Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory
Abstract
We develop a theory of chirality and racemization on isotopy classes of finite loops, formulated intrinsically within the loop isotopy groupoid understood in the categorical sense. Motivated by earlier work on quasigroups \cite{InoueQuasiChirality} and by the classical medical paradigm of mirror-related enantiomers, we restrict admissible mirror transitions to those generated by intrinsic, unit-preserving symmetries. Within this framework, racemization is modeled as a two-state dynamics on isotopy classes, with an effective rate determined by the presence of mirror-isotopisms. Our main result shows that this rate vanishes if and only if no loop isotopism exists between a loop and its opposite, providing a structural criterion for chirality. A strengthened variant based on translation-generated symmetries is discussed in the appendix.
Cite
@article{arxiv.2602.21690,
title = {Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory},
author = {Takao Inoué},
journal= {arXiv preprint arXiv:2602.21690},
year = {2026}
}
Comments
11 pages. A loop-internal theory of chirality and racemization on isotopy classes is developed using the loop isotopy groupoid. By restricting admissible mirror transitions to unit-preserving symmetrie (autotopisms), a structural vanishing criterion for chirality is obtained. This work provides a closed refinement of earlier quasigroup-based results