**Core Concept**
The question is testing the student's knowledge of breast pathology, specifically the identification of a characteristic histopathological feature associated with a particular type of breast cancer. The correct answer requires an understanding of the pathophysiology of invasive breast carcinoma.

**Why the Correct Answer is Right**
The mammography and histopathological examination (HPE) findings suggest a diagnosis of invasive ductal carcinoma (IDC) of the breast. The presence of an irregularly shaped lump with hard consistency and nipple retraction are clinical features of IDC. The histopathological examination typically shows a malignant proliferation of ductal epithelial cells invading the surrounding stroma. The characteristic feature of IDC is the presence of "comedo necrosis" within the ducts, which is a hallmark of this type of cancer.

**Why Each Wrong Option is Incorrect**
* **Option A:** This option is incorrect as it does not describe a characteristic feature of invasive ductal carcinoma.
* **Option B:** This option is incorrect as it is more characteristic of lobular carcinoma in situ (LCIS) or invasive lobular carcinoma (ILC), which typically does not show comedo necrosis.
* **Option C:** This option is incorrect as it describes a feature of fibroadenoma, a benign breast tumor.
* **Option D:** This option is incorrect as it describes a feature of fibrocystic changes, a common benign condition of the breast.

**Clinical Pearl / High-Yield Fact**
Comedo necrosis is a characteristic histopathological feature of invasive ductal carcinoma, and its presence should raise suspicion for this type of cancer in a breast biopsy. It is essential to recognize this feature to make an accurate diagnosis.

**Correct Answer:** A. Comedo necrosis.