A simple brace outside asymmetric products of two trivial braces

Abstract

A full negative answer to Cedo Problem 9 in Oberwolfach Report 51/2019, printed p.3228 (problem 30004311; OWR-17294-011). An explicit nontrivial simple left brace of order 20,000 has additive group C5^4 x C2^5 and metabelian multiplicative group (C5^3 semidirect C2) x (C2^4 semidirect C5), with all multiplicative Sylow subgroups abelian. It is not an asymmetric product of two trivial braces. The five-page research note gives all operations, the group isomorphism, a proof covering every nonzero ideal, and a universal presentation-independent obstruction. Under the full admissible normalized symmetric-cocycle definition, such two-factor products with abelian multiplicative Sylow subgroups have zero primary triple star products. The example instead has (x*x)*x = x^3 nonzero in its 5-primary subbrace. Established radical-ring, matched-product, asymmetric-product, truncated logarithm and cyclic orthogonal methods are credited; no new general construction formalism is claimed. The source archive retains the original proof, three independently authored mathematical audits, extensive current priority and implication audits, four unchanged standard-library verification programs, exact results and per-seed certificates. Finite computation supplements the universal written proofs. A bounded primary-source audit on 17 October 2026 identified no prior full answer or verified prior full implication; exhaustive priority and historical novelty are uncertified. The full bodies of Rump 2020 and 2022 and several final journal versions, including the 2025 Cedo-Okniński paper, were unavailable. Broader consequences are not excluded by abstracts. Citation coverage and the query-defined recent arXiv screen are incomplete; all inspection boundaries are documented. AI tools were used extensively in construction, proof development, computation, adversarial verification, literature research and writing. Unrefereed; AI adversarial reviews are not external human peer review, journal acceptance or formal proof-assistant certification.