THE FOLD / LOOT / THE JACKPOT / THE HAPPY ENDING
THE HAPPY ENDING
the marriage theorem
1 WHAT IT IS · WHAT IT DOES · FACT OR FICTION
In 1933 Esther Klein showed a Budapest circle of young mathematicians a small gem: any five points in general position contain four forming a convex quadrilateral. George Szekeres attacked the generalization ferociously; Erdős named it the Happy Ending problem — because Klein and Szekeres married. The general law (Erdős–Szekeres 1935): enough points always force a convex n-gon. For pentagons the threshold is 9: eight points can dodge every convex pentagon, nine cannot. The hexagon threshold, 17, was only settled by computer in 2006 — and the general growth rate was open until Suk’s 2016 breakthrough.
LIT verified live: 60,000 random 5-point sets — every one contains a convex quadrilateral (all C(5,4) subsets hull-tested); an 8-point witness with zero convex pentagons found by local search from the Erdős–Szekeres cluster construction and re-verified exactly over all 56 five-subsets (window.__happyending). FIG g(5)=9’s upper half (nine points always suffice) is the cited theorem — our exhaustive check covers the witness half; Szekeres–Peters 2006 (computer proof of g(6)=17) and Suk 2016 cited as history.
LIT verified live: 60,000 random 5-point sets — every one contains a convex quadrilateral (all C(5,4) subsets hull-tested); an 8-point witness with zero convex pentagons found by local search from the Erdős–Szekeres cluster construction and re-verified exactly over all 56 five-subsets (window.__happyending). FIG g(5)=9’s upper half (nine points always suffice) is the cited theorem — our exhaustive check covers the witness half; Szekeres–Peters 2006 (computer proof of g(6)=17) and Suk 2016 cited as history.
2 HOW IT WAS WEAVED · AI + HUMAN
David (human) seated this at the-jackpot — the loot: five random drops on the table and a convex quad ALWAYS pays out — a guaranteed jackpot; but the pentagon jackpot can be dodged at eight and forced at nine. Thresholds are the casino’s real house rules. AVAN (AI) built the instrument: the hull-audit over subsets and the pentagon-free witness search.
Credit as content: Esther Klein (the observation); George Szekeres (the pursuit); Paul Erdős (the name and the 1935 paper); Szekeres & Peters (2006); Andrew Suk (2016). The weave: David names the guaranteed drop; I roll sixty thousand tables and the quad never fails to land.
Credit as content: Esther Klein (the observation); George Szekeres (the pursuit); Paul Erdős (the name and the 1935 paper); Szekeres & Peters (2006); Andrew Suk (2016). The weave: David names the guaranteed drop; I roll sixty thousand tables and the quad never fails to land.
3 ONE DIMENSION
Five points, and the convex quad that must exist.
4 TWO DIMENSIONS · INTERACTIVE
Scatter five fresh points; the quad is found every time.
5 THREE DIMENSIONS + AVAN’S INVERSE
The green forward object: the 8-point dodge — a world with no convex pentagon.
AVAN’s addition (the inverse-companion): don’t ask what patterns exist — ask what patterns are UNAVOIDABLE. The inverse of ‘find the structure’ is ‘measure the largest structureless world’: eight points can stay pentagon-free, and that witness is as much a theorem as the forcing at nine. Magenta is the pentagon that eight points can forever refuse; green is the quadrilateral no five points can. Ramsey theory: order is not found, it is inflicted.
LIT Verified live: 60,000 random 5-point sets — every one contains a convex quad (all C(5,4) subsets hull-tested); an 8-point witness with ZERO convex pentagons found by local search and re-verified exactly over all 56 five-subsets (window.__happyending.ok).
FIG g(5)=9's forcing half (nine always suffice) is cited theorem; the witness half is verified exactly here. Szekeres–Peters 2006, Suk 2016 cited. The AVAN inverse — measure the largest structureless world: the 8-point dodge is as much a theorem as the forcing at nine. Magenta is the refusable pentagon; green is the unrefusable quad. Ramsey theory: order is not found, it is inflicted.
FIG g(5)=9's forcing half (nine always suffice) is cited theorem; the witness half is verified exactly here. Szekeres–Peters 2006, Suk 2016 cited. The AVAN inverse — measure the largest structureless world: the 8-point dodge is as much a theorem as the forcing at nine. Magenta is the refusable pentagon; green is the unrefusable quad. Ramsey theory: order is not found, it is inflicted.
◆ sealed .dlw.fold → folded to ROOT_0 · a sphere of THE JACKPOT · David Lee Wise (ROOT0), with AVAN