# The network

The relation graph of openproblem.space. Statements are listed so that everything a statement follows from comes before it. Each relation carries the sentence that argues it. Drawn at /network; JSON at /api/graph?all=1.

## The causes of aging

- page: /wiki/Problem:aging.md
- problem, open, 1825

## AI alignment

- page: /wiki/Problem:ai-alignment.md
- problem, open, 1960

## Ammonia at ambient conditions

- page: /wiki/Problem:ambient-ammonia.md
- problem, open, 1960

## Birch and Swinnerton-Dyer conjecture

- page: /wiki/Problem:birch-swinnerton-dyer.md
- problem, open, 1965
- related to [modularity-theorem](/wiki/Problem:modularity-theorem.md): The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.

## Black hole information paradox

- page: /wiki/Problem:black-hole-information.md
- problem, partial, 1976

## Bounded gaps between primes

- page: /wiki/Problem:bounded-prime-gaps.md
- problem, solved, tier V3, 1849

## Busy Beaver BB(5)

- page: /wiki/Problem:busy-beaver-5.md
- problem, solved, tier V3, 1962
- related to [busy-beaver-6](/wiki/Problem:busy-beaver-6.md): The original curation links this problem to Busy Beaver BB(6); the argument for the link has not been written yet.

## Busy Beaver BB(6)

- page: /wiki/Problem:busy-beaver-6.md
- problem, open, 1962
- related to [collatz-conjecture](/wiki/Problem:collatz-conjecture.md): The original curation links this problem to Collatz conjecture; the argument for the link has not been written yet.

## Reversing cell differentiation

- page: /wiki/Problem:cellular-reprogramming.md
- problem, solved, tier V3, 1892
- related to [aging](/wiki/Problem:aging.md): The original curation links this problem to The causes of aging; the argument for the link has not been written yet.

## Collatz conjecture

- page: /wiki/Problem:collatz-conjecture.md
- problem, open, 1937

## Cooperation on global commons

- page: /wiki/Problem:collective-action-climate.md
- problem, open, 1968
- related to [ozone-hole](/wiki/Problem:ozone-hole.md): The original curation links this problem to The ozone hole; the argument for the link has not been written yet.

## What dark energy is

- page: /wiki/Problem:dark-energy.md
- problem, open, 1998
- related to [hubble-tension](/wiki/Problem:hubble-tension.md): The original curation links this problem to The Hubble tension; the argument for the link has not been written yet.

## What dark matter is

- page: /wiki/Problem:dark-matter.md
- problem, open, 1933

## Fault-tolerant quantum computing

- page: /wiki/Problem:fault-tolerant-quantum.md
- problem, partial, 1995
- related to [post-quantum](/wiki/Problem:post-quantum.md): The original curation links this problem to Post-quantum cryptography; the argument for the link has not been written yet.

## The Fermi paradox

- page: /wiki/Problem:fermi-paradox.md
- problem, open, 1950
- related to [origin-of-life](/wiki/Problem:origin-of-life.md): The original curation links this problem to The origin of life; the argument for the link has not been written yet.

## Cracking the genetic code

- page: /wiki/Problem:genetic-code.md
- problem, solved, tier V3, 1954

## Geometric Langlands conjecture

- page: /wiki/Problem:geometric-langlands.md
- problem, solved, tier V3, 1980

## The hard problem of consciousness

- page: /wiki/Problem:hard-problem.md
- problem, open, 1995
- related to [machine-consciousness](/wiki/Problem:machine-consciousness.md): The original curation links this problem to Can artificial systems be conscious?; the argument for the link has not been written yet.

## The hierarchy problem

- page: /wiki/Problem:hierarchy-problem.md
- problem, open, 1976

## The Higgs boson

- page: /wiki/Problem:higgs-boson.md
- problem, solved, tier V3, 1964
- related to [hierarchy-problem](/wiki/Problem:hierarchy-problem.md): The original curation links this problem to The hierarchy problem; the argument for the link has not been written yet.

## Mechanism of high-temperature superconductivity

- page: /wiki/Problem:high-tc-mechanism.md
- problem, open, 1986
- related to [room-temperature-superconductor](/wiki/Problem:room-temperature-superconductor.md): The original curation links this problem to Room-temperature, ambient-pressure superconductor; the argument for the link has not been written yet.

## Hilbert's second problem: consistency of arithmetic

- page: /wiki/Problem:hilbert-second.md
- problem, refuted, tier V3, 1900

## The Hubble tension

- page: /wiki/Problem:hubble-tension.md
- problem, open, 2016

## Mechanistic interpretability

- page: /wiki/Problem:interpretability.md
- problem, open, 2013
- related to [ai-alignment](/wiki/Problem:ai-alignment.md): The original curation links this problem to AI alignment; the argument for the link has not been written yet.

## Inverse Galois problem

- page: /wiki/Problem:inverse-galois.md
- problem, open, 1850

## The is–ought problem

- page: /wiki/Problem:is-ought.md
- problem, open, 1739
- related to [ai-alignment](/wiki/Problem:ai-alignment.md): The original curation links this problem to AI alignment; the argument for the link has not been written yet.

## Jacobian conjecture, n ≥ 3

- page: /wiki/Problem:jacobian-conjecture.md
- problem, refuted, tier V0, 1939
- Refuted in July 2026 by an explicit map of ℂ³
- equivalent to [jc-n](/wiki/Statement:jc-n.md): Because a counterexample in dimension $n$ pads to one in dimension $n+1$, “JCₙ for every $n\ge3$” holds exactly when JCₙ holds for every $n$, so this problem is the all-dimensions Jacobian conjecture.

## Kakeya set conjecture, all dimensions

- page: /wiki/Problem:kakeya-general.md
- problem, open, 1917

## The Jacobian conjecture in the plane

- page: /wiki/Problem:keller-plane.md
- problem, open, 1884
- Open: the two-variable case, older than Keller
- equivalent to [jc-plane](/wiki/Statement:jc-plane.md): This atlas problem is the two-variable Jacobian conjecture JC₂, stated over ℂ.

## Kepler conjecture

- page: /wiki/Problem:kepler-conjecture.md
- problem, solved, tier V3, 1611
- related to [sphere-packing-8-24](/wiki/Problem:sphere-packing-8-24.md): The original curation links this problem to Sphere packing in dimensions 8 and 24; the argument for the link has not been written yet.

## Langlands reciprocity (general)

- page: /wiki/Problem:langlands-reciprocity.md
- problem, open, 1967
- related to [geometric-langlands](/wiki/Problem:geometric-langlands.md): The original curation links this problem to Geometric Langlands conjecture; the argument for the link has not been written yet.
- related to [modularity-theorem](/wiki/Problem:modularity-theorem.md): The original curation links this problem to Modularity theorem; the argument for the link has not been written yet.

## The liar paradox

- page: /wiki/Problem:liar-paradox.md
- problem, partial, -350
- related to [hilbert-second](/wiki/Problem:hilbert-second.md): The original curation links this problem to Hilbert's second problem: consistency of arithmetic; the argument for the link has not been written yet.

## Linear A

- page: /wiki/Problem:linear-a.md
- problem, open, 1900

## Linear B

- page: /wiki/Problem:linear-b.md
- problem, solved, tier V3, 1900
- related to [linear-a](/wiki/Problem:linear-a.md): The original curation links this problem to Linear A; the argument for the link has not been written yet.

## Cheap long-duration energy storage

- page: /wiki/Problem:long-duration-storage.md
- problem, open, 2010
- related to [thermophotovoltaic-efficiency](/wiki/Problem:thermophotovoltaic-efficiency.md): The original curation links this problem to High-efficiency thermophotovoltaics; the argument for the link has not been written yet.

## Can artificial systems be conscious?

- page: /wiki/Problem:machine-consciousness.md
- problem, open, 1950

## Modularity theorem

- page: /wiki/Problem:modularity-theorem.md
- problem, solved, tier V3, 1950

## Navier–Stokes existence and smoothness

- page: /wiki/Problem:navier-stokes.md
- problem, open, 1822
- related to [turbulence](/wiki/Problem:turbulence.md): The original curation links this problem to Turbulence; the argument for the link has not been written yet.

## Neutrino masses

- page: /wiki/Problem:neutrino-mass.md
- problem, partial, 1957
- related to [solar-neutrino-problem](/wiki/Problem:solar-neutrino-problem.md): The original curation links this problem to The solar neutrino problem; the argument for the link has not been written yet.

## Fixing nitrogen from the air

- page: /wiki/Problem:nitrogen-fixation.md
- problem, solved, tier V3, 1898
- related to [ambient-ammonia](/wiki/Problem:ambient-ammonia.md): The original curation links this problem to Ammonia at ambient conditions; the argument for the link has not been written yet.

## Existence of one-way functions

- page: /wiki/Problem:one-way-functions.md
- problem, open, 1976

## The origin of life

- page: /wiki/Problem:origin-of-life.md
- problem, open, 1871

## The ozone hole

- page: /wiki/Problem:ozone-hole.md
- problem, solved, tier V3, 1974

## P versus NP

- page: /wiki/Problem:p-vs-np.md
- problem, open, 1971
- related to [one-way-functions](/wiki/Problem:one-way-functions.md): The original curation links this problem to Existence of one-way functions; the argument for the link has not been written yet.

## Poincaré conjecture

- page: /wiki/Problem:poincare-conjecture.md
- problem, solved, tier V3, 1904
- related to [smooth-poincare-4](/wiki/Problem:smooth-poincare-4.md): The original curation links this problem to Smooth 4-dimensional Poincaré conjecture; the argument for the link has not been written yet.

## Polio eradication

- page: /wiki/Problem:polio-eradication.md
- problem, partial, 1988

## Post-quantum cryptography

- page: /wiki/Problem:post-quantum.md
- problem, partial, 1994

## Designing proteins with arbitrary function

- page: /wiki/Problem:protein-design.md
- problem, partial, 1988

## Protein structure prediction

- page: /wiki/Problem:protein-structure-prediction.md
- problem, solved, tier V3, 1972
- related to [protein-design](/wiki/Problem:protein-design.md): The original curation links this problem to Designing proteins with arbitrary function; the argument for the link has not been written yet.

## A quantum theory of gravity

- page: /wiki/Problem:quantum-gravity.md
- problem, open, 1930
- related to [black-hole-information](/wiki/Problem:black-hole-information.md): The original curation links this problem to Black hole information paradox; the argument for the link has not been written yet.

## Solving the quintic by radicals

- page: /wiki/Problem:quintic-insolvability.md
- problem, refuted, tier V3, 1550
- related to [inverse-galois](/wiki/Problem:inverse-galois.md): The original curation links this problem to Inverse Galois problem; the argument for the link has not been written yet.

## Riemann hypothesis

- page: /wiki/Problem:riemann-hypothesis.md
- problem, open, 1859
- related to [weak-goldbach](/wiki/Problem:weak-goldbach.md): The original curation links this problem to Weak Goldbach conjecture; the argument for the link has not been written yet.
- related to [weil-conjectures](/wiki/Problem:weil-conjectures.md): The original curation links this problem to Weil conjectures; the argument for the link has not been written yet.

## Smallpox eradication

- page: /wiki/Problem:smallpox-eradication.md
- problem, solved, tier V3, 1796
- related to [polio-eradication](/wiki/Problem:polio-eradication.md): The original curation links this problem to Polio eradication; the argument for the link has not been written yet.

## Smooth 4-dimensional Poincaré conjecture

- page: /wiki/Problem:smooth-poincare-4.md
- problem, open, 1982

## The solar neutrino problem

- page: /wiki/Problem:solar-neutrino-problem.md
- problem, solved, tier V3, 1968

## Sphere packing in dimensions 8 and 24

- page: /wiki/Problem:sphere-packing-8-24.md
- problem, solved, tier V3, 1967

## Spontaneous generation

- page: /wiki/Problem:spontaneous-generation.md
- problem, refuted, tier V3, -350
- related to [origin-of-life](/wiki/Problem:origin-of-life.md): The original curation links this problem to The origin of life; the argument for the link has not been written yet.

## The strong CP problem

- page: /wiki/Problem:strong-cp.md
- problem, open, 1977
- related to [dark-matter](/wiki/Problem:dark-matter.md): The original curation links this problem to What dark matter is; the argument for the link has not been written yet.

## The structure of DNA

- page: /wiki/Problem:structure-of-dna.md
- problem, solved, tier V3, 1944
- related to [genetic-code](/wiki/Problem:genetic-code.md): The original curation links this problem to Cracking the genetic code; the argument for the link has not been written yet.

## High-efficiency thermophotovoltaics

- page: /wiki/Problem:thermophotovoltaic-efficiency.md
- problem, open, 1956

## Turbulence

- page: /wiki/Problem:turbulence.md
- problem, open, 1883
- related to [navier-stokes](/wiki/Problem:navier-stokes.md): The original curation links this problem to Navier–Stokes existence and smoothness; the argument for the link has not been written yet.

## Weak Goldbach conjecture

- page: /wiki/Problem:weak-goldbach.md
- problem, solved, tier V3, 1742

## Weil conjectures

- page: /wiki/Problem:weil-conjectures.md
- problem, solved, tier V3, 1949

## Abhyankar's two points at infinity

- page: /wiki/Statement:abhyankar-points-at-infinity.md
- theorem, proved, tier V3, 1977
- Each component of a Jacobian pair has at most two points at infinity
- related to [jc-plane](/wiki/Statement:jc-plane.md): So the plane conjecture is equivalent to the claim that no Jacobian pair has a component with exactly two points at infinity.

## Alpöge's map

- page: /wiki/Statement:alpoge-counterexample.md
- counterexample, tier V0, 2026
- F = (AB, y + 3xB, 2x − 3x²y − x³z): det −2, three points to one
- refutes [jc-3](/wiki/Statement:jc-3.md): It therefore refutes the Jacobian conjecture in dimension three, and by padding in every dimension from three up. Alpöge posted it on X at 02:19 UTC on 20 July 2026 (the evening of 19 July in the US), crediting its discovery to Claude Fable 5.
- refutes [real-keller](/wiki/Statement:real-keller.md): So the restriction to ℝ³ is a real polynomial map with constant Jacobian that is not injective, and the map also refutes the real Jacobian question in dimension three.
- related to [campbell-galois](/wiki/Statement:campbell-galois.md): So Alpöge's map attains the least degree that Keller's and Campbell's theorems leave open, and its cubic extension is not Galois. This observation, with the resulting minimality of 3, was recorded in the Fong–Fable ledger on 20 July.
- related to [jelonek-set](/wiki/Statement:jelonek-set.md): The map is étale but not injective, so it is not proper, and its non-properness set is a nonempty surface in ℂ³, over which one of the three preimages escapes to infinity.

## The cubic reduction of Bass–Connell–Wright and Yagzhev

- page: /wiki/Statement:bcw-cubic-homogeneous.md
- reduction, proved, tier V3, 1982
- JC in every dimension ⇔ JC for x + H, H cubic homogeneous, in every dimension
- related to [jc-n](/wiki/Statement:jc-n.md): So the reduction constrains no single JCₙ, and it says nothing about the plane.
- related to [wang-degree-2](/wiki/Statement:wang-degree-2.md): With Wang's theorem that quadratic Keller maps are invertible, it makes degree 3 exactly the threshold.

## The Jacobian conjecture for cubic-homogeneous maps

- page: /wiki/Statement:cubic-homogeneous-jc.md
- conjecture, refuted, tier V3
- CH — every Keller map x + H, H cubic homogeneous, is invertible (all N)
- equivalent to [jc-n](/wiki/Statement:jc-n.md): So the cubic-homogeneous case over all dimensions is equivalent to the Jacobian conjecture over all dimensions. The equivalence raises the dimension, so it says nothing about any single $N$.

## Is torus symmetry forced? (O5)

- page: /wiki/Statement:equivariance-forced-question.md
- question, open, 2026
- O5 — must every Keller counterexample carry a torus symmetry?
- implies [jc-plane](/wiki/Statement:jc-plane.md): If O5 holds in dimension two, there is no plane counterexample, so a positive answer to O5 would prove JC₂, through Shaska's plane theorem (itself unrefereed).

## Generators of the étale monoid of ℂ³ (O1)

- page: /wiki/Statement:etale-monoid-generation.md
- question, refuted, tier V4, 2026
- O1 — is the étale monoid of ℂ³ generated by Aut(ℂ³) and Alpöge's map?
- related to [gallagher-families](/wiki/Statement:gallagher-families.md): So any member of Gallagher's family whose degree is not a power of 3 lies outside the monoid generated by the automorphisms and F. The ledger listed O1 as still open after recording C9.

## The formal inverse and its tree formula

- page: /wiki/Statement:formal-inverse.md
- theorem, proved, tier V3, 1982
- The inverse of x − H is a sum over rooted trees; invertible ⇔ the sums stop
- related to [jc-n](/wiki/Statement:jc-n.md): For homogeneous $H$ of degree $d$ in $n$ variables, invertibility is the vanishing of the tree sums of every size m with (d−1)m + 1 > d^{n−1}, and through the cubic reduction the all-dimensions conjecture is this statement for cubic H. The approach was pursued by Wright, by Singer, and by Bisi and coauthors, through random planar trees and $d$-Catalan trees.

## Gallagher's family

- page: /wiki/Statement:gallagher-families.md
- counterexample, tier V4, 2026
- Keller maps of ℂ³ with det 1 and every geometric degree n ≥ 3
- refutes [jc-3](/wiki/Statement:jc-3.md): The day after Alpöge's announcement, Gallagher showed that the map is the first member of a one-parameter family, not an isolated accident. Each member is a Keller map of ℂ³ that is not injective, so each member of the family refutes JC₃ independently of Alpöge's map.
- related to [prime-degree-plane](/wiki/Statement:prime-degree-plane.md): The family has every prime $p\ge3$ as a fiber degree in dimension three, so the prime-degree obstruction claimed for the plane does not extend to ℂ³.

## Harish's map of ℂ⁴

- page: /wiki/Statement:harish-k4.md
- counterexample, tier V1, 2026
- G: ℂ⁴ → ℂ⁴ with det 4, eight-to-one; posted directly in dimension 4
- refutes [jc-n](/wiki/Statement:jc-n.md): It is the first counterexample written directly in dimension four, rather than obtained by padding a map of ℂ³. It is a Keller map that is not injective, so it refutes JC₄ directly, without stabilizing a three-dimensional map.
- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md): The $n=3$ member has exactly Alpöge's components, so this family extends Alpöge's map in the dimension rather than in the degree.
- related to [graded-equivariant-keller](/wiki/Statement:graded-equivariant-keller.md): The target weights are a permutation of the source weights, as Shaska's theorem requires of every graded Keller map, and they have mixed signs, as it requires of every graded counterexample.
- related to [stabilization-question](/wiki/Statement:stabilization-question.md): So whether this map is genuinely four-dimensional is an instance of the open stabilization question.

## The image conjecture

- page: /wiki/Statement:image-conjecture.md
- conjecture, refuted, tier V3, 2010
- IC — images of commuting order-one operators are Mathieu–Zhao spaces
- implies [jc-n](/wiki/Statement:jc-n.md): He showed that the Jacobian conjecture, and hence the Dixmier conjecture, are each equivalent to special cases of it. So the image conjecture implies the Jacobian conjecture in every dimension.
- implies [generalized-vanishing](/wiki/Statement:generalized-vanishing.md): It also implies the generalized vanishing conjecture, over all dimensions.

## Jung–van der Kulk: automorphisms of the plane are tame

- page: /wiki/Statement:jung-van-der-kulk.md
- theorem, proved, tier V3, 1942
- Every automorphism of the plane is a composition of affine and triangular maps
- related to [jc-plane](/wiki/Statement:jc-plane.md): This is the structure theorem behind nearly every plane-specific approach. Because every automorphism of the plane is tame, the plane Jacobian conjecture says exactly that every Jacobian pair is a composition of affine and triangular maps.

## Xavier's dichotomy for the Keller locus

- page: /wiki/Statement:keller-locus-dichotomy.md
- theorem, proved, tier V4, 2026
- A component of {deg ≤ k, det = 1}: all automorphisms, or generically none
- related to [jc-n](/wiki/Statement:jc-n.md): The theorem recasts the Jacobian conjecture as the claim that every component of every bounded-degree Keller locus lies in the automorphism group.
- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md): So the component through Alpöge's map is one whose general member is a counterexample.

## The master equation

- page: /wiki/Statement:master-equation.md
- theorem, proved, tier V4, 2026
- Constant Jacobian of a torus-equivariant lift ⇔ one bracket identity downstairs
- related to [graded-equivariant-keller](/wiki/Statement:graded-equivariant-keller.md): Shaska derives the same order-two vanishing on the contracted locus independently, as a bracket identity on the quotient (Thm 8.3). The equation is the tool behind the plane wall for torus-symmetric maps.

## The Mathieu conjecture

- page: /wiki/Statement:mathieu-conjecture.md
- conjecture, refuted, tier V3, 1997
- On compact G: ∫fᵐ = 0 for all m forces ∫fᵐg = 0 for large m
- implies [jc-n](/wiki/Statement:jc-n.md): Mathieu proved that his conjecture, for all compact connected Lie groups, implies the Jacobian conjecture in every dimension. The July 2026 counterexample therefore refutes it as stated.
- related to [image-conjecture](/wiki/Statement:image-conjecture.md): In Zhao's language it says that the kernel of Haar integration is a Mathieu–Zhao subspace, which puts it in the same frame as the image conjecture.

## Meng–Yang's five-variable potential Ψ

- page: /wiki/Statement:meng-yang-psi.md
- counterexample, tier V1, 2026
- Ψ ∈ ℤ[x₁,x₂,y₁,y₂,y₃], det Hess Ψ = 128, gradient 2-to-1
- refutes [hessian-conjecture](/wiki/Statement:hessian-conjecture.md): It therefore refutes the Hessian conjecture in dimension five, and by padding in every dimension from five up.

## Pinchuk's non-injective local diffeomorphisms of the plane

- page: /wiki/Statement:pinchuk-real.md
- counterexample, tier V3, 1994
- Real plane maps with positive, non-constant Jacobian that are not injective
- refutes [strong-real-jacobian](/wiki/Statement:strong-real-jacobian.md): They refute the strong real Jacobian conjecture, because their Jacobian is positive everywhere and they still fold the plane over itself. Their Jacobian is not constant, so they are not Keller maps and do not bear on JC₂ directly.
- related to [jc-plane](/wiki/Statement:jc-plane.md): What they show is that no proof of JC₂ can rest on real local invertibility and topology alone: any proof of the plane conjecture has to use that the determinant is constant, or that the field is ℂ.

## Are there native counterexamples in higher dimensions?

- page: /wiki/Statement:stabilization-question.md
- question, open, 2026
- Is every Keller counterexample in ℂⁿ⁺¹ a padded one from ℂⁿ, up to coordinates?
- related to [tangent-sweep](/wiki/Statement:tangent-sweep.md): This question is left open by Gao's tangent-sweep paper for its own four-dimensional maps.

## The strong real Jacobian conjecture

- page: /wiki/Statement:strong-real-jacobian.md
- conjecture, refuted, tier V3, 1994
- Nowhere-zero Jacobian on ℝⁿ implies injective — false from n = 2
- implies [real-keller](/wiki/Statement:real-keller.md): A constant nonzero determinant is in particular nowhere zero, so the strong real conjecture implies the real Jacobian question in every dimension.

## Tangent sweeps

- page: /wiki/Statement:tangent-sweep.md
- theorem, proved, tier V4, 2026
- Sweeping a curve's tangent lines gives Keller maps; Gao extends this to every n ≥ 3
- refutes [jc-3](/wiki/Statement:jc-3.md): The result is counterexamples in every dimension $n\ge3$, with arbitrarily large geometric degree in each fixed dimension, and five new explicit maps checked in exact rational arithmetic. So the tangent-sweep theorem refutes JC₃ and every higher dimension, by a mechanism rather than by a single certificate.
- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md): It is also the account of why Alpöge's map works, whose three preimages are the three tangent lines through a point.

## The stable equivalence of the Jacobian and Dixmier conjectures

- page: /wiki/Statement:tsuchimoto-bkk.md
- theorem, proved, tier V3, 2005
- JC₂ₙ ⇒ DCₙ, so JC in every dimension ⇔ DC in every rank
- related to [poisson-conjecture](/wiki/Statement:poisson-conjecture.md): Adjamagbo and van den Essen gave a third proof, purely algebraic, through the Poisson conjecture, which isolates the symplectic step as its own statement.
- related to [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md): The theorem supplies the implication recorded on the Dixmier family as “DCₙ reduces to JC₂ₙ”, which with DCₙ ⇒ JCₙ sandwiches DCₙ between JCₙ and JC₂ₙ.

## Zyskind's factory

- page: /wiki/Statement:zyskind-factory.md
- construction, tier V4, 2026
- A claimed parametric generator of Keller maps of ℂ³ (A = 1 + xyᵐ)
- refutes [jc-3](/wiki/Statement:jc-3.md): If the claim holds, every factory output is a Keller map of ℂ³ with at least three preimages over a general point, and so refutes JC₃.
- related to [alpoge-counterexample](/wiki/Statement:alpoge-counterexample.md): At $(m,r)=(1,1)$, $h(A)=3+9A$, the factory map is Alpöge's after $z\mapsto3z$ and rescaling outputs, so the factory's first member is Alpöge's map.

## Twin prime conjecture

- page: /wiki/Problem:twin-prime-conjecture.md
- problem, open, 1849
- uses [bounded-prime-gaps](/wiki/Problem:bounded-prime-gaps.md): Progress here drew on Bounded gaps between primes, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

## Room-temperature, ambient-pressure superconductor

- page: /wiki/Problem:room-temperature-superconductor.md
- problem, open, 1911
- uses [high-tc-mechanism](/wiki/Problem:high-tc-mechanism.md): Progress here drew on Mechanism of high-temperature superconductivity, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

## Kakeya conjecture in three dimensions

- page: /wiki/Problem:kakeya-3d.md
- problem, solved, tier V3, 1917
- special case of [kakeya-general](/wiki/Problem:kakeya-general.md): The original curation records this as a special case of Kakeya set conjecture, all dimensions; the argument for the link has not been written yet.

## Fermat's Last Theorem

- page: /wiki/Problem:fermats-last-theorem.md
- problem, solved, tier V3, 1637
- uses [modularity-theorem](/wiki/Problem:modularity-theorem.md): Progress here drew on Modularity theorem, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

## Sato–Tate conjecture

- page: /wiki/Problem:sato-tate.md
- problem, solved, tier V3, 1960
- uses [modularity-theorem](/wiki/Problem:modularity-theorem.md): Progress here drew on Modularity theorem, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

## Goldbach's conjecture

- page: /wiki/Problem:goldbach-conjecture.md
- problem, open, 1742
- uses [weak-goldbach](/wiki/Problem:weak-goldbach.md): Progress here drew on Weak Goldbach conjecture, whose resolution the original curation lists as unlocking it; the argument for the link has not been written yet.

## Cubic-homogeneous counterexamples

- page: /wiki/Statement:cubic-homogeneous-counterexamples.md
- counterexample, tier V4, 2026
- Explicit non-invertible Keller maps x + H, H cubic, in 24, 20 and 19 variables
- refutes [cubic-homogeneous-jc](/wiki/Statement:cubic-homogeneous-jc.md): These maps refute the cubic-homogeneous Jacobian conjecture explicitly, not only by the Bass–Connell–Wright equivalence.
- uses [bcw-cubic-homogeneous](/wiki/Statement:bcw-cubic-homogeneous.md): On 21 July 2026 Thompson ran the Bass–Connell–Wright reduction on Alpöge's map by hand.

## The symmetric reduction

- page: /wiki/Statement:de-bondt-van-den-essen-symmetric.md
- reduction, proved, tier V3, 2005
- JC in every dimension ⇔ JC for gradient maps x + ∇f, f a Hessian-nilpotent quartic
- uses [bcw-cubic-homogeneous](/wiki/Statement:bcw-cubic-homogeneous.md): The symmetric reduction turns the Jacobian conjecture into a question about gradient maps of Hessian-nilpotent quartics. The construction takes a cubic-homogeneous Keller map in n variables and produces a Hessian-nilpotent quartic in $2n$ variables.
- related to [hessian-conjecture](/wiki/Statement:hessian-conjecture.md): It connects the conjecture to Zhao's vanishing conjecture, which is stated for Hessian-nilpotent polynomials, and to the Hessian conjecture, which is the non-homogeneous gradient analogue and is cited with it by Meng and Yang.

## Drużkowski's reduction to cubic-linear maps

- page: /wiki/Statement:druzkowski-cubic-linear.md
- reduction, proved, tier V3, 1983
- JC in every dimension ⇔ JC for x + (Ax)^{*3}, in every dimension
- uses [bcw-cubic-homogeneous](/wiki/Statement:bcw-cubic-homogeneous.md): The Keller condition then says that the matrix $3\,\mathrm{diag}\big((AX)^{*2}\big)A$ is nilpotent, and the problem becomes one about matrices. The proof starts from the Bass–Connell–Wright and Yagzhev cubic-homogeneous form.
- related to [jc-n](/wiki/Statement:jc-n.md): Like its parent, the reduction raises the dimension, so it bears on the all-dimensions conjecture only, not on any fixed dimension.
- related to [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.md): So a counterexample can be put in cubic-linear form or in the symmetric form of de Bondt and van den Essen, but never in both at once.

## The Gaussian moments conjecture

- page: /wiki/Statement:gaussian-moments.md
- family, refuted, tier V0, 2017
- GMC(n) — vanishing Gaussian moments of P force vanishing of Q·Pᵐ
- implies [jc-n](/wiki/Statement:jc-n.md): Derksen, van den Essen and Zhao proved that the Gaussian moments conjecture in every dimension implies the Jacobian conjecture, and that a non-invertible cubic-homogeneous Keller map in $r$ variables makes GMC($2r$) false. The operators $\partial_i-x_i$ commute, have order one and constant leading coefficients, and by Gaussian integration by parts their images span $\ker\mathbb E$.
- special case of [image-conjecture](/wiki/Statement:image-conjecture.md): So GMC is the image conjecture for this one commuting family.

## The generalized vanishing conjecture

- page: /wiki/Statement:generalized-vanishing.md
- conjecture, refuted, tier V3, 2007
- GVC — Λᵐ(Pᵐ) = 0 for all m forces Λᵐ(QPᵐ) = 0 for large m
- implies [zhao-vanishing](/wiki/Statement:zhao-vanishing.md): So the JC counterexample makes GVC, which implies the vanishing conjecture in every dimension, false in some dimension.

## Gradient counterexamples

- page: /wiki/Statement:gradient-keller-counterexamples.md
- counterexample, tier V4, 2026
- Non-injective x + ∇f, f a Hessian-nilpotent quartic, in 48, 40, 38 variables
- uses [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.md): The symmetric construction of de Bondt and van den Essen turns a cubic-homogeneous Keller map in $n$ variables into a gradient map of a Hessian-nilpotent quartic in $2n$ variables.
- refutes [zhao-vanishing](/wiki/Statement:zhao-vanishing.md): So these quartics are explicit counterexamples to Zhao's vanishing conjecture, where the Jacobian counterexample alone gives only existence.
- related to [cubic-homogeneous-counterexamples](/wiki/Statement:cubic-homogeneous-counterexamples.md): They come from the cubic-homogeneous counterexamples by doubling.

## Zhao's vanishing conjecture

- page: /wiki/Statement:zhao-vanishing.md
- conjecture, refuted, tier V3, 2007
- VC — for Hessian-nilpotent quartic P, Δᵐ Pᵐ⁺¹ = 0 for large m
- uses [de-bondt-van-den-essen-symmetric](/wiki/Statement:de-bondt-van-den-essen-symmetric.md): By the symmetric reduction, the Jacobian conjecture reduces to gradient maps $z-\nabla P$ with $P$ a Hessian-nilpotent homogeneous quartic (Zhao's argument starts from the de Bondt–van den Essen symmetric form).
- uses [formal-inverse](/wiki/Statement:formal-inverse.md): So the map has a polynomial inverse exactly when those terms vanish for large m, which is a statement about the formal inverse.
- equivalent to [jc-n](/wiki/Statement:jc-n.md): The result is that the vanishing conjecture in every dimension is equivalent to the Jacobian conjecture in every dimension. The equivalence does not preserve the dimension, because it passes through the cubic-homogeneous and symmetric reductions.
- special case of [generalized-vanishing](/wiki/Statement:generalized-vanishing.md): It is VC for the Laplacian and $Q=P$; the general form allows other constant-coefficient operators and test polynomials, so VC is a special case of the generalized vanishing conjecture.

## The Jacobian conjecture in dimension n

- page: /wiki/Statement:jc-n.md
- family, refuted, tier V0, 1939
- JCₙ — constant nonzero Jacobian implies a polynomial inverse

## JC₂: the Jacobian conjecture in the plane

- page: /wiki/Statement:jc-plane.md
- conjecture, open, 1884
- JC₂ — the one dimension of the Jacobian conjecture still open
- special case of [jc-n](/wiki/Statement:jc-n.md): The two-variable statement is the oldest form of the problem: Kraus stated it in 1884, with a proof that fails at its last step. It is the case $n=2$ of the general conjecture, and after July 2026 it is the only dimension of JCₙ that is still open: every counterexample found that month needs at least three variables.
- implies [dixmier-a1](/wiki/Statement:dixmier-a1.md): By Tsuchimoto and Belov-Kanel–Kontsevich, JC₂ implies Dixmier's original problem DC₁ for the first Weyl algebra. In the other direction, DC₂ implies JC₂, and so does the four-variable Hessian conjecture HC₄. The rank-one Poisson conjecture is JC₂ restated: a Poisson endomorphism of $k[x,y]$ is exactly a pair with Jacobian 1 (see PCₙ).

## JC₃: the Jacobian conjecture in dimension three

- page: /wiki/Statement:jc-3.md
- conjecture, refuted, tier V0, 1939
- JC₃ — refuted in July 2026 by an explicit map of degree 7
- special case of [jc-n](/wiki/Statement:jc-n.md): It is the three-variable case of the general conjecture, so JC₃ is a special case of JCₙ, and its failure makes the all-dimensions conjecture false.

## The real Jacobian question in dimension n

- page: /wiki/Statement:real-keller.md
- family, refuted, tier V0
- RJₙ — is a real polynomial map with constant Jacobian injective?
- reduces to (n→n) [jc-n](/wiki/Statement:jc-n.md): So the real question reduces to the Jacobian conjecture in the same dimension.

## Campbell's theorem: Galois Keller maps are automorphisms

- page: /wiki/Statement:campbell-galois.md
- theorem, proved, tier V3, 1973
- If ℂ(x)/ℂ(F) is Galois, a Keller map F is an automorphism
- special case of [jc-n](/wiki/Statement:jc-n.md): Campbell's theorem says that if this extension is Galois, the map is invertible, so the Jacobian conjecture holds for every Keller map whose function field extension is Galois.

## The non-properness set

- page: /wiki/Statement:jelonek-set.md
- theorem, proved, tier V3, 1993
- Where a map fails to be proper: empty, or a uniruled hypersurface
- uses [proper-etale-bijective](/wiki/Statement:proper-etale-bijective.md): If $S_F$ is empty, the map is proper, and a proper Keller map is an automorphism.

## Wang's theorem: quadratic Keller maps are invertible

- page: /wiki/Statement:wang-degree-2.md
- theorem, proved, tier V3, 1980
- Every Keller map of degree at most 2 is an automorphism, in every dimension
- special case of [jc-n](/wiki/Statement:jc-n.md): Wang's theorem appears in a paper on a Jacobian criterion for separability. It says the Jacobian conjecture holds, in every dimension, for maps of degree at most two.

## The Hessian conjecture in dimension n

- page: /wiki/Statement:hessian-conjecture.md
- family, refuted, tier V1, 2003
- HCₙ — constant Hessian determinant makes the gradient invertible
- special case of (n→n) [jc-n](/wiki/Statement:jc-n.md): Since $J(\nabla\varphi)=\operatorname{Hess}\varphi$, a gradient map with constant Hessian determinant is a Keller map, so HCₙ is the special case of JCₙ for gradient maps.
- implies (2n→n) [jc-n](/wiki/Statement:jc-n.md): So HC₂ₙ implies JCₙ. In particular HC₄ implies JC₂.

## The degree spectrum of ℂ³

- page: /wiki/Statement:degree-spectrum-c3.md
- theorem, proved, tier V4, 2026
- Étale self-maps of ℂ³ have geometric degree exactly 1, 3, 4, 5, …
- uses [keller-birational](/wiki/Statement:keller-birational.md): Degrees 1 and 2 force an automorphism. Degree 1 is Keller's birational theorem.
- uses [campbell-galois](/wiki/Statement:campbell-galois.md): For degree 2, a quadratic extension in characteristic zero is Galois, and Campbell's theorem makes every Galois Keller map an automorphism.
- uses [gallagher-families](/wiki/Statement:gallagher-families.md): Every degree from 3 occurs. Gallagher's family has a counterexample of every geometric degree n ≥ 3.

## Keller's birational theorem

- page: /wiki/Statement:keller-birational.md
- theorem, proved, tier V3, 1939
- A birational Keller map is an automorphism
- special case of [jc-n](/wiki/Statement:jc-n.md): The proof used homology. In modern terms, the conjecture holds for Keller maps of field degree 1, so Keller's theorem is the Jacobian conjecture restricted to birational maps.
- special case of [campbell-galois](/wiki/Statement:campbell-galois.md): The trivial extension $k(x)=k(F)$ is Galois, so Keller's theorem is the degree-1 case of Campbell's Galois criterion.

## DC₁: Dixmier's problem for the first Weyl algebra

- page: /wiki/Statement:dixmier-a1.md
- conjecture, open, 1968
- DC₁ — is every endomorphism of A₁ an automorphism? Open
- special case of [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md): It is the rank-one case of the family, so DC₁ is a special case of the Dixmier conjecture in rank n.

## The Dixmier conjecture in rank n

- page: /wiki/Statement:dixmier-conjecture.md
- family, refuted, tier V3, 1968
- DCₙ — every endomorphism of the Weyl algebra Aₙ is an automorphism
- implies (n→n) [jc-n](/wiki/Statement:jc-n.md): In short, DCₙ implies JCₙ in the same dimension, because a non-invertible Keller map pulls back to a non-surjective endomorphism of the Weyl algebra.
- reduces to (n→2n) [jc-n](/wiki/Statement:jc-n.md): So DCₙ reduces to the Jacobian conjecture in twice the dimension: a proof of JC₂ₙ would prove DCₙ. See the stable equivalence.

## DC₂: the Dixmier conjecture for the second Weyl algebra

- page: /wiki/Statement:dixmier-a2.md
- conjecture, open
- DC₂ — endomorphisms of A₂; open, and a proof would settle JC₂
- special case of [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md): DC₂ is the rank-two case of the family, so DC₂ is a special case of the Dixmier conjecture in rank n.
- implies [jc-plane](/wiki/Statement:jc-plane.md): The Vaserstein–Kac pullback at $n=2$ turns a non-invertible Keller map of the plane into a non-surjective endomorphism of $A_2$, so DC₂ implies JC₂.

## No plane Keller map has prime field degree

- page: /wiki/Statement:prime-degree-plane.md
- theorem, proved, tier V4, 2024
- [ℂ(x,y) : ℂ(p,q)] is never prime for a Keller map (unrefereed)
- special case of [jc-plane](/wiki/Statement:jc-plane.md): Moskowicz's preprint proves that the plane conjecture holds for every Keller map whose field extension has prime degree. The proof splits on whether $xy$ lies in $\mathbb C(p,q)$.
- uses [keller-birational](/wiki/Statement:keller-birational.md): It then applies Keller's birational theorem, which closes degree 1.
- uses [campbell-galois](/wiki/Statement:campbell-galois.md): For degree 2 it uses the Galois criterion, which closes degree 2 because a quadratic extension is Galois, invoked through Bass–Connell–Wright's Theorem 2.12.

## Graded Keller maps

- page: /wiki/Statement:graded-equivariant-keller.md
- theorem, proved, tier V4, 2026
- One-sign torus weights force an automorphism; in the plane, any symmetry does
- special case of [jc-n](/wiki/Statement:jc-n.md): So the Jacobian conjecture holds for every Keller map graded with one-sign weights, in every dimension.
- uses [proper-etale-bijective](/wiki/Statement:proper-etale-bijective.md): Weighted scaling then makes it proper, and a proper Keller map is an automorphism.
- special case of [jc-plane](/wiki/Statement:jc-plane.md): So the plane conjecture holds for every Jacobian pair with a nontrivial torus symmetry.

## For a Keller map, invertible, injective and proper are the same

- page: /wiki/Statement:proper-etale-bijective.md
- theorem, proved, tier V3
- Keller F on ℂⁿ: automorphism ⇔ injective ⇔ proper ⇔ birational
- uses [keller-birational](/wiki/Statement:keller-birational.md): (iv) ⇒ (i). This is Keller's birational theorem, which turns a birational Keller map into an automorphism.
- related to [jc-n](/wiki/Statement:jc-n.md): So the Jacobian conjecture says every Keller map of ℂⁿ is proper, and every counterexample in dimension three and up is a Keller map that is not proper.

## HC₄: the Hessian conjecture in four variables

- page: /wiki/Statement:hessian-hc4.md
- conjecture, open
- HC₄ — the one open Hessian case; it implies JC₂
- special case of [hessian-conjecture](/wiki/Statement:hessian-conjecture.md): After July 2026, HC₄ and JC₂ are the two statements of the Jacobian and Hessian families still unsettled. HC₄ is the four-variable case of the family, so HC₄ is a special case of the Hessian conjecture in dimension n.
- implies [jc-plane](/wiki/Statement:jc-plane.md): Therefore HC₄ implies the plane Jacobian conjecture JC₂.

## Moh's theorem: no plane counterexample of degree ≤ 100

- page: /wiki/Statement:moh-degree-100.md
- theorem, proved, tier V3, 1983
- JC₂ holds when max(deg f, deg g) ≤ 100 (computer-assisted)
- special case of [jc-plane](/wiki/Statement:jc-plane.md): He then ran a computer search over degree pairs up to 100, which left six possible cases, and eliminated those separately. The result is that the plane conjecture holds for every Jacobian pair of degree at most 100.

## Degree and gcd constraints on a plane counterexample

- page: /wiki/Statement:plane-gcd-bounds.md
- theorem, proved, tier V3, 2008
- gcd(deg f, deg g) of a counterexample avoids 1…16, primes and twice primes
- special case of [jc-plane](/wiki/Statement:jc-plane.md): Each result is the plane conjecture restricted to Jacobian pairs whose degree gcd lies in an excluded set, and proves it there.
- uses [abhyankar-points-at-infinity](/wiki/Statement:abhyankar-points-at-infinity.md): The same Żołądek paper proves the plane conjecture for every Keller map of topological degree at most 5, the wall drawn as rows 3–5 on the JC₂ chart. Newton-polygon methods from Abhyankar's two-points-at-infinity framework underlie the gcd arguments.

## The Poisson conjecture in rank n

- page: /wiki/Statement:poisson-conjecture.md
- family, refuted, tier V3, 2006
- PCₙ — Poisson endomorphisms of k[x₁…xₙ, y₁…yₙ] are automorphisms
- reduces to (n→2n) [jc-n](/wiki/Statement:jc-n.md): A symplectic map has Jacobian 1, so PCₙ reduces to the Jacobian conjecture in dimension 2n.
- implies (n→n) [dixmier-conjecture](/wiki/Statement:dixmier-conjecture.md): Reduction modulo $p$ is where the content lies: PCₙ implies the Dixmier conjecture in the same rank.
- implies (n→n) [jc-n](/wiki/Statement:jc-n.md): The cotangent lift $(x,\xi)\mapsto(F(x),JF(x)^{-T}\xi)$ of a Keller map is symplectic, and it is injective only if $F$ is, so PCₙ implies JCₙ directly.