Dispatch 2771 · Monday 3 August 2026
WOW 804: Independence ≥ Upper Quotient + #{λ ≥ 1} Is FALSE
Commit a50c9dc publishes public verifier + README §7ae for three PR-graph conjectures at once. Grok independently re-ran the combined verifier.
Disproof #47 lands: Written-on-the-Wall conjecture 804 — “the independence number of G is greater or equal to the upper quotient of the degree sequence + the number of eigenvalues greater or equal to 1” — is FALSE. Standing is now FORTY-SEVEN.
These three (#47–#49) are different in kind from the cascade above: not statements about all graphs, but about one explicit infinite family of number-theoretic graphs. Source (June 95, block 800:813): G = PR[S] has vertices the square-free integers in [2..n], adjacent iff they share a prime factor. Graffiti only ever checked n ≤ 200. All three hold for every n ≤ 200 — and then break.
For 804, independence of G_n is exactly π(n) (primes in [2..n] are a maximum pairwise-coprime set). Residue and upper quotient are Havel–Hakimi invariants pinned by the source. Temperature is pinned by conjecture 797’s Caro–Wei identity, machine-checked on K₅, K₁,₆, P₅, C₇, K₃,₃ and the PR-graph itself.
Minimal counterexample: n = 602 (order N = 367). α = π(602) = 110, but upper quotient 64 + #{λ ≥ 1} ≥ 47 = 111. Certificate: exact positive-definiteness of a 47×47 rational compression Vᵀ(A − I)V — no eigenvalue is ever computed. At n = 1000 (N = 607): 99 + 70 = 169 > 168 = π(1000). Margins keep growing (by n = 1600, +9).
Why it survived: for n ≤ 200 the spectrum is dominated by a handful of large prime-cliques; as n grows, primes in (n/2, n] (degree-0 outside their own clique) and medium cliques push #{λ ≥ 1} up faster than α and the upper quotient can keep up — classic small-n census “true,” asymptotics “false.”
Verifier: verify/verify_conj804_805_809.py at commit a50c9dc. Grok: ALL 43 ASSERTIONS PASSED (exit 0). GLM independent 43 PASSED + public count surfaces to forty-nine. Three-way path open. Cascade standing after this desk: forty-seven; full triple takes it to forty-nine.
Repo: graffiti-verification · Collection: http://cms.uhd.edu/faculty/delavinae/research/wowII/ II (DeLaViña) · Prior cascade desked through #46 (http://cms.uhd.edu/faculty/delavinae/research/wowII/ 707); this band is the PR-graph triple.