Banky Adebajo
21 July 2026

How to fold space
without a crease

Try to fold a sheet of paper without making a crease. You can't. Folding means creasing. For 87 years, mathematicians believed space itself obeyed the same rule. Then seven lines of algebra folded space three layers deep, with no crease anywhere.

This post assumes no math background. Nothing below is a recording: every picture is computed live, and the last three demos run the actual formula itself, so you can poke at things as you scroll.

A machine that moves space

This whole story is about one kind of object: a machine that picks up every point of space and sets it down somewhere new, following a fixed formula. Here is a simple one, acting on a flat grid.

However hard you bend it, the grid lines never cross. No two points ever land on the same spot.

how hard the machine runs0.7
This machine also has a perfect undo: press the button and a second machine puts every point back exactly where it started. Machines built from nothing but plus and times, like this one, are called polynomial maps. That's the only vocabulary you'll need.

The squash meter

Not every machine treats space equally. This one crushes the left side of the grid and stretches the right. You can see the grid lines bunching up as they approach the pink dashed wall.

To measure this, we carry a little test patch around. Its area tells us everything. Area unchanged means space is merely reshaped. Area shrinking toward zero means neighboring points are being crushed together. We'll call this reading the squash meter.

Drag the patch toward the pink wall and watch both the patch and the meter.

drag the patch
Here, space is reshaped but not squashed.
squash meter
1.4
The rule of thumb that powers everything ahead: two points can only be crushed onto the same spot where the meter reads zero. Wherever the meter reads anything but zero, nearby points stay apart, guaranteed. (The meter's formal name is the Jacobian determinant.)

Watch a machine fold space

Now a very different machine. Below is a disk of space, drawn as rays out of the center, like a paper fan. The top half of the fan is solid cyan, the bottom half is dashed rose.

When the machine runs, it opens the fan. It stretches each half around the circle until that half, all by itself, covers the whole disk.

Run it, then look at any single ray: it's cyan with rose dashes on top.

run the machine0
Before: cyan covers the top half, rose covers the bottom half. Every spot is covered exactly once.
Two layers means collisions are guaranteed. Pick any landing spot and two different starting points now land on it: one from the old cyan half, one from the old rose half. Two strangers, forced to share an address. But what did this folding cost the machine?

Every fold has a crease

Same folding machine, with the fold already made. Space is two layers deep now, so every landing spot has two originals. Drag the cyan point anywhere: a rose stranger on the other layer moves too, and the both land here panel shows them landing on the exact same spot.

Your point also carries the amber test patch from earlier, and the both land here panel shows how the patch lands. The squash meter reports exactly that: a big landed patch means a healthy reading, and a patch crushed to a speck means zero.

The challenge: try to steer the two of them into each other.

drag the cyan point
both land here
The only place the pair can possibly meet is the glowing center: the crease of the fold. Head there, and watch the landed patch and the meter.
squash meter
4.45
This is the punchline of the first half of the story. To fold, the machine had to crush space flat along the crease, and the meter catches it red-handed. A fold always has a crease, and the meter always finds the crease. For 87 years, this felt like a law of nature.

Three layers, one pin

Machines can fold more than twice. This one lays space over itself three times. Drag the striped landing spot, and three colored travelers move in the other panel. Every one of them lands exactly on your dot.

Press the button to send the landing spot one full lap around the center, and watch which color ends up where.

drag the landing spot
three travelers
laps: 0
squash meter
2.8
Where is that fabric stitched together? Drag the landing spot toward the center and watch the meter collapse. The center is the pin holding all three layers, and the meter finds it, just like it found the crease. Layers always seem to cost a stitch.

The rule everyone believed

What we've seen so far:

Machines bend space, and some have a perfect undo.
The squash meter detects crushing. If it never reads zero, no two points can ever land on the same spot.
Every fold we built needed a crease or a pin: a spot where the meter dies.

Put those together and one conclusion feels unavoidable. In 1939, Ott-Heinrich Keller wrote it down:

If a plus-and-times machine never squashes (its meter frozen at a single value, never zero), then it can never make two points collide, and it must have a perfect undo.
the Jacobian conjecture (O.-H. Keller, 1939)

Every test agreed. The simplest cases were outright proved. Computers checked enormous families of machines and found no exceptions. Dozens of famous theorems were shown to stand or fall with this rule. Attempted proofs kept cracking; the rule never did.

On 19 July 2026, Levent Alpöge posted seven lines of algebra, found by Claude, Anthropic's Fable model. Let's meet the machine those lines build.

The machine that breaks the rule

Here is the actual formula, squeezed onto four lines so it fits your screen. You don't need to read it. Just notice it's built from nothing but plus, minus, and times. One change from our earlier machines: this one runs on three-dimensional space, so every point has three coordinates (x, y, z) instead of the flat sheet's two:

writing u = 1 + xy:
a = u3z + y2u(4 + 3xy)
b = y + 3xu2z + 3xy2(4 + 3xy)
c = 2x − 3x2y − x3z

The claim: this machine never squashes. Its meter is frozen at −2, everywhere in space. (The minus sign just means it also mirrors space. What matters is that it never touches zero.)

A probe is sprinting around three-dimensional space right now. The grey line is one of the machine's inner gauges going wild. The amber line is the squash meter.

probe at (0.8, -0.5, 1.1)
squash meter
-2.0000
By everything we've learned, this machine cannot fold. No crease, no pin, no stitch, nowhere for layers to be sewn together. So it should be unable to make two points collide.

Three points, one destination

Time to test it. Just like the three-layer machine, you drag the landing spot, and the who lands there panel answers: which points of space land on it? (The machine lives in three dimensions, so we've frozen one of its three dials to fit everything on a flat screen; nothing important hides in the frozen one.)

Drag anywhere in the amber zone. You will always find three solid dots: three genuinely different points of space, all landing on your spot.

drag a landing spot
who lands there
● a real point of space, meter here: -2.00
● a real point of space, meter here: -2.00
● a real point of space, meter here: -2.00
Read that meter column again: −2 at every one of the three colliding points. Collisions with no squashing, anywhere. This is the thing that was supposed to be impossible.
So the machine folds space three layers deep after all. The amber zone is triple-covered, the blue zone single-covered. In the blue zone, two of the dots turn into dashed ghosts: travelers that have slipped off the visible map into math's hidden backstage, the complex numbers. Drag back to amber and they return. And the two zones share a border, where layers appear and disappear.

What happens at the border?

When the colliding pair in the folding machine came together, they met at the crease. This machine has two extra travelers that exist in amber and vanish in blue. So as the landing spot crosses the border, those two travelers must somehow merge and disappear.

Before you press the button, take a second to guess what the two extra travelers will do.

drag the crossing
eyes here
3 real travelers
You can also drag the crossing by hand and creep up on the border slowly. The closer you get, the farther and faster the two travelers fly. Halve the distance to the border and they land roughly 1.4 times farther out.

The shape of the fold

One more picture before the reveal: everything the last two demos showed, seen from outside. The floor is the landing map you've been dragging across, amber zone and blue zone. Above every landing spot we stack its travelers, and the height of each layer shows how far out in space that traveler sits: three layers over amber, one over blue.

Spin it, and watch the outer two layers near the border. They rise faster and faster and dissolve toward the two ∞ marks, off the top and bottom of the picture.

drag sideways to spin
The rose layer dives toward one infinity, the amber layer climbs toward the other, and at no height you could ever draw do they meet. No meeting, no crease, and the meter reads −2 at every point of every layer you can see. So where did the crease go?

The crease is at infinity

Everyone was almost right, for 87 years. Layers really do need a stitch, and the meter really does find every stitch located somewhere in space. This machine's three layers are sewn together too. But the stitch has been pushed past the edge of the world:

edge of space∞
The travelers you chased above were racing toward this stitch, and never arriving, because it's infinitely far away.

That's why the meter never complained. At every actual point of space, nothing is being crushed. The squashing that "should" happen is deferred forever, always farther out. Space gets folded three layers deep, and the meter serenely reads −2 at every point, including at the collisions themselves.

the whole story
Machines move space; some bend it and can undo it.
The squash meter detects crushing. Two points can only land on the same spot where it reads zero.
Every fold we could build paid for its layers with a crease or a pin.
This machine folds three layers with its meter frozen at −2, because its stitch is at infinity.

The aftermath is just beginning. Keller's rule is dead in every dimension from three up. The original two-dimensional version survives, for now, and has overnight become the most interesting open question in the field. And dozens of results that were proved to stand or fall with the rule have now fallen with it.

Announced by Levent Alpöge on 19 July 2026; the map was found by Claude (Anthropic's Fable model). The collision structure and border curve follow David Speyer's writeup on the Secret Blogging Seminar (20 July 2026), and the geometric picture follows Will Sawin's discussion. Decoder ring: the squash meter is the Jacobian determinant, the backstage is the complex numbers, the border is the discriminant curve, and a machine is a polynomial map. Every demo on this page is computed live; the last three run the counterexample map itself.