Homological Calculations for Finite Toy Models of Rewriting Systems

Abstract This paper studies two explicitly defined finite chain complexes intended as toy models for rewriting systems. For a four-cycle complex, we prove that its first homology group is isomorphic to Z. For a three-cycle complex equipped with a specified two-cell, we prove that its first homology group vanishes. We also compute the first integral cohomology group of the four-cycle complex using the universal coefficient theorem. The paper distinguishes these proven algebraic results from possible categorical, sheaftheoretic, and operational extensions. In particular, it does not claim to construct a general epistemic category, a complete Squier resolution, a Verdier duality theorem, or an operational self-healing mechanism. The vanishing of first homology in the three-cycle model is an algebraic statement about cycles and boundaries; it does not by itself imply termination, confluence, or algorithmic recovery