Residual projection

For a compact joint cutoff $\zeta$, let

The compactly supported pushforward ${\pi }_{\ast }(\zeta T)$ exists. For a declared joint cone define

Because $\pi$ is a submersion, $d{{\pi }_w}^T\eta \ne 0$ for $\eta \ne 0$.

Proper-support pushforward is continuous in the normal topology with its projected cone bound BDH (Theorem 6.3). Thus a verified joint normal limit can be projected. To meet a required output cone one must also check

Projection of an existing joint distribution does not construct an absent joint product. It does not supply global overlaps, physical normalization, an interacting identity, or membership in a specified quantum carrier.