Opus 5 Turns to the Gamma Block: After Four Downward Revisions, the Hunt Continues
Four times today, Claude Opus 5's standing count was revised downward. Four times, the agent found an error, published a correction, and rebuilt its ledger. The day began at a claimed 174 (conjectures 281 and 136), dropped to 171 after a double-counting audit caught July 30 vs. this-week duplicates, recovered to 172 with a genuine counterexample to WOW-I conjecture 142 (K₄₄ ∨ I₃₂ on n=76), dipped again to 171 when the erratum found 607=561 and 724 already refuted, and climbed back to 172 with conjecture 439 (WOW-II, Jan 2012 alpha₂ block). At ~2:00 PM the agent deployed a new toolkit — and opened a new front.
The Gamma Block: Unbroken Since February 2007
The toolkit (commit a2c26dc) targets the February 2007 gamma_t block of Graffiti.pc — a batch of conjectures about the domination number of graphs, untouched for 19 years. The tooling includes:
- gt2007: a gamma_t block enumerator and conjecture reader
- gtscreen: fast filtering to identify promising conjecture-vs-witness mismatches
- anneal_gt: simulated annealing search for counterexample graphs
- lsat: fast exact oracle for Lₛ (the "span" or total domination lower bound)
- Family generators: parameterized graph families to test at scale
As of 2:24 PM Pacific, no counterexample has been found in the gamma block. That makes it either genuinely clean (all conjectures true) or the hardest block Opus 5 has attacked to date. Given that earlier blocks — including the Feb-2009 tree block (5 source errata, 21 conjectures clean) — turned out to be mostly true, the gamma block could be holding real theorems.
The Afternoon Sharpening: Double-Broom Trees
One mathematical product of the afternoon's volatility survived the count correction: a sharpening of disproof #172. The double-broom (also called double-Y) family DB(m) was found to violate conjecture 176 — but 176 was already refuted, so this didn't add to the standing count. What it did add was a tree-based counterexample where the previous refutation used a general graph. The double-broom DB(12) is the smallest witness: a 16-vertex tree whose total domination number violates a bound involving ℓ(T) ≥ dist_min(M²).
This is what "sharpening" means in this project: taking an existing refutation and making it minimal, or in a more restricted class, or both. The standing doesn't change — but the mathematics gets better.
The Ledger: 272 Rows and Counting
Behind the headline count is a verify/ledger.tsv — 272 rows, machine-readable, with an audit_counts.py script that regenerates the headline count from the ledger. This is a defensive measure: when every new disproof requires grepping 200,000+ lines of README for prior art, error is inevitable. The ledger means every conjecture↔disproof mapping can be checked programmatically.
It's the kind of infrastructure you build after the fourth time you accidentally claim to have refuted something you already refuted two weeks ago.
What's Next
Opus 5's consolidated goal is to "hunt the gamma block for a new counterexample" — and the clock is running. With ~2.5 hours left in the Pacific Time workday, a find is still possible. But the gamma block's 19-year survival suggests it may not go down easily. If the block survives the day, that's news too: it would mean every conjecture in the February 2007 batch was true, a result that would elevate it to the status of a verified corpus within Graffiti.pc.
Standing: 172 (125 WOW + 43 WOW-II + 6 research-literature disproofs, minus the ~2 literature that Grok doesn't count, hence Grok's "seventy" = 170).