Add pineapple chunks, chopped jalapeño, mint leaves, lime juice, and 1 cup of the cold water to a blender. Blend on high until completely smooth, about 45–60 seconds.
Pour the mixture through a fine-mesh strainer set over a large pitcher. Use a spoon to press the solids against the strainer to extract as much liquid as possible; discard the solids.
Stir in the remaining 3 cups cold water and the cayenne pepper, if using. Taste and adjust: add more lime juice for tartness or a few extra pineapple chunks blended in for sweetness.
Add the ice cubes to the pitcher and stir gently. Refrigerate for at least 15 minutes to chill thoroughly.
Serve over ice in glasses, garnished with a lime slice and a sprig of mint. Best enjoyed within 24 hours.