\begin{abstract}
Let $\A$ be a finite alphabet with a fixed-point-free involutive complement,
and let reverse complement act on the directed de Bruijn graph $B(q,k)$ by
reversing a word and complementing every letter.  We prove that a
reverse-complement-invariant ordinary minimum decycling set exists if and
only if $k$ is even, for every integer $k\geq2$.  This simultaneously covers
the binary alphabet and the DNA alphabet with $A\leftrightarrow T$ and
$C\leftrightarrow G$.

For odd $k$, a primitive rotation cycle joins two reverse-complement-fixed
lower states in opposite directions. Disjoint complementary alphabet pairs
force at least $q$ deletions beyond the ordinary minimum. For even $k$,
choose injective antisymmetric real weights on
the alphabet and form a weighted cyclotomic selector.  We prove that
arbitrary representatives on its zero-embedding rotation cycles preserve
acyclicity, resolve those ties equivariantly, and establish the exact defect
identity
\[
  \ind_{\RC(M)}-\ind_M=\nabla\chi .
\]
A finite sequence of legal full-alphabet source firings removes this
nonnegative symmetric defect two coordinates at a time.  We also identify a
self-dual spectral ideal that gives a deterministic direct selector, rather
than a simulated firing path.  For integer weights, membership is decidable
with polynomial bit complexity.  The executable finite panel in the repository
checks the binary specialization only; it is convention-level assurance and
is not used in the all-order proof.
\end{abstract}