It may be useful to look at some more arbitrary statements which might actually be true: “Easy Truth has red hair”; “StrictlyLogical is 6 ft. tall”, “Invictus2017 owns a Ford Explorer”. Each of these statements does, on linguistic grounds, either describe a fact, or else it describes a non-fact – they are objectively true or false. But I personally have no basis in knowledge for making those statements, and they do not contitute the recognition of a fact of reality. They differ from Peikoff’s parrot or sand message examples where there is no proposition (the thing you see or hear merely physically resembles what could be speech or writing in another context). His savage math example needs to be modified since it is unclear what his point is, so I’ll rewrite that as an illiterate and innumerate person uttering the sentence “the fourth power of 3 is 81” (you can say this based on experience, without understanding what it means, since in English, you can put words like “second, fourth” before “power” and follow that with another number). This statement too is arbitrary, and in that context it is like the parrot utterance in that the person utters the word “power” without grasping what that term refers to. In fact, I would not even call the sand / parrot / savage math examples “statements”.
So compare my examples to Peikoff’s “soul survives”, “fate determined by date of birth”, “sixth sense” and “convention of gremlins”. In those examples, the arbitrariness of the statement largely depends on the fact that the statements presuppose the existence of entities for which there is no evidence. In my examples, all of the concepts involved do unquestionably exist: I just made up relations between actual existents, without any factual basis for claiming those relationships. Arbitrary statements are not necessarily utterly devoid of relationship to reality, because they can refer to actual existents and invoke no mythical entities.
In How we know, Binswanger has an extended analysis of “arbitrary”, which you may find clarifies the nature of the arbitrary.
"Global warming" (which is nowadays not even a statement, it's just a noun phrase assumed to represent some statement), is an example of the arbitrary: it is asserted as self-evident, needing no evidence.