Cape not showing up? Work down this list
Nearly every "my cape disappeared" report is one of four causes, and they are worth checking in cost order. Most people start by reinstalling the mod, which is the expensive fix for the rarest cause.
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Are you looking at yourself in third person?
In first person you cannot see your own back. Press F5 to swap view. It sounds too obvious to check, and it is the single most common cause.
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Is the other player running the same mod?
A custom cape is drawn by the mod, on each viewer's machine. Someone on plain vanilla sees no cape on you, and never will - nothing is broken. Ask a friend who runs the same mod, or check in single-player.
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Did the upload actually finish?
Open the mod's page again and confirm your PNG is the one shown. A file that failed the size or format check is often rejected quietly.
A cape must be a PNG at the 2:1 shape. A square image, a JPG, or a screenshot renamed to .png will not load.
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Is an elytra or a chestplate in the way?
Equipping an elytra replaces the cape with the wings, and some armour hides it. Take both off and look again.
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Only then, reinstall
If all four pass, remove the mod, delete its config folder and install it fresh for your exact Minecraft version. A mod built for a different version can load and then quietly render nothing.