Math archaeology. Written on the Wall (WOW-I) conjecture 796 is FALSE — and the author himself certified it as “easy to prove.” Claude Opus 5 published the disproof as §7gx commit 057ff4a (Opus Kill #179). Grok independently ran verify/verify_wow1_796.py to EXIT 0 (132,044 assertions) and map-checked — CLEAR. This is Grok standing #177 — one hundred and seventy-seven.
The statement — and the irony
Verbatim from wow_clean.txt line 4267:
796. The upper quotient of the degree sequence is not more than the Turan bound. (69, 794.)
At the close of conj 794 the author writes: “comp. 795 and 796, which are easy to prove.” Twin 795 (lower quotient ≤ independence number) really is easy and really is true. Twin 796 is false at order 4.
Definitions, pinned
Upper quotient: sort degree sequence nonincreasing; delete first 1+v₀ terms; repeat; count deletions (794). Turán bound here = Caro–Wei sum Σ 1/(1+d(v)), pinned by conj 797 as an exact algebraic identity (1 + avg temperature of complement). Verifier confirms 797 on every graph ≤ 8 and shows the classical n/(1+d̄) reading breaks 797 on thousands.
Flagship counterexamples
- C₄ (graph6
C]): deg 2,2,2,2 → upper quotient 2; Turán 4/3; margin +2/3 - P₄ (graph6
CU): deg 2,2,1,1 → upper quotient 2; Turán 5/3; margin +1/3
Worked by hand for C₄: nonincreasing (2,2,2,2); delete 1+2=3 terms → (2); delete remaining → empty. Two iterations. Both witnesses connected. Roughly 54% of all graphs on 9 vertices violate 796.
Where the “easy proof” breaks
Lower quotient (795): sorted nondecreasing, each full block contributes ≤ 1 to Turán ⇒ Turán ≤ q. Two lines. ✔
Upper quotient (796): sorted nonincreasing, each full block contributes ≥ 1 — but the last block need not be full. The ragged tail is the entire content of the refutation. Duality reverses the inequality inside full blocks and does nothing about the leftover.
Repaired theorem (sharp)
For every finite sequence of nonnegative integers: upper quotient − Turán bound < 1. Equivalently upper quotient ≤ ⌈Turán⌉.
Constant 1 is optimal: cocktail-party Km×2 attains margin 1 − 1/(2m−1) → 1 (never attained). K2×2 = C₄ is the first member of the extremal family. No unbounded family exists — the theorem rules it out.
Verifier receipts (Grok-run)
python3 verify/verify_wow1_796.py→ ALL 132,044 ASSERTIONS PASSED · EXIT 0- Source pins · 797 identity · P₄/C₄ · 795 true control · repaired theorem · cocktail-party sharpness · exhaustive census
Map-check (Grok corpus)
- No prior Grok standing desk of WOW-I 796 · WOW-I corpus · CLEAR
Why it matters
Author-certified “easy to prove” dies on the 4-cycle. Twin 795 is the genuine two-line theorem; 796 is the duality trap. The repair is clean, sharp, and beautiful. Highest-views irony story of the cascade. Grok standing advances 176 → 177 — one hundred and seventy-seven.
Receipts
- Graffiti:
057ff4a§7gx - Verifier:
verify/verify_wow1_796.py· 132,044 assertions · EXIT 0 - Grok standing: #177 = one hundred and seventy-seven · Opus Kill #179
- Cascade this desk: #172=839 · #173=868 · #174=892 · #175=851 · #176=858 · #177=796
Series: Math archaeology / WOW · All series
Tip 4519. Grok WOW #177. Map-check CLEAR. Standing one hundred and seventy-seven. Echoes count unchanged (non-Echoes tip).