The coefficient ideal controls the support after restriction

The two inclusions at the initial stage. The inclusion $\mathrm{supp}(\mathcal{I},\mu )\cap S=\mathrm{supp}(C(\mathcal{I},\mu ))\cap S\subseteq \mathrm{supp}(C(\mathcal{I},\mu ){\mathrm{\mid }}_S)$ is [F2, F3]; note that the summands of $C(\mathcal{I},\mu )$ have supports containing $\mathrm{supp}(\mathcal{I},\mu )$, and the restriction can only raise orders. Conversely, if $x\in \mathrm{supp}(C(\mathcal{I},\mu ){\mathrm{\mid }}_S)$, then ${\mathrm{ord}}_x({\mathcal{D}}^i(\mathcal{I}){\mathrm{\mid }}_S)\ge \mu -i$ for all $i\le \mu -1$, so in the notation of [F4] every coefficient $c_{\alpha f}{\mathrm{\mid }}_S$ has order at least $\mu -\mathrm{\mid }\alpha \mathrm{\mid }$; reading the Taylor development of $f$ along $S$ gives ${\mathrm{ord}}_x(f)\ge \mu$ for every local section, i.e. $x\in \mathrm{supp}(\mathcal{I},\mu )\cap S$. Hence the supports agree.

Track the coefficients of the original generators throughout the sequence. Let $J_{r,i}$ be the transform on $S_i$ of $({\mathcal{D}}^r(I){\mathrm{\mid }}_S,\mu -r)$, and write a transformed generator as $f_i={\sum }_{\alpha }{c}_{\alpha ,i}{x}_i^{\alpha }$ in completed adapted coordinates. Initially $c_{\alpha ,0}\in J_{\mathrm{\mid }\alpha \mathrm{\mid },0}$ by [F5]. In a nonempty restricted blowup chart with exceptional equation $a$, $x_{i+1}=x_i\mathrm{/}a$ and $f_{i+1}=a^{-\mu }{\sigma }^{\ast }{f}_i$, hence $c_{\alpha ,i+1}=a^{-(\mu -\mathrm{\mid }\alpha \mathrm{\mid })}{\sigma }^{\ast }{c}_{\alpha ,i}$. This is exactly the transform with mark $\mu -\mathrm{\mid }\alpha \mathrm{\mid }$, so the coefficient membership persists. At each stage keep the normal generators $x_i$ fixed while adapting the center by a change of parameters along $S_i$ in the completed coefficient ring. This is possible because the center is a regular subscheme of $S_i$. Such a change transports the coefficient ideals and does not alter the stated membership; no change mixing normal and tangent parameters is needed. A point in the support of the transformed coefficient sum lies in every $J_{r,i}$ support by [F1]; the persistent coefficient membership therefore gives $\mathrm{ord}(c_{\alpha ,i})\ge \mu -\mathrm{\mid }\alpha \mathrm{\mid }$, and the Taylor expansion gives $\mathrm{ord}(f_i)\ge \mu$. This proves the inclusion from restricted coefficient support to ambient support. Conversely, [F2] gives equality of ambient supports for the transformed $C(I,\mu )$ and $I$; restriction raises order by [F3], so ambient support on $S_i$ lies in the restricted coefficient support.

The converse direction. Conversely, let $(S_i)$ be a multiple test blow-up of $C(\mathcal{I},\mu ){\mathrm{\mid }}_S$ with centers $D_i\subseteq S_i$. Lifting the center $D_i$ to $X_i$ and blowing up $X_i$ there is legitimate because $D_i\subseteq \mathrm{supp}[C(\mathcal{I},\mu ){\mathrm{\mid }}_S{]}_i=\mathrm{supp}({\mathcal{I}}_i,\mu )\cap S_i\subseteq \mathrm{supp}({\mathcal{I}}_i,\mu )$ by step 2.1, and its SNC position with the restricted boundary, together with the parameter equations defining $S_i$, makes it SNC with the ambient boundary under the omission convention of [F3]; the blow-up of $X_i$ at $D_i$ restricts to the blow-up of $S_i$ at $D_i$ along the strict transform, and the chart computation of [F4] read in this direction shows that the equality of supports of step 2.1 persists at every stage. Hence $(S_i)$ defines a multiple test blow-up $(X_i)$ of $(\mathcal{I},\mu )$ with all centers contained in the strict transforms of $S$, which is clause (3).