Fat necrosis: This is the most likely diagnosis given the specific mention of "FAT NECROSIS" in the question. Fat necrosis is a benign condition that occurs when there is damage to the fatty tissue, often due to trauma, surgery, or radiation, leading to the formation of firm, painless masses.

Lipoma: A common benign tumor composed of fat tissue, which could present as a palpable mass similar in size to the one described.
Breast cyst: While typically fluid-filled and fluctuant, a breast cyst could potentially feel solid on examination, especially if it is complex or has undergone hemorrhage.
Fibroadenoma: A benign breast tumor that could present as a firm, mobile mass, although they are more common in younger women.

Breast cancer (e.g., liposarcoma, invasive ductal carcinoma): Although less likely, it is crucial to rule out malignancy, especially in cases where the diagnosis is not clear or the patient has risk factors for breast cancer. Liposarcoma, a type of cancer that arises in fat cells, could present similarly to fat necrosis.
Infection or abscess: An infectious process could cause a mass and must be considered, especially if there are signs of inflammation or systemic infection.

Hamartoma: A rare, benign breast tumor that can contain fat, glandular, and fibrous elements, presenting as a palpable mass.
Phyllodes tumor: Although more commonly presenting as a large, rapidly growing mass, phyllodes tumors can occasionally be smaller and could be considered in the differential diagnosis of a breast mass.
Granular cell tumor: A rare tumor that can occur in the breast, typically presenting as a firm, painless mass, although it is more commonly found in other locations.