= Solution
Before intersection, the characteristic map is $x=\xi+\rho_0(\xi)^3t$, with derivative $1+3\rho_0^2\rho_0't$. Because the initial profile decreases, higher-<density> characteristics behind move faster than lower-<density> characteristics ahead. Compression therefore forms <shocks>, which absorb more characteristics. The far states remain one and zero, so a <shock> joining their limiting traces has speed tending to $[q]/[\rho]=1/4$.
The precise large-scale interpretation uses $\zeta=x/t$. The rescaled initial transition tends to a step, and entropy comparison for the convex <flux> gives
$$
\boxed{\rho(\zeta t,t)\longrightarrow1\text{ for }\zeta<1/4,\qquad \rho(\zeta t,t)\longrightarrow0\text{ for }\zeta>1/4.}
$$
One way to see the comparison is to choose distant initial points beyond which the profile is within $\delta$ of each far state. Step profiles with those nearby states bracket it; their <shock> speeds tend to $1/4$ as $\delta\downarrow0$, while their fixed initial locations disappear after division by $t$.
A fixed translation of the <shock> can persist. If the deviation from a reference step is integrable, its signed <mass> fixes a phase shift. Without such a tail assumption one should assert convergence on the $x/t$ scale and $S(t)/t\to1/4$, rather than claim that the exact line $x=t/4$ is the unshifted asymptote for every smooth initial transition.
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