A Four-Point Counterexample to Excision for Directed Cubical Homology of Closure Spaces

Bubenik and Milićević defined cubical singular homology theories of Čech closure spaces from an interval and either the product or inductive product. Excision is known for the product theories and was left open for the three inductive theories. We give a four-point counterexample for the directed interval J₊. For X = J₊ ⊡ J₊, an interior cover {U,V} has H_1(J₊,⊡)(V,U ∩ V; ℤ) ≅ ℤ, generated by the difference of the two coordinate edges. Under inclusion into (X,U), this class is the boundary of the identity square. Hence the excision map is not injective. The calculation is integral, exact, and accompanied by two independently implemented finite-chain checks. This is an unrefereed note and makes no absolute priority claim. Research status: Unrefereed preprint released for independent mathematical scrutiny. Publication on Zenodo does not constitute peer review. AI-use disclosure: AI-assisted tools supported literature search, computation, proof auditing, and manuscript preparation. The author remains responsible for all claims and the final text.