= Solution
The <joint spectrum> $\operatorname{Spec}(n)$ consists of the tuples of simultaneous <eigenvalues> of $X_1,\ldots,X_n$ occurring in irreducible complex $S_n$-modules. A <Gelfand–Tsetlin basis> is a <Gelfand–Tsetlin basis>: its vectors lie in the one-dimensional joint eigenspaces selected by successive restrictions. The weight of $v$ is $\alpha=(a_1,\ldots,a_n)$ when $X_jv=a_jv$ for every $j$.
Define $\operatorname{Cont}(n)$ to be the integer tuples satisfying the <content vector of a standard Young tableau> conditions: $a_1=0$; for $j>1$, at least one of $a_j-1,a_j+1$ occurs earlier; and between any two occurrences of the same integer $a$, both $a-1$ and $a+1$ occur. We will justify their tableau interpretation in part (iv).
Here is the <local spectral rules for Young–Jucys–Murphy elements> calculation used in this question. Average a positive definite <Hermitian inner product> over $S_n$, making the representation unitary. The $X_j$ are commuting self-adjoint operators because <transpositions> are self-adjoint involutions. Put $s_i=(i\ i+1)$ and $d=a_{i+1}-a_i$. The group-algebra relations are
$$
X_{i+1}s_i-s_iX_i=1,\qquad
X_is_i-s_iX_{i+1}=-1,\qquad
s_iX_j=X_js_i\quad(j\ne i,i+1).
$$
For a unit weight vector $v$, the first relation gives $d\langle v,s_iv\rangle=1$. Thus $d\ne0$, $|d|\geq1$, and
$$
w=s_iv-d^{-1}v
$$
has the weight obtained by swapping $a_i,a_{i+1}$, with $\|w\|^2=1-d^{-2}$. If $d=\pm1$, then $w=0$ and $s_iv=\pm v$. If $|d|>1$, then $w\ne0$, providing an admissible spectral swap in the same <irreducible module>.
For the proposed three consecutive weights $(a,a+1,a)$, these rules give $s_iv=v$ and $s_{i+1}v=-v$. But the <braid relation in a Coxeter group> would imply
$$
s_is_{i+1}s_iv=-v,\qquad
s_{i+1}s_is_{i+1}v=v,
$$
a contradiction. Hence \b[the proposed tuple is not in $\operatorname{Spec}(n)$]. The same argument, with the signs reversed, excludes $(a,a-1,a)$ as well.
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