Finite-budget undamped self-consistent projector iteration

Input: $n,r$, a response matrix map $F:U\to {\mathbb{H}}_n$ and its occupied projector map $\mathrm{\Phi }$ from The occupied-projector response map on its strict-gap domain, an invariant set $K\subseteq U$, an initial projector $P_0\in K\cap {\mathcal{P}}_r$, a required gap $g_{min}>0$, residual tolerance $\epsilon \ge 0$, update budget $N\in {\mathbb{N}}_0$, and a Hermitian eigensolver. On each input $F(P_k)$ the eigensolver must return either a certified ascending eigenvalue list and, when the gap is strict, the corresponding occupied projector $\mathrm{\Phi }(P_k)$, or a failure to certify the spectrum; assume $\mathrm{\Phi }(P)\in K$ whenever $P\in K$ and its gap is at least $g_{min}$. Use the Frobenius metric and the fixed-point interface of Fixed-point iteration with a residual stop and finite budget.

For $k=0,1,\dots ,N$:

1. Call the eigensolver on $F(P_k)$. If it fails to certify the ordered spectrum, return $(P_k,\mathtt{\text{eigensolve-failure}},\mathrm{\varnothing },k)$. If the certified gap satisfies ${\lambda }_{r+1}(F(P_k))-{\lambda }_r(F(P_k))<g_{min}$, return $(P_k,\mathtt{\text{gap-loss}},\mathrm{\varnothing },k)$.
2. Set $P_{k+1}^{raw}$ to the returned occupied projector $\mathrm{\Phi }(P_k)$ and compute ${\rho }_k=\mathrm{\parallel }{P}_k-P_{k+1}^{raw}{\mathrm{\parallel }}_F$. If ${\rho }_k\le \epsilon$, return $(P_k,\mathtt{\text{residual-tolerance-met}},{\rho }_k,k)$.
3. If $k=N$, return $(P_k,\mathtt{\text{iteration-budget-exhausted}},{\rho }_k,k)$. Otherwise set $P_{k+1}=P_{k+1}^{raw}\in K$ and continue.

It reports a residual stopping status, gap loss, eigensolver failure, or budget exhaustion separately. A finite budget does not guarantee that the residual tolerance will be met, and the definition asserts no convergence from damping or from a positive gap alone.