I have a field that's filled automatically by a script, and all a user should be able to do is view its value on the issue page.
How can I get it to be visible there, but not on any screen (such as "create issue")? From what I can tell, having a field visible on the issue page requires it to also be visible on the "default screen".
Jira uses "screens" to select what fields appear for various actions on issues. Create, Edit, View and all the transitions in workflows can use them.
Sounds like you need a screen which has the field, plus another which does NOT. You'd want the first screen for "view" and the other screen for create and edit (although if it's a read-only field, it won't appear on create or edit anyway)
Have a look through https://confluence.atlassian.com/display/JIRA/Configuring+Fields+and+Screens - you're not too worried about field config or workflow, it's the screens and screen schemes you need to work through
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