Finite-budget restarted Arnoldi and block LOBPCG return a status under finite-return primitives

For thick or implicit restarted Arnoldi, one cycle has at most $m$ Arnoldi steps. The thick restart processes finite matrices and blocks using finite exact rank-revealing orthogonalization loops, and its block-Arnoldi queue processes at most $m$ basis columns once each; an unavailable target selection, a missing target rule, or a wrong-size or dependent target block returns rank-loss. An implicit restart with no locked pairs uses exactly $p=m-r$ shifts and finite QR steps; an unavailable or wrong-length shift list returns shift-count-unavailable. The implicit-mode thick fallback with locked pairs uses the same finite target and orthogonalization steps. The Arnoldi projected-solver contract makes its call finite-return; the finite shape, residual and coefficient checks either produce a valid current Ritz list or return one of its two listed projected-solve failure statuses. Under [A1] an exact returned eigensolve passes those checks. Assumption [A1] makes the other invoked rules and projected eigensolves finite-return. Thus each cycle either returns a listed status or reaches its single restart-counter update.

For block LOBPCG, every pass through steps 1--4 is finite: its trial block has finitely many columns, its $B$-orthogonalization processes at most those columns, and the projected eigensolve and finite-dimensional operations return in finite work by [A1]. A pass either returns a status or completes one update and increments $k$.

The counter begins at $0$ and increases by one before each further cycle. After at most $N_{\max }$ such increments, the next completed cycle reaches the test $c=N_{\max }$ and returns budget-exhausted unless an earlier acceptance or breakdown status has already returned. The number of cycles is therefore at most $N_{\max }+1$, so the Arnoldi execution terminates.

Starting from $k=0$, no more than $N_{\max }$ updates can occur. After those updates the next pass tests $k=N_{\max }$ and returns budget-exhausted, unless convergence or rank loss already returned. Hence the block LOBPCG execution terminates after finitely many operations.