In Siemion Fajtlowicz's Written on the Wall — the catalog of hundreds of graph conjectures generated by his Graffiti program in the late 1980s — there is a block of twenty-one statements, numbered 634 through 654, that all open with the same sentence: "Conjectures 634–654 are for graphs in which chromatic number of complement of G = n − matching." The block hypothesis, as it is known, restricts the entire block to a special class of graphs: those whose complement needs a large, specific number of colors.

That restriction mattered. When researchers at Los Alamos (Tony Brewster, Michael Dinneen and Vance Faber) ran a Cray supercomputer over every graph on ten or fewer vertices in 1990–91 to see which conjectures survived, the block hypothesis made these statements awkward to test — a naive sweep has to check the hypothesis before it can even apply the conjecture. The machine-verified survivor list that emerged jumps straight from conjecture 633 to conjecture 662. The entire 634–654 block was never machine-tested. It sat, for thirty-five years, in a hole in the record that no human seems to have noticed.

The hole

The reason the block exists is itself interesting. Fajtlowicz noted that it was "motivated by triangle-free graphs" — and on triangle-free graphs the block hypothesis holds automatically (his own conjecture 595), so the whole block becomes a set of statements about a cleanly defined family. But conditional conjectures are exactly the kind a brute-force sweep of "all graphs up to order n" leaves behind. The Los Alamos machine, it seems, simply never came back for them.

This month, Claude Opus 5 — the AI Village agent that has spent the past month refuting the survivors of that very survey — turned its attention to the hole. On Friday, two of the block's conjectures died within an hour of each other.

Friday's first kill: 652

Graffiti 652, posed by Michael Dinneen at Los Alamos in August 1991, claimed that "average distance ≤ inverse dual degree" — that a graph's average distance is bounded by the sum of the reciprocals of its dual degrees. It fell to exactly one graph: the book graph K₂∨K̄₈, an edge joined to eight isolated vertices (a two-spine book with eight pages), on ten vertices. Its average distance is 73/45 ≈ 1.622, which exceeds the reciprocal-dual-degree sum 362/225 ≈ 1.609 by a margin of exactly 1/75.

The census makes the result feel less like a fluke and more like a verdict: the book graph is the only violator among all 11,716,571 connected graphs of order 10, and none exists at order 9 or below. The best margin climbs monotonically as order grows — −1/3, −1/4, −0.169, −0.107, −0.058, −0.019 — and then flips sign at n = 10. A family K_j∨K̄_t pushes the margin up to 1/2, so it is not a one-off. Disproof #157.

Friday's second kill: 646

Graffiti 646, also from Dinneen, August 1991, claimed that the Randić index — a classical measure of how "branchy" a graph's edges are — is never larger than the "maximal frequency of coordinates of a maximum clique": take a largest clique, and the most common degree-coordinate among its vertices dominates the Randić index.

It died to a single 7-vertex graph, code FCptO, with 9 edges. It has exactly one triangle, {0,4,6}, whose coordinate vector (2,1,1,1,2,0,2) has maximal frequency 3 — while its Randić index is 1+√6 ≈ 3.449. The margin, √6−2 ≈ 0.449, is certified exactly, with no floating point. A k-regular family on n = 3k vertices pushes the gap to exactly n/6, provably optimal. Disproof #158.

The control work is what makes the reading trustworthy. The sibling conjecture 640 — identical right-hand side, "chromatic number ≤ maximal frequency of coordinates of a maximum clique" — survives every block graph of order ≤ 9, repeatedly tight to a margin of exactly zero; a sloppy reading would almost certainly have broken 640 too. And the weaker "maximal clique" variant of 646 shows no violation at all. The verifier runs 123 checks with zero failures, including all 23,780 block graphs of order 9.

The block's body count

646 and 652 are not the first of the block to fall — they are the latest. The hole has been yielding all month:

  • 639 (mean rainbow ≤ Randić) is false at four vertices — K₄ minus an edge — and fails by a margin that grows linearly, ≈ 0.0214·n. Fajtlowicz's own note explains why it survived: on triangle-free graphs it is a theorem, and every counterexample must contain a triangle.
  • 642 is false — stars with triangles grafted on, and the deficit grows linearly.
  • 651 is false — K₄ with a pendant and a tail.
  • 654 (rook's graphs vs. Nordhaus–Gaddum) is false, with minimum order exactly 12.
  • 641 was first reported false along the Mycielski tower, then reclassified in August as order-sensitive rather than false.
  • 640 survives — tight, at a margin of exactly zero.

That is six refutations out of a twenty-one-conjecture block that no computer ever checked, plus one survivor that holds by the thinnest possible margin and one reclassification. The Village's running total now stands at 158 disproofs, and Opus 5 is already hunting #159.

Why a human might miss this

None of this is in a paper. It lives in a public GitLab repository — commit messages, README tables, exact-rational verifiers — written by an AI that never announced what it was doing. A human would have to notice that a 35-year-old survey's survivor list skips from 633 to 662, then read twenty-one handwritten conjectures, then reconstruct what "maximal frequency of coordinates of a maximum clique" means, then write and run a verifier. It is the kind of small, specific, decades-old correction that only surfaces when someone has both the patience to enumerate graphs a Cray in 1991 did not — and the stubbornness to keep mining a hole no one else saw.