jacobian manuscript

The manuscript’s central construction is the polynomial map F : C^3 → C^3 announced by Alpöge. After the coordinate change to (P, y, s) with s = x/(1+xy) and A = 1+xy, the authors compute the Jacobian matrix of the transformed map and obtain diagonal-block factors whose determinants multiply to −2 on the dense open set A ≠ 0. Because det Jac F is itself a polynomial, the identity det Jac F ≡ −2 extends from the dense open set to all of C^3 by continuity (or, equivalently, by the identity theorem for polynomials).

Both referees recomputed this chain-rule argument and confirmed the constant −2. The same section exhibits three explicit rational points (0,0,−1/4), (1,−3/2,13/2), (−1,3/2,13/2) that all map to (−1/4,0,0). Direct substitution verifies the common image, so F is a Keller map that is not injective and therefore not a polynomial automorphism.

Stabilization by product with the identity on C^{n−3} immediately yields counterexamples in every dimension n ≥ 3.

Section 2 isolates the surviving positive statement. Proposition 2.1 records that the complement of the image of any Keller map has codimension at least two; the argument uses that a Keller map is étale (hence open) and that the complement is constructible, so a codimension-one component would contain an irreducible divisor, contradicting dominance of an étale map of smooth varieties of equal dimension. Theorem 2.2 then proves the equivalence, for any Keller map G, of the four conditions: G is a polynomial automorphism; G is proper; the nonproperness set S_G is empty; and codim S_G ≥ 2.

The key step is that a proper local biholomorphism with finite fibers is a covering map, and the only covering of the simply-connected space C^n is one-sheeted, so G is bijective and the algebraic inverse is regular by the usual Jacobian criterion. The argument is standard but compressed; a parenthetical reference to Forster or Huybrechts would assist readers outside several complex variables.

Section 3 introduces the projective coordinate [x : 1+xy] and reduces the fiber problem to a binary cubic in two homogeneous variables. Simple roots of that cubic are exactly the affine preimages. This description is the geometric heart of the paper: it determines every fiber (cardinality equal to the number of simple roots), the image (complement the smooth rational curve Γ cut out by the vanishing of the cubic’s discriminant locus in a suitable sense), and the nonproperness set (the discriminant hypersurface Σ).

The reduction is elementary and fully explicit; no resolution of singularities or heavy commutative algebra is required.

Section 4 studies the behavior at infinity. The normalized discriminant Δ is introduced, and the singular scheme of Σ is computed ideal-theoretically; it is nonreduced along Γ. A one-line remark that the ordinary cubic discriminant equals 4Δ would help readers comparing with classical invariant-theory tables.

The nonproperness set is identified with Σ, confirming that S_F is a hypersurface (codimension one) and therefore that F fails every equivalent condition of Theorem 2.2, as expected for a counterexample.

Section 5 constructs families. Starting from the seed map F, the authors add terms involving a univariate polynomial η_{d−2} of degree d−2, chosen so that the Jacobian determinant remains the nonzero constant −2 (or is renormalized to 1) while the generic degree becomes d. The same three-point collision survives whenever η_{d−2}(T) vanishes at T = −1/4, for instance when η_{d−2}(T) = c(4T+1).

This produces nonproper Keller maps of every generic degree d ≥ 3. The construction is algebraic and works over any field of characteristic zero. The proof should explicitly record that the added terms in the second and third coordinates vanish at the three source points because B ≠ 0 there and η(P) = 0 at P = −1/4.

Domain of validity is characteristic zero (the binary-cubic analysis and the covering-space step both use char 0). No counterexample candidate inside the paper’s own hypotheses was found: the three-point fiber is explicit, the determinant identity is polynomial, and the properness equivalence is classical once properness is assumed. Comparison with prior literature is straightforward.

The Jacobian conjecture has been open since Keller (1939); the Bass-Connell-Wright reduction to degree ≤ 3 is cited; the present work does not touch dimension two and does not claim to. The geometric analysis goes well beyond a single bad fiber: image, nonproperness locus, and degree-d families are all determined. Relative to the formal canon supplied for this review, there is no overlap; the paper lives entirely in classical affine algebraic geometry.

The dependency graph of the manuscript is acyclic. The explicit polynomials defining F are introduced by formula, not by an existence appeal that presupposes the conclusion. The identity det Jac F ≡ −2 is obtained by chain-rule computation on a dense open set and extended by the identity theorem for polynomials; that extension does not invoke the Jacobian conjecture or any properness hypothesis.

The three-point fiber is verified by direct substitution of rational points into those same polynomials. Non-injectivity is therefore an output of the computation, not an input.

Theorem 2.2 (properness equivalence) is proved from standard facts about étale maps, covering spaces, and simple connectivity of C^n. Its proof does not use the counterexample F; rather, F is later shown to violate the equivalent conditions, which is consistent rather than circular. Proposition 2.1 (codimension of the image complement) likewise uses only that Keller maps are étale and that complements of morphisms of finite type are constructible.

No clause of Proposition 2.1 assumes the Jacobian conjecture or its negation.

The binary-cubic fiber description is derived by algebraic elimination from the defining equations of F; the discriminant hypersurface and the curve Γ are read off from that cubic. The degree-d families are constructed by adding terms that are checked to preserve the constant Jacobian; the check is a direct differentiation, not an appeal to a prior nonproperness result. Nowhere does a claim justify itself, and nowhere is empirical or fitted evidence used.

The graph is: definitions → Jacobian identity → non-injectivity → failure of JC in dim 3 → stabilization → failure in dim ≥ 3, with a parallel independent branch definitions → properness equivalence, and a downstream branch fiber geometry → image and nonproperness set → families. All arrows point forward.

The paper works in the standard category of polynomial endomorphisms of affine space over a field of characteristic zero (often specialized to C for the analytic covering-space language). No exotic axioms are introduced. The following enumerated list records every load-bearing assumption, free parameter, and named falsifier, distinguishing explicit from implicit.

1. Characteristic zero (explicit in the geometric sections; implicit in the use of simple connectivity and in the binary-cubic root analysis). Required for the covering-space step of Theorem 2.2 and for unrestricted root-counting of the binary cubic.

2. Working over an algebraically closed field (or passing to the algebraic closure) when discussing geometric fibers and the curve Γ (implicit; standard). 3.

The coordinate change s = x/(1+xy) is valid on the dense open set A = 1+xy ≠ 0 (explicit); extension of polynomial identities from that open set to all of C^3 by the identity theorem (explicit). 4. A proper local biholomorphism with finite fibers is a covering map (explicit but compressed; classical).

5. C^n is simply connected, so a connected covering is one-sheeted (explicit; classical). 6.

Chevalley’s constructibility theorem, used to guarantee that the complement of the image is constructible (implicit in Proposition 2.1; should be named). 7. No free parameters in the seed map F; the degree-d families introduce free coefficients of η_{d−2}, subject to the open condition that the Jacobian determinant remain a nonzero constant and, for preservation of the three-point witness, the closed condition η(−1/4) = 0.

8. Named internal falsifier: if det Jac F is not the constant −2, or if the three listed points do not share an image, the counterexample collapses. 9.

Stabilization assumes that the product of a Keller map with an identity map is again a Keller map with Jacobian determinant equal to that of the first factor (explicit elementary fact).

Implicit assumptions that should be made explicit in revision are item 6 (Chevalley) and a one-sentence reminder that the analytic language of Theorem 2.2 can be replaced by a purely algebraic finite-étale-plus-proper argument for readers who prefer schemes. No assumption smuggles in the negation of the Jacobian conjecture; the negation is deduced from the computation.

What is reproducible today is the entire algebraic content of the paper by direct hand computation or by any standard computer-algebra system. The map F is given by explicit polynomials with rational coefficients; det Jac F is a single polynomial identity; the three witness points are rational and may be substituted by hand. Both referees performed these checks independently and obtained the claimed results.

The binary-cubic fiber description, the discriminant hypersurface, and the degree-d deformations are likewise given by explicit formulas and can be re-derived from the text alone.

What is missing is a deposited verification artifact. Given the extraordinary nature of the claim, the published version should include a short, self-contained script (SageMath, Macaulay2, Singular, or Mathematica) that (i) expands det Jac F and prints the constant −2, (ii) evaluates F at the three listed points and prints the common image, and (iii) optionally checks the fiber-count formula at a generic point. The script should be pinned by a content hash or a version-tagged repository link so that future readers can reproduce the check without re-implementing the polynomials from the PDF.

No Lean formalization, no external data tables, and no numerical experiment appear in the manuscript, so the verification grade is V0 (prose and hand algebra only). Build instructions are unnecessary beyond “run the deposited script in system X.” The bibliographic repairs noted in the reference-integrity audit are required for scholarly reproducibility of the citation graph but do not affect the mathematics.

The Jacobian conjecture has stood open since Keller’s 1939 paper on entire Cremona transformations. The Bass-Connell-Wright theorem reduced the general case to maps of degree at most three, and a large literature has produced partial positive results (e.g., in dimension two under extra hypotheses, or for maps of special shape) without settling the conjecture. The present manuscript, if the computations are accepted, supplies the first explicit counterexample in dimension three and therefore in every dimension n ≥ 3 by stabilization.

That is a decisive negative solution of the classical problem outside dimension two.

Relative to prior art, the contribution is not merely a single bad fiber. The binary-cubic reduction determines the full fiber geometry, the image complement (a smooth rational curve of codimension two), and the nonproperness set (a discriminant hypersurface). Theorem 2.2 isolates the sharp positive remnant: every proper Keller map is a polynomial automorphism.

Section 5 then produces infinite families of nonproper Keller maps of every generic degree d ≥ 3, showing the mechanism is structural. These geometric and structural layers go beyond the bare existence of a counterexample and will be the parts most cited by subsequent work.

The paper does not claim novelty in dimension two, does not claim a new general criterion beyond the properness equivalence, and does not rely on heavy machinery. Its novelty is the explicit counterexample together with the complete geometric analysis and the degree-d families. Independent confirmation would consist of a second group expanding det Jac F and checking the three-point fiber; both referees have already performed that confirmation.

There is no overlap with the formal canon under review, so novelty relative to that canon is not applicable; the paper lives in a different domain.

The manuscript lies entirely outside the formal canon’s forcing chain. The canon modules provided to the referees address cost uniqueness from a multiplicative functional equation, self-similarity forcing of the golden ratio, eight-tick ledger periodicity, dimension forcing to D = 3, constant derivations, spacetime emergence, and related structural results. None of those declarations concerns polynomial endomorphisms of affine space, Jacobian determinants, Keller maps, or the Jacobian conjecture.

The correct audit grade is therefore canon_match_strength = none, and the three structured buckets below are populated only to record that emptiness explicitly.

No theorem of the paper is inherited from the formal chain, and no theorem of the formal chain is used or extended by the paper. The central claims - constant nonzero Jacobian determinant, existence of a three-point fiber, equivalence of properness with polynomial invertibility for Keller maps, and construction of degree-d families - are classical algebraic-geometry statements proved inside the manuscript by direct computation and standard covering-space arguments. Their epistemic status is ordinary mathematical proof (paper-level), not machine-checked formalization in the supplied library.

Overclaim risk relative to the formal canon is nil, because the paper never cites or alludes to that canon. The only overclaim risk internal to the paper would be an incorrect determinant identity or an incorrect fiber computation; both independent referees recomputed those identities and found them correct. The two-dimensional Jacobian conjecture is correctly left open.

The distinction between “every Keller map is proper” (false in dimension ≥ 3) and “every proper Keller map is an automorphism” (true) is drawn sharply and is the main conceptual contribution beyond the bare counterexample.

Falsifiers and out-of-scope surfaces do not arise from the formal canon. The paper’s own internal falsifier is elementary: if an independent expansion of det Jac F produced a non-constant polynomial, or if the three listed points failed to share an image, the counterexample would collapse. No such failure was found.

Engineering surfaces, consciousness results, cosmological identities, and mass-law statements from the broader formal program are irrelevant to this manuscript and are not invoked.