What was already known
Fisher covered tournaments with n=2δ+1. Earlier dense-graph cases restricted the shape of missing edges; this result allows up to n/2 missing edges without such a restriction.
AI involvement
GPT-5-family models found the fixed-target capacity and double-counting argument. The human author initiated, directed, edited and verified the final proof.
GPT-5-family models supplied the decisive dense-case argument under the author’s direction; stronger autonomy is not documented.
Importance
The theorem adds a structurally unrestricted dense case and sharpens the lower bound on possible counterexample size.
Limitations
The n=2δ+3 case and the general conjecture remain open. A Claude audit covered an intermediate draft, while the author reports checking the final proof.
Technical verification
The author expressly reports verifying the proofs of the final dense-case theorem. This is a source-reported complete human proof check; the Claude audit of an intermediate draft is not counted as human or machine checking.
Formal checking
Author verification of final manuscript proofs
- Checker
- Paper author (sole human author)
- Scope
- Theorem 1.1 for n=2δ+2 and its corollary with Fisher’s theorem
- Basis
- Reported by the linked checker
Checking a formal statement does not establish that it matches the original problem.
Human understanding
The author’s check and editing do not separately document a human explanation of the fixed-target capacity mechanism. Few is the evidence default.
Correction and publication history
The theorem adds a structurally unrestricted dense case and sharpens the lower bound on possible counterexample size.
Read this published revision