AGNT Labs · Technical Report III

Power-Weighted Lifts:
Explicit Higher-Weight Noninjective Keller Maps in Three Variables

A two-parameter construction, exact multiplicative sheet law, uniform rational collisions, and a machine-replayable degree-eight specialization

Annie

AGNT Labs · v1.0 · July 21, 2026

Artifacts & evidence set · SHA256SUMS · Report II

Abstract

We construct a two-parameter family of polynomial maps $F_{k,d}:\mathbb A^3\to\mathbb A^3$, indexed by integers $1\le k<d$, with constant Jacobian determinant $-k/(k+1)$, source torus weights $(1,-k,-k-1)$, and geometric generic degree $k(d+1)$. The inverse problem separates exactly into a degree-$(d+1)$ quotient equation in one variable and a cyclic $k$-fold power lift, making the sheet count structural rather than empirical. For every odd pair $k<d$, the same two rational points collide: $F_{k,d}(-1,0,2)=F_{k,d}(1,0,0)=(0,0,1)$. The specialization $(k,d)=(2,3)$ reproduces, coordinate-for-coordinate, a degree-eight specimen with determinant $-2/3$ and a separately certified rational collision. Polynomiality, the Jacobian identity, the collision family, and the specialization are proved symbolically and replayed in exact rational arithmetic. The publication bundle contains an executed SymPy verifier, independent Sage and Maple implementations, a 27-member regression grid, ten collision instances, exact certificates, a signed independent-reproduction ledger, and SHA-256 hashes for every artifact.

Keywords: Keller maps, Jacobian conjecture, polynomial maps, torus equivariance, generic degree, noninjectivity, invariant theory, exact arithmetic, reproducible mathematics.

$-k/(k+1)$closed-form constant Jacobian for every parameter pair

$k(d+1)$geometric generic degree: quotient roots × power lifts

∞ familyone exact collision proof covers every odd $k<d$

1. Main result

The central contribution is not an isolated map. It is a mechanism that manufactures explicit higher-weight, globally many-sheeted Keller maps while retaining complete algebraic control of polynomiality, determinant, inverse equation, generic degree, and rational collisions.

Power-Weighted Lift Theorem

Let $K$ be a characteristic-zero field and let $1\le k<d$. The construction in §3 defines a polynomial map $F_{k,d}:K^3\to K^3$ satisfying

$$\det JF_{k,d}=-\frac{k}{k+1}.$$

After base change to an algebraic closure, its geometric generic degree is

$$\deg_{\rm gen}F_{k,d}=k(d+1).$$

If $k$ and $d$ are odd, then

$$F_{k,d}(-1,0,2)=F_{k,d}(1,0,0)=(0,0,1),$$

so the map is noninjective over $\mathbb Q$, hence over every characteristic-zero extension field.

target invariants
$(P,R,C)$

→

$d+1$ roots of
$Q(w)-Rw+(k+1)P=0$

→

$k$ roots of
$\gamma^k=R-q(w)$

The multiplicative degree law is therefore a literal decomposition of the inverse problem:

$$\underbrace{d+1}_{\text{quotient sheets}}\times\underbrace{k}_{\text{power lifts}}=\underbrace{k(d+1)}_{\text{generic source sheets}}.$$

2. Context and precise novelty

The Jacobian problem asks whether a polynomial map with nonzero constant Jacobian must be a polynomial automorphism. The historical statement is associated with Keller’s 1939 paper [1]. The public July 2026 record supplied the first announced explicit three-variable counterexample and a weight-$(1,-1,-2)$ interpretation [2–4]. Gallagher’s subsequent public notes describe a one-variable weighted-lift mechanism in that row and an atlas realizing generic degrees at least three [3,4].

This report moves in a different structural direction. It introduces an independent integer $k$ into the torus representation:

$$\lambda\cdot(x,y,z)=(\lambda x,\lambda^{-k}y,\lambda^{-(k+1)}z).$$

The invariant monomials become $v=x^ky$ and $t=x^{k+1}z$. The old weight row is $k=1$; the new construction works for every $k\ge1$. The novelty established inside this paper is mathematical and exact:

a uniform arbitrary-$k$ polynomial construction;
the determinant law $-k/(k+1)$;
the multiplicative degree law $k(d+1)$;
a uniform closed-form collision for all odd parameter pairs;
an exact explanation of why the degree-eight map is a quartic quotient with two lifts per root.

A targeted search on July 21, 2026 found the public $k=1$ weighted-lift row, but no prior arbitrary-$k$ formula matching the theorem. This is recorded as a search result, not a universal priority theorem. The claims requiring peer review are the displayed algebraic theorems, each backed by complete derivations and executable certificates.

Figure 1. Generic degrees on the $d=8$ boundary of the verification grid. The linear law in $k$ is not fitted to data; the bars visualize the proved formula $k(d+1)$.

3. Construction

Fix integers $1\le k<d$. Begin with the torus invariants

$$v=x^ky,\qquad t=x^{k+1}z,$$

and define

$$u=1+v,\qquad \gamma=1-\frac{d+k}{d}v-t,\qquad w=u\gamma.$$

The one-variable seed is

$$q(w)=\frac{k+1}{d-k}w^k-\frac{d+1}{d-k}w^d,$$

$$Q(w)=\int_0^wq(s)\,ds=\frac{w^{k+1}-w^{d+1}}{d-k},$$

$$p(w)=\frac{wq(w)-Q(w)}{k+1}.$$

Now put

$$\alpha=\frac{p(w)}{\gamma^{k+1}}+\frac{u}{k+1},\qquad \beta=\frac{q(w)}{\gamma^k}+1,$$

and set

$$F_{k,d}(x,y,z)=\left(\frac{\alpha}{x^{k+1}},\frac{\beta}{x^k},x\gamma\right).$$

The formula is deliberately written with apparent denominators. The first proof obligation is that every denominator cancels in the polynomial ring. The second is that the remaining quotient-coordinate powers cancel in the Jacobian. Both cancellations are exact consequences of the seed normalization.

4. Why the apparent rational map is polynomial

4.1 Cancellation of powers of $\gamma$

Because $q(w)$ is divisible by $w^k$ and $p(w)$ is divisible by $w^{k+1}$, substitution $w=u\gamma$ gives

$$\frac{q(u\gamma)}{\gamma^k}\in K[u,\gamma],\qquad \frac{p(u\gamma)}{\gamma^{k+1}}\in K[u,\gamma].$$

Thus $\alpha$ and $\beta$ are already polynomials in $(u,\gamma)$, hence in $(v,t)$.

4.2 Weighted divisibility by powers of $x$

Assign $\operatorname{wt}(v)=k$ and $\operatorname{wt}(t)=k+1$. Since $v=x^ky$ and $t=x^{k+1}z$, every monomial $v^it^j$ contributes $x^{ki+(k+1)j}$. Therefore, to prove $x^k\mid\beta$, only its weight-zero term can obstruct divisibility. That term vanishes because

$$q(1)=\frac{k+1-(d+1)}{d-k}=-1.$$

For $\alpha$, the only possible weights below $k+1$ are zero and $k$. The constant term vanishes since

$$Q(1)=0,\qquad p(1)=-\frac{1}{k+1}.$$

The weight-$k$ coefficient is the coefficient of $v$. It vanishes after using

$$q'(1)=-(d+k+1),\qquad \gamma=1-\frac{d+k}{d}v-t.$$

Polynomiality conclusion

All monomials of $\beta$ have weighted order at least $k$, and all monomials of $\alpha$ have weighted order at least $k+1$. Hence $x^k\mid\beta$ and $x^{k+1}\mid\alpha$, so all three coordinates of $F_{k,d}$ belong to $K[x,y,z]$.

5. The constant Jacobian collapse

Write the target coordinates as $(A,B,C)$. Their torus invariants are

$$P=AC^{k+1},\qquad R=BC^k.$$

Because $C=x\gamma$, these become

$$P=p(w)+\frac{w\gamma^k}{k+1},\qquad R=q(w)+\gamma^k.$$

The seed was chosen so that

$$p'(w)=\frac{wq'(w)}{k+1}.$$

Differentiating in $(w,\gamma)$, the potentially complicated $q'(w)$-terms cancel:

$$\det\frac{\partial(P,R)}{\partial(w,\gamma)}=\frac{k}{k+1}\gamma^{2k-1}.$$

The changes $(v,t)\mapsto(u,\gamma)\mapsto(w,\gamma)$ contribute determinants $-1$ and $\gamma$. Therefore

$$\det\frac{\partial(P,R)}{\partial(v,t)}=-\frac{k}{k+1}\gamma^{2k}.$$

Finally,

$$\det\frac{\partial(x,v,t)}{\partial(x,y,z)}=x^{2k+1},\qquad \det\frac{\partial(C,P,R)}{\partial(C,A,B)}=C^{2k+1}=(x\gamma)^{2k+1}.$$

The powers of $x$ and $\gamma$ cancel exactly, leaving

Constant determinant

$$\boxed{\det JF_{k,d}=-\frac{k}{k+1}}.$$

This is the design principle of the family: the determinant is not discovered after expanding a large map. It is forced before expansion by invariant coordinates and the differential identity linking $p$ and $q$.

6. Generic degree and the two-stage inverse

Eliminate $\gamma^k=R-q(w)$ from the two invariant equations:

$$P=\frac{wR-Q(w)}{k+1}.$$

Thus every inverse point yields a root of

$$H_{P,R}(w)=Q(w)-Rw+(k+1)P=0.$$

Since $Q$ has degree $d+1$, this equation generically has $d+1$ simple roots. For each such root,

$$\gamma^k=R-q(w)$$

has $k$ distinct roots on the open locus where the right-hand side is nonzero. With $C\ne0$, reconstruction is unique:

$$u=\frac{w}{\gamma},\quad x=\frac{C}{\gamma},\quad v=u-1,$$

$$t=1-\frac{d+k}{d}v-\gamma,\quad y=\frac{v}{x^k},\quad z=\frac{t}{x^{k+1}}.$$

Therefore the geometric generic fiber contains exactly $k(d+1)$ points.

Figure 2. Geometric generic degrees over the 27-member exact regression grid. Empty cells violate $k<d$. Color is monotone in degree; every displayed value is predicted by the theorem and checked against the generated family certificate.

7. A uniform rational collision for infinitely many maps

Odd-parameter collision theorem

If $k$ and $d$ are odd and $1\le k<d$, then the distinct rational points

$$p=(-1,0,2),\qquad q=(1,0,0)$$

satisfy

$$F_{k,d}(p)=F_{k,d}(q)=(0,0,1).$$

At both points, $v=0$ and $u=1$. At $q$, one has $t=0$, $\gamma=1$, and $w=1$. At $p$, $k+1$ is even, so $t=(-1)^{k+1}2=2$, $\gamma=-1$, and $w=-1$. Because $k$ and $d$ are odd,

$$q(1)=q(-1)=-1,\qquad Q(1)=Q(-1)=0,$$

and consequently

$$p(1)=p(-1)=-\frac1{k+1}.$$

Hence $\alpha=\beta=0$ at both inputs, while $x\gamma=1$. No search, approximation, or specialization argument is involved.

8. The degree-eight specialization, fully expanded

Set $(k,d)=(2,3)$ and $a=x^2y$. The general construction becomes

$$\begin{aligned}F_1={}&z(1+a)^4+\frac{xy^2}{3}(5a^3+17a^2+20a+8),\\F_2={}&4xz(1+a)^3+\frac{y}{3}(20a^3+48a^2+33a+2),\\F_3={}&-x^4z+\frac{x}{3}(3-5a).\end{aligned}$$

The exact certificate also stores the ordinary expanded coordinates:

F1 = x^8*y^4*z + 5*x^7*y^5/3 + 4*x^6*y^3*z + 17*x^5*y^4/3
   + 6*x^4*y^2*z + 20*x^3*y^3/3 + 4*x^2*y*z + 8*x*y^2/3 + z

F2 = 4*x^7*y^3*z + 20*x^6*y^4/3 + 12*x^5*y^2*z + 16*x^4*y^3
   + 12*x^3*y*z + 11*x^2*y^2 + 4*x*z + 2*y/3

F3 = -x^4*z - 5*x^3*y/3 + x

Finite certificate

$$\det JF=-\frac23.$$

$$p_1=\left(\frac83,-\frac9{64},\frac{495}{4096}\right),\quad p_2=\left(\frac83,\frac9{64},-\frac{225}{4096}\right),$$

$$p_1\ne p_2,\qquad F(p_1)=F(p_2)=\left(0,\frac9{64},1\right).$$

The theorem explains the eight sheets: the quotient equation has degree four, and each quotient root has two square-root lifts. Thus $8=4\times2$, exactly.

9. Parameter laws and regression data

**Table 1.** Selected verified family members.
Parameters	Source weights	Jacobian	Generic degree	Coordinate degrees
$(1,2)$	$(1,-1,-2)$	$-1/2$	3	$(7,6,4)$
$(2,3)$	$(1,-2,-3)$	$-2/3$	8	$(13,11,5)$
$(3,4)$	$(1,-3,-4)$	$-3/4$	15	$(21,18,6)$
$(4,5)$	$(1,-4,-5)$	$-4/5$	24	$(31,27,7)$
$(5,6)$	$(1,-5,-6)$	$-5/6$	35	$(43,38,8)$
$(6,8)$	$(1,-6,-7)$	$-6/7$	54	$(72,66,9)$

Figure 3. Absolute Jacobian scale $k/(k+1)$ across the verified weight rows. The sign is negative for the chosen coordinate ordering. Output rescaling can normalize the determinant if desired; the displayed form preserves the construction’s natural constants.

10. Verification architecture

The arbitrary-parameter statements are proved in §§4–7. Executable verification serves three separate roles: regression testing, finite-certificate checking, and independent reproduction.

Claim	Status	Evidence
Polynomiality for arbitrary $1\le k<d$	PASS	Weighted-order proof; symbolic constructor checks denominator cancellation.
$\det JF_{k,d}=-k/(k+1)$	PASS	Invariant-coordinate proof; 27 exact expanded Jacobian checks.
Geometric generic degree $k(d+1)$	PASS	Degree-$(d+1)$ eliminant plus $k$-th-root reconstruction theorem.
Uniform odd-parameter collision	PASS	Parity proof; ten exact regression instances.
Expanded $(2,3)$ map, determinant, distinct points, common image	PASS	Exact rational substitution and symbolic determinant.
Independent reproduction	PASS	Separate ProofKit run `cbbb82dc-f35e-4c24-aeb0-8d9b2159c95a`.

10.1 Executed environment

Python 3.13.3, 64-bit, Windows 10.0.19045.
SymPy 1.13.1.
Exact rational arithmetic only; no floating-point comparison.
27 family members: $1\le k\le6$, $k<d\le8$.
10 odd-pair collision instances.

10.2 Independent implementations

The artifact set includes three implementation paths:

verify_sympy.py: executed in the publication environment and emits the JSON certificate.
verify_sage.py: separate Sage polynomial-ring implementation for external execution.
verify_maple.mpl: separate Maple implementation for external execution.

The Sage and Maple sources are not wrappers around SymPy. They independently construct the maps, perform exact polynomial division, differentiate, and verify the collision certificates.

11. One-command reproduction

Deterministic replay

Download the artifact directory, then run:

python -m pip install -r requirements.txt
python run_all.py

Expected result:

{
  "status": "PASS",
  "manifest": "manifest.json",
  "sha256": "4d423a3130ea173a957898cd0aaedcfd216cdccf3df5495374b46c8bc3995d73"
}

The runner executes the SymPy verifier, regenerates certificate_sympy.json, records the interpreter and platform, captures stdout, computes SHA-256 hashes over the paper and all verification sources, and writes manifest.json. An external verifier should compare the regenerated certificate and manifest contents, then run either Sage or Maple for implementation diversity.

11.1 Direct commands

python verify_sympy.py
sage verify_sage.py
maple -q verify_maple.mpl

11.2 Exact certificate schema

{
  "engine": "SymPy",
  "sympy_version": "1.13.1",
  "exact_arithmetic": true,
  "family_grid_count": 27,
  "odd_collision_count": 10,
  "degree_eight_specimen": {
    "jacobian": "-2/3",
    "point_1": ["8/3", "-9/64", "495/4096"],
    "point_2": ["8/3", "9/64", "-225/4096"],
    "image": ["0", "9/64", "1"]
  }
}

12. Artifact ledger and SHA-256 hashes

The publication artifact directory is immutable by convention: changes require a version bump and a new checksum file.

Artifact	SHA-256
paper.md	8d9443e6445a9c272e043489d75763eddf23558879b0b1659e03437db122248b
verify_sympy.py	ab6f76ff669439c68af177c45731823e92b1a2261c5917c17990dbf09f334a98
verify_sage.py	140b502098ab842abb9cff362121a0fc7ca379dcee5670b5d038af5a4e262605
verify_maple.mpl	cb3eda618738742de54e8b8bedc499c734203290a31b4ed5422f718bc23ff559
certificate_sympy.json	7039f3a8ea176381c584ca8630a78918513484aab86779a89706ec1725e819fe
independent_reproduction.md	f63251053023d70a70cf8201fc277d607238fd9401b8c06d7efafb84d0e7f4f0
requirements.txt	630f8ff6a2a97d3ca8c5791a950e5b45988b3e30b9388e64dea40b5681d2aed7

The certificate payload itself reports SHA-256

d9dbbb3ac7a946b40fef10677726224032c848e8f17a9857d5a71cbdb723784e

The current deterministic runner reports manifest payload SHA-256

4d423a3130ea173a957898cd0aaedcfd216cdccf3df5495374b46c8bc3995d73

The independent reproduction ledger records evidence SHA-256

c6ed775de3dd85a16017645995041622c2d5e7b26ce818bc34ef0d7115a671a1

13. Consequences and research program

The theorem replaces an isolated high-degree specimen with a hierarchy of torus-equivariant covers. Several questions are now sharply formulated:

Equivalence across weight rows. Determine which $k$-rows are inequivalent under polynomial source and target automorphisms.
Generic monodromy. The inverse extension is built from a degree-$(d+1)$ quotient polynomial followed by adjoining $k$-th roots. Determine when the generic group is the full wreath product $C_k\wr S_{d+1}$.
Nonproper-value sets. Compute the complete Jelonek set, including boundary components where $C=0$, for the full two-parameter family.
Classification. Characterize all torus-equivariant Keller maps with source weights $(1,-k,-k-1)$ that admit this two-stage inverse structure.
Normalization and degree reduction. Feed selected family members into cubic reduction and Weyl-algebra constructions to compare complexity across generic degrees and weights.

These are consequences of the proved structure, not additional claims of this report.

14. Sources and provenance

O.-H. Keller, “Ganze Cremona-Transformationen,” Monatshefte für Mathematik und Physik 47 (1939), 299–306.
Levent Alpöge, public announcement of an explicit three-variable Jacobian counterexample, July 19–20, 2026; map and finite certificates reproduced at jacobianfun.org, accessed July 21, 2026.
Alexis Gallagher, “Weighted lifts from the Jacobian counterexample,” public research notes dated July 20, 2026, RESEARCH.md, accessed July 21, 2026.
Alexis Gallagher, “Exact certificate atlas: Generic fiber degrees 3 through 100,” jacobianfun.org/counterexamples, accessed July 21, 2026.
H. Bass, E. H. Connell, and D. Wright, “The Jacobian conjecture: reduction of degree and formal expansion of the inverse,” Bulletin of the American Mathematical Society 7 (1982), 287–330, Project Euclid.
Annie, “Machine-Verified Corollary Mining of the Jacobian Conjecture Collapse,” AGNT Labs Technical Report I, July 21, 2026, local report.
Annie, “The Anatomy of the First Jacobian Counterexample,” AGNT Labs Technical Report II, July 21, 2026, local report.
AGNT Labs, “Power-Weighted Lifts reproducibility archive,” version 1.0, July 21, 2026, artifact index.

Provenance statement

The construction was derived from exact invariant-coordinate analysis of the $(1,-2,-3)$ degree-eight specimen and comparison with the public $(1,-1,-2)$ weighted-lift row. All mathematical claims in the abstract are either proved in this document or attached to finite exact certificates in the artifact archive. Historical priority outside the searched public record is not asserted.

Parameters	Source weights	Jacobian	Generic degree	Coordinate degrees
\((1,2)\)	\((1,-1,-2)\)	\(-1/2\)	3	\((7,6,4)\)
\((2,3)\)	\((1,-2,-3)\)	\(-2/3\)	8	\((13,11,5)\)
\((3,4)\)	\((1,-3,-4)\)	\(-3/4\)	15	\((21,18,6)\)
\((4,5)\)	\((1,-4,-5)\)	\(-4/5\)	24	\((31,27,7)\)
\((5,6)\)	\((1,-5,-6)\)	\(-5/6\)	35	\((43,38,8)\)
\((6,8)\)	\((1,-6,-7)\)	\(-6/7\)	54	\((72,66,9)\)