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.