$\omega_+>\omega_-$ and is finite in a right neighbourhood of $\omega_+$, which by convexity of $\kappa$ implies $0<\kappa'(\omega_+)<\infty$.

#### Intrinsic area measure.

Thanks to , the process $$\begin{aligned}
        \mathcal{M}(n):=\sum_{|u|=n}\chi_u(0)^{\omega_-},\quad n\geq 0
    \end{aligned}$$ is a uniformly integrable martingale, see Lemma 2.4 in [@BBCK16]. Its terminal value $\mathcal{M}:=\lim_{n\to\infty}\mathcal{M}(n)$ is called the *intrinsic area* of $\mathbf{X}$ and we point out that, by construction, it does not depend on $\alpha$.

Endowed with the distance $d(\ell,\ell'):=\exp(-\sup\{n\geq 0:\ell(n)=\ell'(n)\})$, $\partial\mathbb{U}$ is a complete metric space. The *intrinsic area measure* $\mathcal{A}$ is then the unique measure on $\partial\mathbb{U}$ such that the mass of the subsets of leaves having ancestor $u\in\mathbb{N}^n$ at generation $n\geq 0$ satisfies $$\begin{aligned}
        \mathcal{A}\left(\left\{\ell\in\partial\mathbb{U}:\ell(n)=u\right\}\right)=\lim_{k\to\infty}\sum_{|v|=k}\chi_{uv}(0)^{\omega_-}.
    \end{aligned}$$ Note that the total mass of $\mathcal{A}$ is $\mathcal{M}$. Denoting by $\zeta_\ell$ the height of $\ell\in\partial\mathbb{U}$, that is the time at which the lineage of this leaf goes extinct in terms of the growth-fragmentation’s time, we define the area of the ball with radius $t\in\mathbb{R}_+$ centered at the origin by $$\begin{aligned}
        A:t\mapsto\mathcal{A}\left(\left\{\ell\in\partial\mathbb{U}:\zeta_\ell\leq t\right\}\right).
    \end{aligned}$$ The increasing function $A$ will be the object of interest of the rest of the paper. It satisfies the following Markov-branching type property that we shall use all along this work.

\[Lemma Markov-branching property of A\] For 