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arXiv · 2603.02243

Chirality and Racemization on Isotopy Classes of Quasigroups

Takao Inoué
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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.

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Takao Inoué. 2026-02-21. Chirality and Racemization on Isotopy Classes of Quasigroups. https://arxiv.org/abs/2603.02243

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