Cubic Geometric Inflation with Radiation and a Minimally Coupled Scalar Field: Dynamical-Systems Analysis and Inflationary Diagnostics

We investigate the background dynamics of cubic Geometric Inflation coupled with radiation and a minimally coupled canonical scalar field with an exponential potential, using the cubic truncation as a minimal test of geometric-to-scalar inflationary cascades. We formulate the cosmological equations as an autonomous dynamical system and analyze its finite critical points and representative Hubble-flow trajectories. The cubic sector can generate an initial geometrically accelerated phase, but its duration grows only logarithmically with the initial curvature hierarchy and is insufficient to solve the horizon problem for moderate initial conditions. The subsequent scalar-field-dominated stage obeys the power-law relation r≃8(1−ns) under the standard GR-based first-order mapping, placing it in strong tension with current bounds on primordial tensor modes. Moreover, no value of the constant exponential-potential slope yields a graceful exit into a standard radiation-dominated epoch. The regular branch requires trans-Planckian radiation densities in the asymptotic past, whereas the alternative branch contains a finite-curvature dynamical degeneracy. Thus, although cascade-like background trajectories exist, the minimal cubic model does not satisfy the basic background requirements of a viable inflationary cosmology.