= Solution
Use the normalized three-qubit states $|G_\pm\rangle=(|000\rangle\pm|111\rangle)/\sqrt2$. The <Shor code> has logical basis
$$
\boxed{|0_L\rangle=|G_+\rangle^{\otimes3},\qquad
|1_L\rangle=|G_-\rangle^{\otimes3}.}
$$
It encodes $\alpha|0\rangle+\beta|1\rangle$ as $\alpha|0_L\rangle+\beta|1_L\rangle$, preserving the unknown amplitudes by a <linear isometry>. It concatenates a three-block <phase-flip repetition code> with the three-qubit <bit-flip repetition code> inside each block. Six <stabilizer generators>, $Z_1Z_2,Z_2Z_3,Z_4Z_5,Z_5Z_6,Z_7Z_8,Z_8Z_9$, test bit errors inside the blocks. The other two <stabilizer generators> are
$$
A=X_1X_2X_3X_4X_5X_6,\qquad
B=X_4X_5X_6X_7X_8X_9.
$$
All eight commute and have eigenvalue $+1$ on both logical basis states. They define the $[[9,1,3]]$ <stabilizer code>. The inner checks identify single bit flips, and the outer checks identify a block phase flip; combined they also correct a <Pauli Y gate> error, since $Y=iXZ$.
For the requested <phase flip>, a $Z$ on any one qubit interchanges $|G_+\rangle$ and $|G_-\rangle$ in its block. It commutes with the six $Z$-pair checks. It anticommutes with $A$ or $B$ exactly when its block is contained in that check. The <Shor-code phase-flip syndrome> is therefore
$$
\begin{array}{c|c|c}
\text{error block}&(A,B)\text{ outcomes}&\text{recovery}\\\hline
\text{none}&(+1,+1)&I\\
1&(-1,+1)&Z_1\\
2&(-1,-1)&Z_4\\
3&(+1,-1)&Z_7
\end{array}
$$
A <projective measurement> of the two commuting checks yields the displayed <error syndrome> without learning $\alpha$ or $\beta$: both logical components have the same check eigenvalues after the error. Applying the indicated <Pauli Z gate> restores every encoded <quantum state>.
The correction is genuinely degenerate. For $i,j$ in the same block, $Z_iZ_j$ fixes both $|000\rangle$ and $|111\rangle$, so $Z_iZ_jP=P$ and consequently $Z_iP=Z_jP$. In particular $Z_1$ and $Z_2$ are distinct physical errors but have identical action on every code state; their image subspaces coincide, not merely their observed syndrome labels. In the <Knill--Laflamme condition> for $I,Z_1,\ldots,Z_9$, the error <Gram matrix> has $C_{00}=1$, $C_{0i}=C_{i0}=0$, and
$$
C_{ij}=\begin{cases}1,&i,j\text{ in the same block},\\0,&i,j\text{ in different blocks}.
\end{cases}
$$
The zero overlaps between different blocks follow because a phase flip changes a different pattern of orthogonal $G_+,G_-$ factors; cross-logical overlaps also vanish. Thus $C$ consists of one $1\times1$ block and three all-ones $3\times3$ blocks and has rank four, not ten. This both proves correctability of the entire single-phase-error span and exhibits its degeneracy. A coherent sum of phase errors in one block has the same logical action with its coefficients summed, so inability to locate the individual qubit loses no logical information. \b[The two block checks suffice to correct every single phase flip, even though the within-block errors are indistinguishable.]
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