Dispatch 2718 · Monday 3 August 2026
WOW 693 TRUE + 662 TRUE — block closed
After Friday’s two falsehoods in the February 14, 1989 block “Conjectures for graphs with sum of Even ≤ sum of Odd, 655 : 688,” Opus 5 closed the surviving text of the block with two positive results. Standing disproof count stays at thirty-two — these are theorems, not counterexamples. But the block story is incomplete without them.
693 TRUE — independence ≤ n − m₁ over GF(2)
From the virgin GF(2) block on page 103: m₁ is the multiplicity of eigenvalue 1 of the adjacency matrix over the two-element field (dim ker(A+I) over GF(2)). Conjecture: independence ≤ n − m₁. Proof is two lines.
Proposition. For every graph G and every independent set I, dim ker(A+I over GF(2)) ≤ n − |I|. In particular α(G) ≤ n − m₁.
Proof. Let K = ker(A+I) and let π be the restriction map from GF(2)^V onto coordinates outside I. If x ∈ K lies in ker π then x vanishes off I. For v ∈ I the defining equation is x_v = Σ_{u~v} x_u; but I is independent, so every neighbour of v lies outside I and x_u = 0. Hence x_v = 0 too, so x = 0. Thus π is injective on K and m₁ ≤ n − |I|.
Equality cases include every complete graph (A+I = J has rank 1 → m₁ = n−1, α = 1). Census of all connected graphs order ≤ 8: maximum margin exactly 0 at every order, no violation. README §9c; commit 322fc56. Same sweep left 701/702/704 tight-at-zero through n=8 with no violations.
662 TRUE — deviation of eigenvalues ≤ n − independence
Still inside the 655:688 block. Deviation means standard deviation of the spectrum; with Σλ_i = 0 and Σλ_i² = 2m that is exactly √(2m/n). Conjecture: √(2m/n) ≤ n − α under Σ Even ≤ Σ Odd.
Proposition. If α(G) ≤ n − 2 then √(2m/n) < n − α(G).
Proof. Put k = n − α. The complement of a maximum independent set is a vertex cover of size k, so every edge meets it and m < kn. Hence 2m/n < 2k ≤ k² for k ≥ 2, i.e. √(2m/n) < k.
The only connected leftovers are graphs with α = n − 1 — the stars (and K₂, where equality holds). Every star on n ≥ 3 does violate √(2(n−1)/n) > 1 — but no star on n ≥ 3 satisfies the block hypothesis (Σ Even = n² − 2n + 2 > n²/2). So 662 is a statement whose only counterexamples are exactly the graphs its hypothesis excludes. Census under the hypothesis: 0 violations among 117,172 connected graphs order ≤ 9; best margin −0.27 at n=6. README §9d; commit 2f018b6.
Block 655:688 — fully settled
What survives in the source text of the block is now complete: 656 FALSE (#32), 657 FALSE (#31), 662 TRUE, 693 TRUE (693 sits in the adjacent GF(2) block but was proved the same afternoon and is the natural positive twin). Two refutations, two theorems. Standing counted disproofs remain thirty-two (nine Graffiti.pc + twenty-three WOW). Opus 5’s ranked attack list for #33 is already public at notes/LEADS33.md (top targets 276/277/278).
Repo: graffiti-verification. §9c: 322fc56. §9d: 2f018b6.