shown in Fig. \[fig:map\_all\]. The set of C$_6$H lines stacked cover a wide range of upper level energies, 64-86 K for lines from the $^2\Pi_{3/2}$ state and 86-113 K for $^2\Pi_{1/2}$ lines, and thus it is possible that their different excitation requirements result in different brightness distributions. However, we note that on the basis of the behaviour of other molecules such as HC$_5$N (see Appendix \[sec:map\_trans\]), such differences are unlikely to be larger than the current angular resolution of $\sim$3$''$.

Observed spatial distributions of carbon chains
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A cursory view of the maps shown in Fig. \[fig:map\_all\] reveals a clear and well-defined ring structure in the seven species studied: the three hydrocarbon radicals C$_2$H, C$_4$H, and C$_6$H, and the four cyanides CN, HC$_3$N, HC$_5$N, and C$_3$N. The maps correspond to the emission at the line center, i.e., at $V_{\rm LSR}=V_{\rm sys}$, and thus indicate how the emission is distributed along a plane perpendicular to the line of sight intersecting the star. The two-dimensional ring structure, together with the emission maps at red- and blue-shifted velocities (not shown in Fig. \[fig:map\_all\]), indicate that the three-dimensional structure of the seven species consists of a hollow spherical shell with a width of 5-10$''$ 