Stroke: The combination of headache, left arm numbness, and left arm weakness is highly suggestive of a stroke, particularly if the symptoms are sudden in onset. The headache could be indicative of a hemorrhagic stroke, while the numbness and weakness are classic signs of cerebral ischemia affecting the motor and sensory areas of the brain.

Transient Ischemic Attack (TIA): Often referred to as a "mini-stroke," TIAs have similar symptoms to a stroke but are temporary, usually lasting less than 24 hours. The symptoms of headache, arm numbness, and weakness could be indicative of a TIA, which is a warning sign for a future stroke.
Migraine with Aura: Some migraines are accompanied by an aura, which can include neurological symptoms such as numbness, weakness, and visual disturbances. If the headache is severe and has features of a migraine (e.g., unilateral, throbbing, accompanied by nausea or sensitivity to light), and the arm symptoms are part of the aura, this could be a consideration.
Multiple Sclerosis (MS): MS is an autoimmune disease that affects the brain and spinal cord, leading to a wide range of neurological symptoms, including numbness, weakness, and vision problems. While less common, MS could present with these symptoms, especially if there are other signs such as optic neuritis or previous episodes of neurological dysfunction.

Subarachnoid Hemorrhage: A severe headache (often described as "the worst headache of my life") accompanied by sudden neurological deficits could indicate a subarachnoid hemorrhage, which is bleeding into the space surrounding the brain. This is a medical emergency.
Brain Tumor: Although less likely, a brain tumor could cause these symptoms, especially if the tumor is located in an area that affects motor and sensory function, such as the frontal or parietal lobe.
Cervical Disc Herniation: A herniated disc in the cervical spine could compress nerves leading to the arm, causing numbness and weakness. If the herniation is significant, it could also cause spinal cord compression, leading to more severe neurological deficits.

Arteritis (e.g., Giant Cell Arteritis): Inflammation of the arteries, particularly those supplying the brain, can lead to headache and neurological symptoms. This is more common in older adults.
Mitochondrial Myopathies: Rare genetic disorders that affect the mitochondria, leading to muscle weakness and other neurological symptoms, could potentially present with these symptoms, although they would typically be part of a broader clinical picture.
Neurosyphilis: A late manifestation of syphilis infection, neurosyphilis can cause a wide range of neurological symptoms, including headache, numbness, and weakness, although it is relatively rare in areas with access to modern healthcare.