What was already known
The cited best randomized upper bound before this work was 2.5; a universal lower bound approached 2.1126 for at least three candidates.
AI involvement
Nisarg Shah supplied directions, literature connections and proof strategies. He credits Codex using GPT-5.6 Sol with deriving all proofs, and says he checked, simplified and explained them.
The author explicitly credits Codex with deriving every mathematical proof; he supplied directions and strategies and later verified and explained them.
- Supporting source (arxiv.org) — Abstract; Theorem 1; §5; Appendix B.7; AI discussion
- Supporting source (arxiv.org) — Submission history: v1 29 August and v2 6 September 2026
Importance
Narrows the known gap for randomized ordinal metric voting rules.
Limitations
The v2 proof uses a different potential argument from v1. Its exact-rational certificate checks seven polynomial sign problems, while Shah reports checking the full mathematical reduction. The optimum remains unknown.
Technical verification
For v2 Theorem 1, Shah expressly reports verifying every mathematical detail. The repository reports a frozen exact-rational Bernstein certificate for seven polynomial sign problems, with 192 accepted boxes, and archives a separate exact certificate for v1. Machine checking covers the polynomial inequalities, while the author check covers the full argument. Neither checker was rerun here.
Independent checking
author verification of all mathematical details and simplification of proofs
- Checker
- Nisarg Shah
- Scope
- v2 Theorem 1 exact 2.14401485306 distortion guarantee; v1 2.3282 argument checked separately
- Basis
- Reported by the linked checker
Independent checking
frozen exact rational Bernstein subdivision certificate; validates polynomial identities, box coverage, and coefficient signs
- Checker
- certificate_distortion.py in Shah’s repository
- Scope
- v2 Theorem 1 polynomial inequalities for the exact rational 2.14401485306 bound
- Basis
- Reported by the linked checker
archived exact rational Bernstein subdivision certificate for 56 polynomial problems
- Checker
- old/certificate_distortion_v1.py in Shah’s repository
- Scope
- v1 Theorem 1 polynomial inequalities for the 2.3282 mixed-rule bound
- Basis
- Reported by the linked checker
Human understanding
Shah explicitly simplified and explained the potential criterion, telescoping argument, and exact-certificate connection; no unaccounted central mechanism is documented.
Correction and publication history
Narrows the known gap for randomized ordinal metric voting rules.
Read this published revision