Finite-budget relaxed density-matrix mixing

Input: $D_0\in {\mathcal{D}}_r$ from The fixed-particle admissible Hermitian density domain, a Hermitian response matrix map $F:U\to {\mathbb{H}}_n$ whose domain contains every entered density iterate, and its occupied-projector map $\mathrm{\Phi }$, defined at the strict-gap points of $U$ as in The occupied-projector response map on its strict-gap domain, a Hermitian eigensolver that for each $F(D_k)$ returns a certified ordered spectrum and occupied projector when the gap is strict, a certified no-gap result, or a failure status, a fixed mixing parameter $\alpha \in [0,1]$, a fixed-point residual tolerance $\epsilon \ge 0$, and a finite update budget $N\in {\mathbb{N}}_0$. For $k=0,\dots ,N$, first call the eigensolver on $F(D_k)$. If it fails, return $(D_k,\mathtt{\text{eigensolve-failure}},\mathrm{\varnothing },k)$; if the gap is absent, return $(D_k,\mathtt{\text{gap-loss}},\mathrm{\varnothing },k)$. Otherwise use the returned occupied projector as $\mathrm{\Phi }(D_k)$ and compute $r_k=\mathrm{\parallel }{D}_k-((1-\alpha )D_k+\alpha \mathrm{\Phi }(D_k)){\mathrm{\parallel }}_F\mathrm{.}$ If $r_k\le \epsilon$, return $(D_k,\mathtt{\text{residual-tolerance-met}},r_k,k)$. If $k=N$, return $(D_k,\mathtt{\text{iteration-budget-exhausted}},r_k,k)$. Otherwise update $D_{k+1}=(1-\alpha )D_k+\alpha \mathrm{\Phi }(D_k)$ and continue.

By Density mixing preserves admissibility but need not preserve idempotence, each successful update stays in ${\mathcal{D}}_r$; it need not be idempotent. This algorithm uses the residual and finite-budget update pattern of Fixed-point iteration with a residual stop and finite budget for the relaxed map, with a gap check before each response evaluation. It makes no convergence claim unless the derivative-bound and invariance hypotheses in A uniform derivative bound certifies contraction of relaxed density mixing are verified. If $\alpha =0$, then the relaxed residual is identically zero and the update does not move; that stop status alone says nothing about the unmixed response residual.