What was already known
A 2025 study searched graphs through 100 vertices without finding a counterexample to the conjecture.
AI involvement
Claude agents proposed the dumbbell witness in John Erlbacher’s directed Demonstrandum campaign. Erlbacher audited its verification gates.
Claude agents supplied the witness under Erlbacher’s direction; individual-result autonomy is not documented.
Importance
Resolves a longstanding spectral graph conjecture with a small exact witness.
Limitations
The result addresses the stated variance and distance readings. It does not establish a global smallest counterexample; a larger corroborating witness uses floating-point checks only.
Technical verification
Two exact spectral routes and a separate rebuild reportedly certify the finite counterexample.
Independent checking
exact computation
- Checker
- checker_g143.py exact SymPy and Berkowitz/Sturm routes, with verify_kills.py rebuild
- Scope
- Dumbbell graph certificates violating Graffiti 143 under both average-distance readings
- Basis
- Reported by the linked checker
Human understanding
No named person explains this individual construction; Few is the evidence default.
Correction and publication history
The 39-vertex witness concerns positive-eigenvalue variance in Graffiti 143.
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