Final answer:
The principal diagnosis is metastatic carcinoma of the lung to the brain.
Step-by-step explanation:
The principal diagnosis in the case of a 65-year-old patient with a history of lung cancer, ataxia, syncope, and a fractured arm as a result of falling, who undergoes a closed reduction of the fracture and a complete workup for metastatic carcinoma of the brain, and is found to have metastatic carcinoma of the lung to the brain and undergoes radiation therapy to the brain is metastatic carcinoma of the lung to the brain.