The TNM classification system for p16-positive oropharyngeal cancer is a framework used to stage cancer based on tumor size, lymph node involvement, and distant metastasis. This classification system takes user inputs regarding the staging parameters and systematically determines the cancer stage.

Inputs required are:
1. Tumor Classification:
   - Tx: Primary tumor cannot be assessed
   - T0: No evidence of primary tumor
   - T1: Tumor <= 2 cm in greatest dimension
   - T2: Tumor > 2 cm but <= 4 cm in greatest dimension
   - T3: Tumor > 4 cm in greatest dimension or extension to the lingual surface of epiglottis
   - T4: Tumor invading larynx, extrinsic muscle of tongue, medial pterygoid, hard palate, or mandible or beyond; mucosal extension to the lingual surface of epiglottis from primary tumors of the base of tongue; vallecula does not constitute an invasion of larynx
   
2. Lymph Node Classification:
   - Nx: Regional lymph nodes cannot be assessed
   - N0: No regional lymph node metastasis
   - N1: Metastasis in <= 4 lymph nodes
   - N2: Metastasis in > 4 lymph nodes
   
3. Metastasis Classification:
   - M0: No distant metastasis
   - M1: Distant metastasis present
   
The calculation involves the following logic: 
- Stage I is assigned when the tumor is classified within T0 to T2 and lymph node involvement is either absent or limited to four nodes (N0 or N1) with no distant metastasis (M0).
- Stage II is determined if either a T0 to T2 tumor with more than four lymph nodes involved (N2) and no metastasis (M0), or a T3 or T4 tumor with no more than four lymph nodes involved (N0 or N1) and no metastasis (M0).
- Stage III involves a T3 or T4 tumor with more than four lymph nodes involved (N2) and no metastasis (M0).
- Stage IV is declared if there is any level of tumor classification or lymph node involvement with distant metastasis present (M1).
- If inputs cannot be fully assessed or selections are incomplete, the stage cannot be fully assessed.

Table of conditions:
| Tumor Condition          | Lymph Nodes Condition | Metastasis Condition | Result          |
|--------------------------|-----------------------|----------------------|-----------------|
| T0, T1, T2               | N0, N1                | M0                   | Stage I         |
| (T0, T1, T2 and N2) OR   |                       |                      |                 | 
| (T3, T4 and N0, N1)      |                       | M0                   | Stage II        |
| T3, T4                   | N2                    | M0                   | Stage III       |
| Any                      | Any                   | M1                   | Stage IV        |

The output is a string indicating the cancer stage based on the user inputs, or informing if the inputs are incomplete or cannot be fully assessed. This classification method aids clinicians in diagnosing and formulating appropriate treatment plans for patients with p16-positive oropharyngeal cancer.

The TNM classification for p16-positive oropharyngeal cancer is a clinical staging system developed by the American Joint Committee on Cancer (AJCC). It is designed for the stratification of patients with p16-positive oropharyngeal cancer, a subtype of head and neck cancer associated with human papillomavirus (HPV) infection. The system aids in predicting prognosis, guiding treatment decisions, and facilitating communication among healthcare providers.

The primary components of this classification system are tumor size and extent (T), regional lymph node involvement (N), and distant metastasis (M). Each of these components is assigned a numerical value based on the severity or extent of disease. The T category is determined by the size of the primary tumor and its extent of invasion into surrounding tissues. The N category is based on the number, size, and location of regional lymph nodes involved. The M category indicates whether distant metastasis is present or absent.

The final TNM stage is determined by combining the T, N, and M categories. This stage provides a comprehensive picture of the patient's disease status and is crucial in determining the most appropriate treatment strategy. The TNM classification for p16-positive oropharyngeal cancer has been specifically tailored to reflect the unique clinical behavior and prognosis of this HPV-associated malignancy, which differs significantly from HPV-negative head and neck cancers.