Theorem: Full-tail diagonal realization after transformation

Let (eq:fulltail) hold for calibrated sources and fix $g\in {\mathsf{P}}_{+}^{\uparrow }$. There are experiments on the target substrate at every sufficiently large scale $n$, with

If two source families have matched skeletons and differ only on payload footprints contained in $(K^{\prime }{)}_{{\rho }_j}$, where ${\rho }_j\to 0$, their transformed target families can use matched skeletons and differ only in a neighborhood of $g{K}^{\prime }$ whose radius tends to zero.

Proof. For each fixed source $j$, Theorem thm:T supplies a threshold beyond which its fine error is at most $1/j$. Choose strictly increasing integers $M_j$, with $M_j\ge j$, so that this statement holds for every $n\ge M_j$.

For $n\ge M_1$, define

The maximum is finite, and $j(n)\to \infty$. Transport source $j(n)$ to target scale $n$. Then

which tends to zero. The acquisition estimate is identical in form. Exact instrument preservation gives

For matched modifications, choose each $M_j$ to work for the common skeleton and its changed-footprint union. The target changed footprints lie within

of $g{K}^{\prime }$. This radius tends to zero. Payload-independent transport supplies the common target skeleton.