Sep 2026· Zenodo (CERN European Organization for Nuclear Research)
Geometric and Algebraic Topology
Abstract
The Collision Cover Walkers, Shards, and Finite Sheets over the Fivefold Algebra C.J Tully Independent Researcher Engineer Australia https://orcid.org/0009-0007-5661-7332 https://doi.org/10.5281/zenodo.22997306 Abstract The Broken Mirror Theorem forbids a golden diagonal in any associative lift of A = C[Z5]. What remains is a covering graph of the Cayley graph of Z5, with a marked class of edges that record self-collisions. A Walker is a point of this cover. A shard is a fibre of closed walks that return in fold and disagree in history. We classify the finite-sheeted covers, compute the collision cocycle on short words and on cyclic families, and isolate the invariants that distinguish shards. The fivefold product is never deformed. The cover is the memory of the walk.
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