Cooperative Boundary Switching of Mesoscopic Real Spectra in a Non-Hermitian Chain Family
At fixed bulk criticality, every fixed nonsingular real single-end potential preserves a growing local population of real eigenvalues in an open analytic family of non-Hermitian chains. An open region of two-end real potentials removes this population. A boundary-amplitude criterion fixes the minimum number of controlled ends. The population grows as $\sqrt N$, where $N$ is chain length, although its normalized spectral weight vanishes and the local limiting bands remain unchanged. For a representative chain's four zero-detuning settings, an independent passive Hermitian double chain converts this distinction into separated resonances at one fixed local port under a common coupling scan at fixed probe carrier. Its absolute reflection contrast and stability margin both scale as $N^{-1}$, with explicit sufficient resource budgets. Boundary amplitudes thus fix a minimum control requirement that limiting bands and thermodynamic spectral density do not specify.