Chirality and Racemization on Isotopy Classes of Quasigroups

TL;DR

This paper introduces a gauge-invariant Markov model for studying chirality and racemization in quasigroups, linking isotopy classes to dynamical stability and symmetry properties, with a concrete example of order 7.

Contribution

It develops a novel dynamical framework for chirality in quasigroups using gauge theory and isotopy, revealing conditions for chiral stability and symmetry-based classifications.

Findings

Convergence to racemic equilibrium with symmetric mirror transitions

Chiral stability characterized by zero transition rate $k([Q])=0$

Existence of structurally chiral quasigroup classes of order 7

Abstract

We develop a structural and dynamical theory of chirality for quasigroups formulated at the level of isotopy classes. Interpreting isotopy as a gauge symmetry of re-coordinatization and mirror parastrophy as handedness reversal, we introduce a gauge-invariant continuous-time two-state Markov model in which transitions occur only between a quasigroup and its mirror. We prove that this dynamics descends to the isotopy quotient, yielding a reduced generator governed by a single class-dependent rate $k([Q])$. Symmetric mirror transitions lead to convergence toward a racemic equilibrium, whereas the vanishing condition $k([Q])=0$ characterizes dynamical chiral stability. By restricting admissible transitions to those generated by intrinsic symmetries, we show that $k([Q])=0$ is equivalent to the absence of mirror-isotopisms. A concrete example of order $7$ demonstrates the existence of structurally chiral quasigroup classes.