Gram variance and permutation information in Chollet's permanent inequality
Year: 2026
Type: preprint
Abstract: \section*{Declaration of Interests}This research is not funded by any third party. The author therefore has no competing interests to declare. \section*{Data Availability Statement}No empirical datasets were collected or analyzed in this study. The code, exact inputs and documentation for reproducing the finite verifications will be provided as supplementary material along with the submission of the main article. Online Resource~1 (\texttt{ESM\_1.zip}, v0.2.0) provides Python codefor the finite calculations listed in \texttt{proof\_map.csv}, includingselected curvature inputs in F.1--F.3 used in Theorem~\ref{thm:new-all-mixed-exposure-payment}and the moving-reference comparisonsin Lemmas~\ref{lem:mixed-reference-cap-lower} and~\ref{lem:mixed-reference-cap-upper}.The archive includes the execution instructions, dependency versions, and review-use terms. \smallskip\noindent\textbf{Online Resource 1.}Python source and instructions for reproducing the finite calculations indexed in\texttt{proof\_map.csv}. \section*{Author's Contribution Statement}As the sole author of this article, the author declares that no other authors have explicitly or implicitly participated in this research or should be responsible for the content of this article. Throughout the research process, the author does not consult any human experts for their advice. The author constructs the proof architecture, writes and proofreads the entire article, extends the results, and takes full responsibility. \section*{Disclosure of AI Usage}During the revisions of this manuscript, the author used OpenAI Codex to assist with LaTeX typesetting in the 10-point amsart document class, including equation layout and the correction of compilation errors. The tool (served mostly like “grammarly”) was also used to suggest keywords and Mathematics Subject Classification codes, identify grammatical errors, and propose alternative phrasings to improve the clarity and precision of author-written text. Some grammatical suggestions recommended by Codex are arguably more lucid, better structured, and more understandable to the readers, so the author does not intentionally preserve the human-specific “naturalness of expressions” in some parts of this manuscript. Technical reviews and revisions of selected mathematical arguments were moderately applied (the author expects this ratio to be much less than 10 percents). This included correcting quantifier errors, identifying proof steps requiring further justification and assisting with the corresponding revisions, identifying stronger conclusions supported by the existing arguments, and checking selected key lemmas before submission. No substantial arguments or contents is generated by any sort of LLM. The author evaluated the suggestions incorporated into the manuscript, personally verified the validity of the adopted mathematical revisions, and takes full responsibility for the accuracy and integrity of the final manuscript. We give special thanks to Openai. In the verification process, Chatgpt(Openai) helps developing and checking the supplementary Python code. Before the final submission, Chatgpt assisted with the literature search and drew the author's attention to the 2026 preprint by Qingfeng Li, Qirui Zhao and Wenhao Luo, \textit{Chollet’s Permanent Conjecture Through Order Six via Border Induction}, which reports a proof of Chollet’s conjecture for complex Hermitian positive semidefinite matrices of orders up to six. This assistance tremendously helped clarify the scope of previously reported results and the positioning of the present work. The author takes full responsibility for the selection, interpretation and citation of the literature and for the mathematical content of this article. Finally, the author run an AI-check at https://nlplab.cs.duke.edu/snapjudge/ established by researchers at Duke University. The author uploads the pdf version to the website, and the result shows that 90 percent of the article is "Human Written" and 10 percent is "AI Generated." The report is somehow biased since the website even flagged the table of contents as "AI-Generated." Therefore, the author advice the referees to treat this report cautiously.
Author: Liu, Zhouding
Institution: University of Wisconsin–Madison
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225791102
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