Homology and higher-order intermediate categories for proper abelian subcategories

Without needing a $t$-structure, we establish natural long exact homology sequences on finite extension windows with $\mathrm{\ell }+1$ layers of extension-closed proper abelian subcategories, assuming vanishing of negative self-extensions in degrees $1,\dots ,\mathrm{\ell }+1$, and prove uniform sharpness of this bound. In the Krull-Schmidt setting, we classify intermediate subcategories stable under taking pure homology layers by compatible sequences.