**Core Concept**
Digital gangrene of the third and fourth fingers in a young patient with hypertension and palpable peripheral pulses suggests a condition that affects the small arteries or arterioles. The absence of systemic inflammation, as indicated by negative ANA, Anti-dsDNA, and ANCA, points towards a non-inflammatory cause.

**Why the Correct Answer is Right**
The most likely diagnosis in this case is **Buerger's disease** (Thromboangiitis obliterans). This condition is characterized by inflammation and thrombosis of small and medium-sized arteries and veins, typically in the limbs of young smokers. The digital gangrene in this patient, along with the palpable peripheral pulses, is consistent with Buerger's disease. The absence of systemic inflammation and the presence of hypertension may be secondary to the disease process.

**Why Each Wrong Option is Incorrect**
**Option A:** **Raynaud's phenomenon** typically presents with episodic vasospasm of the digital arteries, leading to color changes and pain in the fingers. The absence of systemic inflammation and the presence of digital gangrene make this diagnosis less likely.

**Option B:** **Buerger's disease** is the most likely diagnosis, as explained above.

**Option C:** **Scleroderma** is a systemic autoimmune disease that can cause digital gangrene due to vasculopathy and fibrosis. However, the absence of systemic inflammation and the presence of hypertension make this diagnosis less likely.

**Option D:** **Diabetes mellitus** can cause digital gangrene due to neuropathy and microangiopathy. However, the absence of systemic inflammation and the presence of palpable peripheral pulses make this diagnosis less likely.

**Clinical Pearl / High-Yield Fact**
Buerger's disease is a rare condition that affects young smokers, and its diagnosis is often delayed due to the lack of systemic symptoms. A high index of suspicion and a thorough history of smoking are essential for making this diagnosis.

**Correct Answer:** B. Buerger's disease.