Finite-budget certified SCF with safeguarded DIIS and a contractive fallback

Input: an open domain $U$, a Hermitian response matrix map $F:U\to {\mathbb{H}}_n$, and its gapped occupied-projector map $\mathrm{\Phi }$ from The occupied-projector response map on its strict-gap domain; a nonempty complete invariant set $K\subseteq U\cap {\mathcal{P}}_r$ with $\mathrm{\Phi }(K)\subseteq K$ and certified contraction factor $0\le q<1$; an initial projector $P_0\in K$; a certified initial error bound $E_0\ge \mathrm{\parallel }{P}_0-P_{\ast }{\mathrm{\parallel }}_F$ for the unique fixed point $P_{\ast }$; residual tolerance $\epsilon \ge 0$; finite update budget $N\in {\mathbb{N}}_0$; memory $m\ge 1$; coefficient cap $C\ge 1$; a Hermitian eigensolver that, for a matrix $M$, returns a certified ordered spectrum and occupied projector when the $r$th gap is strict, a certified no-gap result, or a failure to certify the spectrum; an approximate response evaluator that at $X\in K$ returns $\tilde{\mathrm{\Phi }}(X)\in K$ and $\eta (X)\ge 0$ with $\mathrm{\parallel }\tilde{\mathrm{\Phi }}(X)-\mathrm{\Phi }(X){\mathrm{\parallel }}_F\le \eta (X)$, or a failure status; and a constrained least-squares solver that either returns the certified global minimizer with the minimum-norm tie-break for the residuals and cap $C$, or reports failure. Every called solver returns one of its specified results.

The terminal output is $(P_k,\mathtt{\text{status}},b,k)$, where $P_k$ is the current projector and $b$ is its certified error bound. The current budget $E_k$ remains valid at gap, eigensolver, response-evaluation, or invariance-certificate failure statuses.

For $k=0,\dots ,N$:

1. If no certified response evaluation is cached for $P_k$, compute $F(P_k)$ and call the eigensolver on it. If the eigensolver certifies no strict gap, return $(P_k,\mathtt{\text{gap-loss}},E_k,k)$; if it fails to certify the spectrum, return $(P_k,\mathtt{\text{eigensolve-failure}},E_k,k)$. With a certified strict gap, call the approximate response evaluator. If it fails, return $(P_k,\mathtt{\text{response-evaluation-failure}},E_k,k)$. Cache $(F(P_k),\tilde{\mathrm{\Phi }}(P_k),\eta (P_k))$ at exactly $P_k$. Whether this evaluation is new or cached, check that the certified baseline output belongs to $K$; if not, return $(P_k,\mathtt{\text{invariance-certificate-failure}},E_k,k)$. Retain the current pair $(P_k,F(P_k))$ for this proposal step. Set ${\bar{\rho }}_k=\mathrm{\parallel }{P}_k-\tilde{\mathrm{\Phi }}(P_k){\mathrm{\parallel }}_F+\eta (P_k)$. If ${\bar{\rho }}_k\le \epsilon$, return $(P_k,\mathtt{\text{residual-tolerance-met}},{\bar{\rho }}_k\mathrm{/}(1-q),k)$.
2. If $k=N$, return $(P_k,\mathtt{\text{iteration-budget-exhausted}},E_k,k)$.
3. Call Finite-memory capped DIIS projector proposal with fallback with the up-to-$m-1$ preceding stored pairs, the current pair $(P_k,F(P_k))$, the constrained least-squares solver, the eigensolver, $m$, and $C$. For a candidate $Y$, certify $Y\in K$, compute $F(Y)$, certify its strict gap with the eigensolver, and obtain one approximate response evaluation $\tilde{\mathrm{\Phi }}(Y)$ with error $\eta (Y)$. Accept $Y$ only if all certificates succeed and $\frac{\mathrm{\parallel }Y-\tilde{\mathrm{\Phi }}(Y){\mathrm{\parallel }}_F+\eta (Y)}{1-q}\le q{E}_k\mathrm{.}$ If accepted, set $P_{k+1}=Y$ and $E_{k+1}=q{E}_k$.
4. If step 3 does not accept a candidate for any reason, use the certified baseline evaluation already obtained and checked in step 1. In particular, this includes a least-squares failure, coefficient-cap or affine-constraint failure, failed candidate $K$-membership, failed candidate gap, failed candidate response-error certificate, or failed candidate fixed-point residual test. Set $P_{k+1}=\tilde{\mathrm{\Phi }}(P_k)$ and $E_{k+1}=q{E}_k+\eta (P_k)$.
5. Append the current pair $(P_k,F(P_k))$ to the preceding-iterate history, retaining at most the newest $m-1$ preceding pairs. If step 3 accepted $Y$, cache its certified evaluation $(F(Y),\tilde{\mathrm{\Phi }}(Y),\eta (Y))$ as the next iteration's current evaluation; if the baseline branch was used, evaluate its new iterate when the next loop begins. Never attach a response matrix evaluated at one density to a different density.

The DIIS commutator objective is only a proposal score. Acceptance uses the separate fixed-point residual $\mathrm{\parallel }Y-\mathrm{\Phi }(Y){\mathrm{\parallel }}_F$ with its additive eigensolver error certificate. The algorithm makes no unconditional success claim: finite budget exhaustion and each certificate failure are explicit outcomes.