These categories are general. Not every statement in a description will apply to a given solution or proof marked with a particular category, but many statements will apply, and the general theme of the category will hold.
0: The solution/proof is missing.
OR, The solution/proof makes no sense or is unrelated to the assigned problem.
1: The solution/proof contains serious logical flaws, and lacks adequate justification or explanation. There may be misuses of notation that significantly hinder communication of reasoning. There are few, if any, complete sentences. Little understanding of mathematical ideas and reasoning is shown. Computations present are significantly flawed. No structure is apparent in the solution/proof. The problem/theorem is not restated and there is no conclusion.
OR, The bulk of the assigned solution/proof is not present.
2: The solution/proof has some gaps in reasoning. The argument does not form a coherent whole. There are many flaws of mathematical grammar. The problem/theorem is not restated appropriately. The individual statements of the solution/proof are not connected, or presented out of logical sequence. Computations present are flawed. If present, diagrams are improperly drawn. There is no concluding statement.
OR, The solution/proof might be correct, but this is not demonstrated by the presentation of the solution/proof; a good solution/proof may be read, by a generous reader, into what is presented.
OR, A significant chunk of the solution/proof is not present.
3: The solution/proof is correct or nearly correct and logically coherent. The reader is required to fill in some details that should have been explained or justified. There may be some explanations, arguments, ore justifications that are correct but unclear, cumbersome, needlessly complicated, or awkward. Notation is used appropriately, with few errors, and mathematical grammar is substantially correct. Computations present are correct. If appropriate, diagrams are present. There is a concluding statement.
4: The solution/proof is correct and flows logically. The presentation uses correct mathematical grammar and uses notation correctly. The structure of the solution/proof is apparent and the bulk of the details are easy to follow (even for a fellow student). Links are made to appropriate definitions and previously known theorems. Computations present are correct. There is a clear conclusion. If appropriate, diagrams are present and clear. No details are missing, and also no details are belabored. The solution/proof may be elegant, and may also contain very minor flaws.