The condition characterized by a white, non-removable patch on the lateral border of the tongue is Hairy leukoplakia.

- Hairy leukoplakia typically appears as white, corrugated or hairy-appearing plaques on the lateral borders of the tongue.
- These lesions are not scrapable, which helps differentiate them from other white patches caused by infections like oral candidiasis.
- Hairy leukoplakia is closely associated with Epstein-Barr Virus (EBV) infection and is commonly seen in immunocompromised patients, especially those with HIV/AIDS.

Let's briefly contrast this with the other options:
- Oral candidiasis presents as white patches that are often removable by scraping and may leave a raw, erythematous surface. It commonly affects the buccal mucosa, tongue, and palate but typically involves removable plaques.

- Lichen planus exhibits a lace-like (reticular) white pattern or sometimes erosive white lesions, usually on the buccal mucosa or tongue, but it is usually not a thick, hairy patch and is often bilateral.

- Geographic tongue manifests as map-like, depapillated, erythematous areas with white borders that change location over time. It does not present as a permanent white patch.

In summary, the key distinguishing features of hairy leukoplakia are:
- Location: lateral border of the tongue
- Appearance: white, hairy or corrugated plaques
- Non-removable: not scrapable
- Association: immunocompromised states and EBV infection

Reference: Oral Medicine and Pathology, Volume 2, Chapter 15, p. 342-345