The Low-Component Cases for Total Cut Complexes of Disconnected Graphs

For total cut complexes of disconnected graphs, we resolve the three previously open low-component cases k = d, d + 1, and d + 2 under the stated component hypotheses. Together with the previously known high-component range and an independent argument for the required d = 2 case, this completes the wedge-of-spheres formula for every k ≥ d. The proof uses explicit triangular fillings, Alexander duality, joins, deformation retractions, and a nerve-lemma argument. Status: Public Beta v0.1; internally verified candidate proof; external mathematical review pending.