The coefficient ideal is equivalent to the marked ideal

The multiple test blow-ups coincide. By [F2], a multiple test blow-up of the coefficient sum is a simultaneous multiple test blow-up of its summands; since the $i=0$ summand is $(\mathcal{I},\mu )$, every such sequence is a multiple test blow-up of $(\mathcal{I},\mu )$. Conversely, [F3] shows that every multiple test blow-up of $(\mathcal{I},\mu )$ is a multiple test blow-up of every derivative summand, hence of their sum. Thus the two families coincide in every characteristic.

Equal supports at every stage. At any stage $k$, [F4] shows that the support of $({\mathcal{I}}_k,\mu )$ is contained in the support of every derivative summand. Their intersection therefore contains $\mathrm{supp}({\mathcal{I}}_k,\mu )$, using the transformed-derivative inclusion in [F3]; the reverse inclusion follows because the $i=0$ summand is exactly $({\mathcal{I}}_k,\mu )$. By [F2], this intersection is the support of the coefficient sum's controlled transform. Hence the two supports agree at every stage, and together with step 1.1 and [F5] this proves $C(\mathcal{I},\mu )\simeq (\mathcal{I},\mu )$ in every characteristic.