.

This is an independent working note, not a peer-reviewed publication and not a blanket claim of priority. The original map must be credited to the announcement linked below. Several structural observations also appeared in same-day public notes; the paper records those overlaps explicitly.

Why this repository is worth sharing

The repository is more than a determinant check, but it should not be presented as a second breakthrough comparable to the counterexample itself. Its public value is the combination of exact reproducibility, a unified geometric explanation, explicit calculations that can be reused by other researchers, and an adversarial record of which tempting generalizations fail.

The strongest responsible description is:

A reproducible research audit and structural study of the announced map, containing several proved map-specific deductions, reusable general lemmas, explicit failure examples, and a carefully separated frontier of open problems. Historical priority of the newer deductions remains unverified.

Contribution hierarchy

The following hierarchy ranks mathematical status and likely usefulness, not personal credit or guaranteed historical novelty.

Level 0 — the originating breakthrough, not claimed here

• The displayed polynomial map, its constant Jacobian, and its global noninjectivity originate with Levent Alpoge's announcement and the discovery history credited there.
• The inverse cubic, degree-three structure, and parts of the monodromy picture overlap same-day public discussions.

Level 1 — independently certified foundations

• det(DF)=-2 and the displayed three-point collision are verified exactly.
• The inverse cubic, discriminant, complete fiber classification, image, nonproperness set, and finite completion are reconstructed in exact arithmetic.
• The repository provides executable checks and regression tests rather than relying on numerical agreement or authority.

Level 2 — strongest map-specific structural contributions

• The full off-diagonal collision space is identified with an explicit affine complement and proved smooth and factorial, with trivial Picard group and explicitly computed units.
• The finite completion, boundary, torsor structure, inverse-Jacobian frame, quotient obstruction, and plane-slice pole obstruction are organized into one coherent model.
• Escape at infinity is quantified by contact order and Newton polygons; an explicit degenerate family proves where the first Newton polygon stops being sufficient.

These are the most plausible contributions for specialist attention. They are proved in the working note, but priority remains to be assessed by experts.

Level 3 — reusable general deductions prompted by the map

• A residue-ratio theorem prevents polynomial cancellation of poles; its unequal-pole weighted form follows from a dense weighted-residue map.
• Complete constant directions in an inverse-Jacobian frame form a vector space contained in the translation stabilizer of the image.
• Morse inverse equations of the form R(w)-P*w+Q have full symmetric monodromy, yielding a radical-inversion obstruction in degree at least five.

The methods are broader than this example, although parts are applications of standard valuation, group-theoretic, and covering-space techniques.

Level 4 — established-theory consequences and conditional arithmetic

• Related-conjecture consequences use published implication or equivalence theorems and should be credited to those theorems.
• Finite-field S_n fiber statistics follow from Chebotarev only at good reductions with geometrically connected maximal-monodromy closure, along extension fields. They are not exact statements for every prime.

Level 5 — open research program

• The repository proposes testable next problems: collision-complement topology, full Newton trees, completeness criteria, exact higher-degree fiber enumerators, minimal ordinary degree, optimized stable reductions, and the two-dimensional filling obstruction.
• These are research directions, not announced solutions.

Five recently audited directions

Each direction was reconstructed from definitions, subjected to failure searches, and checked independently where finite computation could help.

1. Residues and divisorial filling. If two functions have equal simple poles and the ratio of their leading residues is transcendental over the ground field, every nonconstant polynomial in them retains its full pole. A counterexample over R shows that transcendence cannot be replaced by the ambiguous phrase “nonconstant” over arbitrary fields. A weighted version is proved under a dense weighted-projective residue hypothesis.
2. Factorial collision geometry. The off-diagonal collision variety is a smooth affine factorial threefold with trivial Picard group. Its units modulo constants form Z, generated in the projective model by D^3/E^2.
3. Translation symmetry and inverse flows. Complete constant combinations of inverse-Jacobian vector fields must preserve the image by translation. The omitted curve has no nonzero translation symmetry, so every nonzero such direction for this map is incomplete. No converse is claimed.
4. Quantitative escape at infinity. Contact order k at a smooth discriminant point gives pole order k/2. At the cusp, the first Newton polygon gives min(alpha,2*beta/3) under a separability condition. An explicit degenerate family shows that later Newton steps can create arbitrarily larger half-integral escape orders. In that family the root displacement is (N-1)/2, while derivative vanishing and inverse-coordinate blow-up are (N+1)/2; half-integrality occurs for even N>3.
5. Maximal monodromy and arithmetic complexity. Full S_n monodromy rules out generic radical inversion for n>=5. At suitable finite-field reductions, Chebotarev predicts the fixed-point distribution of S_n for rational fiber sizes over extension fields.

The detailed proofs, corrections, tests, and confidence assessments are in FIVE_DIRECTIONS.md and REFEREE_AUDIT_FIVE_RESULTS.md.

Strongest results developed here

The paper proves, with exact symbolic checks where applicable:

• an explicit ordered-root/Galois-closure model as the complement of a smooth hypersurface in PGL_2, including its product structure, Grothendieck class, and finite-field point counts;
• full symmetric-group monodromy S_n for every generic degree in the contemporaneous one-variable weighted-lift family, via a general theorem for R(w)-Pw+Q when R is a Morse polynomial;
• exact escape exponents at infinity: two inverse branches escape like epsilon^(-1/2) across a smooth discriminant point, while three escape like epsilon^(-2/3) at the omitted cusp;
• a strong plane-slice obstruction: k[Q_p,R_p] \cap k[x,y]=k, so no nonconstant polynomial in the two outputs fills across the deleted hyperbola, even after polynomial target postcomposition;
• a general square law for the Jacobian of weighted quotients, together with an explicit collapsed critical divisor in this example;
• commuting, divergence-free inverse-Jacobian vector fields that form a global algebraic frame, plus an explicit finite-time escaping trajectory and a completeness criterion relevant to invertibility;
• a finite smooth completion with boundary A^2, its affine-bundle/torsor description, and uniqueness of the nontrivial torsor up to bundle automorphism;
• ordinary-degree growth formulas for the all-degree family, exact real and finite-field fiber statistics, and a consequences ledger separating proved implications from open questions.

These are proof-level results about this example and its associated families, not a claim of a second breakthrough comparable to the counterexample itself. Some may be independently useful or new, but historical priority requires expert literature review. LIMITATIONS.md and the paper state the known overlaps and non-claims.

Contents

• paper/main.md: readable paper in Markdown.
• paper/main.tex: typeset LaTeX source.
• output/pdf/jacobian_counterexample_audit.pdf: built paper, when present.
• src/verify.py: exact symbolic certificate using SymPy.
• tests/test_map.py, tests/test_five_directions.py, and tests/test_referee_audit.py: exact regression and adversarial tests for the main identities and the five-direction research audit.
• CONSEQUENCES.md: claim-by-claim ledger of implications and non-implications.
• FIVE_DIRECTIONS.md: five new deep investigations, with general theorems, proofs, realistic applications, and explicit next conjectures.
• REFEREE_AUDIT_FIVE_RESULTS.md: hidden-hypothesis audit, counterexamples to overbroad formulations, independent checks, and confidence assessments.
• FACTORIAL_COLLISION_CERTIFICATE.md: standalone derivation of the collision complement's class group, Picard group, factoriality, and unit group, with dependencies and characteristic hypotheses made explicit.
• RESEARCH_FRONTIER.md: proved new deductions, rejected routes, and the highest-value next theorem targets.
• PROMPT_AUDIT.md: fulfillment check against the original project request.
• paper/references.bib: bibliography.
• LIMITATIONS.md: scope, status, provenance, and non-claims.

Reproduce the exact checks

Python 3.11 or later is recommended.

python3 -m venv .venv
. .venv/bin/activate
python -m pip install -r requirements.txt
make verify
make test

The verifier uses symbolic expansion and exact rational arithmetic. No floating-point equality is used for a theorem-level assertion.

The verification suite covers the core certificate and the machine-checkable identities used by the structural arguments. The paper distinguishes these computer-checked identities from arguments proved in prose.

To build the PDF, install a LaTeX distribution containing latexmk, pdflatex, amsmath, amsthm, hyperref, and booktabs, then run:

make paper

Primary public source and status

• Levent Alpoge's announcement, 20 July 2026 (UTC).

As of 21 July 2026, this audit found an announcement and several same-day technical discussions, but no peer-reviewed paper by the announcer. The polynomial identities themselves are finite exact calculations and do not depend on peer review. Broader claims of novelty, historical priority, and community acceptance do.

Attribution

Credit for the displayed counterexample belongs to Levent Alpoge and the discovery history stated in his announcement. This repository was prepared as an independent, user-directed, AI-assisted audit. Add maintainer authorship before archival citation if appropriate; do not erase the discovery credit or the AI-assistance disclosure.

Code is MIT licensed. Paper and explanatory text are CC BY 4.0.