Real spectra for the separated spherical Robin operators

We prove that the separated spherical-strip operators with PT-symmetric Robin conditions have real spectrum for every half-width $0<a<\pi/2$, every real boundary parameter and every integer Fourier mode. The proof establishes strict interlacing between the associated real Robin and Dirichlet spectra at every index. A positive Laplace representation of the Fourier coefficients of $\sec^2x$ gives the comparison for nonzero integer modes, while a first-order factorization treats the zero mode. Canonical products then give a strict half-plane sign for the characteristic ratio and a bounded inverse at every nonreal spectral parameter.