= Solution
\b[Assume additionally that the internal modus-ponens condition requested in part (c) holds for this $R$.] If <positive introspection> also held for every sentence, then $R$ would satisfy all three <Hilbert-Bernays-Löb derivability conditions>: external <necessitation rule> was proved in part (c), <modus ponens rule> is the added condition, and introspection is the supposition.
The diagonal lemma applies to $R$ just as to any <arithmetic> formula. The usual <Gödel second incompleteness theorem> for a predicate with these conditions would then imply
$$
PA\nvdash\neg R(\ulcorner\bot\urcorner)
$$
when $PA$ is consistent. Part (b) proves the opposite. Therefore the three conditions cannot all hold, and under the added modus-ponens hypothesis there must be some sentence $X$ such that
$$
\boxed{PA\nvdash R(\ulcorner X\urcorner)\to R(\ulcorner R(\ulcorner X\urcorner)\urcorner).}
$$
This is the intended contradiction argument and requires no identification of a particular $X$.
Without that extra hypothesis, the valid conclusion is only that \b[at least one of internal <modus ponens rule> and <positive introspection> fails]. Rosser predicates can be constructed that satisfy <positive introspection>, so a failure of that condition cannot be inferred from the printed general assumptions alone. https://arxiv.org/abs/1902.06863[Rosser provability and the second incompleteness theorem: constructions with different derivability conditions]{ref}. The distinction concerns proof coding, not a missing page of the exam.
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