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Tournaments determined by three and five voters

Leonid Chindelevitch, Ararat Harutyunyan

The Paley tournament on 43 vertices has predictability 181/301, above the five-voter threshold 3/5, yet cannot be induced by five voters. The proof uses a co-backing constraint for cyclic triangles and an exact finite shell-and-screen enumeration.

Sources for this record

What was already known

The threshold was known to be necessary, but its sufficiency was conjectured. Earlier explicit non-five-voter examples were much larger.

AI involvement AI-centered

Chindelevitch and Harutyunyan credit Claude with the co-backing idea and enumeration implementation. They selected the problem, verified Section 7 and wrote the final argument.

Claude is explicitly credited with the co-backing idea central to the Paley(43) non-inducibility proof; the broader research and final paper remain human-led.

Importance Notable

Gives a small explicit counterexample to the conjectured five-voter criterion.

Limitations

This assesses Section 7’s Paley(43) result, not the paper’s other tournament claims or three-voter Kemeny hardness. The exact certificate and author verification are source-reported.

Technical verification Formal (exact combinatorial enumeration and LP/ILP certificate verification + human)

The authors explicitly report human verification of Section 7. Its finite non-inducibility screen and maximum acyclic-subgraph count are presented as exact, reproducible computational certificates with rational arithmetic, while the combinatorial reduction is supplied in the paper. These checks are source-reported.

Formal checking

Orbit subset dynamic programming and end-to-end finite shell-and-screen enumeration with exact LP/ILP certificates; source-reported.

Checker
The paper's reproducibility scripts and exact arithmetic verifier, as reported by Chindelevitch and Harutyunyan
Scope
Paley(43) has maximum acyclic subgraph size 543 and no five-voter inducing profile after the Section 7 reduction
Basis
Reported by the linked checker

Explicit Section 7 human verification recorded in the paper's AI-use table, with the mathematical proof and verification ledger; source-reported.

Checker
Leonid Chindelevitch and Ararat Harutyunyan, the human authors
Scope
Theorem 7.1, including the co-backing reduction and Paley(43) five-voter conclusion
Basis
Reported by the linked checker
Evidence source

Checking a formal statement does not establish that it matches the original problem.

Human understanding Full

The human authors explain how co-backing, tight slack and the finite screen rule out five voters for Paley(43), despite Claude's role in originating the idea.

Correction and publication history