A counterexample to ideal closure of property-(s) elements in skew braces

Abstract

This concise research note gives an explicit counterexample to Colazzo's unrestricted question asking whether the elements with property (s) in a skew left brace form an ideal. On the abelian additive group Z x Z/4Z, a standard semidirect multiplication gives exactly S = Z x {0,1}. The element (0,1) has both defining indices equal to two; its additive double has both indices infinite. The proof is self-contained over all integers and also shows failure of lambda-invariance. The finite brace, semidirect construction and high-bit action have established antecedents in Bachiller, Smoktunowicz-Vendramin and Dietzel. Prior finite-conjugacy characterizations and simple infinite-module variations of Di Matteo-Ferrara and Mondal-Shumyatsky-Trombetti-Yadav are credited. Alper Ferudun's unrefereed October 11, 2026 preprint, DOI10.5281/zenodo.23292254, gives exactly the same counterexample. GitHub records our original PR156 on September 30, 2026 at12:24:37UTC; its mathematical proof is unchanged. This formal preprint is not backdated. We describe these as contemporaneous separate discoveries, meaning separate disclosures in the same period. Mutual independence, literally simultaneous release, historical novelty, first and exclusive priority are not certified. The bounded priority audit found no earlier explicit negative answer in the complete bodies read, but several publisher bodies were unavailable; the exact gaps and available versions are recorded. The deposit contains the manuscript PDF and a support archive with standalone LaTeX, exact standard-library controls, ordinary and optimized outputs, deliberately broken negative controls, source locators, chronology evidence and priority audits. Finite computations supplement the written infinite-coset proof and cannot prove infinitude. AI tools were used extensively in solving, drafting, source comparison, reproduction and adversarial verification. This preprint is unrefereed and has not undergone conventional human peer review. The author is responsible for its contents.