Thanks, a lot of this helps.

Having said that, if

    - y is a binary "ironic"(1), "not ironic"(0) label and
    - x is the content of a tweet and
    - LLM(x) is meant to represent some probability range [0,1] representing p(y=1|x)

then it seems that p(y=1|x) === LLM(x) already.

Why bother with p(y=1|x) = σ( α + β*LLM(x) ) as in the article? You are right that I missed the case where LLM(x) = 0, but that just means that p(y=1|x) has exactly 2 values when defined as above and β -> inf.

    - when LLM(x) = 0 then p(y=1|x) = 1/(1+e^-α)
    - when LLM(x) = (0,1] then p(y=1|x) = 1/(1+e^-inf) = 1

This is substantially lossy and converts a continuous output LLM(x) of range [0,1] to a step function not even defined as a set {0,1} but instead the set {1/(1+e^-α),1} for unclear gain. It also loses all sensitivity to LLM(x) near 0, and only detects/changes value when LLM(x)=0 exactly. It might make more sense if LLM(x) is not limited to [0,1] like you have claimed because then at least the logistic regression is clamping the output to [0,1]. If you were wrong about the range of LLM(x), this would come back to asking authors to actually define their terms.

My point is that when math is used to justify something or communicate something, it should be explained or very apparently right. If when someone goes to try to understand the math it doesn't match the claims being made in the article ("recovering the LLM") then it throws the rest of the article into doubt.