Finite Schur–Parlett recurrence with block merging and inconclusive statuses

Input a finite complex matrix $A\in {\mathbb{C}}^{n\times n}$, $n\ge 1$, a holomorphic function $f$ on a neighborhood of $\sigma (A)$ with the finite derivative data required by the diagonal block evaluators, Schur controls, a Sylvester separation threshold $\tau \ge 0$, an initial partition of $\{1,\dots ,n\}$ into nonempty contiguous intervals, and a finite table of exact Jordan certificates indexed by contiguous diagonal block intervals. The partition and certificates are supplied inputs; a missing certificate for any current block returns missing-jordan-data.

1. Call Finite-budget computed Francis Schur reduction with explicit statuses in its complex branch. Propagate any non-Schur status. Treat its returned $Q,T$ as candidates.
2. Verify by exact arithmetic that $Q^{\ast }Q=I$, $A=QT{Q}^{\ast }$, and $T$ is upper triangular. If any identity fails, return schur-certificate-failure. The computed Schur routine's residual diagnostic alone does not pass this test.
3. Start from the supplied contiguous block partition of $T$. At each start or restart, initialize every block of $F$ to zero. For each block $T_{ii}$ call Finite analytic evaluation from a supplied exact Jordan decomposition with its exact Jordan certificate. On status exact, store its returned matrix as $F_{ii}$. If a certificate or required derivative is missing, return missing-jordan-data; propagate any other non-exact status without storing a diagonal value.
4. Visit block pairs by increasing distance $j-i$. Form $C_{ij}=F_{ii}{T}_{ij}-T_{ij}{F}_{jj}+{\sum }_{i<k<j}(F_{ik}{T}_{kj}-T_{ik}{F}_{kj})\mathrm{.}$ Call Finite-budget Bartels-Stewart solve for a Sylvester equation on $T_{ii}{F}_{ij}+F_{ij}(-T_{jj})=C_{ij}$ with threshold $\tau$. If it returns success, store its exact solution as $F_{ij}$. If it returns singular-pivot or pivot-separation-failure and the partition has multiple blocks, merge the contiguous blocks from $i$ through $j$, discard their old diagonal data, and restart step 3. Each merge reduces the block count, so there are at most $n-1$ merges. Any other non-success status is returned as inconclusive-sylvester with its underlying status.
5. Return the factored value $QF{Q}^{\ast }$ with status parlett-success.

The procedure makes no floating-point stability or error-bound claim. Its success branch requires exact Schur and local solve certificates.