Finite Riccati candidate, Newton-Kleinman, and doubling wrappers return a status

Each candidate-extraction branch calls only its matching finite Schur or QZ routine. The finite controls and declared status behavior are covered by [F1]–[F2]. Any reduction, regularity, range or budget failure immediately propagates its inconclusive status; no convergence-until-success loop is present.

Exact Newton–Kleinman performs at most $N$ policy evaluations. Each admitted Bartels–Stewart call has the complete per-pass control packet and returns a finite success or failure status by [F6]; failure stops immediately and success is followed by one finite gain update. Hence exact mode returns within its declared outer budget.

Each safeguarded outer pass clears finitely many positive integer denominators from the rational current-cost system, preserving its solution while making the stored point-call data dyadic, or returns range-failure. It inspects the finite list of $L+1$ gains and makes one current-cost point-certificate call. Each trial then uses finitely many outward operations to verify its inverse, residual and stability bounds. By [A1] each call and outward check returns its enclosure or status in finite work, with certified balls interpreted by [F4] and [F6]. A trial passes the certified acceptance inequality or is rejected; if no trial passes, the routine returns safeguard-inconclusive without changing the current gain. At most $N$ outer passes therefore make at most $N(L+1)$ trial inspections and $N$ point-certificate calls. No assertion of convergence for arbitrary inexact proposals follows from this finite search.

Each SDA pass $j<N$ requests one solve with $I+G_j{H}_j$ and, in finite mode, at most $n$ point-certificate calls for its columns. The finite-mode initialization solves the order-$m$ system $RV=B^{\ast }$ on finite dyadic encodings by finite exact rational elimination, forms $BV,C^{\ast }C$, and rounds finitely many components or returns its declared failure; it makes no point-certificate call. At initialization and after each update, form $S_j=R+B^{\ast }{H}_{j}B$ exactly. In exact mode its possibly non-dyadic entries are handled by the finite exact elimination rule, as are the recurrence solves; the residual is then evaluated exactly. In finite mode $S_j$ is dyadic, so its order-$m$ residual solve uses at most $m$ point-certificate calls followed by outward residual evaluation. By [A1] and [F6] all these operations return a result or failure in finite work. An unavailable residual is reported without changing recurrence status. Failure of the recurrence solve immediately returns its declared range/inverse status; success is followed by the fixed finite matrix updates. Thus every pass and final report terminate.

On successful reduction, reordering inspects finitely many blocks and returns ordered or an explicit status by [F3]. On an ordered return its first required factor columns are the selected subspace of the represented computed form, with only the original unstructured residual certificate transported. Under [A1] the graph-block bound and verified solve [F4] terminate; missing invertibility or a failed solve takes the stated inconclusive branch. Residual, symmetry, positivity and closed-loop diagnostics use finitely many matrix/enclosure operations, and failure or unavailable evidence takes its explicit return. The enclosure interpretation in [F4] does not itself produce a terminating verifier; that is the supplied contract [A1]. The QZ diagnostic retains precisely the limitation in [F5]. Thus the wrapper returns in finite work with ordering and equation-level acceptance kept separate.

There are at most $N$ such passes. The final symmetry defects and, when its solve is available, the standard DARE residual use finite-dimensional arithmetic under [A1] and the conventions [F7]. Therefore the exact and finite-arithmetic paths return their declared output/status by the cap. The recorded raw matrices and defects supply no rounded PSD or structured-backward-error claim without separate checks, just as the candidate and safeguarded branches retain their original acceptance restrictions.