First, I would mention that a growing part of academic readership would IMHO be faster to understand your model from the R formula than from the math notation, so - depending on your target audience - doing away with the math notation completely might be feasible, though a bit riskier as it is not (yet) commonly done.
Definitely also check out
Understanding the Mathematical Model and Mathematical description of a brms model where a similar problem is discussed
I see a few weird things:
- I thought you were using
cumulative("logit")family, but using \Phi suggest you usecumulative("probit") - Categorical variables would be encoded via dummy coding with a reference level sou you would have a coefficient for each level of category, condition and category:condition, except for the reference level.
- The \sigma_s, \sigma_m, \sigma_t should probably be the sd on the varying intercepts - those are however currently missing from the models, you should have something like so you would need to explicitly write those out. An example of how that might look like is on page 8 of my recent publication
And there are potentially other issues.
You might also want to simplify and just to state the model in matrix notation ((https://en.wikipedia.org/wiki/Mixed_model)) and then rely on the brms formula for details.
But if you need to have the full math notation, feel free to get back for additional clarification :-)