Splenic Infarct with Systemic Emboli: A splenic infarct can lead to systemic emboli, which may cause a cerebral vascular accident (CVA) presenting with right lateral gaze and left-sided weakness. The profound anemia (Hgb 4) could be due to bleeding from the infarcted spleen.

Cerebral Vasculitis: Inflammation of the blood vessels in the brain could lead to symptoms mimicking a CVA, including weakness and gaze abnormalities. Systemic vasculitis could also involve the spleen, leading to infarction.
Hemorrhagic Stroke: A hemorrhagic stroke could present with sudden onset of weakness and gaze abnormalities. The profound anemia could be due to bleeding from the stroke itself.
Severe Sepsis: Sepsis can cause a systemic inflammatory response, leading to organ dysfunction, including cerebral and splenic involvement. The anemia could be due to sepsis-induced bleeding or hemolysis.

Aortic Dissection: An aortic dissection can lead to splenic infarction and cerebral emboli, presenting with the described symptoms. Missing this diagnosis could be fatal.
Pulmonary Embolism: A large pulmonary embolism can cause systemic hypoxia, leading to cerebral and splenic hypoperfusion. The anemia could be due to bleeding from anticoagulant therapy or underlying conditions.
Cardiac Tumor (e.g., Myxoma): A cardiac tumor can embolize to the brain and spleen, causing infarction and presenting with the described symptoms.

Thrombotic Thrombocytopenic Purpura (TTP): TTP is a rare disorder characterized by thrombocytopenia, microangiopathic hemolytic anemia, and organ dysfunction. It could present with cerebral and splenic involvement.
Antiphospholipid Syndrome: This autoimmune disorder can cause arterial and venous thrombosis, leading to splenic infarction and cerebral vasculature involvement.
Sickle Cell Disease: In patients with sickle cell disease, splenic infarction and cerebral vasculature involvement can occur due to sickling of red blood cells.