No, it’s genuinely impossible to know how much of Buckmaster’s Codex data is in OpenAI’s training set.
First, the conversations are anonymized, so there's no simple way to inspect the training dataset and identify which specific conversations belong to Buckmaster.
Second, OpenAI uses these anonymized chats to generate synthetic training data, i.e. they fabricate new conversations based on specific conversation patterns where the model performs poorly, and uses these synthetic conversations as training data for future models. The synthetic data could potentially contain some of selections of Buckmaster's original chats, but it is unknowable how his specific writing could have influenced these synthetic data sets or what portion belongs to him. This information is untraceable and effectively double anonymized.
Third, OpenAI explicitly uses user feedback (the thumbs up or thumbs down ratings), as RLHF to train models. However, this feedback is anonymized and stripped of user identifiers. It's not possible to trace a specific feedback to Buckmaster, nor do we know if Buckmaster ever used this feature. I doubt Buckmaster recalls or can provide a list of every time he used this feature over the past year. OpenAI doesn't have one.
Note that the first and second only happen if Buckmaster "Improve the model for everyone" setting enabled, which I find unlikely. But that doesn't exclude option three from this list.
You seem to think that it is some "gotcha" that OpenAI refuses to make a blanket denial, but they cannot do so in good faith, because they have a genuine understanding of their own system. They don't know where the data they have came from.
This situation meets the requirement of Russell's teapot, since neither party has enough evidence to prove nor disprove what information is actually in OpenAI's training set.