```json
{
  "TragedyDegree": 2,
  "ComedyDegree": 4
}
```

Explanation:

- **Tragedy Degree**: The story contains very few tragic elements. Emma faces challenges and the pressure of proving her findings, but these setbacks are not emotionally overwhelming or deeply tragic. Therefore, a score of 2 is appropriate.

- **Comedy Degree**: The story is relatively light-hearted with Emma ultimately achieving a significant academic breakthrough and being acknowledged by the headmaster. While it is not filled with strong comedic aspects or an extremely happy ending, it does contain positive and uplifting elements. Thus, a score of 4 reflects the moderate presence of light-heartedness and a satisfying conclusion.
