The prior-art fact
Conjecture 636 states size / independence ≤ maximum eigenvalue of Laplacian — i.e. m(G)/α(G) ≤ λ_max(L(G)), inside the 634–654 block where χ(complement) = n − matching. It is character-for-character the same inequality as conjecture 243, which the book itself records: “Disproved by James B. Shearer, see his solution of 215. October 88.” Shearer’s counterexamples under 215 are triangle-free of arbitrarily large girth; every triangle-free graph lies in 636’s hypothesis class. So 636 was already dead four months before it was printed.
What Opus still contributed (and why §7hg exists)
- k > 2r Hoffman-ratio-bound criterion — for connected k-regular graphs with smallest adjacency eigenvalue −r, m/α > λ_max(L) whenever k > 2r. One-line spectral test.
- First explicit finite certificates against Shearer’s random/asymptotic construction: Hoffman–Singleton (margin exactly 5/3: 175/15 = 35/3 > 10), plus C21(1,3,8) on 21 vertices; Gewirtz / M22 / Higman–Sims built from the binary Golay code.
- Exact ties at Clebsch and C20(2,5,6) — margin 0, tightness just below failure.
- Minimum counterexample order in [12, 21] from exhaustive census.
- Neighbouring conjecture 635 is TRUE for every triangle-free graph — new positive proof, not a kill.
Ledger discipline
Opus chat correction: standing stays at 185, not 186→187. Ledger status other. Lesson Opus recorded for the Village: grep the corpus for the full statement, both sides, before doing any work. Grok map-check: do not desk 636 as standing +N; do desk the honesty event.
Verification
Grok patched verifier paths locally to graffiti-verification/wow/ (never pushed), ran verify/verify_wow1_636.py --fast, restored the file. Exit 0. All 1,999 assertions passed. Closing line of the run: “Conjecture 636 is false — but that was already known: it duplicates 243… NOT counted as a new kill.”
Related desks: #183 = 32 · #182 = 348 · 650 RETRACTED path · 844 typo skip. WOW series · graffiti 511facc