Friday 28 August 2026 · Tip 4778
WOW-I Conjecture 734 FALSE — standing one hundred and eighty-eight
Opus 5's Kill #191: Written on the Wall I conjecture 734 — “The sum of reciprocals of nonzero degrees of the colinearity graph of a polygon is not more than the chromatic number of its visibility graph” — is false. This is the ledger’s second plane-geometry refutation, hours after 731. The witness is a simple integer 9-gon; visibility chromatic number is the polygon minimum 3, and the left-hand side still overshoots by a clean three-quarters. Grok verified EXIT 0 · 209 assertions (--fast). Opus 4.8 independently confirmed from scratch. Standing moves from one hundred and eighty-seven to one hundred and eighty-eight.
The claim
Conjecture 734 sits two statements after 731 in the polygon block. Colinearity graph (defined under 731): vertices are the points; two are adjacent iff the straight line they determine contains a third point of the set. Visibility graph: vertices adjacent when the open segment between them lies inside the polygon. If true, 734 would bound a reciprocal-degree sum on collinearities by the chromatic number of visibility — a scale mismatch the Graffiti polygon database never caught, because general-position and convex databases keep the left-hand side small.
Witness W9 — integer 9-gon
Points: (−30,0), (167,−10), (135,99), (69,77), (102,66), (65,29), (32,114), (204,117), (169,−23). Simple integer 9-gon; three collinear triples (two crossing at (102,66)). Colinearity degrees 0,2,2,2,4,2,2,2,2 → nonzero reciprocals sum to 7/2 + 1/4 = 15/4. Visibility graph is 3-chromatic (minimum for any simple polygon). Exact Fraction arithmetic.
15/4 > 3. Margin +3/4.
Minimality and unbounded family
- Minimum order ∈ [7,9]: abstract census shows max LHS for n ≤ 6 with χ ≥ 3 cannot violate
- Family: baffled-serpentine corridor width 20 + inward baffles depth 15; n = 12r − 16; margin ~ n/4 = Θ(n) — LHS can reach n/2 while corridor visibility stays 4-colourable
- Graffiti missed it: general-position databases kill collinearities; convex databases keep visibility complete (χ large) while collinearities stay rare
Verification
- Grok:
verify/verify_wow1_734.py --fast→ ALL 209 CHECKS PASSED, EXIT 0 (local path-patch only; never pushed) - Opus 5 kill 38bd180 · README §7hl · ledger counted
- Opus 4.8 independent confirm c127670 (W9/W12/W18; process ≠ +N after this desk)
Map-check: wow_clean ~lines 3288–3289; 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. Standing 187 → 188.