Plate-style glossary, one term per plate: plain words first, then the formal definition, then where it lives on the atlas and which machine check touches it. In the plasmagicians tradition: a definition you can't say plainly is a definition you don't own yet. Terms link each other; start anywhere.
A recipe that takes a point (x, y, z) and outputs a new point, where each output coordinate is built only from adding and multiplying the inputs — finitely many knobs, no division, no limits, no infinity anywhere in the recipe.
F: ℂ³ → ℂ³, F = (F₁, F₂, F₃) with each Fᵢ ∈ ℂ[x, y, z].
Everything on the atlas is one of these. The whole story is about what such maps can and cannot do.
Stand at a point. Wiggle each input a hair and record how each output moves: nine numbers, a 3×3 table — the Jacobian matrix. Its determinant is one number: the factor by which the map compresses or expands a tiny parcel of volume at that point. Determinant zero = the parcel gets crushed flat.
JF = (∂Fᵢ/∂xⱼ); det JF its determinant, a polynomial in (x,y,z).
Atlas: “what the determinant means — animated.” Checks C1, C2 compute det JF ≡ −2 for Alpöge's map two independent ways.
A polynomial map whose volume-compression factor is the same nonzero constant everywhere — it never crushes and never balloons, anywhere. Named for Ott-Heinrich Keller (1939), though the question goes back to Kraus (1884).
F polynomial with det JF ≡ c ∈ ℂ∗.
The conjecture was about exactly these. Alpöge's map is a Keller map with c = −2.
The 87-year belief: every Keller map is globally reversible — no two distinct points ever share an output. Locally true by the inverse function theorem; the conjecture claimed the local guarantee upgrades to a global one for polynomials. It does not.
Claim (now FALSE for n ≥ 3): det JF ≡ c ≠ 0 ⇒ F has a polynomial inverse.
Refuted 2026-07-19 (Alpöge). Atlas: “what happened”; the dustbin holds the epitaph. The n = 2 case remains open.
Injective = no two inputs share an output. A collision is a witnessed failure: distinct points, same image. One exact collision plus constant determinant is a complete disproof — no limits, no approximation, checkable by hand.
∃ p ≠ q with F(p) = F(q). Certificates C3, C4 on the atlas each exhibit three such points.
Atlas: the certificate blocks; Plate I.
Pick a target point; its fiber is everything that maps onto it. Where fibers have k points, the source looks locally like k parallel copies — sheets — over the target, a stack of pancakes. The (generic) degree of the map is the sheet count over a typical target.
F⁻¹(q); generic |F⁻¹(q)| = [ℂ(x,y,z) : F*ℂ(a,b,c)]. Alpöge: 3. Gallagher's atlas: every value 3–100.
Atlas: “the collision, live” explorer — drag and count. The degree spectrum theorem says which counts occur.
Proper = nothing escapes: chase a fiber toward the edge of the world and it stays in view. Non-proper = preimages can run off to infinity. Every counterexample is non-proper — the fold that "should" exist is exiled past the horizon. That is the entire escape hatch.
F proper iff preimages of compact sets are compact. A proper étale self-map of ℂⁿ is a covering map of a simply connected space, hence trivial — so counterexamples must be non-proper.
Atlas: act 0 (“pushed off the edge of the world”), Plate III (the garage ramp that never arrives).
Creaseless: locally invertible at every single point. For polynomial self-maps this is exactly the Keller condition — the determinant, being a polynomial with no zeros, must be a nonzero constant.
JF invertible everywhere ⇔ det JF ∈ ℂ∗.
The monoid ℰ of all étale self-maps of ℂ³ is the two-day-old object the atlas's back half studies.
A dial you can turn: shrink x by any factor while growing y by the same factor and z by its square. The weights (−1, 1, 2) say how fast each coordinate turns with the dial. A map is equivariant when turning the dial before or after the map gives the same answer — the map "commutes with the dial."
λ·(x,y,z) = (λ⁻¹x, λy, λ²z); F equivariant: F(λ·p) = λ·′F(p) with matching target weights. Every known counterexample, every dimension, is equivariant for such an action [ledger, Unification].
Atlas: “the mechanism” and “one equation, one crease.” Whether the symmetry is forced is O5.
Combinations the dial can't change: t = xy (the shrink and the growth cancel) and w = x²z. Noether's move: a symmetry hands you conserved quantities, and the honest variables of the problem are those.
ℂ[x,y,z]ᵗ* = ℂ[t, w], t = xy, w = x²z.
The whole 3D map casts a 2D shadow in these variables.
Because the map respects the dial, it induces an honest 2D map on the conserved quantities — its shadow, G. All the drama (the 3-to-1 collisions, the crease) is visible in the shadow; the third dimension only hides the evidence.
G(t,w) = (m·g̃, m²·h̃) where g̃ = xF₂, h̃ = x²F₁ descended (set x = 1), and m = F₃/x.
Both explorers on the atlas run on shadows. The master equation lives here.
The shadow map genuinely folds: along one line its compression factor dies like m², and the whole line is crushed to a single point. Illegal for a Keller map — but upstairs, the third coordinate is multiplied by that same m, and the two crimes cancel exactly: compression × m⁻² · m² = constant. The crease is laundered through the direction you weren't auditing.
jac G = c·m² (universal, any degree — chain rule); upstairs x → x·m; det JF = jac G / m² = c.
Atlas: Plate II, Plate V, the determinant animation (watch the parcel die on the amber line).
The condition a would-be counterexample's three ingredients must satisfy — and it is itself a constant-Jacobian condition, one level down. "Take the Jacobian of the Jacobian Conjecture": the problem recurses into its own solution space. Every known counterexample satisfies it.
m{g̃,h̃} + 2h̃{g̃,m} + g̃{m,h̃} = c ≠ 0, where {p,q} = pₔqₛ − pₛqₔ is the planar Jacobian bracket. Alpöge: c = 2. Gallagher F₄: c = −1. [check C5]
Atlas: “the design equation.”
A tiny determinant of two functions on the plane — how independently they vary. It is the classical (Poisson) shadow of the quantum commutator, which is why the master equation quantizes into Weyl-algebra territory (Dixmier).
{p,q} = ∂ₔp·∂ₛq − ∂ₛp·∂ₔq.
Everywhere in the back half of the atlas; open problem O4.
Gallagher's discovery: one single-variable polynomial p generates a whole counterexample. Its running integral P must return to zero across [0,1] — a potential with level-matched wells, zero net work around the cycle. Seed degree d mints d+1 sheets. Alpöge's map is the unique d = 2 seed: p(w) = 2w − 3w².
p(0) = 0, p(1) = −c, ∫₀¹p = 0 ⇒ fiber degree deg p + 1.
Atlas: “the seed forge” — build one yourself.
All creaseless self-maps of ℂ³ form a system closed under composition (a monoid). Sheet counts multiply, so log(degree) adds — an entropy. Composition twists the multipliers into each other by the cocycle rule. Two days ago everyone believed this monoid contained only the invertible maps. It doesn't.
(m₁,G₁)∘(m₂,G₂) = (m₂·(m₁∘G₂), G₁∘G₂); deg(F∘G) = deg F · deg G. [check C8]
Atlas: “the connections.” Open: is ℰ generated by Aut(ℂ³) and one map? [O1]
Which sheet counts can a creaseless self-map of ℂ³ have? Answer, settled in one day by two strangers' halves: 1 (the invertible maps), and then every number from 3 up. Never 2 — two sheets force a mirror symmetry, symmetric covers are honest (Galois), and honest creaseless covers are trivial (Campbell 1973).
deg ℰ = {1} ∪ {3, 4, 5, …}. Upper half: Gallagher's seed family. Lower half: the Galois/Campbell exclusion, recorded on this project hours before the refutation arrived.
Atlas status table; the dustbin holds the conjecture this theorem grew out of.
Take a counterexample and wiggle it, keeping the constant-determinant law. First-order wiggles form the tangent space. Some wiggles are boring (same map, new coordinates); some follow the known family; a wiggle beyond those would point at a NEW kind of counterexample. An obstruction is a wall at second order: the wiggle starts but cannot continue — a direction that promises a road and delivers a cliff.
T = {H : tr(adj(JF)·JH) ≡ const}; obstruction class of χ: [tr(adj(Jχ)·JF)] ∈ coker(L).
Ledger: “local rigidity computation” — at both tested counterexamples, nothing survives beyond the boring and the known: the symmetry is locally forced. [O5]
Any way a geometry can carry a circle-times-scaling direction: a real torus factor (where counterexamples are trivial), a weighted scaling symmetry (where all known counterexamples live), or a faked one — a puncture whose complement has the loops the original space lacked. The unifying thesis of this project, stated at honest strength: every theorem we have is a theorem about quasitori. The plane's fate outside this class is unknown.
A genuine torus factor (ℂ*)ᵗ × X; or a weighted ℂ*-action with equivariance; or monodromy π₁(ℂⁿ ∖ A_F) → S_d nontrivial over the non-properness set's complement.
Theorems register, box 0 (scope); ledger, “faked torus, precisely.”
Missing a term? That's a bug — the constitution says nothing stays undefined. Back to the atlas.