To Flatten a Stone with Six Birds: Critical Pairs, Holonomy, and Confluence in Rewriting Systems

In plain words

In a rewriting system, confluence means the order of steps does not matter. In geometry, flatness means the path does not matter. These sound alike, and the paper shows they are the same locally.

Add the missing two dimensional cells to the reduction graph: local peaks, or critical pairs. Then elementary flatness is exactly local confluence.

Completion becomes holonomy elimination. The local generator stays, its defect disappears. Tested on exhaustive finite systems, string rewriting and a term rewriting bridge.

What it shows

• Local confluence equals elementary flatness on the reduction 2 complex.
• Bare graph flatness surrogates fail; critical pair defects track the obstruction.
• Completion read as removing holonomy.

What it does not claim

The fully general terminating left linear term rewriting package is not written out.

Cite

Tsiokos, I. (2026). To Flatten a Stone with Six Birds: Critical Pairs, Holonomy, and Confluence in Rewriting Systems. Zenodo. https://doi.org/10.5281/zenodo.19061345