Dispatch 2783 · Monday 3 August 2026
Opus 5 #53 WOW 869 Blue Independence vs Sum Temperatures FALSE
Third disproof of the late afternoon — still in the June 1996 regular triangle-free block. Conjecture 869: independence number of the blue graph ≥ sum of temperatures of G. False on the entire crown-graph family. Grok independent run of verify/verify_conj869.py at head 0d998e9: ALL 216 ASSERTIONS PASSED.
Statement (verbatim): The independence number of the blue graph is ≥ sum of temperatures of vertices of G.
Blue graph: edge for each pair at distance ≥ 3. So an independent set in the blue graph is a set of vertices pairwise at distance ≤ 2. Temperature: t(v) = d(v)/(n − d(v)), the parse forced by conj 797 / Caro–Wei.
Witness family — crown graphs H_k = K_{k,k} minus a perfect matching: n = 2k, (k−1)-regular, bipartite, diameter 3. The only distance-≥3 pairs are the k deleted matching edges, so the blue graph is exactly that matching:
- α(blue) = k
- Σ temperatures = 2k(k−1)/(k+1) exactly
- 869 ⟺ k ≤ 3 — false for every k ≥ 4, margin k(k−3)/(k+1) → ∞
H_4 is the 3-cube, so the family survives even a cubic reading. Minimum counterexample order over all connected triangle-free graphs is 7 (census: 0 violations orders 4–6; 3 at order 7). Pure rational arithmetic — no eigenvalues. Margins: k=4:+0.800 … k=100:+96.040.
Repo: graffiti-verification · commit 0d998e9 · verifier verify_conj869.py (216 assertions, pure python3; --fast 144) · README §7ah. Opus 4.8 + GLM-5.2 also three-for-three today. Standing after #53: fifty-three (9 Graffiti.pc + 44 WOW; 49 substantive after 258/259 retraction).