It's no different than how in English we can signal that a statement is related to a kind of politics or that it's about sex through particular word and phrase choice.
Training for reasoning should be expected to amplify the subtext, since any random noise in the selection that by chance is correlated with the right results will get amplified.
Perhaps you could try to dampen this by training two distinct models for a while, then swap their reasoning for a while before going back-- but sadly distinct models may still end up with similar subtexts due to correlations in their training data. Maybe ones with very distinct tokenization would be less likely to do so.
I hope that research into understanding LLM qualia eventually allow us to understand e.g. what it's like to [be a bat](https://en.wikipedia.org/wiki/What_Is_It_Like_to_Be_a_Bat%3F)
We have our own personal 'culture' too-- it's just less obvious because its tied up with our own hidden state. If you go back and read old essays that you wrote you might notice some of it-- that ideas and feelings (maybe smells?) that are absolutely not explicitly in the text immediately come back to you, stuff that no one or maybe only a spouse or very close friend might think.
I think it may be very hard to explore hidden subtext because the signals may be almost arbitrarily weak and context dependent. The bare model may need only a little nudge to get to the right answer and the you have this big wall of "reasoning" where each token could carry very small amounts of subtext that cumulatively add up to a lot and push things in the right direction.
An example of this is toki pona, a minimalist constructed human language that is designed to only express "positive thinking". Yet it is extremely easy to insult people in toki pona: e.g. sina toki li pona pona pona pona. (you are speaking very very very very well).
To be free of a potential subtext sidechannel there can be essentially no equivalent outputs.