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Precise statement
In metric social choice, voters rank candidates by distance in an unknown metric space, while a randomized voting rule must use only these rankings. The paper introduces random-size stable lotteries and proves that, by mixing a suitably chosen random-size stable lottery with Integrated Veto, one obtains a randomized voting rule with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$. The proof combines infinite-dimensional conic linear-programming duality, heuristic nonlinear optimization, and exact rational verification using polynomial nonnegativity in the Bernstein basis.
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-5.6 Sol; Claude Opus 5.0
GPT-5.6 Sol derived all mathematical proofs in the paper from research directions, literature connections, proof and search strategies, and inspiration supplied by Nisarg Shah. It autonomously introduced stable-lottery ingredients, developed progressively stronger bounds, and derived the proofs leading to $2.3282$. Shah then generalized one proposed lottery to random-size stable lotteries, guided the search over distributions, verified all final mathematical details, and rewrote and simplified the exposition with GPT-5.6 Sol and Claude Opus 5.
The disclosure is in the paper itself, not only in this entry: "All the proofs in this document were obtained using GPT-5.6-Sol with guidance from the author."
Unreviewed. The author states that he verified all final mathematical details; by this site's ladder an author's own check does not move the tier, however expert, and Shah is among the leading researchers on metric distortion. The bound $11641/5000$ rests, per the abstract, on an exact rational verification via polynomial nonnegativity in the Bernstein basis, which is checkable in principle but has not been re-run here. No referee and no formalization.
The paper proves that there exists a randomized voting rule using only ordinal rankings with metric distortion at most $11641/5000=2.3282$. This improves the previous best upper bound of $2.5$ and closes about $44\%$ of the gap to the known asymptotic lower bound of approximately $2.1126$. It does not determine the optimal randomized metric distortion: for $m\ge4$, the exact optimum and its asymptotic limit remain open.
Known method families
argument (source-reported)
Source-reported tools: argument.
Independent: unknown · difference confidence: 0
What remains uncertain
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VibeMath has not independently audited the mathematical statement, proof, or novelty claim.