Invertibility in the mis\`ere multiverse

TL;DR

This paper proves that every universe in restricted misère play has the conjugate property and characterizes invertible elements, advancing understanding of invertibility in combinatorial game theory.

Contribution

It establishes the conjugate property for all universes and provides a characterization of invertible elements, addressing longstanding challenges in misère game analysis.

Findings

Every universe has the conjugate property.

Characterization of invertible elements in each universe.

Open problems on non-trivial invertible elements.

Abstract

Understanding invertibility in restricted mis\`ere play has been challenging; in particular, the possibility of non-conjugate inverses posed difficulties. Advances have been made in a few specific universes, but a general theorem was elusive. We prove that every universe has the conjugate property, and also give a characterisation of the invertible elements of each universe. We then explore when a universe can have non-trivial invertible elements, leaving a slew of open problems to be further investigated.