Resolution of the Swallowtail Branch Locus of the Alpöge–Mathew–Fable Jacobian Counterexample

A Three-Level Mathematical Comedy in the "Awfully Sophisticated Proofs" Tradition, with Lamport-Style Verification and a Swallowtail Visualization.

Paul Pajo — Independent Researcher, De La Salle College of Saint Benilde, Manila paulamerigo.pajojr@benilde.edu.ph July 2026

Abstract

On July 19–20, 2026, Levent Alpöge announced that Fable (Anthropic) had found a polynomial map $F : \mathbb{C}^3 \to \mathbb{C}^3$ with $\det J_F \equiv -2$ that is not injective, refuting the Jacobian conjecture [1]. The map was found by sweeping a $\mathbb{C}^*$-equivariant ansatz [2]

$$F = \left(\tfrac{h_1(u,v)}{x^2},; \tfrac{h_2(u,v)}{x},; x,h_3(u,v)\right), \qquad u = xy,; v = x^2 z,$$

which forces $\det J_F$ to depend only on $(u,v)$ and reduces the search to a linear equation in $h_1$.

The map is a forgetful morphism from a normalized cubic-factorization space [5]: $F = (2c_3, 2c_1, c_0)$, forgetting which linear factor was chosen. Its swallowtail branch locus $B = V(\Delta) \subset \mathbb{C}^3$, where

$$\Delta = -\tfrac{27}{4} a^2 c^2 - \tfrac{1}{4} a b^3 + \tfrac{9}{2} a b c + \tfrac{1}{4} b^2 - 4c,$$

is the discriminant hypersurface of the covering's preimage cubic. $B$ has $A_2$ (cuspidal) singularities along the rational curve $\mathrm{Sing}(B) = {(a, 4/(3a), 4/(27 a^2)) : a \in \mathbb{C}^*}$, the locus of triple-root fibers.

We execute Hironaka's theorem [7] in full on $B$: Hilbert–Samuel function, characteristic polyhedron, resolution invariant, permissible center, blow-up, strict transform, smoothness verification. Every step is formalized in Lamport hierarchical proof style [10].

The result is a three-level comedy in the MathOverflow tradition [11]:

• Level 1: 218 pages to resolve a swallowtail that Zariski resolved in 50 pages [12].
• Level 2: 1964 Fields Medal machinery applied to a surface synthesized in 2026 by an AI.
• Level 3: Hironaka smooths the wreckage of a conjecture dissolved by a 47-character prompt.

The machinery is heavier. The result is lighter. It actually works.

1. Introduction and Motivation

This paper proves that a specific algebraic surface is smooth after one blow-up. The surface exists because a famous conjecture is false. The conjecture was killed by an AI following a 47-character prompt. We use 218 pages of algebraic geometry to smooth the debris.

1.1 The Alpöge–Mathew–Fable Counterexample

The Jacobian conjecture [8] asked: if $F : \mathbb{C}^n \to \mathbb{C}^n$ is polynomial with $\det J_F \in \mathbb{C}^*$, is $F$ an automorphism? It was open since 1939 for all $n \geq 2$.

On July 19–20, 2026, Levent Alpöge posted [1]:

"hello there the jacobian conjecture is false thanx"

crediting Akhil Mathew for posing the problem and Fable (Anthropic's AI) for finding the map. The counterexample was formalized in Lean within hours by Paul Lezeau.

1.2 The Ansatz: 47 Characters of Deep Structure

The map's algebraic structure is illuminated by Salberger's community-reconstructed ansatz [2]:

$$F = \left(\tfrac{h_1(u,v)}{x^2},; \tfrac{h_2(u,v)}{x},; x,h_3(u,v)\right), \qquad u = xy,; v = x^2 z.$$

This is not a blind guess. It encodes a $\mathbb{C}^$C∗ action: under $\lambda \cdot (x,y,z) = (\lambda x, \lambda^{-1} y, \lambda^{-2} z)$, the substitutions $u = xy$ and $v = x^2 z$ are $\mathbb{C}^$-invariant (weight 0), and $(F_1, F_2, F_3)$ have $\mathbb{C}^$C∗-weights $(-2, -1, +1)$. The Jacobian $\det J_F$ has weight $(-2)+(-1)+(+1) = -2$ from the components, plus weight $+2$ from the coordinate measure $dx \wedge dy \wedge dz$. Total: weight 0, so $\det J_F$ is $\mathbb{C}^$-invariant — it depends only on $(u,v)$. Furthermore, $\det J_F$ is linear in $h_1$, converting the search to a linear equation. Fable swept $h_3$ affine and $h_2$ cubic over small integer coefficients, solved for $h_1$, and found the map.

1.3 The Map as a Forgetful Morphism

The derivation document [5] reconstructs the geometry post hoc. Writing $F = (2c_3, 2c_1, c_0)$ in terms of coefficients of $L(t)Q(t) = c_3 t^3 + c_2 t^2 + c_1 t + c_0$ (linear $L$, quadratic $Q$, with $c_2 = 1$ and $\rho = 1$ fixed), $F$ is the map that forgets which linear factor was chosen. Three generic choices exist, so $F$ is generically 3-to-1.

Figure 1. The swallowtail branch locus $B = V(\Delta)$ and its resolution. Top left: cross-sections of $B$ at fixed $a$, showing the classical swallowtail shape evolving with parameter. Top centre: $B$ in $(a,b,c)$-space with $\mathrm{Sing}(B)$ (orange) and the special point $p_0 = (1, 4/3, 4/27)$ (yellow). Top right: image of a grid in $\mathbb{A}^3$ under $F$, showing three overlapping sheets (colored by original $z$). Bottom row: the ansatz weight structure; discovery timeline; before/after resolution table.

1.4 The Mathematical Value of This Paper

The main theorem (Theorem 6.1) is a routine corollary of Hironaka. The mathematical value lies elsewhere:

1. $B$ is the swallowtail surface — the classical catastrophe-theoretic discriminant of a cubic family. This identification is new; the paper makes it explicit for the first time for the Alpöge–Mathew–Fable map.
2. $\mathrm{Sing}(B)$ is the triple-root locus — the exact set of target values where all three sheets of the covering collapse simultaneously. Computing it explicitly ($b = 4/(3a)$, $c = 4/(27 a^2)$) is new.
3. The $A_2$ singularity type is verified — confirming the expected catastrophe-theoretic structure of the counterexample's branch locus.
4. The ansatz encodes the $\mathbb{C}^*$ action — making explicit the equivariant structure that allowed the AI to find the map.
5. First singularity-theory note on a counterexample that is 8 days old at time of writing.

1.5 Before and After

| Before: $B = V(\Delta)$ | After: $\widetilde{B}$

-68.267.847-113-73.952-191-73.952z"/> | | --- | --- | | Equation: $\Delta = -\tfrac{27}{4} a^2 c^2 - \cdots$ (degree 4) | Equation: $\widetilde{\Delta}$, strict transform | | Singular locus: ${(a, 4/(3a), 4/(27 a^2))}$, rational curve | Singular locus: $\varnothing$ | | $\nu^*(\Delta, p) = 2$ for $p \in \mathrm{Sing}(B)$ | $\nu^*(\widetilde{\Delta}, p) = 1$ everywhere | | $H_{B,p}(k) = 2k$ | $H_{\widetilde{B},p}(k) = k$ | | $A_2$ cuspidal singularity along $\mathrm{Sing}(B)$ | Smooth: $\nabla \widetilde{\Delta} \neq 0$ everywhere | | Origin: 47-char prompt + AI sweep | Resolution: 218-page theorem |

2. The Swallowtail: Definitions and Computations

Setup 2.1 (Ambient data). Let $F = (F_1, F_2, F_3) : \mathbb{C}^3 \to \mathbb{C}^3$ be the Alpöge–Mathew–Fable map, in the Alpöge form [1]:

$$\begin{aligned} F_1 &= (1 + xy)^3 z + y^2(1 + xy)(4 + 3xy), \ F_2 &= y + 3x(1 + xy)^2 z + 3 x y^2 (4 + 3xy), \ F_3 &= 2x - 3x^2 y - x^3 z. \end{aligned}$$

Write target coordinates as $(a, b, c)$. The preimage cubic at $(a, b, c)$ is

$$P(t; a, b, c) = \tfrac{a}{2} t^3 + t^2 + \tfrac{b}{2} t + c.$$

Lemma 2.2 (The swallowtail). The branch locus of $F$ is $B = V(\Delta) \subset \mathbb{C}^3_{(a,b,c)}$ where

$$\Delta(a,b,c) = -\tfrac{27}{4} a^2 c^2 - \tfrac{1}{4} a b^3 + \tfrac{9}{2} a b c + \tfrac{1}{4} b^2 - 4c.$$

This is the swallowtail surface: the discriminant locus of the cubic family ${P(\cdot; a, b, c)}_{(a,b,c) \in \mathbb{C}^3}$. $B$ has degree 4.

Proof. $F$ is a generically 3-to-1 covering whose fiber over $(a, b, c)$ consists of the three factorizations of $P(t; a, b, c)$. The fiber degenerates iff $P$ has a repeated root, iff $\mathrm{disc}(P) = 0$. Computing:

$$\mathrm{disc}!\left(\tfrac{a}{2} t^3 + t^2 + \tfrac{b}{2} t + c\right) = 18 \cdot a^2 \cdot \tfrac{b}{2} \cdot c - 4 \cdot a^2 \cdot \tfrac{b^3}{8} - 4c + \tfrac{b^2}{4} - 27 \cdot \tfrac{a^4}{4} \cdot c^2 = \Delta(a,b,c).$$

The total degree of $\Delta$ is 4 (monomials $a^2 c^2, a b^3, a b c, b^2, c$ have degrees 4, 4, 3, 2, 1). The classical discriminant of a one-parameter cubic family is the swallowtail catastrophe surface [3]. $\square$

Remark 2.3. The Alpöge showcase fiber maps $(0, 0, -1/4)$, $(1, -3/2, 13/2)$, $(-1, 3/2, 13/2)$ all to $(-1/4, 0, 0)$. Direct computation: $\Delta(-1/4, 0, 0) = 0$. So the showcase target is on $B$: the preimage cubic $-\tfrac{1}{8} t^3 + t^2 = t^2(1 - t/8)$ has a double root at $t = 0$. The "three preimages" correspond to the three factorizations of a cubic with a double root — two share the root at $0$, one takes the simple root at $8$. This is the swallowtail geometry directly.

Lemma 2.4 (Singular locus). $\mathrm{Sing}(B) = {(a, 4/(3a), 4/(27 a^2)) : a \in \mathbb{C}^*}$. Equivalently, $\mathrm{Sing}(B) = V(3ab - 4,; 27 a^2 c - 4) \cap {a \neq 0}$. This is the triple-root locus: the set of target values whose preimage cubic has a triple root (all three sheets collapse).

Proof. The gradient system $\Delta = \partial_a \Delta = \partial_b \Delta = \partial_c \Delta = 0$. From $\partial_c \Delta = 0$ ($a \neq 0$): $c = (9ab - 8)/(27 a^2)$. Substituting into $\partial_b \Delta = 0$: $-\tfrac{3}{4} a b^2 + 2b - \tfrac{4}{3a} = 0$, with unique solution $b = 4/(3a)$, hence $c = 4/(27 a^2)$. All four conditions verified at $(a, 4/(3a), 4/(27 a^2))$ for all $a \neq 0$. At $a = 0$: $\partial_c \Delta|_{a=0} = -4 \neq 0$, so no singular points. The preimage cubic at $(a, 4/(3a), 4/(27 a^2))$ is $a^2 (t - (-2/(3a)))^3$: a triple root, confirming the geometric interpretation. $\square$

Lemma 2.5 (Smoothness of $\mathrm{Sing}(B)$). $\mathrm{Sing}(B)$ is a smooth rational curve.

Proof. The $2 \times 3$ Jacobian matrix of $(3ab - 4, 27 a^2 c - 4)$ at $(a, 4/(3a), 4/(27 a^2))$ has the minor $(3a)(27 a^2) = 81 a^3 \neq 0$ for $a \neq 0$, so the ideal defines a smooth curve. The parametrization $a \mapsto (a, 4/(3a), 4/(27 a^2))$ is a closed immersion $\mathbb{C}^* \hookrightarrow \mathbb{C}^3$. $\square$

Lemma 2.6 (Singularity type). At each $p_0 \in \mathrm{Sing}(B)$, the singularity of $B$ is of type $A_2$ (cuspidal swallowtail): after local analytic coordinates, $B \cong V(u^2 + v^3)$ locally.

Proof. At $p_0 = (1, 4/3, 4/27)$, set $A = a - 1$, $B' = b - 4/3$, $C' = c - 4/27$. The initial form $\mathrm{in}${p_0}(\Delta) = -\tfrac{4}{27} A^2 - \tfrac{2}{3} A B' + 2 A C' - \tfrac{3}{4} B'^2 + \tfrac{9}{2} B' C' - \tfrac{27}{4} C'^2inp0​​(Δ)=−274​A2−32​AB′+2AC′−43​B′2+29​B′C′−427​C′2. The Hessian matrix of this quadratic form has eigenvalues ${-413/54, 0, 0}$ and rank 1, so $\mathrm{in}${p_0}(\Delta) = \lambda L^2 for the linear form $L = -\tfrac{4}{27} A - \tfrac{1}{3} B' + C'$. A rank-1 initial form, together with a nondegenerate cubic term, gives $A_2$ by the standard classification [3, Ch. 1]. This is consistent with the swallowtail: the discriminant of a one-parameter cubic family has $A_2$ cusps along the triple-root locus. $\square$

3. Hironaka Invariants

Lemma 3.1 (Hilbert–Samuel and multiplicity). For $p \in \mathrm{Sing}(B)$: $\nu^*(\Delta, p) = 2$ and $H_{B,p}(k) = 2k$.

Proof. $\Delta$ vanishes at $p$ and its gradient vanishes at $p$ (Lemma 2.4), so $\nu^* \geq 2$. The initial form is the nonzero quadratic $\lambda L^2$, so $\nu^* = 2$ exactly. The rank-1 initial form gives local-ring behavior $H_{B,p}(k) = 2k$ (cuspidal: one smooth branch of multiplicity 2). $\square$

Lemma 3.2 (Characteristic polyhedron). With $L$ as main variable, the characteristic polyhedron is $\Gamma(\Delta; u, L) = {1} \subset \mathbb{R}^1$.

Proof. Writing $\Delta = \lambda L^2 + (\text{degree-3 terms})$ with $L$ as main variable: the Newton polygon vertex in perpendicular coordinates is at normalized exponent $2/2 = 1$. $\square$

Lemma 3.3 (Permissibility). $Z = \mathrm{Sing}(B)$ is a permissible blow-up center.

Proof. $Z$ is smooth (Lemma 2.5). $H_{B,p}(k) = 2k$ is constant on $Z$ (Lemma 3.1), so $B$ is normally flat along $Z$. By Hironaka [7, Def. 3.4], $Z$ is permissible. $\square$

4. The Blow-Up

Lemma 4.1 (Total transform). In local coordinates $(A, s, \tau)$ near $p_0$, with $s$ and $\tau s$ the transverse coordinates to $Z = \mathrm{Sing}(B)$:

$$\pi^*(\Delta) = s^2 \cdot \widetilde{\Delta}, \quad \text{where } \widetilde{\Delta}|_{s=0} = -\tfrac{27}{4}!\left(A + \tfrac{1}{2}\right)^{!2} \tau^2 + \tfrac{9}{2}!\left(A + \tfrac{1}{2}\right)\tau - A - \tfrac{3}{4}.$$

Proof. Substituting $B' = s$, $C' = \tau s$ in the local expansion of $\Delta$ near $p_0$, the minimum power of $s$ in the transverse directions is $s^2$ (from $\nu^* = 2$). The strict transform formula gives $\widetilde{\Delta} = \pi^*(\Delta)/s^2$; setting $s = 0$ yields the displayed expression. Direct computation verified. $\square$

Lemma 4.2 (Smooth strict transform). $\mathrm{Sing}(\widetilde{B}) = \varnothing$.

Proof. Along $E = {s = 0}$: $\partial_\tau(\widetilde{\Delta}|_{s=0}) = -\tfrac{27}{2}(A + 1/2)^2 \tau + \tfrac{9}{2}(A + 1/2)$. Setting this to zero (with $A \neq -1/2$): $\tau = 1/(3(A + 1/2))$. Substituting into $\widetilde{\Delta}|${s=0} = 0Δ∣s=0​=0 and $\partial_A \widetilde{\Delta}|${s=0} = 0 simultaneously: no solution (the resultant is nonzero, verified by direct computation). Away from $E$: $\widetilde{B} \cong B \setminus \mathrm{Sing}(B)$, which is smooth. $\square$

5. The Main Proposition (Lamport Proof)

Proposition 5.1 (Resolution of the swallowtail branch locus). The swallowtail branch locus $B$ of the Alpöge–Mathew–Fable Jacobian counterexample admits a resolution of singularities: there exists a smooth variety $\widetilde{B}$ and a proper birational morphism $\pi : \widetilde{B} \to B$, an isomorphism over $B \setminus \mathrm{Sing}(B)$. Concretely: one blow-up of $\mathbb{C}^3$ along $\mathrm{Sing}(B)$ produces this resolution.

Proof sketch. Identify $B$ as the swallowtail (Lemma 2.2). Find $\mathrm{Sing}(B)$: the triple-root rational curve (Lemma 2.4). Compute $\nu^* = 2$, verify permissibility (Lemmas 3.1, 3.3). Blow up along $\mathrm{Sing}(B)$; the strict transform is smooth (Lemma 4.2); $\nu^*$ drops to 1; Hironaka terminates.

The entire argument is superseded by: $B$ is a variety over $\mathbb{C}$ (char 0), so Hironaka [7] applies.

Lamport hierarchical proof.

• (1)1. $B$ is an algebraic variety over $\mathbb{C}$, $\dim B = 2$, characteristic 0. By: $B = V(\Delta)$, $\Delta \in \mathbb{C}[a,b,c]$, $\mathrm{char},\mathbb{C} = 0$. $B$ is a hypersurface in $\mathbb{A}^3$, so $\dim B = 2$. By Lemma 2.2.

• (1)2. $B$ is singular along the smooth rational curve $Z = \mathrm{Sing}(B)$, with $A_2$ singularity type. By: Sub-proof (2)1–(2)4.

  • (2)1. $Z = {(a, 4/(3a), 4/(27 a^2)) : a \in \mathbb{C}^*}$. By: Lemma 2.4: unique solution to $\Delta = \nabla \Delta = 0$ for $a \neq 0$.
  • (2)2. $Z$ is a smooth curve. By: Lemma 2.5: Jacobian rank 2 everywhere.
  • (2)3. Singularity type along $Z$ is $A_2$. By: Lemma 2.6: Hessian rank 1 (eigenvalues ${-413/54, 0, 0}$), giving $\lambda L^2$. With nondegenerate cubic term: $A_2$ by [3, Ch. 1].
  • (2)4. q.e.d. $B$ is singular along the smooth $A_2$-curve $Z$. By: (2)1, (2)2, (2)3.

• (1)3. $Z$ is a permissible blow-up center. By: Lemma 3.3: $Z$ smooth, $H_{B,p}$ constant on $Z$.

• (1)4. Total transform: $\pi^$(\Delta) = s^2 \cdot \widetilde{\Delta}π∗(Δ)=s2⋅Δ. By: Lemma 4.1: local computation with $\nu^$ = 2.

• (1)5. Strict transform $\widetilde{B} = V(\widetilde{\Delta})$ is smooth. By: Lemma 4.2: no singular points on $E = {s = 0}$; smooth away from $E$ since $\widetilde{B} \cong B \setminus Z$.

• (1)6. Resolution invariant drops: $\nu^$(\widetilde{\Delta}) = 1 < 2 = \nu^(\Delta). By: (1)5: $\widetilde{B}$ smooth $\Rightarrow \nu^* = 1$. Lemma 3.1: $\nu^*(\Delta) = 2$. Hironaka termination criterion [7, Thm. 4.1] fires.

• (1)7. $\pi : \widetilde{B} \to B$ is proper, birational, iso over $B \setminus Z$. By: Blow-up of $\mathbb{C}^3$ along smooth $Z$ is proper; birational and iso over $\mathbb{C}^3 \setminus Z$ by construction.

• (1)8. q.e.d. $\pi : \widetilde{B} \to B$ is a resolution. By: (1)5 smooth; (1)7 proper birational; (1)6 invariant drops. Hironaka [7, Def. 1.1]. $\square$

6. The One-Sentence Proof

Theorem 6.1 (Main theorem). $B$ admits a resolution of singularities.

Proof. $B$ is an algebraic variety over $\mathbb{C}$ (characteristic 0). By Hironaka [7]. $\square$

Everything in §§2–5 is a 212-page elaboration of this sentence.

7. Corollaries

Corollary 7.1. The swallowtail surface of any degree-3 polynomial family in $\mathbb{C}^3$ admits a resolution of singularities.

Proof. Apply Hironaka [7]. $\square$

Corollary 7.2. The wreckage of the Jacobian conjecture in dimension 3 is smooth after one blow-up.

Proof. One blow-up along $\mathrm{Sing}(B)$, as in Proposition 5.1. $\square$

Corollary 7.3. If $\mathrm{JC}(3)$ had been true, the branch locus $B$ would be empty, and the resolution of the empty variety would be the empty variety. That proof is shorter.

8. Discussion: A Three-Level Comedy

This paper places itself in the MathOverflow tradition of "awfully sophisticated proofs for simple facts" [11]. The joke operates on three levels.

Level 1. Zariski [12] resolved surface singularities in 1939 in 50 pages. A direct blow-up along $\mathrm{Sing}(B)$ works in two pages. We use 218 pages.

Level 2. Hironaka's theorem is from 1964. The surface $B$ was synthesized in July 2026 by an AI (Fable, Anthropic) executing a search specified by a mathematician (Akhil Mathew) in a 47-character prompt. The 1964 Fields Medal machinery is applied to a 2026 AI-generated artifact.

Level 3. The 47-character prompt encoded a $\mathbb{C}^*$-equivariant ansatz that reduced the search for a $\mathrm{JC}(3)$ counterexample to a linear equation. Fable solved the linear equation. The counterexample's branch locus turned out to be the swallowtail — a classical catastrophe-theoretic object studied since Thom (1972). Hironaka's theorem resolves it.

The entire chain:

47-char prompt $\to$ AI sweep $\to$ Counterexample $\to$ Swallowtail branch locus $\to$ Hironaka resolution $\to$ Smooth manifold

spans 62 years of algebraic geometry, one AI system, one mathematician's insight, and one tweet.

The machinery is heavier. The result is lighter. It actually works.

The ansatz as compressed mathematics

The ansatz $F = (h_1/x^2, h_2/x, x h_3)$ with $u = xy$, $v = x^2 z$ is not a brute-force search key. It is a theorem in compressed form: "any $\mathbb{C}^*$-equivariant polynomial map of these weights will have constant Jacobian if the linear equation in $h_1$ is satisfied, and the covering's branch structure will be determined by the discriminant of a cubic." The AI found the specific $h_1, h_2, h_3$; the mathematicians found the structure class.

Acknowledgements

Thanks to Levent Alpöge and Fable (Anthropic) for the counterexample [1]; to Akhil Mathew for posing the problem; to O. Salberger for the structural ansatz [2]; to Heisuke Hironaka for a theorem of sufficient generality; to René Thom for the swallowtail; to the MathOverflow community [11]; to Leslie Lamport for making verbosity structural; and to Claude Sonnet 4.6 for drafting, formatting, and solutioning.

References

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