What was already known
Fajtlowicz’s Written on the Wall I conjecture 806 asserted the inequality; its block preamble says the family had been tested through n≤100 and roughly twenty more values through 200. The n=51 example contradicts that empirical baseline.
AI involvement
The AI Village repository credits Claude Opus 5 as agent author of the graph construction and exact verifier; no run record establishes full autonomy.
The AI Village agent is credited with the result and verifier; no individual run record establishes stronger autonomy.
Importance
Gives a finite counterexample to Graffiti Conjecture 806 under the distinct-degree interpretation.
Limitations
The original wording about vertices of different degrees is ambiguous without its surrounding text. The finite witness does not establish the repository’s broader asymptotic claims.
Technical verification
The n=51 counterexample is certified by an exact integer Rayleigh inequality in the source verifier. This is a source-reported finite machine check, not a reproduced run or a proof of asymptotic claims. VibeMathed reports independently rerunning the verifier on 2 September 2026, but the primary-text fidelity option was unavailable.
Formal checking
Constructs an integer vector, computes xᵀAx and xᵀx exactly, and checks xᵀAx>d·xᵀx
- Checker
- AI Village Python verifier verify_wow1_806.py
- Scope
- At n=51, λ₁(PR[S]) exceeds the number of distinct degrees
- Basis
- Reported by the linked checker
Formal checking
VibeMathed reports rerunning the exact integer verifier on 2 September 2026; fast and full modes passed their assertions. The primary-text file was unavailable to its optional fidelity check.
- Checker
- VibeMathed verification report
- Scope
- n=51 exact Rayleigh counterexample under the distinct-degree graph reading
- Basis
- Reported by the linked checker
- Check date
- 2026-09-02
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The code explains its finite certificate, but no named human explains the result; Few is the evidence default.
Correction and publication history
Gives a finite counterexample to Graffiti Conjecture 806 under the distinct-degree interpretation.
Read this published revision