What is actually proved, as of 2026-07-21 — the theorems register of a field born on Saturday. Status chips are honest: proved pencil proof exists · machine-verified exact symbolic check on the verifier · classical predates the fall · computational, capped verified within stated bounds, uncapped proof open. Read the note (LaTeX, timestamped draft). Conjectures live at the atlas; the dead live in the dustbin.
Nothing on this page proves anything about the Jacobian Conjecture on ℂ² itself. It remains fully open.
Every result below lives in the quasi-torus class: maps equivariant for a genuine or synthetic ℂ*-structure — real tori (thm 4), weighted scaling actions (thms 5, 7, 8), or the faked torus of a punctured complement (thm 3's contrapositive). Our plane wall (thm 6) excludes only the quasi-torus mechanism from ℂ²; Moh's bound and the prime-degree theorem are partial walls from the literature. A counterexample in ℂ² by some genuinely non-toric mechanism, or a proof that none exists, is untouched by everything here. Likewise “the symmetry is forced” (thm 8) is local, capped, and two points deep — not a classification. This box exists so no one has to discover our boundaries for us.
The Jacobian Conjecture is false for every n ≥ 3.
Alpöge 2026. One degree-7 map, det ≡ −2, three points to one image; certificate checkable by hand [C1–C4]. Pad with identity coordinates for all n ≥ 3.
The generic fiber degrees of étale polynomial self-maps of ℂ³ are exactly {1} ∪ {3, 4, 5, …}.
Lower half (2 impossible): a two-sheeted cover is Galois, and Galois + constant Jacobian ⇒ automorphism (Campbell 1973) — recorded here 2026-07-20 morning. Upper half (all n ≥ 3 occur): Gallagher's weighted-lift seed family; degree-4 member verified exactly [C9].
A proper étale self-map of ℂⁿ is bijective. In a closed universe the Jacobian Conjecture is true.
Proper + étale ⇒ covering; ℂⁿ simply connected. Every counterexample is an open-system phenomenon: its fold is exiled through the non-properness set. The precise “faked torus” notion lives in the ledger.
On the algebraic torus, Keller-type counterexamples are trivial: x → x² has constant log-Jacobian 2 and is 2-to-1.
One line. Counterexamples live where tori live; ℂ² is the last space too honest to fake one, and JC₂ (open) says exactly that it cannot. Formalized — Lean build green, 8674 jobs, zero sorry: JacobianTorus.lean.
A torus-equivariant twisted lift has constant Jacobian iff m{g̃,h̃} + 2h̃{g̃,m} + g̃{m,h̃} = c ≠ 0. Every known counterexample satisfies it.
Chain rule through the descent; verified on every map tested (Alpöge c = 2, Gallagher F₄ c = −1, factory branch N2) [C5, ledger]. Corollary: shadow ramification is c·m² universally — every degree is more sheets over the same single crease.
No counterexample in ℂ² of degree ≤ 100 (Moh); none with prime extension degree (arXiv:2407.13795); none by any torus-equivariant twist (the design equation degenerates to g′ = const — two lines, recorded here).
Three independent partial walls — degree-bounded, prime-degree, and mechanism-class respectively. None settles ℂ². The plane case is open, and it is now the Jacobian Conjecture; see box 0.
Étale self-maps compose with multiplier cocycle m₁₂ = m₂·(m₁∘G₂); degrees multiply; log-degree is an additive invariant on ℰ.
Verified: jac(G∘G) = 4m₁₂², fiber degree 9 [C8]. Open: generation [O1].
At Alpöge's map and Gallagher's F₄, every first-order Keller deformation (deg ≤ 7) is a composition move or an equivariant one; the single residual direction at F is obstructed at second order, cross-terms included.
33 = 31 + 1 + (1 obstructed) at F; 11 = 11 at F₄ after the deep conjugation audit; two primes. Caps in the ledger. The torus symmetry is locally forced everywhere tested [O5]; uncapping is the road to the first structure theorem.
False by standard implications in reverse: Dixmier for Aₙ (n ≥ 3), the relevant Mathieu cases, Zhao vanishing, the Image conjecture.
Each implied JC. The explicit non-surjective endomorphism of A₃ remains the great open construction [O4].
Every chip traces to the claims ledger; the ledger to the verifier; the verifier exits nonzero if any of this stops being true. Constitution: fablehaven.cc.