A joint exposure law over all assets reproduces each pairwise systematic covariance on its diagonal coupling, and the induced portfolio quadratic form equals the second moment of the aggregated loading \(B_w=\sum _i w_iB_i\). Coherence is therefore automatic once the object is a single joint law, and the quadratic form is positive semidefinite for that reason alone. This is the honest route around ??prop:pairwise-not-psd.

With a centred isotropic factor and residuals cross-orthogonal to the exposures, the covariance of factor-driven returns equals the systematic covariance of the exposure laws, and the residual cross term vanishes. This is what licenses reading \(\kappa _{ij}\) as a return covariance rather than only as a loading-space inner product; without the orthogonality hypothesis the whole-return claim does not follow.