  We present a detailed abundance analysis of 23 elements for a newly
discovered carbon-enhanced metal-poor (CEMP) star, HE 0414-0343, from the
Chemical Abundances of Stars in the Halo (CASH) Project. Its spectroscopic
stellar parameters are Teff = 4863 K, log g = 1.25, vmic = 2.20 km/s, and
[Fe/H] = -2.24. Radial velocity measurements covering seven years indicate HE
0414-0343 to be a binary. HE 0414-0343 has [C/Fe] = 1.44 and is strongly
enhanced in neutron-capture elements but its abundances cannot be reproduced by
a solar-type s-process pattern alone. Traditionally, it could be classified as
"CEMP-r/s" star. Based on abundance comparisons with AGB star nucleosynthesis
models, we suggest a new physically-motivated origin and classification scheme
for CEMP-s stars and the still poorly-understood CEMP-r/s. The new scheme
describes a continuous transition between these two so-far distinctly treated
subgroups: CEMP-sA, CEMP-sB, and CEMP-sC. Possible causes for a continuous
transition include the number of thermal pulses the AGB companion underwent,
the effect of different AGB star masses on their nucleosynthetic yields, and
physics that is not well approximated in 1-D stellar models such as proton
ingestion episodes and rotation. Based on a set of detailed AGB models, we
suggest the abundance signature of HE 0414-0343 to have arisen from a >1.3 Msun
mass AGB star and a late-time mass transfer, that transformed HE 0414-0343 into
a CEMP-sC star. We also find the [Y/Ba] ratio well parametrizes the
classification and can thus be used to easily classify any future such stars.
