Coefficient lens
The explicit formula
Best for checking the determinant and the collision with a computer algebra system—or by hand.
Publicly posted by Levent Alpöge, with Fable credited in the post.
A constant-Jacobian map with a three-point fiber
Three roots.
One forgotten mark.
This explicit polynomial map’s derivative is invertible everywhere, yet three different points have the same image. Its hidden mechanism is unexpectedly simple: mark one root of a cubic, then forget which root was marked.
- Jacobian
- det JF = −2
- Generic fiber
- 3 points
- Dimension
- 3 (and all n ≥ 3)
- Dimension 2
- still open
Artwork: an abstract view of three marked-root sheets converging on one coefficient space.
The idea in 30 seconds
A cubic remembers its roots, but not their names.
A typical cubic polynomial has three distinct roots. Pair the polynomial with a choice of one root: that is a marked cubic. Now erase the mark. There are usually three possible inputs—one for each chosen root—but only one unmarked output.
- 1Start with three roots.
Together they determine a cubic.
- 2Choose one root.
The choice is extra information—a mark.
- 3Normalize the mark.
A geometric scaling makes the source exactly three-dimensional affine space.
- 4Forget it.
Locally nothing folds, but globally three choices can share one image.
The concrete map
No hidden numerical approximation.
Every claim is an exact polynomial identity. For complex coordinates (x,y,z), set:
F₁ = (1 + xy)³z + y²(1 + xy)(4 + 3xy)
F₂ = y + 3x(1 + xy)²z + 3xy²(4 + 3xy)
F₃ = 2x − 3x²y − x³z
Explore the real slice
See a pencil of curves—and a surface mesh—transform.
Important: the theorem concerns complex 3-space (six real dimensions). These plots show carefully chosen real curve families and sampled real surface meshes.
The 3D canvas is unavailable in this browser. The exact coordinates remain available in the table below.
Shared view
| Branch | Source (x, y, z) | Target F(x, y, z) |
|---|
One map, three lenses
The approaches are equivalent—but they do different explanatory work.
They are not three unrelated counterexamples. They are three presentations of the same marked-root morphism, after explicit source and target coordinate changes.
Projective lens
Mark, then forget a root
Best for seeing degree three, the ramification locus, and the special affine slice.
Posted by Andy Jiang (@davikrehalt), explicitly labeled “GPT”.
Resultant lens
Factor and fix the scale
Best for a structural differential proof: write a cubic as a linear factor times a quadratic and normalize their resultant.
Derived and documented by Aaron Lou (@aaron_lou).
| Lens | Input remembers | Main strength | Coordinate cost |
|---|---|---|---|
| Explicit F | Three affine numbers | Immediate exact certificate | Geometry is hidden by expansion |
| Marked root | A cubic and one simple projective root | Fibers become obvious | Requires projective language |
| Factor/resultant | A factorization C = ℓq with Res(ℓ,q)=1 | Étaleness is structural | Requires choosing an affine chart |
The proof spine
Four statements carry the whole argument.
-
1
Package each output as a binary cubic.
For a target (a,b,c), form Q(U,V)=cU³−2U²V+bUV²−2aV³.
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2
Find the marked root inside the source.
At (U,V)=(1+xy,x), exact identities give Q=0, Qᵤ=2V, and Qᵥ=−2U. The root is automatically simple.
-
3
Forgetting the root is locally invertible.
The gradient normalization removes the scaling ambiguity. Differentiation—or the resultant-one model—then gives det JF=−2 everywhere.
-
4
Globally, a generic cubic offers three choices.
Each simple root gives one source point. Hence a generic fiber has three points; repeated-root limits make inverse branches escape rather than create affine ramification.
Guided explainer
Follow the construction, one moving idea at a time.
This 4:37 Manim film connects the roots, the projective picture, the exact formula, and the escaping branches. The embedded audio is a reproducibly configured offline Piper draft; its captions and the transcript below use the same canonical narration text. A human recording remains the preferred final voice.
Read the exact narration transcript
This is the complete narration, in scene order. Scroll this panel for all 14 scenes. Three-dimensional scenes show only the real slice R³ inside the complex map on C³.
Provenance
Three public descriptions
This map reached the public in three distinct forms. Levent Alpöge posted the explicit polynomial formula, crediting Akhil for the question and Fable for the work. Later, Andy Jiang, at davikrehalt, posted the projective marked-root description and explicitly presented it as GPT output. Aaron Lou then gave a resultant-based derivation and global affine chart. This repository compiles, explains, visualizes, and checks those contributions.
The puzzle
Local invertibility can fail globally
The Jacobian condition is local: at every finite point, the derivative is invertible. Imagine a tiny neighborhood being moved without being pinched flat. The surprise is global. Different distant points can still acquire the same image if inverse branches disappear through infinity. That is the mechanism we will see.
Levent and Fable
The explicit polynomial certificate
Alpöge's presentation begins with three explicit polynomials in x, y, and z. Exact symbolic expansion gives determinant minus two everywhere. The formula is the fastest certificate that the map is locally invertible, but by itself it gives little clue why these particular terms fit together.
Levent and Fable
Three points, one value
Now substitute these three source points. They are distinct, yet every one maps to negative one quarter, zero, zero. This already proves noninjectivity. The remaining descriptions explain the hidden object that each point is recording.
The common idea
Mark one root, then forget the mark
A cubic has three projective roots when multiplicity is counted. Temporarily mark one root and leave the other two unordered. For a cubic with three distinct roots, there are exactly three possible markings. If we now forget the mark, those three choices become one cubic. Generic degree three is therefore built into the construction.
Jiang and GPT
The projective marked-root map
Jiang's post packages this as a projective map. A point p and an unordered pair q, r are sent to the unordered triple p, q, r. In symbols, projective one-space times its second symmetric power maps to its third symmetric power. This is simply the operation: forget which root was distinguished.
Jiang and GPT
Delete only bad marked roots
The forgetful map ramifies only when the marked root collides with one of the two residual roots. Delete exactly that ramification divisor. The unmarked roots may still collide with each other. After deletion, the differential is invertible everywhere that remains, so the restricted map is étale.
Jiang and GPT
The affine-space miracle
Next choose a hyperplane tangent, but not osculating, to the small diagonal of triple roots. Its complement in projective three-space is affine three-space. Remarkably, the corresponding source complement is also affine three-space. Choosing coordinates on these two spaces turns the geometric forgetful map into Alpöge's polynomial map.
Lou
A marked root is a factor
Lou's description replaces a marked root by a factorization of the binary cubic: a linear factor ell for the marked root, times a quadratic factor q for the other two. But the factorization has a redundant scale. Multiplying ell by s and q by s inverse leaves the cubic unchanged.
Lou
The resultant fixes the scale
The linear-quadratic resultant detects whether the marked root is simple. It is nonzero exactly when ell and q have no common root. Under the redundant scaling, the resultant is multiplied by s. Requiring resultant one therefore chooses one and only one representative on every simple-root scaling orbit. This is the resultant gauge.
Equivalence
Three languages for one arrow
These are not competing maps. They are three coordinate systems for one arrow. Alpöge gives the direct formula and collision certificate. Jiang and GPT expose the marked-root geometry. Lou supplies the normalized factor space and global affine chart. Lou's coordinate map G equals an output reversal after F and a sign change in the source. That is left-right equivalence, not dynamical conjugacy.
Visualization
Exact curves on the real slice
Complex three-space has six real dimensions, so here we visualize only its real slice. Three elementary source curves are mapped exactly: the x-axis becomes a target line, the y-axis becomes a parabola, and the z-axis becomes another target line. The same formulas drive both this animation and the browser artifact.
Visualization
Two inverse branches escape
A more revealing target curve represents cubics with roots zero, s, and one minus s. Each possible marked root defines a different source curve, but all three map to the same target curve. As s approaches zero, two marked roots collide. Their two source branches run toward infinity, while the branch marking the remaining simple root stays finite. No finite critical point appears.
Conclusion
The proof spine and its scope
The proof now has a short spine. Both chosen complements are affine three-space. Removing marked-root ramification makes the forgetful map étale. A general cubic has three simple roots, so the map has generic degree three and cannot be injective. Constant Jacobian, three-point fibers, and escape at infinity are three views of the same geometry. Adding identity coordinates extends the construction to higher dimensions, while dimension two remains a separate problem.
Narration text and captions are CC BY 4.0; linked sources retain their own terms.
4:37 offline draft voice: Piper TTS 1.4.2, en_US-libritts-high, speaker 0. The voice model was trained on LibriTTS train-clean-360. Voice, dataset, and software notices. A human-recorded final narration is planned.
Provenance, without ambiguity
Formula, explanation, and compilation are distinct contributions.
The repository preserves the public sequence and cites each contribution where it is used. It does not transfer discovery credit to the compiler.
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Levent Alpöge posts the explicit map.
The earliest formula post located by this audit states the map, determinant, and three-point collision; it credits Akhil for the question and Fable in connection with the work.
Formula post by Levent Alpöge -
Andy Jiang posts the projective marked-root construction.
The post, explicitly introduced as “GPT:”, explains the map as forgetting a marked point in P¹ × Sym²(P¹) → Sym³(P¹), after removing ramification and choosing a tangent-but-not-osculating affine slice.
Projective post by Andy Jiang (@davikrehalt) -
Aaron Lou (@aaron_lou) publishes a factorization–resultant derivation.
Lou’s note gives a direct affine chart for a linear factor times a quadratic, normalized by resultant one. The audit identifies its exact left–right coordinate equivalence to the original formula.
Read Aaron Lou’s derivation (PDF) -
Bartosz Naskrecki compiles and independently audits the record.
Bartosz Naskrecki is the repository compiler and landing-page author: exact certificates, comparison, documentation, visualization, and reproducible artifacts. He does not claim discovery of the formula or its geometric derivations.
Open the public repository