Is there a phrase that means "thinking out loud", but on paper? [closed]

Brainstorming or freewriting is a term used by scholars and authors.

From the University of North Carolina at Chapel Hill:

When you freewrite, you let your thoughts flow as they will, putting pen to paper and writing down whatever comes into your mind.

I would say you are writing stream of consciousness. This makes it clear you're just writing whatever comes to your head.

Thinking out loud in English connotes that the person has not sufficiently formed the thought completely or thought the entire idea through, and so is merely processing the idea audibly without knowing its conclusion or validity. Doing something like this on paper (meaning writing it) might be referred to as "drafting" an idea or "sketching," or compiling notes. Additionally a "rough sketch" or a "rough draft" shows that the writing is not yet polished, edited, or completed.

Think it, jot it down TFD

to make a note of something

I keep reading your question and wanting to coin new phrases.

writing out loud
mind writing

But that doesn't help if you're looking for an already existing word or phrase.

I've come across several ideas that express thinking aids (such as mind mapping and the already-mentioned brainstorming), but they aren't necessarily related to putting words on paper (or doing so on your own). Also, if you are "free associating," your writing will not be constrained, but if you are actually "working out a problem" (something specific) it's likely that what you write will have some purpose or framework behind it.

An alternative is to use a qualifier with a different term: a handwritten brain dump:

the act or an instance of comprehensively and uncritically expressing and recording one's thoughts and ideas (as on a particular topic)