**Core Concept**
The greater sciatic foramen is a large opening in the pelvis through which several important structures pass. It is a key anatomical landmark in the lower limb and plays a crucial role in the innervation and vascular supply of the lower limb.

**Why the Correct Answer is Right**
The correct answer is the structure that does not pass through the greater sciatic foramen. The greater sciatic foramen is bounded by the greater sciatic notch of the ilium and the sacrotuberous and sacrospinous ligaments. The structures that pass through it include the sciatic nerve, the internal pudendal vessels and nerve, the nerve to obturator internus, and the superior gluteal vessels and nerve.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect because the sciatic nerve does pass through the greater sciatic foramen. It is the largest nerve in the body and is responsible for innervating the lower limb.
* **Option B:** This option is incorrect because the internal pudendal vessels and nerve do pass through the greater sciatic foramen. They are responsible for supplying blood and nerve fibers to the external genitalia.
* **Option C:** This option is incorrect because the nerve to obturator internus does pass through the greater sciatic foramen. It is a branch of the sacral plexus that innervates the obturator internus muscle.
* **Option D:** This option is incorrect because the superior gluteal vessels and nerve do pass through the greater sciatic foramen. They are responsible for supplying blood and nerve fibers to the gluteal region.

**Clinical Pearl / High-Yield Fact**
The greater sciatic foramen is a common site for injury to the sciatic nerve, which can result in a condition known as sciatica. Sciatica is characterized by pain, numbness, and tingling in the lower limb, and can be caused by a variety of factors including trauma, compression, and inflammation.

**Correct Answer: D. The superior gluteal vessels and nerve do not pass through the lesser sciatic foramen (they pass through the greater sciatic foramen).**