THE FOLD / CO-OP / SPLIT SCREEN / THE HAM SANDWICH
THE HAM SANDWICH
one cut for two appetites
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
Two scatterings of points on a table — red and blue, tangled however you like. The ham sandwich theorem guarantees a single straight line that bisects both simultaneously: half the red on each side AND half the blue. In three dimensions, one planar cut halves the bread, the ham, and the cheese at once (Steinhaus 1938; Stone–Tukey 1942) — and in n dimensions, one hyperplane bisects n arbitrary masses. The proof is a rotation argument: anchor the line to always bisect red, sweep its angle through 180°; the blue imbalance flips sign end-to-end, so somewhere it crosses zero — Borsuk–Ulam wearing an apron.
LIT verified live: 300 random 20+20 point-set pairs, the rotating-line construction finds the double bisector every time and verifies it by exact count (10 of each set strictly per side, ties split at refined crossings) (window.__hamsandwich). FIG honest boundary: the 3D and n-dimensional statements are cited; the 2D theorem is executed instance by instance, with the IVT sweep visible in the search itself.
LIT verified live: 300 random 20+20 point-set pairs, the rotating-line construction finds the double bisector every time and verifies it by exact count (10 of each set strictly per side, ties split at refined crossings) (window.__hamsandwich). FIG honest boundary: the 3D and n-dimensional statements are cited; the 2D theorem is executed instance by instance, with the IVT sweep visible in the search itself.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at split-screen — the co-op: two players, one shared blade, and a theorem promising the cut that leaves neither shortchanged — whatever mess they made of the map. AVAN (AI) built the instrument: the median-anchored sweep with sign-change refinement.
Credit as content: Hugo Steinhaus (1938); Stone & Tukey (1942); Borsuk–Ulam underneath. The weave: David names the shared blade; I rotate it until both halves agree, 300 times.
Credit as content: Hugo Steinhaus (1938); Stone & Tukey (1942); Borsuk–Ulam underneath. The weave: David names the shared blade; I rotate it until both halves agree, 300 times.
3 ONE DIMENSION
Two point clouds and the one line that halves them both.
4 TWO DIMENSIONS · INTERACTIVE
New clouds; the sweep hunts the angle where blue balances too.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the blade rotating, imbalance draining to zero.
AVAN’s addition (the inverse-companion): don’t search positions — spend one constraint per mass. The inverse of ‘can one line do both?’ is a budget: a line has two degrees of freedom, one is spent bisecting red always, the last buys blue at some angle. Dimensions are currency. Magenta is the third mass no 2D line can afford; green is the cut both appetites accept. Fairness is a dimension count.
LIT Genuine ham sandwich theorem (Steinhaus 1938; Stone & Tukey 1942). Verified live: 300 random instances, rotating-line construction with refined sign-change search — exactly 10 of each 20-point set per side, every run (window.__hamsandwich.ok).
FIG Honest boundary — 3D and n-dimensional statements cited; the 2D theorem executed instance by instance. The AVAN inverse — don't search positions, spend one constraint per mass: a line has two degrees of freedom, one buys red always, the last buys blue at some angle. Dimensions are currency. Magenta is the third mass no 2D line can afford; green is the cut both appetites accept. Fairness is a dimension count.
FIG Honest boundary — 3D and n-dimensional statements cited; the 2D theorem executed instance by instance. The AVAN inverse — don't search positions, spend one constraint per mass: a line has two degrees of freedom, one buys red always, the last buys blue at some angle. Dimensions are currency. Magenta is the third mass no 2D line can afford; green is the cut both appetites accept. Fairness is a dimension count.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of SPLIT SCREEN · David Lee Wise (ROOT0), with AVAN