for any strictly hyperbolic system in the form if all Rankine-Hugoniot discontinuities $(u_L,u_R,\sigma)$ entropic for $\eta$ lie on an i-shock curve for some $i$ and the extended Lax admissibility condition holds: $$\begin{aligned}
    \label{extended_lax_admissibility_condition}
    \lambda_{i-1}(u_L) \leq \sigma \leq \lambda_{i+1} (u_R),\end{aligned}$$ where $\lambda_0\coloneqq -\infty$ and $\lambda_{n+1}\coloneqq\infty$. Moreover, we only use the first inequality in and the fact that $\lambda_1(u)\leq \lambda_{i-1}(u)$ for all $u\in\mathcal{V}$ and for all $i>1$.

    Furthermore, note that for *any* strictly hyperbolic system in the form , if $u_R$ and $u_L$ live in a fixed compact set, then there exists $\delta>0$ such that will hold if $\abs{u_R-u_L}\leq\delta$. Similarly, for any strictly hyperbolic system endowed with a strictly convex entropy, all Rankine-Hugoniot discontinuities $(u_L,u_R,\sigma)$ entropic for $\eta$ will locally be in the form $S^i_{u_L}(s)=u_R$ for some $s>0$, and where $S^i_{u_L}$ is the i-shock curve issuing from $u_L$. See [@lefloch_book Theorem 1.1, p. 140] and more generally [@lefloch_book p. 140-6]. For the full Euler system , $(\mathcal{H}3)$ will hold regardless of the size of the shock $(u_L,u_R)$.

-   Fix $B,\rho>0$. Then, for all $u\in\mathcal{V}$ with $\abs{u}\leq B$ and for all $s\in[\rho,B]$, we have $$\begin{aligned}\label{s_shock_strength_comparable_part1}
    (s-\rho)\inf_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))&\leq\eta(u|S_u^1(s))\\
    &=\int\limits_\rho^s \frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))\,\mbox{d}t \leq (s-\rho)\sup_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t)).
    \end{aligned}$$ Note that $$\begin{aligned}
    0<\inf_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))\end{aligned}$$ and $$\begin{aligned}
    0<\sup_{{\substack{u\in\mathcal{V},\hspace{.02in}\abs{u}\leq B\\t\in[\rho,B]}}}\frac{\mbox{d}}{\mbox{d}t}\eta(u|S_u^1(t))<\infty\end{aligned}$$ due 