**Core Concept**
In the context of spinal cord injuries, the clinical presentation of isolated loss of proprioception and vibratory sensation on one side of the body is characteristic of damage to the posterior columns of the spinal cord. This is due to the specific tracts involved in these sensory modalities being located in the posterior white matter of the spinal cord.

**Why the Correct Answer is Right**
The posterior columns of the spinal cord are responsible for transmitting proprioceptive and vibratory information from the body to the brain. The fasciculus gracilis and fasciculus cuneatus are the specific tracts involved in this process. Damage to these tracts, as in the case of a unilateral spinal cord injury, results in the loss of proprioception and vibratory sensation on the same side of the body. This selective loss of function is due to the decussation of these tracts at the medulla oblongata, where the sensory information from the contralateral side of the body is transferred to the opposite side of the spinal cord.

**Why Each Wrong Option is Incorrect**
**Option A:** Damage to the lateral corticospinal tract would result in weakness or paralysis of the muscles on the same side of the body, which is not consistent with the clinical presentation described.
**Option B:** Injury to the spinothalamic tract would cause loss of pain and temperature sensation on the same side of the body, which is not the primary complaint in this case.
**Option C:** Damage to the anterior horn cells would result in weakness or paralysis of the muscles on the same side of the body, which is not consistent with the clinical presentation described.

**Clinical Pearl / High-Yield Fact**
The posterior columns of the spinal cord are responsible for transmitting proprioceptive and vibratory information from the body to the brain, making damage to these tracts a classic cause of "stumbling gait" and loss of proprioception in patients with spinal cord injuries.

**Correct Answer: C. Brown-Séquard syndrome**