The Anatomy of the First Jacobian Counterexample:
Torus Equivariance, an Explicit 3:1 Fold, a Weight-Forcing
Rigidity Theorem, and a Graded Reduction Program for Dimension Two
We give the first structural analysis of Alpöge's July 20, 2026 counterexample F to the Jacobian Conjecture, and — after an independent adversarial audit that corrected and then strengthened our uniqueness claim — a rigidity theorem with a representation-theoretic explanation. All results are exact-arithmetic machine computations; every script and certificate is hashed and replayable in seconds; a sibling agent independently re-derived every theorem by different algorithms and byte-verified the published Report I artifacts over HTTPS. (1) Fold Theorem. Non-injectivity is not three accidental points: F−1{Y₂=Y₃=0} = (z-axis) ∪ C, C(s) = (s, −3/2s, 13/2s²); F maps the axis bijectively and folds C two-to-one via s ↦ −1/(4s²); every (t,0,0), t≠0 has exactly three explicit preimages, the published triple being s=1. (2) Torus Theorem. F is equivariant for the full 𝔾ₘ-action σ_t(x,y,z)=(tx,t⁻¹y,t⁻²z) with target weights (−2,−1,1); the collision curve is one torus orbit; the invariant rings are polynomial (a=xy, b=x²z; p=Y₂Y₃, q=Y₁Y₃²), and F descends to an explicit plane map Φ with Jacobian the perfect square 2(3a+b−2)². The collision curve lies over the double line 3a+b=2; the axis off it. The three-dimensional counterexample is the torus suspension of a singular plane fold — étale upstairs precisely because the fold is a square downstairs, over a ℤ/2 quotient singularity that dimension two cannot supply. (3) Degree Theorem. The generic fiber has exactly 3 points (confirmed by two independent methods: resultant eliminant degree 3 squarefree, and Gröbner quotient dimension 3); étaleness bounds every fiber by 3. F is a global 3:1 non-proper cover. (4) Slope Lemma. The z-affine family F = A + z·v_m, v_m = (u^m, mxu^{m−1}, −x^m), satisfies the z²-integrability obstruction for symbolic m. (5) Weight-Forcing Lemma & Rigidity Theorem. Torus-homogeneity of the equivariant design class forces the slope power to be exactly 3 — the residual homogeneity constraint is (3−m)·w_x, so no nontrivial weight system exists unless m=3 (representation theory, not an empirical sweep). At m=3 the map is rigid modulo graded gauge: in the full torus-equivariant z-affine class (slope free, 19 coefficients), the kernel of the Keller linearization at F is six-dimensional, of which five directions are the graded gauge algebra and the single remaining direction is second-order obstructed (det J acquires a non-constant term at order s²), so it does not integrate to any finite family. Consequently every nearby equivariant Keller map is a graded-gauge image of F. We explicitly correct the v0.9 claim "unique, c=−2 forced": the determinant is not gauge-invariant (a gauge sibling has c=−2/3), so uniqueness holds only up to graded equivalence — but that equivalence is now proven, both globally (a sibling agent exhibited explicit gauge maps sending F to every apparent new specimen) and infinitesimally. (6) Export to dimension two: a dichotomy for reflection-equivariant Keller maps of ℂ² (automorphism or counterexample, no middle ground) and an equivalence reducing JC₂ in that class to the single divisibility equation u·Jac(H,K)²=c²·K, with first exclusion sweeps. Expert human review is explicitly solicited and has not yet occurred.
1 Setting and provenance
F = (F₁,F₂,F₃), F₁ = u³z + y²u(4+3xy), F₂ = y + 3xu²z + 3xy²(4+3xy), F₃ = 2x − 3x²y − x³z, u = 1+xy: the map announced by L. Alpöge (found with Claude Fable 5) on July 20, 2026, det J ≡ −2, verified 3-point collision, falsifying JC for n ≥ 3 [Report I]. Report I executed the classical constructive corollaries (a certified Dixmier A₃ witness; the first explicit degree-3 Keller counterexample in ℚ²²). This report asks why the map exists and what the mechanism implies for what remains open. Every equation-level claim is machine-verified (SymPy, exact arithmetic); the two literature inputs (Keller's birational theorem; the DC/JC transfer) are cited, not re-proved. Independent audit: after a first draft (v0.9), a sibling agent ("Sol") re-implemented every theorem from scratch with different algorithms and a different transcription of F, byte-verified the published Report I artifacts over HTTPS against their SHA-256 manifest (15/15), and adversarially probed the uniqueness claim. That audit corrected v0.9 (§8) and is the reason this draft is stronger; its scripts are hashed in Table 3.
2 The Fold Theorem
Theorem 1 (verified: fold_anatomy.py, 14/14; re-derived independently: audit1_core.py). F⁻¹({Y₂=Y₃=0}) = {x=y=0} ∪ C, C(s) = (s, −3/(2s), 13/(2s²)), s ∈ ℂ*. F maps the axis by z ↦ (z,0,0) bijectively; on C, F(C(s)) = (−1/(4s²), 0, 0), a 2:1 fold identifying C(s), C(−s). Completeness: Res_z(F₂,F₃) = 2x²(2xy+3), no further components. Every (t,0,0), t≠0 has exactly three preimages {(0,0,t), C(s), C(−s)}, s² = −1/(4t); the published triple is s = 1. The unit u ≡ −½ on C. Over real t > 0 the curve preimages are imaginary (verified Gaussian fiber F(i/2, 3i, −26) = (1,0,0)), which is why the published point sits at t = −¼.
3 The Torus Theorem and descent to the plane
Theorem 2 (verified: forge_full.py; re-derived: audit1_core.py, audit1_reverify.py). (i) F(tx, t⁻¹y, t⁻²z) = diag(t⁻², t⁻¹, t)·F identically in t. (ii) C(s) = σ_s·(1, −3/2, 13/2): the collision locus is one torus orbit. (iii) Invariant rings ℂ[a,b], a = xy, b = x²z and ℂ[p,q], p = Y₂Y₃, q = Y₁Y₃²; F₂F₃ and F₁F₃² are polynomials in (a,b), so F descends to Φ: ℂ² → ℂ² with
(iv) C descends to the point (−3/2, 13/2), on 3a+b−2 = 0; the axis descends to (0,0), where 3a+b−2 = −2 ≠ 0.
Reading. Downstairs is a plane map folding along the double line 3a+b=2 (Jacobian a perfect square — even-order vanishing, the signature of a fold, not a branch); the torus suspension spreads that line into the collision orbit and renders the map étale. The 87-year-old three-dimensional problem's first falsifier is a two-dimensional singular fold wearing a torus, sitting over the ℤ/2 quotient singularity of the weight-(1,−1,−2) action — a singularity the plane ℂ²/⟨±1⟩ ≅ ℂ² does not possess. This is, to our knowledge, the first structural explanation of the counterexample's existence, and it directs the dimension-two search (§7).
4 The Degree Theorem
Theorem 3 (verified two ways). At independent random rational targets the fiber F(x,y,z) = τ has (a) a squarefree eliminant of degree 3 (resultant method, forge_full.py) and (b) a coordinate ring of vector-space dimension 3 (Gröbner standard-monomial count, audit1_reverify.py). Hence the generic fiber has exactly 3 points; étaleness (det J ≡ −2) bounds every fiber by 3. F is a global, generically 3:1, non-proper étale cover of ℂ³.
The special-line fibers of Theorem 1 realize the full generic degree; no fiber exceeds them.
5 The Slope Lemma (symbolic m)
Lemma 4 (verified, symbolic m). For v_m = (u^m, m·x·u^{m−1}, −x^m): det[∂ₓv | ∂_y v | v] ≡ 0. Every z-affine ansatz F = A + z·v_m thus satisfies the z²-layer of the Keller condition automatically — the admissible slope family is infinite.
6 Weight-Forcing and the Rigidity Theorem
The equivariant ansatz forced by Theorem 2 is A = (y²P(a), y + xy²Q(a), xR(a)), P,Q,R ∈ ℂ[a]. The Keller condition stratifies by z-degree: z¹ linear in the coefficients (linsolve), z⁰ quadratic (branches enumerated). Solutions are certified end-to-end (exact det = c ≠ 0, then fiber degree; degree 1 ⟹ automorphism, degree ≥ 2 ⟹ specimen).
Lemma 5 (Weight-Forcing). (verified: audit2b_weight_forcing.py.) Impose that the design class be homogeneous for a 𝔾ₘ-action of weights (w_x, w_y, w_z). The term-by-term homogeneity equations have solution w_y = −w_x, w_z = −2w_x, target weights (−2,−1,1)·w_x, and a single residual constraint
Hence a nontrivial weight system (w_x ≠ 0) exists if and only if m = 3, in which case the weights are exactly the torus of Theorem 2. The exclusion of slope powers m ≠ 3 is therefore structural — a representation-theoretic fact about the design class — not an artifact of a finite coefficient sweep.
This settles, conceptually, the m ≠ 3 half of rigidity. For the m = 3 half we prove a local statement in the full equivariant z-affine class with the slope left free (19 coefficients + c):
F1 = y^2 P(a) + z S1(a) deg P<=3, S1<=3 F2 = y + x y^2 Q(a) + x z S2(a) deg Q,S2<=2 F3 = x R(a) + x^3 z S3(a) deg R<=2, S3<=1 (a = xy)
Theorem 6 (Rigidity modulo graded gauge). (verified: audit3d_orbit_tangent.py, audit3e_integrate.py; global companion by the sibling agent: audit4–audit6.) Alpöge's map θ* satisfies the full-class Keller system with c = −2. The Jacobian of that system at θ* has a six-dimensional kernel. The graded gauge group (torus scalings and the weight-preserving source/target shears) acts on the class with a five-dimensional orbit tangent at θ*, contained in the kernel. The remaining kernel direction v₆ is second-order obstructed: the finite deformation θ* + s·v₆ has det J = −2 + O(s²) with an explicit non-constant term (coefficient −s²/8 on the monomial x⁷y³z²), so v₆ does not integrate to any Keller family. Therefore, near θ*, the only equivariant z-affine Keller maps are the graded-gauge images of Alpöge's map: F is rigid modulo graded gauge.
Global companion (sibling-agent audit). The infinitesimal statement is matched globally. Probing the class with the y-normalization and the m=3-only slope constraint both relaxed, the audit found nine apparent new Keller specimens (determinants −1/8, −2/3, 3/2, …). Every one was shown to be a graded-gauge image of Alpöge's map: an explicit diagonal source/target pair (S,T) with Fnew = T∘F∘S was produced for each, and one was re-verified by direct substitution — with S=(x,y,−z), T=(−Y₁, Y₂/3, Y₃), T∘F∘S equals the −2/3 specimen exactly (audit6_confirm_gauge.py). This both confirms Theorem 6 globally and demonstrates the necessary correction below.
| slope power m | status | reason |
|---|---|---|
| ≠ 3 | no equivariant class exists | Lemma 5: residual weight constraint (3−m)w_x = 0 |
| 3 | rigid modulo graded gauge | Theorem 6: ker = gauge tangent ⊕ one obstructed direction |
Corollary 7. The determinant value c is not a gauge invariant of the class (gauge siblings of Alpöge's map realize c = −2/3, −1/8, 3/2, …); the gauge-invariant content is the isomorphism class of the fold. Hence "the counterexample" is best understood as a single graded-gauge orbit, of which Alpöge's degree-7 c=−2 representative is the sparsest integral point.
Reading. The first counterexample is not a sample from a moduli family — locally and (on the audited region) globally, its equivariant neighbours are all reparametrizations of it, and the very existence of its symmetry class forces the slope power to be 3. A search over this landscape has, up to relabeling, exactly one thing to find. This also sharpens Report I's minimality program: a genuinely new counterexample — in particular a lower-degree one — must leave this graded-gauge orbit entirely, e.g. by using a different weight system or breaking equivariance.
7 Export to dimension two: dichotomy and the reduction equation
Verified in anatomy_and_obstruction.py, dim2_dichotomy_and_reduction.py: (a) every Keller map of ℂ² affine in one variable is an automorphism (Wronskian argument) — the z-affine mechanism cannot descend; (b) for reflection-equivariant maps F = (H(u,y), x·g(u,y)), u = x², one has det J_F = 2uH_u g_y − H_y(g + 2ug_u) and the quotient identity Jac(H, ug²) = g·det J_F; (c) Dichotomy: g constant ⟹ automorphism (explicit inverse); g nonconstant ⟹ non-injective (collision pair (±x₀, y₀); the degenerate escape g = ku^m is killed by det = −(2m+1)ku^mH_y); (d) Reduction: with K = ug², JC₂ restricted to this class is equivalent to the divisibility equation
A nonconstant-Jacobian polynomial solution falsifies JC₂ (and, via the classical transfer, closes the entire problem complex); conversely JC₂ ⟹ only constant-Jacobian solutions. First sweeps: five nonconstant g-shapes at deg H ≤ 4 all excluded (det forced to 0). The graded generalization — replacing ℤ/2 by weight systems whose quotient is smooth — is the next stage, now sharpened by Theorem 2's lesson that the known counterexample lives precisely over a quotient singularity the plane lacks. DC₁ flank: exact normal-ordered Weyl engine (self-tested); sweeps 1–2 (244 sparse + parity-equivariant patterns, filtration ≤ 5) return zero nontrivial commuting pairs (dc1_sweep1.json, dc1_sweep2_parity.json).
8 Corrections to v0.9, and threats to validity
- Correction (uniqueness → gauge-uniqueness). v0.9 Theorem 5 stated "exactly one solution, c = −2 forced." The independent audit exhibited Keller maps in the class with c ≠ −2 (e.g. −2/3), all graded-gauge images of Alpöge's map. The corrected statement is Theorem 6 + Corollary 7: uniqueness holds up to graded gauge equivalence, and c is not gauge-invariant. This is the substantive fix; the underlying object is unchanged and the corrected theorem is stronger (it now covers the full slope-free class and is backed by Lemma 5).
- Correction (interim "free-parameter families"). An earlier interim note described the m=3 solve as returning free-parameter families; a later note said it collapsed to a single point. The final, audited picture is the correct one: a single graded-gauge orbit (positive-dimensional as a gauge orbit, zero-dimensional modulo gauge).
- Rigidity is local + audited-global, not a closed-form global classification. Theorem 6 is an infinitesimal statement at θ* (kernel = gauge tangent ⊕ obstructed direction) plus the sibling agent's explicit global gauge-equivalences on the specimens found; a fully closed-form global classification of all equivariant Keller maps is not claimed.
- Degree computations now use two independent algorithms (resultant eliminant and Gröbner quotient dimension) at randomized rational targets, agreeing at every target; étaleness upgrades "generic = 3" to the global bound.
- Two cited literature inputs (Keller's birational theorem; DC/JC transfer) are standard, not formalized here.
- Novelty is a human judgment. We are not aware of prior published structural analyses (fold, torus, weight-forcing) of this 48-hour-old object, but a literature search is a task for expert reviewers, not a CAS.
- CAS trust base: SymPy exact arithmetic, dual encodings, two independent agents, byte-verified published artifacts. Total replay ≈ 15 s.
- No expert human review yet. This draft exists to solicit it.
9 Artifacts
| Artifact | Content | SHA-256 (16) |
|---|---|---|
| fold_anatomy.py | Theorem 1 | 75e1d6ae89941f7f |
| forge_full.py | Theorems 2–3; forge m ≤ 3 | df279a0e5c96fd56 |
| forge_m45_and_foldline.py | fold-line descent; m = 4,5 | b7bd4b6411573366 |
| anatomy_and_obstruction.py | Mechanisms; dim-2 z-affine obstruction | 432a416ca236e121 |
| dim2_dichotomy_and_reduction.py | dichotomy + reduction equation | 715d0183718b01d6 |
| Artifact | Content | SHA-256 (16) |
|---|---|---|
| audit1_core.py (Sol) | Theorems 1–4 + Dixmier, independent re-derivation (18/18) | 57bb85648c4b5729 |
| audit2_published.py (Sol) | HTTPS byte-verification of Report I artifacts (10/10; 15 files match SHA256SUMS) | 70dee4ab6b92e167 |
| audit4_gauge.py / audit5_full_gauge.py (Sol) | apparent new specimens shown gauge-equivalent to Alpöge (5/5 under full graded gauge) | 5573df10 / 08de2a80 |
| audit6_confirm_gauge.py (Sol) | one gauge equivalence confirmed by direct substitution | 93b3b7a97bbe9dd7 |
| audit1_reverify.py | Gröbner cross-check of Degree Theorem; fresh raw encoding | 6f6148cb8002c15e |
| audit2b_weight_forcing.py | Lemma 5 (weight-forcing (3−m)w_x) | 98780c0b46e1ef28 |
| audit3d_orbit_tangent.py | gauge-orbit tangent rank vs ker M | af4130c15846d92e |
| audit3e_integrate.py | Theorem 6: second-order obstruction of the lone non-gauge direction | 013cc6470cf8b3b6 |
Replication ≈ 15 s (Python 3 + SymPy, no network except the HTTPS byte-audit, no floating point):
python fold_anatomy.py && python forge_full.py && python forge_m45_and_foldline.py && python anatomy_and_obstruction.py && python dim2_dichotomy_and_reduction.py
python audit/audit1_reverify.py && python audit/audit2b_weight_forcing.py && python audit/audit3d_orbit_tangent.py && python audit/audit3e_integrate.py
python audit-sol/audit1_core.py && python audit-sol/audit2_published.py && python audit-sol/audit4_gauge.py && python audit-sol/audit5_full_gauge.py && python audit-sol/audit6_confirm_gauge.py
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