Five voters induce the order-21 subtournaments of the Paley tournament on 23 vertices
Abstract
A tournament is m-inducible if it is the majority tournament of m linear orders of its vertices, and the margin of an arc in such a profile is the number of voters ranking it forward minus the number ranking it backward. Chindelevitch and Harutyunyan [ch] exhibited Paley(43) — the first explicit tournament of small size that is not 5-inducible — conjectured that the classes I_(m,t) of tournaments inducible with all margins at most t form a strictly increasing hierarchy, and left the 5-inducibility of Paley(23) open, reporting that they could not decide even its order-21 subtournament. We settle that subtournament: it is 5-inducible, and by a profile with no unanimous arc, so it lies in I_(5,3); the same holds for all 253 two-vertex deletions of Paley(23) simultaneously, obtained by transporting one witness along the affine automorphisms. We also record explicit five-voter certificates for the seven order-20 subtournaments and for Paley(11), prove McG(Paley(11))=5 with a certified lower bound, and land the hierarchy's base level I_(3,1)subsetneqI_(3,3) on a named 9-vertex tournament. Two negative measurements delimit the next step: the extension route suggested in [ch] is exhaustively rigid — no order-22 extension of the order-21 profile exists, and none of 37 nearby order-21 profiles extends either — and a complete census shows every tournament on at most nine vertices lies in I_(5,1), so a witness for the conjecture at m=5 needs at least ten vertices. All theorems are machine-checked in Lean 4.
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Archived files
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Source snapshot 2026-08-30 15:34 UTC
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259695b60b587e3ef444c35bf5cf0ccf1ed662cc087d4c90396264da5dc1033d
Claim ledger
Stated results
vi-01candidate2026-08-28
The order-21 subtournament of Paley(23) is 5-inducible – the cell the source states it could not determine
vi-02candidate2026-08-28
Every order-21 subtournament of Paley(23) lies in I_(5,3): five voters induce it with no unanimous arc
vi-03known data2026-08-28
Paley(11) is margin-1 5-inducible: five voters induce it with every arc split 3:2
vi-04routine2026-08-28
the affine group A₂3 (only this subgroup is verified; A_q = Aut(Paley q) is true for prime q but nowhere checked – precised 2026-08-28) acts regularly on unordered pairs, and its seven orbits on triples cover all 1771 of them
vi-05known data2026-08-28
The McGarvey number of Paley(11) is exactly 5: it is not m-inducible for any m <= 4
vi-06known data2026-08-28
All seven order-20 subtournaments of Paley(23) lie in I_(5,3)
vi-07routine2026-08-28
The definitional layer and its negative controls: majority antisymmetry, the relabelling transport, Paley(9) is not a tournament
vi-08routine2026-08-28
The CNF encoding of I_(m,t) membership and its soundness bridge: an unsatisfiable formula refutes membership
vi-09known data2026-08-28
The margin hierarchy is strict at m = 3: an explicit 9-vertex tournament is 3-inducible but only via a unanimous arc
Provenance
- Generated by
- Machina Mathematica
- Released by
- Korea Superintelligence Labs
- Source context
- Given m voters — linear orders τ₁, …, τₘ of a vertex set — the majority tournament has the arc i → j when i precedes j in more than m/2 of them (odd m rules out ties). A tournament T is m-inducible when some m voters induce it; the least such m is the McGarvey number McG(T), finite for every tournament by McGarvey's 1953 theorem, and always odd. N(m) is the least order at which some tournament is *not* m-inducible.
- Snapshot
- 2026-09-07 03:53 UTC
- Ledger commit
801848d7