PomeCorollary 11.46.22.7C248BB2EDD2 The pullback of $U$ has the pulled-back ordinary type and relative envelope-class membership. Its residual retains the proved regularity across $X_0\times \{ o\} $, and its finite Monge–Ampère mass is $\binom Nm V_0$. The background $H$ is nonclosed and $v_-(H)=0$.
Corollary 11.46.227C248BB2EDD2
Proof. The pullback of $U$ obeys
\[ \mathrm{MA}_ H(U)=\binom Nm\operatorname {pr}_0^*(h_0^ m) \wedge \operatorname {pr}_ Y^*(q\kappa ^ k), \qquad \int _ X\mathrm{MA}_ H(U)=\binom Nm V_0. \]
Indeed the bounded truncation $\max \{ U,-j\} $ is a pullback. The bounded corrected-product expansion of proposition 5.28.2, lemma 5.28.4 leaves only the displayed binomial term by dimensions; multiplication by $1_{\{ U>-j\} }$ and increasing $j$ give the exact formula and finite mass. The residual is the pullback of $R$, so its proved regularity extends across $X_0\times \{ o\} $. It is ordinarily equivalent to the pullback of $u_\alpha $ and belongs to the corresponding relative stretching class by the same explicit $v_ t$.
Also $v_-(H)=0$: bounded $h_0$-psh potentials with masses tending to zero have pullbacks of mass $\binom Nm$ times those masses. The background is genuinely nonclosed, since a closed positive $h_0$ would have all bounded masses equal to $V_0>0$ by Stokes. The zero-lower-mass conclusion retains the stated factor hypothesis; the calculation does not determine an individual density threshold. $\square$
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