Finite-budget shift-invert and harmonic extraction return a status under finite-return primitives

In the shift-invert branch, the Arnoldi loop has exactly $m$ requested solves unless an earlier solve reports unavailable or exact breakdown returns. The Ritz extraction has finitely many candidates (at most $m$); each candidate invokes at most one additional solve. By [A1], the remaining operations return their values in finite work, while each unavailable solve explicitly returns linear-solve-unavailable. Therefore this branch terminates.

In the harmonic branch, forming $W,G,C$ and computing the rank of $W$ use finite-dimensional operations by [A1]. If the rank is below $k$, the algorithm returns projected-pencil-unresolved before QZ. If the rank is $k$, its statement gives a regular projected pencil and calls QZ with the finite budget $N$; [F3, F4] then give a QZ status after finitely many operations under the assumed QZ hypotheses.

After a QZ return, the algorithm maps each possible QZ status to one of its own listed statuses. On qz-computed, it inspects only the finitely many diagonal pairs of the finite projected pencil and either returns on an unresolved pair or completes the candidate list. Thus the harmonic branch also terminates, proving the assertion for both branches.