$\mathcal{P}\mathcal{T}$ symmetric Klein-Gordon oscillators in Lorentz-violating wormholes

We study spin-0 $\mathcal{P}\mathcal{T}$-symmetric Klein-Gordon (KG) oscillator fields in a (3+1)-dimensional Lorentz-violating (LV) traversable wormhole background. The wormhole geometry, characterized by a smooth minimal throat $a$ and a regular lapse sector, induces a curvature-driven deformation of the relativistic quantum dynamics under Lorentz symmetry breaking. A nonminimally coupled non-Hermitian $\mathcal{P}\mathcal{T}$ symmetric scalar bosonic field ${\mathcal{F}}_t(x)=i\tilde{\mathrm{\Omega }}x$, with $\tilde{\mathrm{\Omega }}=\mathrm{\Omega }\mathrm{/}\sqrt{1-\zeta }$, generates a globally regular effective $\mathcal{P}\mathcal{T}$ symmetric KG oscillator free of centrifugal singularities. The radial equation reduces to a confluent Heun system, admitting bound states only under polynomial truncation conditions that impose constraints among $(\mathrm{\Omega },a,\zeta ,n,\mathrm{\ell })$. The resulting energy spectrum is real, discrete and exactly symmetric under $E\to -E$, determined by spacetime curvature, Lorentz-violation parameter $\zeta$, and oscillator strength $\mathrm{\Omega }$, within a conditionally exactly solvable framework.