Finite-memory capped DIIS projector proposal with fallback

Input: the current pair $(P_k,F(P_k))$ with $P_k\in {\mathcal{P}}_r$ and $F(P_k)$ Hermitian; up to $m-1$ earlier pairs $(P_i,F(P_i))$, in chronological order, with $P_i\in {\mathcal{P}}_r$ and $F(P_i)$ Hermitian; memory cap $m\ge 1$; coefficient cap $C\ge 1$; a constrained least-squares solver that either returns a globally minimizing coefficient vector selected by minimum Euclidean norm among all minimizers, with certificates for feasibility and optimality, or reports failure; and a Hermitian eigensolver for mixed Fock matrices. The eigensolver returns a certified ordered spectrum and, when the $r$th gap is strict, its occupied projector, or reports failure. The procedure is one proposal step:

1. Form each commutator residual $R_i=[F(P_i),P_i]:=F(P_i)P_i-P_{i}F(P_i)$ for the earlier pairs followed by the current pair. This gives at most $m$ residuals.
2. Call the constrained solver on these residuals with $c\in {\mathbb{R}}^s$, ${\sum }_i{c}_i=1$, and $\mathrm{\parallel }c{\mathrm{\parallel }}_1\le C$. If it reports failure, including a singular Gram-system failure, return $(P_k,\mathtt{\text{singular-or-uncertified-ls}})$; the caller then performs the baseline SCF update. The returned vector must be the certified global minimizer with the minimum-norm tie-break guaranteed to exist by Capped DIIS coefficients have a unique minimum-norm least-squares selection.
3. If the returned coefficients fail either constraint or their feasibility certificate, return $(P_k,\mathtt{\text{coefficient-cap-failure}})$; the caller then performs the baseline update. Otherwise form $F_{mix}={\sum }_i{c}_{i}F(P_i)$ and call the eigensolver on it. If the eigensolver fails or does not certify a strict $r$th gap, return $(P_k,\mathtt{\text{candidate-gap-failure}})$; the caller then performs the baseline update.
4. If the eigensolver certifies a strict gap, return $(Y,\mathtt{\text{candidate}})$, where $Y$ is its occupied projector ${\mathrm{\Pi }}_r(F_{mix})$.

The commutator least-squares objective defines a proposal; it is not a fixed-point error certificate. The outer safeguarded procedure checks $\mathrm{\parallel }Y-\mathrm{\Phi }(Y){\mathrm{\parallel }}_F$ separately before accepting a candidate. This algorithm makes no bare DIIS convergence claim.