Housing
What size is your tank?
1 gallon
What temperature is your tank?
~76
Does your tank have a filter?
No
Does your tank have an air stone or other type of aeration?
No
Is your tank heated?
No
What tank mates does your betta fish live with?
None
Food
What type of food do you feed your betta fish?
Topfin Pellets, Topfin Bloodworms, Hikari Krill, and Meal Worms
How often do you feed your betta fish?
Twice a day with pellets with varying between krill and worms, fasting once a week or as needed
Maintenance
How often do you perform a water change?
3-4 times weekly
What percentage of the water do you change when you perform a water change?
50%
What type of additives do you add to the water when you perform a water change?
Tetra Aquasafe
Symptoms and Treatment
How has your betta fish's appearance changed?
Not that I can really see yet
How has your betta fish's behavior changed?
No
When did you start noticing the symptoms?
Have you started treating your fish? If so, how?
Just clean water
Does your fish have any history of being ill?
No
How old is your fish (approximately)?
1 year or less
The betta in question:

This picture is before anything happened so it won't be helpful in finding where the piece came off from, but maybe someone could spot something wrong with him to cause this issue.