Minimum reverse-complement-invariant decycling sets

Theorem 1.1 claims, for every finite nonempty alphabet with a fixed-point-free complement and every $k\ge2$, $$\tau_{\rm RC}(q,k)=N_q(k)\quad\Longleftrightarrow\quad k\text{ is even}.$$ For odd $k\ge3$, Theorem 3.1 gives $\tau_{\rm RC}(q,k)\ge N_q(k)+q$, hence an excess of at least two for binary words and four for DNA. For even orders, Sections 6-8 provide a weighted construction and a direct spectral selector; integer weights permit exact polynomial-bit membership testing for a single word. The case $k=2$ is handled separately. This completely classifies attainability of the ordinary minimum and partially answers the external symmetric-minimum question. It does not determine the general exact odd optimum, prove a packing-covering equality, give quantitative residual-path/window bounds, or establish biological performance. The classical ordinary minimum itself is prior work.

Theorem 1.1 claims, for every finite nonempty alphabet with a fixed-point-free complement and every $k\ge2$, $$\tau_{\rm RC}(q,k)=N_q(k)\quad\Longleftrightarrow\quad k\text{ is even}.$$ For odd $k\ge3$, Theorem 3.1 gives $\tau_{\rm RC}(q,k)\ge N_q(k)+q$, hence an excess of at least two for binary words and four for DNA. For even orders, Sections 6-8 provide a weighted construction and a direct spectral selector; integer weights permit exact polynomial-bit membership testing for a single word. The case $k=2$ is handled separately. This completely classifies attainability of the ordinary minimum and partially answers the external symmetric-minimum question. It does not determine the general exact odd optimum, prove a packing-covering equality, give quantitative residual-path/window bounds, or establish biological performance. The classical ordinary minimum itself is prior work.