= Solution
The commuting terms
$$
S_j=X_jX_{j+1},\qquad 1\leq j<n,
$$
are independent <stabilizer group>[stabilizer generators]. Since $J>0$, every ground state has $S_j=+1$. The resulting two-dimensional <stabilizer subspace> is
$$
\mathcal C_n^Z=\operatorname{span}\{|+\rangle^{\otimes n},|-\rangle^{\otimes n}\},
$$
the <phase-flip repetition code>. A convenient pair of logical Pauli operators is
$$
\overline Z=X_1,
\qquad
\overline X=Z_1Z_2\cdots Z_n.
$$
Indeed, they commute with every stabilizer, anticommute with each other, and are not stabilizers.
For phase-flip errors $E_A=\prod_{j\in A}Z_j$ and $E_B=\prod_{j\in B}Z_j$, the operator entering the <Knill--Laflamme condition> is $E_A^\dagger E_B=E_{A\triangle B}$ and has weight at most $2t$. Every nonempty proper product of $Z_j$ anticommutes with some $S_j$ unless it is the full logical operator $\overline X$. The Knill--Laflamme conditions therefore hold whenever $2t<n$, and fail once two allowed errors can differ by $\overline X$. Thus
$$
\boxed{t=\left\lfloor\frac{n-1}{2}\right\rfloor.}
$$
Each physical $X_j$ commutes with all stabilizers, and $X_jX_1$ is a product of stabilizers. Hence every $X_j$ acts on the code as the undetectable logical operator $\overline Z$. The code cannot detect, and therefore cannot correct, even a single bit flip.
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