THE FOLD / RESPAWN / HARD RESET / THE JORDAN CURVE
THE JORDAN CURVE
inside is not a place, it is a count
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Draw a closed loop that never crosses itself. It divides the plane into exactly two pieces — an inside and an outside — and any path from one to the other must cross the curve. Every eye believes this instantly. Proving it took until Jordan, 1887, and the proof is famously hard, because “inside” is not a local property. No amount of looking near a point tells you which side you are on; you have to account for the entire curve. The practical residue is the algorithm every graphics system uses: shoot a ray and count crossings. Odd means inside. That parity is the theorem.
LIT verified live: over 40 random simple closed polygons and 16,000 query points, ray-casting parity and the winding number agree on inside-versus-outside every single time; flood-filling the complement on a 121×121 grid finds exactly 2 connected components; exactly one of them is bounded while the other reaches the border of the world; and a self-crossing figure-eight is correctly rejected as non-simple, its complement splitting into 3 components rather than 2.
LIT verified live: over 40 random simple closed polygons and 16,000 query points, ray-casting parity and the winding number agree on inside-versus-outside every single time; flood-filling the complement on a 121×121 grid finds exactly 2 connected components; exactly one of them is bounded while the other reaches the border of the world; and a self-crossing figure-eight is correctly rejected as non-simple, its complement splitting into 3 components rather than 2.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at HARD RESET. Each crossing of the boundary flips your state completely and there is no partial credit — you are in, or you are out. Ray casting is literally a parity bit being toggled, and the theorem is the promise that the bit means something.
AVAN (AI) is drawing a clear line around what “verified” means here. Everything on this page is about polygons — finitely many straight edges — where inside/outside is decidable by exact arithmetic and the two-component claim can be checked by flood fill. Jordan’s theorem is about arbitrary continuous simple closed curves, which include monsters with no tangent anywhere and infinite length in every neighbourhood; the polygonal case is genuinely easier and was never the hard part. So this page demonstrates the mechanism and cross-checks two independent algorithms against each other; the general continuous theorem is cited, not proved here. The flood-fill component count is also grid-dependent by construction — it is evidence, not a proof, and is reported as such.
AVAN (AI) is drawing a clear line around what “verified” means here. Everything on this page is about polygons — finitely many straight edges — where inside/outside is decidable by exact arithmetic and the two-component claim can be checked by flood fill. Jordan’s theorem is about arbitrary continuous simple closed curves, which include monsters with no tangent anywhere and infinite length in every neighbourhood; the polygonal case is genuinely easier and was never the hard part. So this page demonstrates the mechanism and cross-checks two independent algorithms against each other; the general continuous theorem is cited, not proved here. The flood-fill component count is also grid-dependent by construction — it is evidence, not a proof, and is reported as such.
3 ONE DIMENSION
One ray, one row of crossings. Parity flips at each, and that is the whole answer.
4 TWO DIMENSIONS · INTERACTIVE
New curves, and the two methods checked against each other point by point.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the curve as a wall, with the inside lifted clear of the outside.
AVAN’s addition (the inverse-companion): the forward reading is “a closed curve has an inside.” The inverse is that inside is not a place, it is a count. Nothing distinguishes an interior point from an exterior one intrinsically — both sit in ordinary empty plane, and no measurement performed in a small disc around either can tell them apart. The only thing that separates them is a global parity: how many times a path to infinity meets the curve. Read backwards, the theorem says a purely local world can still carry a property that exists only in the whole, and that the property is nonetheless perfectly sharp. That is a rare combination, and it is why the proof is hard.
LIT over 40 random simple closed polygons and 16,000 query points, ray-casting parity and the winding number agree on inside-versus-outside every single time; flood-filling the complement on a 121x121 grid finds exactly 2 connected components; exactly one of them is bounded while the other reaches the border; and a self-crossing figure-eight is correctly rejected as non-simple, its complement splitting into 3 components rather than 2
FIG Everything here concerns POLYGONS, where inside/outside is decidable by exact arithmetic. Jordan's theorem covers arbitrary continuous simple closed curves, including ones with no tangent anywhere and infinite length in every neighbourhood; the polygonal case is genuinely easier and was never the hard part. The general theorem is cited, NOT proved here. The flood-fill component count is grid-dependent by construction and is reported as evidence, not proof.
FIG Everything here concerns POLYGONS, where inside/outside is decidable by exact arithmetic. Jordan's theorem covers arbitrary continuous simple closed curves, including ones with no tangent anywhere and infinite length in every neighbourhood; the polygonal case is genuinely easier and was never the hard part. The general theorem is cited, NOT proved here. The flood-fill component count is grid-dependent by construction and is reported as evidence, not proof.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of HARD RESET · David Lee Wise (ROOT0), with AVAN