# Arbitrary comparison tournaments with Gaussian moments

Public research note prepared on 14 September 2026 with OpenAI ChatGPT at Tito's request. Historical novelty is unconfirmed and the note has not been peer reviewed.

The theorem constructs arbitrary finite pairwise comparison patterns among real analytic, strongly log-concave distributions arbitrarily close to a standard normal, with any prescribed finite number of Gaussian moments, common median and mode, and optional finitely many prescribed Gaussian quantiles.

## Full content

- [HTML article and complete proof](https://gaussian-comparison-tournaments.titoreinaldo.chatgpt.site/)
- [Full note in Markdown](https://gaussian-comparison-tournaments.titoreinaldo.chatgpt.site/gaussian_comparison_discovery.md)
- [Exact Python verifier](https://gaussian-comparison-tournaments.titoreinaldo.chatgpt.site/verify_gaussian_tournaments.py)

The Markdown file contains the entire theorem, proof, explicit example, limitations, and prior-work references. The verifier uses only the Python standard library. The construction's advantages may be arbitrarily small. It does not establish that all infinitely many Gaussian moments can be matched by distinct distributions.