state our results, we first give a rigorous definition of the involved objects. Call $$S=\left\{ \mathbf{s}=(s_{1},s_{2},\ldots ):s_{1}\geq s_{2}\geq \ldots \geq
0;\sum_{i\geq 1}s_{i}\leq 1\right\} ,$$ and endow it with the topology of pointwise convergence.

A Markovian $S$-valued process $(F(t),t\geq 0)$ starting at $(1,0,\ldots)$ is a ranked self-similar fragmentation with index $\alpha \in \mathbb{R}$ if it is continuous in probability and satisfies the following fragmentation property. For every $t,t^{\prime }\geq 0$, given $F(t)=(x_{1},x_{2},\ldots )$, $F(t+t^{\prime })$ has the same law as the decreasing rearrangement of the sequences $x_{1}F^{(1)}(x_{1}^{\alpha }t^{\prime }),x_{2}F^{(2)}(x_{2}^{\alpha
}t^{\prime }),\ldots $, where the $F^{(i)}$’s are independent copies of $F$.

By a result of Bertoin [@bertsfrag02] and Berestycki [@berest02], the laws of such fragmentation processes are characterized by a $3$-tuple $(\alpha ,c,\nu )$, where $\alpha $ is the index, $c\geq 0$ is an “erosion” constant, and $\nu $ is a $\sigma $-finite measure on $S$ that integrates $\mathbf{s}\mapsto 1-s_{1}$ such that $\nu (\{(1,0,0\ldots )\})=0$. Informally, $c$ measures the rate at which fragments melt continuously (a phenomenon we will not be much interested in here), while $\nu $ measures instantaneous breaks of fragments: a piece with size $x$ breaks into fragments with masses $x\mathbf{s}$ at rate $x^{\alpha }\nu (\mathrm{d}\mathbf{s})$. Notice that some 