  We study the mechanism of topological mass-generation for 3d Chern-Simons
(CS) gauge theories, where the CS term can retain the gauge symmetry and make
gauge boson topologically massive. Without CS term the 3d massless gauge boson
has a single physical transverse polarization state, while adding the CS term
converts it into a massive physical polarization state and conserves the total
physical degrees of freedom. We newly formulate the mechanism of topological
mass-generation at $S$-matrix level. For this, we propose and prove a new
Topological Equivalence Theorem (TET) which connects the $N$-point scattering
amplitude of the gauge boson's physical polarization states ($A^a_{\rm{P}}$) to
that of the transverse polarization states ($A^a_{\rm{T}}$) under high energy
expansion. We present a general 3d power counting method on the leading energy
dependence of $N$-point scattering amplitudes in both topologically massive
Yang-Mills (TMYM) and topologically massive gravity (TMG) theories. With these,
we uncover a general energy cancellation mechanism for $N$-gauge boson
scattering amplitudes which predicts the cancellation $E^4 \to E^{4-N}$ at tree
level. Then, we compute the four-point amplitudes of $A^a_{\rm{P}}$'s and of
$A^a_{\rm{T}}$'s, with which we explicitly demonstrate the TET and establish
such energy cancellations. We further extend the double-copy approach and
construct the four-point massive graviton amplitude of the TMG theory from the
massive gauge boson amplitude of the TMYM theory. With these, we newly uncover
striking large energy cancellations $E^{12}\to E^1$ in the four-graviton
amplitude of the TMG, and establish its new correspondence to the leading
energy cancellations $E^4 \to E^0$ in the four-gauge boson amplitude of the
TMYM.
