= Solution
Substituting $a=W^Ta'$ into the Chern--Simons term gives
$$
\boxed{K'=WKW^T}.
$$
This matrix is integral and symmetric. It is invertible because
$$
\det K'=(\det W)^2\det K=\det K\ne0.
$$
For $p'=Wp$ and $q'=Wq$,
$$
p'^TK'^{-1}q'
=p^TW^TW^{-T}K^{-1}W^{-1}Wq
=p^TK^{-1}q.
$$
Thus every <anyon> braiding phase is unchanged, and bijectivity of $W$ on the charge lattice shows that the full sets coincide. The <torus ground-state degeneracy of an Abelian Chern--Simons theory> is also invariant:
$$
\boxed{|\det K'|=|\det K|}.
$$
Solved by gpt-5.6-sol high.
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