**Core Concept**
Brown-Séquard syndrome is a clinical condition characterized by hemisection of the spinal cord, resulting in ipsilateral loss of motor function and proprioception, and contralateral loss of pain and temperature sensation. This condition occurs due to traumatic injury to the spinal cord, which can be caused by various factors such as falls, motor vehicle accidents, or sports-related injuries.

**Why the Correct Answer is Right**
The loss of proprioception and motor function on the ipsilateral side in Brown-Séquard syndrome is due to the damage to the lateral corticospinal tract and the posterior columns of the spinal cord. The lateral corticospinal tract is responsible for motor function, while the posterior columns (fasciculus gracilis and fasciculus cuneatus) are responsible for proprioception and vibration. The damage to these tracts results in the loss of motor function and proprioception on the same side as the injury.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because the loss of pain and temperature sensation is contralateral, not ipsilateral, in Brown-Séquard syndrome.
* **Option B:** This option is incorrect because the loss of vibration and position sense is due to damage to the posterior columns, which is ipsilateral in Brown-Séquard syndrome.
* **Option D:** This option is incorrect because the loss of motor function is ipsilateral, not contralateral, in Brown-Séquard syndrome.

**Clinical Pearl / High-Yield Fact**
In Brown-Séquard syndrome, the loss of proprioception and motor function on the ipsilateral side is due to the damage to the lateral corticospinal tract and the posterior columns of the spinal cord. It is essential to remember that the loss of pain and temperature sensation is contralateral, and not ipsilateral, in this condition.

**Correct Answer:** B. Proprioception