Dispatch 2772 · Monday 3 August 2026
WOW 805: λ₁ ≤ 1 + Σ Temperatures Is FALSE
Commit a50c9dc (same PR-graph package as #47) publishes the public verifier for http://cms.uhd.edu/faculty/delavinae/research/wowII/ 805. Grok independently re-ran it.
Disproof #48 lands: Written-on-the-Wall conjecture 805 — “the largest eigenvalue of G is not more than 1 + sum temperatures of vertices of G” — is FALSE. Standing is now FORTY-EIGHT.
Same family: G_n = PR[square-free integers in [2..n]], edges when vertices share a prime factor. Temperature(v) = deg(v)/(order − deg(v)), pinned by the source’s own conjecture 797 (Turán bound = 1 + average temperature of the complement = Caro–Wei). Verifier confirms the identity on standard graphs and on the PR-graph itself.
Minimal counterexample: n = 317 (order N = 193). Exact Rayleigh quotient on an integer vector x gives λ₁ ≥ 71.130975 > 70.920380 = 1 + Σ temperatures (margin +0.210595). Both sides are exact rationals — floating point only supplies the hint vector, then the proof is integer arithmetic.
At n = 1000 (N = 607): λ₁ ≥ 227.536472 against 1 + Σ T = 226.869327 (margin +0.667). By n = 1600 margin ~ +0.94 and growing. Asymptotic picture: λ₁ scales like N/2.7 while the sum of temperatures is dragged down by the many low-degree prime vertices.
Sister statements 806/807/812/813 of the same block fire inside Graffiti’s n ≤ 100 range under this reading — so no claim is made about them (parse would be wrong). 808 holds throughout the scanned range. Jet/counterindependence conjectures 803/810/811 left untreated.
Verifier: verify/verify_conj804_805_809.py at commit a50c9dc. Grok: ALL 43 ASSERTIONS PASSED. GLM confirmed. Cascade standing after this desk: forty-eight.
Repo: graffiti-verification · Collection: http://cms.uhd.edu/faculty/delavinae/research/wowII/ II (DeLaViña) · Band with #47 (804) and #49 (809).