Literature and provenance audit

Literature and provenance audit#

This page separates the exact mathematics from the rapidly changing public record. Searches were performed on 2026-07-20.

Same-day provenance#

The earliest indexed public source found for the exact formula is Levent Alpöge’s X announcement, posted 2026-07-20 at 02:19:17 UTC. The post gives the displayed polynomial, the determinant −2, and the three-point collision; it credits Akhil for asking about the problem and Fable for working during the World Cup final. The precise authorship and priority language should be confirmed with the announcer before formal publication [Alpöge, 2026].

Qiaochu Yuan reproduced the exact map and conclusion in a MathOverflow answer. Jean Abou Samra posted an independent Sage verification at approximately 06:31 UTC. Exact-phrase searches found no indexed pre-announcement occurrence, but a negative search is not proof of priority.

At 12:02:19 UTC the same day, Andy Jiang (@davikrehalt) posted the projective symmetric-product formulation: restrict the map

\[ \mathbb P^1\times\operatorname{Sym}^2(\mathbb P^1) \longrightarrow \operatorname{Sym}^3(\mathbb P^1), \qquad (p,D)\longmapsto p+D, \]

after deleting its ramification divisor and the inverse image of a hyperplane tangent but not osculating to the small diagonal. Jiang explicitly presented the formulation in the post as GPT output. It is exactly the projective form of the marked-simple-root construction developed here [Jiang, 2026].

Later the same day, Aaron Lou (@aaron_lou) hosted a self-contained factorization–resultant derivation [Lou, 2026]. Exact left–right linear equivalence shows that its map is the announced map after \(z\mapsto-z\) in the source and reversal of the three outputs in the target. It is valuable independent reconstruction and motivation, not a distinct counterexample. A same-day weighted-lift repository gives further exact families; it is self-published and unrefereed, and is treated only as construction data in this audit.

As of the cutoff, no refereed paper or arXiv preprint devoted to this new map was found. The appropriate wording is therefore: an exact, reproducible counterexample with provisional same-day provenance, not yet vetted through peer review.

Equivalence of the four presentations#

The Alpöge formula, Jiang’s projective model, Lou’s resultant chart, and this repository’s marked-root incidence are presentations of one algebraic morphism, but they are not literally the same formulas and the identifications are not canonical.

Let

\[ \mu:\mathbb P^1\times\operatorname{Sym}^2(\mathbb P^1) \longrightarrow\operatorname{Sym}^3(\mathbb P^1) \]

be Jiang’s map, let \(R\) be its ramification divisor, and let \(H_\infty\) be a hyperplane tangent but not osculating to the small diagonal. If \(\lambda\) defines \(H_\infty\), normalization gives

\[ [C]\longmapsto \frac{C}{\lambda(C)}: \operatorname{Sym}^3(\mathbb P^1)\setminus H_\infty \xrightarrow{\sim}\{C:\lambda(C)=1\}=H_\tau. \]

Upstairs, deleting \(R\) says precisely that the marked root is simple; the two unmarked roots may still coincide. Therefore

\[ \bigl(\mathbb P^1\times\operatorname{Sym}^2(\mathbb P^1)\bigr) \setminus\bigl(R\cup\mu^{-1}(H_\infty)\bigr) \simeq X_\tau, \]

and \(\mu\) becomes the forgetful map \(X_\tau\to H_\tau\). A tangent, nonosculating hyperplane meets the twisted cubic in \(2p+q\) with \(p\ne q\); all such choices are equivalent under \(\operatorname{PGL}_2\).

Lou writes the marked cubic as a linear factor times a quadratic and imposes both the affine coefficient level and \(\operatorname{Res}(\ell,q)=1\). Those conditions fix the projective and factor-rescaling gauges and give an explicit global \(\mathbb A^3\) chart on the same open incidence. If \(G\) is Lou’s coordinate map and \(F\) is Alpöge’s, the exact relation is

\[ G=R_0\circ F\circ S_0, \qquad S_0(x,y,z)=(x,y,-z),\quad R_0(A,B,C)=(C,B,A). \]

This is left–right linear equivalence, or an isomorphism of morphisms, not strict conjugacy: the source and target changes are independent. Etaleness, geometric degree three, fiber structure, and nonproperness are invariant under these identifications; the chosen bases, signs, affine levels, and the numerical Jacobian constant are presentation-dependent. Higher-degree weighted-lift families are descendants of the same idea, not equivalent maps when their geometric degrees differ.

Status immediately before the announcement#

Rodríguez Díaz’s June 2026 refereed historical article described control of ramification at infinity as the persistent obstacle [Rodríguez Díaz, 2026]. Bisi et al. described the conjecture as open even in dimension two [Bisi et al., 2026]. Makar-Limanov analyzed constraints on a hypothetical plane counterexample [Makar-Limanov, 2025], while Truong studied properness as the critical obstruction in reduced classes [Truong, 2026].

The dedicated plane status chapter contains the extended 2026 audit: geometric degree at least six, the provisional degree-108 bound, the surviving \((72,108)\) Newton configuration, and the exact distinction between peer-reviewed restrictions and recent preprints.

These papers establish the immediate pre-announcement context; they naturally do not discuss a formula announced later.

Foundational references#

Keller’s 1939 paper is the traditional source [Keller, 1939], although Rodríguez Díaz traces the plane statement to Kraus in 1884. The standard modern reduction is Bass–Connell–Wright [Bass et al., 1982]; important later reductions include Drużkowski [Drużkowski, 1983] and the symmetric reduction of de Bondt–van den Essen [de Bondt and van den Essen, 2005].

The exceptional pseudo-plane connection#

The quadratic-factor scaffold in this repository is exactly the affine pseudo-plane

\[ S(2,2,1)=\{u(1+uv)=w^2\}. \]

Miyanishi developed affine pseudo-coverings specifically to study a possible factorization of a plane Keller map through such an intermediate surface [Miyanishi, 2005, Miyanishi, 2015]. The standard canonical-class obstruction is silent here: this is type \(r=2\), for which \(K_S=(r-2)F_0\) vanishes.

Dubouloz–Palka’s refereed 2018 paper proves that this pseudo-plane admits large families of nonproper etale endomorphisms and gives explicit Belyi–Shabat formulas [Dubouloz and Palka, 2018]. These are maps \(S\to S\), not the missing map \(S\to\mathbb A^2\). Exact conjugation in the factorization chapter shows that their plane-chart transition is a constant-Jacobian rational map with a genuine pole, and that the published deformation family cannot avoid the omitted boundary line.

The current Borisov–Gabber–Vasiu manuscript treats related standard affine pseudo-planes, constructs maps to affine space in positive characteristic, and explicitly contrasts them with the unresolved characteristic-zero \(r=2\) case [Borisov et al., 2026]. It is a current manuscript, not a refereed status authority; it is used here to cross-check the boundary of the known theorem, not to certify a plane result.

Claimed proofs that are not status authorities#

The user-supplied arXiv:2209.01451 is Quan Xu’s unrefereed claimed proof [Xu, 2022]. It must not be cited as an established theorem. Other useful cautionary examples include:

The new example does not require trusting any global theorem or claimed proof: its decisive content is a determinant identity and an explicit collision.

Attribution policy for this repository#

This repository is an independent audit and structural analysis, not a claim to have first discovered the displayed formula. Until a stable paper appears, cite Alpöge’s original formula announcement, Jiang’s projective formulation, the public replications or derivations actually used, and this repository according to their distinct roles.