Finite SQP status is sound and the exact local branch inherits OPT6 conditions

By [F1], a certified return contains a computed linear solve residual bound $\eta \le \tau$. Applying [F2] gives the exact inequality $\mathrm{\parallel }Ks+F_A(z){\mathrm{\parallel }}_2\le \eta \le \tau$. The theorem supplies no smaller bound unless the reported $\eta$ is smaller.

Write the quadratic objective as $\varphi (x)=\frac{1}{2}{x}^{T}Qx+c^{T}x+c_0$ with symmetric $Q$, and stack all equalities and active inequalities as $c_A(x)=Gx-d$. With their stacked multiplier $\mu$, the algorithm's convention gives $F_A(x,\mu )=(Qx+c+G^T\mu ,Gx-d)$ and $K=\left[\begin{array}{cc}Q& G^T\\ G& 0\end{array}\right]$. Thus $F_A$ is affine in the combined primal-dual variable and $F_A(z+s)=F_A(z)+Ks$. Step 1.1 therefore bounds the returned point's active KKT residual. Affine box inequalities alone do not give this identity when an equality constraint is nonlinear.

In the separate exact local branch, the hypotheses listed in [F3] imply the active KKT Jacobian is nonsingular and locally Lipschitz. The cited OPT6 Newton theorem then applies to the exact full-step active-system iteration started sufficiently near its root and gives quadratic convergence. This conclusion is conditional on those hypotheses and is independent of finite-budget status.