= Solution
Abbreviate $P(\ulcorner A\urcorner)$ by $\Box A$. Since $S\vdash\neg X$, it proves $X\to G$. The theoremhood and distribution conditions give
$$
S\vdash\Box X\to\Box G.
$$
The fixed-point equation gives $S\vdash G\to(\Box G\to X)$, because $G$ implies $\neg\Box G$ and a false antecedent implies any conclusion. Applying theoremhood and distribution yields
$$
S\vdash\Box G\to\Box(\Box G\to X).
$$
<Positive introspection> gives $\Box G\to\Box\Box G$; distribution then gives $\Box G\to\Box X$. Thus $S\vdash\Box G\leftrightarrow\Box X$. Combining with $G\leftrightarrow\neg\Box G$ proves
$$
\boxed{S\vdash G\leftrightarrow\neg\Box X.}
$$
In particular, choosing a fixed logical <contradiction> for $X$ identifies this <Gödel sentence> with the <formal consistency statement> of $S$.
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