Explanation:

When using a tool in image editing software, such as sharpen, blur, dodge, or smudge, it is important to understand how each tool works and the effect it will have on the image. In this case, the question asks about a specific mode where over-application of the tool will produce noise. Let's examine each option to understand why only the "sharpen" tool fits this description.

Sharpen:
- The sharpen tool is used to enhance the details and edges in an image, making it appear sharper.
- When applied in moderation, the tool can improve the clarity and definition of the image.
- However, if the sharpen tool is excessively applied, it can introduce noise or artifacts into the image.
- The over-application of the sharpen tool can create halos and increase the appearance of noise, leading to a degraded image quality.

Blur:
- The blur tool is used to soften or blur areas of an image, reducing the clarity and detail.
- Over-application of the blur tool will not produce noise but rather further increase the blurriness of the image.
- It can be used to create a depth-of-field effect or to smooth out imperfections.

Dodge:
- The dodge tool is used to lighten specific areas of an image.
- Over-application of the dodge tool will not introduce noise but can result in overexposure or loss of detail in the highlighted areas.
- It is commonly used to brighten shadows or selectively lighten certain parts of an image.

Smudge:
- The smudge tool is used to blend or smudge pixels, creating a soft or painterly effect.
- Over-application of the smudge tool will not produce noise but can result in excessive blurring or distortion of the image.
- It is often used for artistic purposes to create a sense of motion or to blend colors seamlessly.

Conclusion:
Based on the explanations above, it is clear that the "sharpen" tool is the only option out of the four mentioned that, when over-applied, can introduce noise into an image. The other tools may cause different effects but do not directly lead to noise.