The following quote from Malcolm Turnbull is actually not what this question is about, but nevertheless it is a good-fit epigraph:
The laws of mathematics are very commendable, but the only law that applies in Australia is the law of Australia.
Though the scenario I give in this question is exaggerated, hypothetical, edge-case one, it is technically valid and showcases the essence of the question.
A recent question/answer seems to be telling us that it is perfectly fine for a judge to reject an argument even implicitly — i.e. without actually discussing it. Such a judgment might only be vulnerable to appeal, but in no way to judicial discipline / misconduct investigation. If not successfully appealed, it becomes case law — binding on all inferior courts. Where rights of appeal do not exist (e.g. the court is the top court in its jurisdiction), the judgment becomes set-in-stone law — until cancelled either by the same court or via legislation.
So, let's consider the same example as in that question, but more specific one:
I am right and Rob is wrong because blablabla... 2×2=4 ...blablabla.
No Bob is wrong, because blablabla... and 2×2 actually equals 5.
But look, lets see what 2×2 equals [math proof follows]. It is 4, you see.
The judge says:
I accept Rob's contention that Bob is wrong because 2×2=5. Bob has no case.
So, the judge implicitly rejects the math proof that 2×2=4 (as their Honour "does not have to address every argument raised in their judgement") and, effectively, makes it case law that 2×2=5.
Is this scenario technically/legally possible? Would the case law that 2×2=5 stand for some time? Will the judge not face any disciplinary consequences but just public outcry and reputation damages?
I appreciate that one might want to say that no judge would accept such an extremely outrageous merit-lacking argument as "2×2=5" and I totally agree (with hope). But the way this question applies to reality is that there is no clear boundary between "2×2=5" and any mundane argument that Rob might argue — as far as its acceptance by a judge is concerned.