Exact fibers of the Keller counterexamples F₄, F₅ and F₆
Abstract
The Jacobian conjecture was refuted in dimension three in July 2026, and a general construction of Gao — sweeping tangent direction fields on parametrised hypersurfaces — has since produced Keller counterexamples in every dimension greater than two. Four of those maps, F₄,…,F₇ in dimensions four and five, have generic fibers of 5, 10, 6 and 12 points, and their complete fiber stratifications are deferred there to a later version, the relevant Gröbner bases being out of reach. We determine several fibers of these maps exactly and constrain every fiber of two of them. The degree-ten fiber polynomial of F₅ has three leading coefficients that do not depend on the target; they force Σᵢ(rᵢ-1/4)²=0 on any fiber whose roots are all real, so at most one real target can admit such a fiber, and that target is rational and explicit. Over it the fiber of F₅ turns out to be empty: every solution of the eliminated fiber system carries γ=0 and is deleted by the monomial twist. The target is therefore an explicit rational point off the image of F₅, certified by a resultant Bézout identity UA+VB =-750 (4w₁-1)¹⁰ valid over every field of characteristic zero. For F₆ the same mechanism gives Σᵢrᵢ²=1 and Σᵢrᵢ³=5/2 for every target, which are incompatible over ℝ: the fiber sextic never splits over the reals, so at most four of the six generic preimages are ever real. Four is attained — we exhibit a rational target of F₆ with four distinct rational preimages and prove that its fiber has exactly six points, the condition γ ≠ 0 at every root being verified exactly rather than numerically. Finally we exhibit a rational target of F₄ with five distinct rational preimages, so that the geometric degree five is attained over ℚ. The certificates behind these statements — the fiber evaluations, the Bézout identity and the two reality arguments — are machine-checked in Lean 4.
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Claim ledger
Stated results
KF1candidate2026-08-28
F₄: an explicit rational target with five distinct rational preimages – the geometric degree 5 realised over Q
KF2candidate2026-08-28
F₅: an explicit rational point off the image – the degree-ten fiber polynomial degenerates to 20480(w₁-1/4)¹0 and the unique root carries gamma = 0
KF3candidate2026-08-28
F₅: if the fiber polynomial splits over R then all ten roots equal 1/4 – the target-independent coefficients pin the whole real locus
KF4candidate2026-08-28
F₆: the fiber sextic never splits over R – at most four of the six generic preimages are ever real
KF5candidate2026-08-28
F₆: an explicit rational target with four distinct rational preimages – the rational maximum of KF4 is attained
KF6known data2026-08-28
F₇: the closed-form transverse solution of the curve-type fiber system in dimension five, and R₀ = w⁵(w-1)(w⁶+w⁵+w⁴-w³-w²-w+1), as polynomial identities
KF7known data2026-08-28
F₄: the divisibility statements C | S₁, C² | S₂, C | S₃ and det J(F₄) = -44/9, as polynomial identities from the construction
KF8known data2026-08-28
F₅: the divisibility statements C | S₁, C² | S₂, C³ | S₃ and det J(F₅) = 160/29, as polynomial identities from the construction
KF9candidate2026-08-28
F₆: at the explicit target (1,-33/8,-1,-1,57/8) the fiber sextic factors as 2w(w+1)(w-1/2)(w-3/2)(w²+5/4) and gamma is nonzero at every one of its six roots – the fiber has exactly six points
KF10candidate2026-08-29
F₆: the two non-rational preimages of the fiber over (1, -33/8, -1, -1, 57/8), explicitly in Q(sqrt(-5)) – with KF5 and KF9 the complete six-point fiber of F₆, and the first non-rational preimage exhibited for any of F₄-F₇
KF11measurement2026-08-29
MEASUREMENT: the price of a Q(sqrt(-5)) fiber point in Lean – 691 lines and 29 s CPU for two modules, no native_decide, a parametric (K, s) carrier instead of a quadratic-field construction, and staging the substitution at every w = gammaᵏ u boundary is 135x less certificate
This ledger entry is reported in prose and is not bound to a Lean theorem.KF12routine2026-08-29
F₆: the stage inverted in closed form on gamma!= 0, so every root of the fiber sextic lifts to an explicit preimage – six distinct points derived rather than exhibited, and the four rational KF5 points re-proved by normₙum without native_decide
KF13routine2026-09-02
F₄: the fiber correspondence in both directions – w2 = N/w1 and gamma = G/w1 from the 2x2 minor -w1, the elimination w1*(third residual) = -R with no side condition, and the triangular stage inverted AND shown unique, so preimages correspond to the roots of the tangency quintic carrying gamma!= 0 at every target with Y2!= 0
KF14candidate2026-09-02
F₄: the fiber over (1,-13/6,-5/18,97/18) is EXACTLY the five rational points of KF1 – the tangency quintic splits as (2/9)(w+5/2)(w+1)(w+1/2)(w-1)(w-2), gamma takes the nonzero values 21/4, -1/2, 5/12, 15/4, -7/6 at the five roots, and there is no sixth preimage over any field of characteristic zero, C included
KF15candidate2026-09-02
F₄: a target-explicit invariant of every fiber – prodⱼ (1 + w₁j²) = 49 + (81/4)(Y₁ + Y₃ - 2/3)² whenever the tangency quintic splits, hence never below 49 and equal to 49 exactly on the plane Y₁ + Y₃ = 2/3; consequently no fiber over a rational target has a sweep parameter with w² = -1, and every totally real fiber has one with |w₁| > 1
KF16candidate2026-09-02
F₅: the first exhibited preimage of any of its fibers – F₅(1, 0, 0, -1/2) = (1, 90/29, 129/58, 55/58) with gamma = 1, w₁ = 1, w₂ = 1/2, so that target IS in the image (the complement of the empty-fiber row KF2), together with the F₅ stage inverted in closed form on gamma!= 0
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- The Jacobian conjecture (Keller 1939) fell in dimension three on 19 July 2026 (Alpöge), with a uniform family the next day (Gallagher) and a geometric explanation — the *tangent sweep* — on 23 July (Speyer). On 31 July 2026 Gao posted arXiv:2608.00222, which generalises the sweep from plane curves to direction fields on hypersurfaces and produces five new explicit Keller counterexamples. Four of them are the subject of this family:
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
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