Bring sugar, water, corn syrup, and salt to a boil in a 1 1/2-quart saucepan, stirring until sugar is dissolved, then boil syrup until reduced to 3/4 cup, about 5 minutes. Stir in cardamom and cool completely.
Stir together persimmon purée, cardamom syrup, milk, half-and-half, and lemon juice and chill, covered, until cold, at least 3 hours.
Freeze mixture in ice cream maker, then transfer to an airtight container and put in freezer to harden.
Persimmon milk can be chilled 1 day. Sherbet keeps 1 week.
