  We present a detailed investigation of the anomalous gravitational amplitude
in a simple two-dimensional model with Weyl fermions. We employ a mathematical
strategy that completely avoids any regularization prescription for handling
divergent perturbative amplitudes. This strategy relies solely on the validity
of the linearity of the integration operation and avoids modifying the
amplitudes during intermediate calculations, unlike studies using
regularization methods. Additionally, we adopt arbitrary routings for internal
loop momenta, representing the most general analysis scenario. As expected, we
show that surface terms play a crucial role in both preserving the symmetry
properties of the amplitude and ensuring the mathematical consistency of the
results. Notably, our final perturbative amplitude can be converted into the
form obtained using any specific regularization prescription. We consider three
common scenarios, one of which recovers the traditional results for
gravitational anomalies. However, we demonstrate that this scenario inevitably
breaks the linearity of integration, leading to an undesirable mathematical
situation. This clean and transparent conclusion, enabled by the general nature
of our strategy, would not be apparent in similar studies using regularization
techniques.
