\section{Legal defect descent and even existence}
\label{sec:even}

Reverse complement carries an ordinary MDS to an ordinary MDS because it
reverses directed cycles bijectively.

\begin{lemma}[Positive-potential firing]
\label{lem:positive-firing}
Let $A$ and $N$ be ordinary MDSs and suppose
\[
  \ind_N-\ind_A=\nabla f
\]
for an integral potential $f\geq0$ with minimum zero.  If $A\neq N$, some
lower state $u$ with $f(u)>0$ admits a valid full-alphabet $F$-move from $A$.
\end{lemma}

\begin{proof}
Assume otherwise.  At every positive state $u$, failure of the source
condition supplies an incoming residual arc $v\to u$.  It may be chosen
nonloop: a residual loop would already contradict that $A$ is decycling.
Since this arc is not in $A$,
\[
  \ind_N(v\to u)=f(v)-f(u)\in\{0,1\},
\]
so $f(v)\geq f(u)>0$.  Iterating the predecessor choice within the finite
positive support eventually produces a directed residual cycle, a
contradiction.
\end{proof}

\begin{lemma}[Defect-paired update]
\label{lem:paired-update}
Suppose an ordinary MDS $A$ satisfies
\[
  \ind_{\RC(A)}-\ind_A=\nabla f,\qquad
  f\geq0,\qquad f\circ\RC=f,\qquad \min f=0.
\]
If $A$ is not invariant, fire an enabled positive state $u$ supplied by
Lemma~\ref{lem:positive-firing}.  The new MDS $A'=F_u(A)$ satisfies the same
conditions with
\[
  f'=f-\ind_u-\ind_{\RC(u)}.
\]
Moreover $u\neq\RC(u)$ and $\sum f'=\sum f-2$.
\end{lemma}

\begin{proof}
The physical firing gives $\ind_{A'}-\ind_A=\nabla\ind_u$.  Reverse
complement reverses lower arcs, hence sends this gradient to
$-\nabla\ind_{\RC(u)}$.  Subtracting the update from its image gives
\[
  \ind_{\RC(A')}-\ind_{A'}
  =\nabla\bigl(f-\ind_u-\ind_{\RC(u)}\bigr).
\]
The lower length $k-1$ is odd, so a reverse-complement-fixed lower word would
have a central letter fixed by complement; therefore $u\neq\RC(u)$.  Symmetry
of $f$ makes both decremented coordinates positive.  Nonnegativity,
symmetry, and minimum zero are preserved, and the total decreases by two.
\end{proof}

\begin{theorem}[Even-order descent]
\label{thm:even-descent}
For every even $k\geq4$, the seed $M_0$ reaches a literal
reverse-complement-invariant ordinary MDS by legal full-alphabet $F$-moves.
Every scheduler that repeatedly uses Lemma~\ref{lem:positive-firing} executes
exactly
\begin{equation}
  \frac12\bigl|\{u\in\A^{k-1}:\lambda_s(u)>0\}\bigr|
  \label{eq:move-count}
\end{equation}
physical firings.
\end{theorem}

\begin{proof}
Theorem~\ref{thm:coboundary} initializes the symmetric nonnegative potential
$f=\chi$.  Its minimum is zero: every homopolymer lower word $a^{k-1}$ has
\[
  \lambda_s(a^{k-1})
  =s(a)\sum_{j=1}^{k-1}\sin(2\pi j/k)=0.
\]
While the current MDS is not invariant, Lemmas
\ref{lem:positive-firing}--\ref{lem:paired-update} supply another legal move
and decrease $\sum f$ by two.  The process terminates.

At an invariant terminal, $\nabla f=0$.  The lower de Bruijn graph is
strongly connected---append the letters of any target word---so $f$ is
constant.  Its preserved minimum is zero, hence $f=0$.  Conversely the
potential cannot vanish before the indicator defect does.  Starting from the
$0$--$1$ potential $\chi$ therefore gives exactly~\eqref{eq:move-count}
moves.
\end{proof}

The theorem asserts neither uniqueness of the terminal, confluence, nor
shortest distance among unrestricted firing paths.  A ``paired update'' is
one physical firing whose defect potential changes at two lower states, not
two simultaneous firings.