Score-Determined Induced Tournament Statistics: an All-Orders Classification

Precise statement

A tournament orients every pair in a round-robin (winner → loser). The score sequence is the sorted win-count list. Reversing a directed 3-cycle never changes scores, so score-equivalent tournaments can look structurally different. Question: Which linear combinations of induced k-subtournament type-counts are score-determined — identical across all tournaments sharing a score sequence, at any host size? Answer: Exactly the linear combinations of degree-multiplicity counts $m_0,\dots,m_{k−1}$, where $m_r$ counts how many of the $k$ chosen vertices have exactly $r$ internal wins. These $k$ functions satisfy one linear relation, so score-determined statistics have dimension $k−1$.

The source statement is reproduced for indexing with attribution. Mathematical correctness requires domain-expert or mechanical review. VibeMath has not independently audited statement fidelity, correctness, priority, or novelty.

What AI did

GPT Sol 5.6

Absolutely everything, i occasionally steered and redirected.

Provider: OpenAI · Prompt public: unknown · Independence: unknown

Verification boundary

unreviewed

Checked by this site on 17 August 2026: the Zenodo reproducibility archive (21959291, Howard-Stanley) is real and carries the manuscript and code. The classification itself was not checked here, no independent review exists, and - flagged plainly - nobody has swept the tournament-theory literature for prior appearances of this question, which for a self-posed problem is the main risk. The disclosure ("absolutely everything, I occasionally steered") is the frankest ai-discovered claim in the catalog.

Correctness: unknown · statement fidelity: unaudited · peer review: none

Timeline

1. 2026-08-14

   Reproducibility Archive for “Score-Determined Induced Tournament Statistics: An All-Orders Classification and a Square-Zero Defect Algebra”

   VibeMathed reports this item as resolved. VibeMath preserves that report as a source assertion and has not independently authored a plain-language mathematical explanation.

Known method families

computation (source-reported)

Source-reported tools: computation.

Independent: unknown · difference confidence: 0

What remains uncertain

VibeMath has not independently audited the mathematical statement, proof, or novelty claim.

• VibeMath has not independently verified the mathematical claim.
• AI-attempt independence and training-data exposure are unknown unless explicitly documented.
• VibeMath has not independently audited the mathematical statement, proof, or novelty claim.