GOLDWALK A Marked-Walk Engine for Cyclic Groups

Abstract

GOLDWALK A Marked-Walk Engine for Cyclic Groups Author: C. J. Tully ORCID: 0009-0007-5661-7332 https://doi.org/10.5281/zenodo.23021217 Version: 0.1 Date: 29 September 2026 License: CC BY 4.0 Resource type: Technical note / Software documentation --- Abstract GOLDWALK is a software and theorem-card package for enumerating closed walks on the Cayley graph of a cyclic group \mathbb{Z}_n with self-steps marked. Given \mathbb{Z}_n, a start fold, and a length, the engine returns every closed word, the gold-length of each word, the shard table, and the generating-function coefficients. The package includes a public test oracle: for n=5, start 1, and lengths 2 through 6, the totals must equal 4, 12, 52, 204, 820. It also ships two theorem-card results: the permutation period of the constant itinerary a^\infty on an m-sheet cover, and the Broken Mirror no-go for associative diagonal-only twists of \mathbb{C}[\mathbb{Z}_n]. This document describes the engine, the public test promise, the theorem card, and the intended use. Keywords: marked walks, cyclic groups, Cayley graphs, group algebras, closed walks, generating functions, shard tables, geometric deep learning, graph neural networks, reproducibility.