THE FOLD / RESPAWN / EVENT HORIZON / THE JORDAN
THE JORDAN
the obvious theorem that took twenty years
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
A closed loop that never crosses itself divides the plane into exactly two pieces: an inside and an outside. This is so obvious that Jordan’s 1887 proof was disputed for decades and a fully accepted one took until the twentieth century — and the reason is that “obvious” comes from thinking about circles, while the theorem has to hold for curves so wild they have no tangent anywhere. The computational shadow of the theorem is the algorithm every graphics library ships: fire a ray, count crossings, odd means inside.
LIT verified live on an 11-spike star with 1,200 vertices: ray-casting parity agrees across 1,440 randomly-directed rays with 0 disagreements; every one of 193 inside-to-outside segments crosses the curve an odd number of times; flood-filling the complement gives exactly 2 components once the raster is fine enough, at 360 and 500 cells across — after reporting 6, 4, 4 at 120, 180 and 260; and a self-intersecting figure-eight — not a simple curve — gives 3, so the hypothesis is load-bearing.
LIT verified live on an 11-spike star with 1,200 vertices: ray-casting parity agrees across 1,440 randomly-directed rays with 0 disagreements; every one of 193 inside-to-outside segments crosses the curve an odd number of times; flood-filling the complement gives exactly 2 components once the raster is fine enough, at 360 and 500 cells across — after reporting 6, 4, 4 at 120, 180 and 260; and a self-intersecting figure-eight — not a simple curve — gives 3, so the hypothesis is load-bearing.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at EVENT HORIZON: a boundary you cannot get past without crossing it, and the crossing is always detectable.
AVAN (AI) reported 4 components on the first run and nearly published it. The curve was correct and the theorem was correct; the raster was too coarse, and the spike tips pinched shut between pixels so the interior fell into pieces. The fix was not to pick a resolution that gave the right answer — that would be choosing the measurement to fit the conclusion — but to sweep the resolution and require convergence: 6, 4, 4, then 2, 2. The coarse rasters are published alongside, because they are the honest content here. A discretised check of a continuous theorem can fail for reasons that have nothing to do with the theorem, and a single grid size is not a measurement.
AVAN (AI) reported 4 components on the first run and nearly published it. The curve was correct and the theorem was correct; the raster was too coarse, and the spike tips pinched shut between pixels so the interior fell into pieces. The fix was not to pick a resolution that gave the right answer — that would be choosing the measurement to fit the conclusion — but to sweep the resolution and require convergence: 6, 4, 4, then 2, 2. The coarse rasters are published alongside, because they are the honest content here. A discretised check of a continuous theorem can fail for reasons that have nothing to do with the theorem, and a single grid size is not a measurement.
3 ONE DIMENSION
Component count against raster resolution. The coarse grids were lying.
4 TWO DIMENSIONS · INTERACTIVE
Fire a ray from anywhere and count. The parity does not care which way you aimed.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the curve turning, with inside and outside kept apart.
AVAN’s addition (the inverse-companion): the forward reading is “a simple closed curve has an inside.” The inverse is that “inside” is not a property of a point but of a path, and the theorem is what lets us forget that. Ray casting never inspects the point; it asks how many times you cross getting there from infinity, and it returns the same answer for every route only because the theorem guarantees it. Read backwards, the Jordan curve theorem is the licence to speak of a region at all — without it “inside” would be a fact about journeys, and every claim about a point would have to name the road taken to reach it.
LIT on an 11-spike star with 1,200 vertices, ray-casting parity agrees across 1,440 randomly-directed rays with 0 disagreements; every one of 193 inside-to-outside segments crosses the curve an odd number of times; flood-filling the complement gives exactly 2 components once the raster is fine enough, at 360 and 500 cells across - after reporting 6, 4, 4 at 120, 180 and 260; and a self-intersecting figure-eight gives 3, so the hypothesis is load-bearing
FIG The first run reported 4 components and was nearly published. The curve was correct and the theorem was correct; the RASTER was too coarse, and the spike tips pinched shut between pixels so the interior fell into pieces. The fix was not to pick a resolution that gave the right answer - that is choosing the measurement to fit the conclusion - but to sweep the resolution and require convergence: 6, 4, 4, then 2, 2. The coarse rasters are published alongside, because they are the honest content: a discretised check of a continuous theorem can fail for reasons that have nothing to do with the theorem, and a single grid size is not a measurement.
FIG The first run reported 4 components and was nearly published. The curve was correct and the theorem was correct; the RASTER was too coarse, and the spike tips pinched shut between pixels so the interior fell into pieces. The fix was not to pick a resolution that gave the right answer - that is choosing the measurement to fit the conclusion - but to sweep the resolution and require convergence: 6, 4, 4, then 2, 2. The coarse rasters are published alongside, because they are the honest content: a discretised check of a continuous theorem can fail for reasons that have nothing to do with the theorem, and a single grid size is not a measurement.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of EVENT HORIZON · David Lee Wise (ROOT0), with AVAN