  The basic objective of this investigation is to explore the impact of a novel
gravitational modification, specifically, the $f(\mathcal{G}, \mathbf{T}^2)$
(where $\mathbf{T}^2 \equiv T_{\alpha\beta}T^{\alpha\beta}$, $T^{\alpha\beta}$
denotes the stress-energy tensor) model of gravitation, upon the complexity of
time-dependent dissipative as well as non-dissipative spherically symmetric
celestial structures. To find the complexity factor $(\mathbb{C}_{\mathbf{F}})$
from the generic version of the structural variables, we performed Herrera's
scheme for the orthogonal cracking of Riemann tensor. In this endeavor, we are
mainly concerned with the issue of relativistic gravitational collapse of the
dynamically relativistic spheres fulfilling the presumption of minimal
$\mathbb{C}_{\mathbf{F}}$. The incorporation of a less restrictive condition
termed as quasi-homologous $(\mathbb{Q}_{\mathbf{H}})$ condition together with
the zero $\mathbb{C}_{\mathbf{F}}$, allows us to formulate a range of exact
solutions for a particular choice of $f(\mathcal{G}, \mathbf{T}^2)$ model. We
find that some of the given exact solutions relax the Darmois junction
conditions and describe thin shells by satisfying the Israel conditions, while
some exhibit voids by fulfilling the Darmois constraints on both boundary
surfaces. Eventually, few expected applications of the provided solutions in
the era of modern cosmology are debated.

General Relativity and Quantum Cosmology;High Energy Physics - Theory