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Induct on the distance between equal entries $a_i=a_j=a$ of a point of the joint spectrum. Consecutive equal entries are impossible by the local spectral rules for Young–Jucys–Murphy elements. It suffices to consider consecutive occurrences of $a$: proving the assertion for each such pair proves it for any wider pair.

Suppose $a-1$ is missing between them. There can be at most one occurrence of $a+1$ in the interval. Otherwise a pair of consecutive $a+1$ entries has a shorter gap, and the induction hypothesis forces an intervening $a$, contrary to our choice of the pair. Move the two endpoint entries $a$ inward past entries different from $a,a-1,a+1$. Every move is an admissible spectral interchange. If there is no $a+1$, this creates consecutive equal entries. If there is one, this creates $(a,a+1,a)$, already excluded by the braid relation in a Coxeter group. Both are impossible.

Interchanging the roles of $a-1$ and $a+1$ excludes the omission of $a+1$ as well. Therefore

First exclude the short patterns $(a,a+1,a)$ and $(a,a-1,a)$ in the joint spectrum. In the first pattern, the local spectral rules for Young–Jucys–Murphy elements make $s_r$ act by $+1$ and $s_{r+1}$ by $-1$ on the same simultaneous eigenvector. The two sides of the braid relation in a Coxeter group then act by opposite signs. The second pattern gives the same contradiction with signs reversed.

Now induct on the length of a spectral vector. A prefix of length $i-1$ is spectral for $S_{i-1}$: decompose the restricted group representation and retain a nonzero component of the simultaneous eigenvector. If $a_i$ differed by neither $+1$ nor $-1$ from every earlier entry, move it left using the allowed adjacent interchanges. Encountering an equal entry would contradict the distinctness of consecutive eigenvalues. Otherwise it reaches the first position, forcing $a_i=0$. For $i>1$ the original first entry was also zero, so an equal entry would indeed have been encountered. ThusThis argument uses only the stated local rules. In particular, iterating the result from $a_1=0$ also shows that all coordinates are integers.

The first Young–Jucys–Murphy element is the empty sum $X_1=0$. Applying it to the nonzero simultaneous eigenvector defining a point of the joint spectrum gives $0=a_{1}v$. HenceFor the remaining spectral arguments use the following local facts, without assuming the restriction branching rule for a symmetric group: consecutive eigenvalues are distinct; a difference $a_{r+1}-a_r=\pm 1$ makes $s_r=(rr+1)$ act on the corresponding line as $\pm 1$; every other adjacent interchange gives another point of the joint spectrum in the same irreducible representation. These are the local spectral rules for Young–Jucys–Murphy elements. They follow from $s_r{X}_r{s}_r+s_r=X_{r+1}$ and the one- or two-dimensional analysis of the two neighboring eigenspaces. They are the appropriate local results used below.