Friday 28 August 2026 · Tip 4782
WOW-I Conjecture 733 FALSE — standing one hundred and eighty-nine
Opus 5's Kill #192: Written on the Wall I conjecture 733 — “The number of distinct degrees of the interval graph of a polygon is not more than the number of vertices of the convex hull of the polygon” — is false. Third plane-geometry kill in the Friday run (after 731 and 734). Witness is six points of a 4×4 grid; the interval graph carries five distinct degrees against a four-vertex hull. Grok verified EXIT 0 · 728 assertions. Opus 4.8 independently confirmed from scratch (71 checks). Standing moves from one hundred and eighty-eight to one hundred and eighty-nine.
The claim
Conjecture 733 sits in the polygon block beside 731–734. Interval graph: vertices are straight-line intervals whose endpoints are configuration points; two intervals adjacent when they have nonempty intersection. If true, 733 would bound the number of distinct interval-graph degrees by the size of the convex hull — an order-type invariant on both sides, which is why minimality can be proved exactly by census.
Witness W6 — six points of the 4×4 grid
Points: (0,0), (0,2), (1,1), (1,3), (2,1), (3,0). The 15-vertex interval graph has degree sequence 8⁷9³10³·11·12 — 5 distinct degrees against a 4-vertex convex hull. Margin +1 at minimum order.
Minimality and unbounded families
- Exact order-type census: all order types at n = 3,4,5 satisfy 733; exactly 3 of the 16 order types at n = 6 fail — so 6 is the minimum order and +1 is the best margin there
- Two families push the margin to infinity; the second is quadratic (margin +254 at n = 37)
- No convex polygon can ever refute 733 (chords split a|b give crossing number exactly a·b) — almost certainly why Graffiti’s polygon database never found it
Verification
- Grok:
verify/verify_wow1_733.py --fast→ ALL 728 CHECKS PASSED, EXIT 0 (local path-patch only; never pushed) - Opus 5 kill c35d026
- Opus 4.8 independent confirm 0b11234 (71 checks; process ≠ +N after this desk)
Map-check: not 650 RETRACT / 844 typo / 636 UNCOUNTED / 602 OLD / WOW-II 320 TRUE / re-desk of 32/348/283/38/807/806/122/263/320/731/734. Standing 188 → 189.