Schrödinger and Klein–Gordon oscillators in Eddington-inspired Born-Infeld gravity: Degree-one confluent Heun polynomial correspondence

We investigate Schrödinger and Klein–Gordon (KG) oscillators in the spacetime of a global monopole (GM) within Eddington inspired Born-Infeld (EiBI) gravity, including, in the relativistic sector, the coupling to a Wu-Yang magnetic monopole (WYMM). By reducing the radial equations to the confluent Heun form and enforcing termination of the Heun series, we obtain conditionally exact solutions in which the radial eigenfunctions truncate to polynomials of degree [Formula: see text]. This truncation imposes algebraic constraints on the oscillator frequency and restricts the allowed maximum value for the orbital angular momentum quantum number [Formula: see text] for each [Formula: see text]. In the lowest nontrivial case [Formula: see text], the degree-one Heun polynomial yields a closed analytic expression for the frequency and determines a finite upper bound on [Formula: see text], dictated jointly by the EiBI deformation and the GM deficit. The resulting parametric correlations reveal a sharp geometric control of the spectrum: EiBI nonlinearities and the angular deficit fix the admissible bound states through polynomial truncation conditions. The confluent Heun correspondence is made explicit, providing a rigorous and reproducible framework for extracting analytical solutions from otherwise non-polynomial Heun structures. Applying the same method to the KG oscillator with a WYMM, we derive conditionally exact particle and antiparticle energies in a closed form. The relativistic spectrum exhibits perfect charge symmetry and a precise dependence on the WYMM strength, the EiBI parameter and the angular momentum constraint. To the best of our knowledge, this constitutes the first unified and fully consistent treatment of conditionally exact Schrödinger and Klein–Gordon oscillator solutions in EiBI gravity based on a degree-one confluent Heun polynomial.