Rank four mod eight has KSp as the unshifted source

In this rank the ungraded fiber Clifford algebra is a quaternionic matrix algebra. Its spinor commutant is ${\mathbb{H}}^{\mathrm{o}\mathrm{p}}$, and the multiplicity objects in [F1] are right-quaternionic bundles. Hence its map is $[E]\mapsto {\mathrm{\Phi }}_V(E{\otimes }_{\mathbb{H}}\mathrm{\Sigma })$, not the ordinary real tensor product by a real line coefficient.

At a point in rank four, a quaternionic line produces the actual quaternionic Hopf generator of ${\widetilde{KO}}^0(S^4)$. Regarding that multiplicity line merely as a four-dimensional real space changes the balanced tensor operation; it does not turn the theorem into a real-line-source formula. The same Clifford residue and balanced evaluation apply in every rank $8m+4$, proving the stated source distinction.