Theorem 13.26.7279D427FE3F2

Proof. Suppose that $p=P_\theta [p]_{\mathcal I}$. By theorem 13.24.3, there are analytic ordinary models $r_ j$ in background $\theta $ with $d_\theta (r_ j,p)\to 0$ and a common positive lower-volume bound. The transports $a_ j=T^\rho _{\eta _ j,\varepsilon _ j}(r_ j)$ have analytic singularities: the convex coefficients are rational, and ordinary projection preserves the singularity type of an analytic potential by proposition 18.4.8. The estimate (13.26.5) gives $d_{\eta _ j}(a_ j,p_ j)\to 0$.

Conversely, assume that such analytic approximants exist, and put $\delta =\underline{\operatorname {vol}}_\theta (p)$. By theorem 13.26.3, the lower volume of $p_ j$ is at least $c_{\varepsilon _ j}^ n\delta \geq 2^{-n}\delta $. The upper-volume deficit from $a_ j$ to $\max (a_ j,p_ j)$ is at most $d_{\eta _ j}(a_ j,p_ j)$, which tends to zero. Applying lemma 13.18.1 with $b=2$ in background $\eta _ j$, and then addition, gives

for all sufficiently large $j$.

Put $\beta _ j=\theta +\varepsilon _ jh$. Lemma 13.26.1, with $\beta =\eta _ j$ and $\gamma =\beta _ j$, and projection invariance in background $\beta _ j$ imply $d_{\beta _ j}(a_ j,p_ j)\to 0$. Also $d_{\beta _ j}(p_ j,p)\to 0$. Indeed, (13.26.6) bounds $\max (p_ j,p)$ above by $c_{\varepsilon _ j}P_{\widehat\theta _{\varepsilon _ j}}[p]$, and $c_{\varepsilon _ j}\widehat\theta _{\varepsilon _ j}\geq \beta _ j$ by the form inequality in the proof of theorem 13.26.3. Hence the upper volume of this maximum in background $\beta _ j$ is at most $c_{\varepsilon _ j}^ n\overline{\operatorname {vol}}_{\widehat\theta _{\varepsilon _ j}}(p)$, which tends to $\overline{\operatorname {vol}}_\theta (p)$. The upper volumes of $p_ j$ and $p$ in background $\beta _ j$ lie between $c_{\varepsilon _ j}^ n\overline{\operatorname {vol}}_\theta (p)$ and this bound, so they have the same limit.

Project $a_ j$, $p_ j$ and $p$ to ordinary models in background $\beta _ j$. Their lower volumes have a common positive bound, and their upper volumes are bounded by $v_+(\Omega )$, since $\beta _ j\leq \Omega $. The three-point estimate (13.22.6) in background $\beta _ j$ therefore gives $d_{\beta _ j}(a_ j,p)\to 0$, with constants independent of $j$. Now theorem 13.25.5, applied with the backgrounds $\beta _ j$, shows that $p$ is an ideal model and gives the stated approximation in the original background.

For another choice of transport, theorem 13.26.3 shows that the two reference models in background $\eta _ j$ have volume comparison tending to zero. The lower-volume bound just proved and the same uniform three-point estimate transfer the convergence condition in both directions. This proves independence of the choice. $\square$