In July 2026 an explicit polynomial map was announced that would refute a conjecture open since Keller 1939, and a small literature followed it within days. This page takes none of it on trust. 11 explicit maps — 8 transcribed or reconstructed from published sources whose bytes this repository holds and re-hashes, 3 built here — are decided in exact rational arithmetic: the Jacobian determinant expanded symbolically and compared coefficient by coefficient, every claimed collision re-evaluated as fractions, distinctness checked. Then the machine throws the published witnesses away and finds the collisions again blind. No float participates in any verdict, and the certificates are checkable by a stranger with a Python interpreter and no code from this repository.
Published, not peer-reviewed, not independently rerun. What is decided here is a property of the explicit maps printed in the certificate — determinants, collisions, distinctness — not a claim about the standing of anyone's paper. The external results are cited as CLAIMS and their bytes are pinned; this page asserts no priority, adjudicates no dispute, and contains no map in dimension 2, so it decides nothing about the plane.
Every row below passed the same three exact checks at this build, twice: once through this repository's instrument and once through a stdlib-Python verifier that shares no arithmetic with it. The lane column is the one to read first — it says where the map came from, and the page never lets those blur.
| certificate | source, as recorded | what the object is | dimension | det J | points | lane |
|---|---|---|---|---|---|---|
| keller-0 | Alpöge 2026-07-19 | the announced map, transcribed from the pinned preprint that prints it | n = 3 | −2 | 3 | transcribed · pinned |
| keller-1 | Alpöge 2026-07-19 | the n=3 map with identity coordinates adjoined — the stabilization, stated once | n = 8 | −2 | 3 | transcribed · pinned |
| keller-2 | tangent-sweep d=3 | our own degree-3 curve, [6, −6, 1] — geometric degree 4 | n = 3 | −2 | 2 | generated here |
| keller-3 | tangent-sweep d=4 | our own degree-4 curve, [52/5, −51/5, 0, 1] — geometric degree 5 | n = 3 | −2 | 2 | generated here |
| keller-4 | tangent-sweep d=5 | our own degree-5 curve, [20, −19, 0, 0, 1] — geometric degree 6 | n = 3 | −2 | 2 | generated here |
| keller-5 | Meng–Yang arXiv:2607.22198 | Hessian determinant of a degree-14 polynomial in 5 variables; its gradient is the map | n = 5 | 128 | 2 | transcribed · pinned |
| keller-6 | Gallagher zenodo.21479195 d=2, fiber degree 3 | the paper's seed [2, −3] rebuilt under its own gauge — generic fiber degree 3 | n = 3 | 1 | 2 | rebuilt from a published seed |
| keller-7 | Gallagher zenodo.21479195 d=3, fiber degree 4 | the paper's seed [3/2, −3/2, −1] rebuilt under its own gauge — generic fiber degree 4 | n = 3 | 1 | 2 | rebuilt from a published seed |
| keller-8 | Gallagher zenodo.21479195 d=4, fiber degree 5 | the paper's seed [17/10, −27/10, 1, −1] rebuilt under its own gauge — generic fiber degree 5 | n = 3 | 1 | 2 | rebuilt from a published seed |
| keller-9 | Gallagher zenodo.21479195 d=5, fiber degree 6 | the paper's seed [9/5, −14/5, 0, 1, −1] rebuilt under its own gauge — generic fiber degree 6 | n = 3 | 1 | 2 | rebuilt from a published seed |
| keller-10 | Gallagher zenodo.21479195 distinct member | the paper's seed [1, 0, −2] rebuilt under its own gauge — generic fiber degree 4 | n = 3 | −1 | 2 | rebuilt from a published seed |
Sources are recorded attributions, transcribed together with the bytes they were transcribed from. This corpus transcribes from 2 held files — gallagher2026.pdf · mengyang2026.pdf — and 8 of the 11 entries carry one; the 3 generated rows carry none, because there is nothing to transcribe. The announced dimension-3 map arrived as a post with no canonical bytes of its own, so its rows pin the preprint that prints it in full — which is the same file the Hessian entry pins. Worth saying plainly: the cross-check in §4 is therefore an ARITHMETIC independence, not a second independent source. Two objects transcribed separately from one preprint, linked by an identity recomputed here from scratch. The certificate carries the machine's whole pin table; the independent verifier re-hashed every entry in it during this build and matched 15 of 15. A drifted source refuses the certificate rather than quietly certifying against a memory of it.
The Jacobian conjecture is a statement about maps whose Jacobian determinant is a nonzero constant — Keller maps. A counterexample in dimension n is one object with three parts, and each part is a finite question about exact rationals:
Those three facts together say the map is a Keller map that is not injective, so it is not an automorphism. That is the whole deduction, and it is the one the certificate makes — a statement about the printed map, decided here, independent of who published it or whether their argument stands.
The float layer is present and powerless. It samples the determinant at one floating-point point to decide whether the symbolic expansion is worth running; nothing about a passing sample is believed, and the screen may only prune. Every verdict on this page comes from the exact layer.
A verifier that has never rejected a plausible forgery has an unknown false-accept rate. This build ran 32 checks with 9 red controls, and refuses if any control goes quiet: one coefficient of the map nudged by 10⁻⁶ (the determinant must stop being constant), one collision point moved by 10⁻⁹ (exact evaluation must catch it), the same point listed twice (distinctness must catch it), a curve whose twist equations are violated by 10⁻⁶ (the generator must REFUSE to ship it rather than emit it), a seed that violates the published gauge, a forged source hash, a source with no recorded pin, a perturbation of the Hessian polynomial, and the fiber hunter's own negative control in §5. The independent verifier carries its own forged-coefficient control and exits nonzero if it passes.
These are not the same act, and a page that blurs them is not worth reading. 3 entries are TRANSCRIBED: somebody published a map and its witnesses, this repository holds the bytes, and the audit re-decides the printed object. 5 are REBUILT: the paper published a recipe and a seed rather than a finished map, so the map is regenerated here from the seed with every gauge condition verified, then decided. 3 are GENERATED: curves chosen here, pushed through the published tangent-sweep construction, producing maps that appear in no paper.
The generator is calibrated before it is believed. At curve degree 2 it must reproduce the announced map polynomial for polynomial — same curve [4, −3], same twist, same coefficients — and the battery fails if a single monomial differs. Only then are the higher degrees admitted, and each one goes through exactly the same exact audit as a published claim, with its collision witnesses constructed from a secant line and then re-evaluated as fractions as if a stranger had supplied them.
| certificate | our curve coefficients | geometric degree | det J | Jacobian monomials | the two witnesses |
|---|---|---|---|---|---|
| keller-2 | [6, −6, 1] | 4 | −2 | 78 | (−2/3, 5/2, 183/8) (2/11, −13/2, 297/8) |
| keller-3 | [52/5, −51/5, 0, 1] | 5 | −2 | 141 | (−5/11, 16/5, 25707/875) (5/41, −46/5, −279907/875) |
| keller-4 | [20, −19, 0, 0, 1] | 6 | −2 | 222 | (−6/25, 31/6, 222325/1512) (6/101, −107/6, −5512277/1512) |
A new curve through a published mechanism is a new instance, not a new mechanism — the construction is Speyer's tangent sweep as written up by Gao (arXiv:2608.00222), and this repository claims none of it. These three maps have also NOT been checked for coordinate-equivalence against the members of the published infinite family: they may or may not be equivalent to something already in print, and nothing here decides that. What is certified is exactly what the certificate says — these polynomials, this determinant, these two points.
The corpus holds two claims about two different conjectures: a map in dimension 3 with det J = −2, and a degree-14 polynomial in 5 variables whose Hessian determinant is claimed to be 128. They are transcribed separately, from separate statements, and they are connected by an identity this build recomputes from scratch rather than cites.
Take the dimension-3 map F, adjoin three new variables y₁, y₂, y₃, and form the single six-variable polynomial φ = y · F — 16 monomials. Its Hessian is a 6×6 matrix of polynomials (126 monomials in total), and its determinant, expanded symbolically over the rationals during this build, comes out as the constant:
Both sides are computed here; neither is transcribed. So the Hessian entry is not a second, unrelated thing to trust: the Hessian construction descends from the same map the first entry certifies, and the identity linking them is checked exactly, every build. If the transcription of either object were wrong, this constant would not be −4. It is a cross-check with no shared arithmetic to hide in — the determinant of a 6×6 polynomial matrix does not accidentally collapse to −(−2)².
Everything above decides witnesses somebody supplied. This section supplies none. For each map and each rational target, a damped multistart Newton hunt looks for preimages in floating point — a hunter that proves nothing and is allowed to miss — and every candidate it reaches is then certified in a Krawczyk box on the EXACT map, with interval-enclosed rational coefficients and strict interior containment. Boxes that are provably disjoint hold provably distinct preimages. The output is a lower bound by construction: more preimages can exist, fewer cannot.
On the announced map, aimed at its own collision image, the hunt certifies 3 preimages without being told any of them — and each of the 3 published rational witnesses lands inside one of the certified boxes, which is the check that the rediscovery is of the same points and not of three others. The published witnesses are the answer key; the hunt never reads it.
| cell | target | where the target came from | witnesses constructed | certified blind | verdict |
|---|---|---|---|---|---|
| alpoge | (−1/4, 0, 0) | the map's own collision image | 3 | 3 | ≥ 3 preimages, certified |
| alpoge-own-target | (0, 1, 0) | chosen here by a fixed enumeration — not a published image | 2 | 3 | ≥ 3 preimages, certified |
| sweep-d3 | (−13/8, −2, 1) | the map's own collision image | 2 | 3 | ≥ 3 preimages, certified |
| sweep-d4 | (−13/5, −16/5, 1) | the map's own collision image | 2 | 2 | ≥ 2 preimages, certified |
| sweep-d5 | (−61/12, −19/3, 1) | the map's own collision image | 2 | 0 | no certified preimage |
| gallagher-d2 | (−4, −4, 1) | the map's own collision image | 2 | 3 | ≥ 3 preimages, certified |
| gallagher-d3 | (−1/2, −1/2, 1) | the map's own collision image | 2 | 3 | ≥ 3 preimages, certified |
| gallagher-d4 | (−11/10, −11/10, 1) | the map's own collision image | 2 | 1 | 1 preimage — nothing proved |
| gallagher-distinct | (0, 0, 1) | the map's own collision image | 2 | 2 | ≥ 2 preimages, certified |
Three things in that table are worth the reader's attention. The self-chosen target. A skeptic can say the collision image was handed to us even if the witnesses were not — so one cell aims at a target picked here by a fixed enumeration of plain rational points, (0, 1, 0), which is nobody's published image and nobody's constructed collision. It certifies 3 preimages. The hunter beating its input. 3 cells certify MORE preimages than the number of witnesses their construction wrote down. The honest failures. 2 cells do not certify non-injectivity at all — the float hunter did not reach enough roots, and the certificate says so instead of rounding up. Those maps are still certified non-injective by §1; the fiber counter simply did not re-find it unaided, and a lower bound that is allowed to be weak is the only kind that can be trusted when it is strong.
A preimage counter that manufactures non-injectivity would find extra roots everywhere, so it is run against a map that provably has none: the triangular automorphism (x + y², y + z³, z), injective by construction with det J = 1. Over 400 starts on the same ladder of scales, the dedup by certified box disjointness returns EXACTLY 1 preimage. The counter can come back with one, and does — which is what makes 3 evidence rather than an artifact.
The certificates are narrow on purpose, and the value of a narrow claim is that it is either true or refuted. Stated plainly, here is everything this page is NOT saying:
The certificate is a detached artifact: certs/keller-certificate.json (sha256 9ee1ff71a81a0a93…) holds every map as an explicit list of monomials with exact rational coefficients, every witness as exact rationals, the claimed determinant, and the sha256 of every source it was transcribed from. Nothing in it is implicit and nothing in it is a float. It is meant to be checked without this repository:
python3 tools/verify_keller.py certs/keller-certificate.json --sources corpus/sources
That file is Python standard library only — fractions.Fraction, hashlib, json — and shares no arithmetic with the machine that wrote the certificate. It re-derives the Jacobian by formal differentiation, expands the determinant, evaluates every witness, re-hashes the pinned sources, and then forges a coefficient and requires the audit to FAIL. At this build it decided 11 entries with 0 failures, matched 15 source hashes, and its red control fired.
Inside the repository, node instruments/keller/battery.js runs the instrument's own gate — 32 checks at this build, 9 of them red controls, including the generator calibration against the announced map and the blind fiber rediscovery. This page refuses to render if the battery fails, if the independent verifier fails, if the detached certificate has drifted by one monomial from the family that produced it, or if the six-variable doubling identity in §4 stops coming out −4.