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

Ioannis Tsiokos

Abstract: Confluence is usually read as order-independence of reduction, while flatness is usually read as path-independence of transport. This paper argues that their local cores coincide once rewriting is equipped with its missing two-dimensional cells. The relevant object is not the bare reduction graph but a reduction 2-complex whose 0-cells are states, whose 1-cells are one-step rewrites, and whose generating 2-cells are local peaks or, in structured rewriting settings, reachable critical pairs. In that setting, elementary flatness is exactly local confluence. We prove this core equivalence and record the supporting fact that confluence plus normalization yields a unique normal form from a chosen start state. We then show computationally that the corrected local/global picture survives exhaustive finite-ARS stress tests in the terminating regime, curated string-rewriting examples, and a curated left-linear term-rewriting bridge with genuine nonroot overlap. These experiments also show that bare graph-flatness surrogates fail, while reachable critical-pair defects align with the observed local obstruction layer. Finally, explicit before/after examples show that completion can be read as holonomy elimination: the local generator persists while its defect disappears. The full finite terminating left-linear TRS theorem package remains to be written in full generality, but the correct object, local generators, and evidence boundary are now fixed.