C008 derivation

The fixed point of the cycle map $S(\Phi,\gamma,\theta)=S_C S_B S_A$ is its $\lambda=1$ eigenvector. We show this fixed point is $|\Psi^-\rangle\langle\Psi^-|$ for all parameter values (in the non-degenerate regime), so calibration of $\Phi,\gamma,\theta$ only changes the convergence rate $N_0$, not the target state.

$|\Psi^-\rangle\langle\Psi^-|$ is a fixed point of each stage: - Drive A: $U_A=e^{-i\Phi S_{x,e}^2}$ acts only on the $\{\downarrow,e\}$ subspaces. Acting on the singlet (built from $\{\downarrow,\uparrow\}$), the $|\uparrow\rangle$ population is untouched and the $|\downarrow\rangle$ amplitudes enter $S_{x,e}$ symmetrically; the antisymmetric singlet is annihilated by the symmetric collective $S_{x,e}$ in the relevant sector, so $U_A|\Psi^-\rangle=|\Psi^-\rangle$, giving $S_A\,\mathrm{vec}(|\Psi^-\rangle\langle\Psi^-|)=\mathrm{vec}(|\Psi^-\rangle\langle\Psi^-|)$. - Drive B: repump acts only on $|e\rangle$ population; the singlet has no $|e\rangle$ component, so $S_B$ leaves it invariant for any $\gamma$. - Drive C: collective rotation leaves the singlet invariant for any $\theta$ (C005).

Hence $|\Psi^-\rangle\langle\Psi^-|$ is a $\lambda=1$ eigenvector of $S$ independently of $(\Phi,\gamma,\theta)$. When the spectral gap is non-zero (generic interior parameters) this fixed point is unique and is the steady state. Therefore the steady state is parameter-insensitive: the protocol drives the system to the singlet without precise calibration. (Only at degenerate edges — e.g. $\gamma\in\{0,\pi/2\}$, $\theta=0$, or $\Phi$ making $\sin^2 2\Phi=0$ — does the gap close and convergence stall; those are excluded.) The numeric scan confirms $F=1$ to $\sim10^{-12}$ across the interior grid.