For every n ≥ 1 and prime $p\gt n+1$, we construct a $2^n$-stage ordered filtration of $L_{n-1}L_{K(n)}S_p^\wedge$ with the classical chromatic-splitting cofibers. The map from the first stage to the target is the canonical localization unit.

For every prime p ≥ 5, we construct an explicit eight-stage filtration of $L_2L_{K(3)}\mathbb S_p^\wedge$ 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.

For every prime p ≥ 5, the canonical map $L_0L_{K(3)}S\to L_0L_{K(2)}L_{K(3)}S$ for the sphere spectrum S is nonzero on π₋₃. Consequently, the height-three strong chromatic splitting formula is false in this range, even as an equivalence of underlying $E(2)$-local spectra without specified summand maps.

At the prime three and height three, the chromatic overlap cannot be constructed from the rational, height-one, and height-two local spheres by finitely many sums, shifts, cofibers, and retracts in the category of $E(2)$-local modules over the derived 3-complete sphere. This gives a negative answer to the ordinary finite-assembly question for chromatic overlaps.

For the derived p-completion of the sphere, the canonical weak chromatic splitting map has no homotopy retraction at height p for every prime p ≥ 5, and at height $p+1$ for every prime p ≥ 7. The classical sphere product $\beta_1^{(p-1)^2}$ gives a nonzero kernel class in both ranges.