The height-three chromatic overlap: an explicit filtration and its attachments

For every prime p ≥ 5, we construct an explicit eight-stage filtration of $L_2{L}_{K(3)}{\mathbb{S}}_p^{\wedge }$ by the local-sphere layers in the height-three chromatic-splitting pattern. The map from its first stage to the overlap is the canonical unit, and two signed fracture formulas identify all attachments for the chosen local maps and compatibility homotopy. A companion canonical-map theorem further implies that the first height-one attachment is nonzero.