Every finite-budget QZ run returns a status and records unitary transforms and explicit deflations

The initialization uses finitely many Givens loops by [F2]. In a QZ sweep, the first left rotation is chosen from the computed first two components of $(\beta H-\alpha T)e_0$, whose only possibly nonzero entries have relative indices $0,1$. The following right rotation removes the new $T_{1,0}$ entry and can create only the bulge $H_{2,0}$.

For an infinite-deflation step with $T_{k,k}=0$, the descending right rotations make $T_{j,j}=0$ for $j=k-1,\dots ,a$. At each such stage, $T_{j+1,j}=T_{j+1,j+1}=0$, so the right rotation does not create a subdiagonal entry in $T$. It can create only $H_{j+2,j}$, which the paired left rotation zeros; that left rotation preserves triangularity because the affected entries of column $j+1$ of $T$ are both zero. The final left rotation zeros $H_{a+1,a}$ while column $a$ of $T$ remains zero, isolating the top one-by-one pair $(H_{a,a},0)$. The mandatory post-isolation scan reports the $(0,0)$ branch as singular-pencil-suspected, so it is not assigned an eigenvalue. The targeted zeros are explicit assignments; applying rounded rotations to other already-zero coordinate pairs preserves them exactly.

There are finitely many initialization loops, at most $2n$ rotations per infinite isolation or per active-block sweep, at most $n$ successful infinite removals, and at most $N$ sweeps over at most $n$ active blocks. An unsuccessful isolation exits rather than repeating indefinitely. Each residual/controller evaluation uses finite prescribed factor/column loops. Every scalar operation either succeeds or exits with range-failure; missing unitarity or arithmetic certificates exit with certificate-inconclusive. Thus every branch terminates with a declared status.

If the current bulge is $H_{j+1,j-1}$, [F3] supplies a left Givens factor on rows $j,j+1$ that zeros it. This can create the single $T_{j+1,j}$ entry; a right Givens factor on columns $j,j+1$ zeros that entry and can create only the next Hessenberg bulge $H_{j+2,j}$. The loop ends at the last active row, so the sweep ends Hessenberg-triangular. Computed target entries are explicitly zeroed with [F5], so this support induction also holds in rounded storage. Structural writes are included in their arithmetic allowance; only separate tolerance writes enter the deflation ledger.

For a run with certified factors, each local update has the form $A_j=Q_j^{\ast }(A_{j-1}+E_j^A+D_j^A)Z_j$ and $B_j=Q_j^{\ast }(B_{j-1}+E_j^B+D_j^B)Z_j$, where $E_j$ includes the structural target write. A separate tolerance write replacing an entry $d$ by zero has identity factors and perturbation $-d$ at that entry, with all other entries zero; these are the algorithm's $D_j^A,D_j^B$. Any post-action perturbation is pulled back as required by [F5]. Applying [F4] to this finite list gives $S=Q^{\ast }(A+{\mathrm{\Delta }}_A)Z$ and $T=Q^{\ast }(B+{\mathrm{\Delta }}_B)Z$ for the accumulated pair $({\mathrm{\Delta }}_A,{\mathrm{\Delta }}_B)$. No bound is asserted without local norm certificates.

Before qz-computed, both arrays are triangular and all diagonal pairs are nonzero. Their homogeneous determinant is ${\prod }_k(\beta S_{kk}-\alpha T_{kk})$, a product of nonzero linear polynomials, hence is not identically zero. The returned pencil is regular; any label with second coordinate zero is an infinite candidate of this returned perturbed pencil. This proves no exact original-input infinite-eigenvalue or forward-error claim.