I compute the solutions
that shouldn't exist.

This study investigates the two-dimensional axisymmetric flow and heat transfer of a Cu-Al₂O₃/water hybrid nanofluid driven by a permeable, radially stretching/shrinking disk in a thermally stratified medium. The analysis incorporates the combined effects of a uniform transverse magnetic field, thermal radiation, velocity slip at the surface, Joule heating, and wall mass suction. The hybrid nanofluid is modeled through the Tiwari and Das framework together with established thermophysical correlations for two dissimilar nanoparticles suspended in water. The governing partial differential equations for momentum and energy are reduced to a coupled system of nonlinear ordinary differential equations by means of suitable similarity transformations and solved numerically using the bvp4c routine in MATLAB, a finite difference scheme based on the three-stage Lobatto IIIa formula. Dual (first and second branch) solutions are found to exist for a specific range of the shrinking parameter provided that sufficient mass suction is imposed. The influence of the magnetic parameter, copper volume fraction, Eckert number, velocity slip parameter, and radiation parameter on the dimensionless velocity and temperature distributions is examined for both solution branches. The results indicate that increasing the magnetic parameter accelerates the flow and reduces the temperature in the first solution, while the second branch exhibits the opposite trend. Higher radiation parameter and Eckert number significantly enhance the thermal field, with the lower branch attaining larger temperatures. The findings are relevant to heat transfer enhancement in cooling technologies, rotating disk systems, and thermally stratified industrial processes.