Final answer:
A star on the horizontal branch of the H-R diagram is undergoing nuclear fusion in the core, synthesizing helium into heavier elements.
Step-by-step explanation:
If a star is on the horizontal branch of the H-R diagram, the correct statement is: it is undergoing nuclear fusion in the core. At this stage, a star, like the Sun, has finished its prolonged adolescence on the main sequence. The star now fuses helium into carbon and oxygen in its core, resulting in a brief period of stability. It is not in the protostar stage (option a), as that occurs before the main-sequence stage. It is also not a red giant (option c), which it becomes after the helium in the core is exhausted and the star expands significantly. Lastly, it is not in the main sequence stage (option d) as it has already evolved away from that phase.