Lemma: Diagonal transport

Assume T. Suppose experiments ${\xi }_j$, on an infinite sequence of source scales, satisfy

in Hausdorff, event, and total-variation topology. For every $g\in {\mathcal{P}}^{\uparrow }$, there is a full-tail realizing family for $(gK,gR,\mu )$. The identity case supplies full-tail filling.

Proof. For each source scale $j$, choose an increasing threshold $m(j)$ so that all target scales $m\ge m(j)$ admit its transport and have $b_{{\xi }_j,g,m}\le 2^{-j}$. For $m(j)\le N<m(j^{\prime })$, with $j^{\prime }$ the next source scale, use the transport of ${\xi }_j$ at scale $N$. As $N\to \infty$, the selected $j\to \infty$. Its support error is $a_j(g)+2^{-j}\to 0$, acquisition events converge to $gR$, and its law is ${\mu }_j\to \mu$.