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Chirality and Racemization on Isotopy Classes of Loops: A Groupoid-Based Structural Theory

Takao Inoué

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.

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.
Paper Structure (13 sections, 6 theorems, 22 equations)

This paper contains 13 sections, 6 theorems, 22 equations.

Table of Contents

  1. Introduction
  2. Loop Isotopy Groupoid and Mirror Loops
  3. Remark on terminology.
  4. Two-State Dynamics and Class Functions
  5. Structural Chirality via Autotopisms
  6. Examples
  7. Groups.
  8. Non-associative loops.
  9. Comparison with the quasigroup case.
  10. Conclusion
  11. Strengthening: Translation-Generated Variant
  12. Translations and inner mappings
  13. Stronger admissibility

Key Result

Lemma 3.5

The operator $\mathcal{L}$ preserves class functions.

Theorems & Definitions (23)

  • Definition 2.1: Loop isotopy groupoid
  • Definition 2.2: Opposite loop
  • Definition 2.3: Mirror-isotopism
  • Definition 3.1: Class function
  • Remark 3.2
  • Remark 3.3
  • Definition 3.4: Two-state generator
  • Lemma 3.5
  • proof
  • Proposition 3.6: Descent
  • ...and 13 more