% Section: Introduction
\section{Introduction}\label{sec:intro}

Zero forcing was introduced by the AIM Minimum Rank -- Special Graphs Work Group \citeyearpar{aim2008zeroforcing} as a combinatorial upper bound on the maximum nullity $M(G)$ of the real symmetric matrices whose off-diagonal pattern is prescribed by a graph $G$: every zero-forcing set $S$ satisfies $M(G)\le |S|$, hence $M(G)\le \Z(G)$, and the inequality is tight for many families \citep{barioli2010parameters,hogben2022book}. Computing $\Z$ is NP-hard in general, and exact values on structured infinite families are usually obtained by pairing an explicit forcing construction with a matching lower-bound argument. Such arguments are delicate: a figure-based case analysis for the generalized Petersen graphs $\PP{n}{3}$ \citep{rashidi2020petersen} was recently shown to fail at $n=12$ \citep{arxiv2607_19412}, and the sharp results for $\PP{n}{3}$ and $\PP{n}{4}$ were only established in 2026 with machine-checked finite cases \citep{arxiv2610_24059}.

The generalized Petersen graph $\PP{n}{k}$ \citep{watkins1969tait}, for $n>2k$, has vertices $u_0,\dots,u_{n-1}$ (outer cycle) and $v_0,\dots,v_{n-1}$ (inner vertices) and edges $u_iu_{i+1}$, $u_iv_i$ and $v_iv_{i+k}$, indices modulo $n$ (Figure~\ref{fig:anatomy}). For $k=5$ the published information was, to our knowledge, the following: $\Z(\PP{n}{5})\le 12$ for all $n$ \citep{alameda2018families,rashidi2020petersen}; $\Z(\PP{11}{5})=6$ \citep{rashidi2020petersen,arxiv2607_19412}; $\Z(\PP{24r}{5})=12$ for $r\ge1$, attributed to \citet{alameda2018families}; a general lower bound of $6$ for $n\ge 15$ \citep{rashidi2020petersen}; and a handful of better upper bounds at $n=12,13,14,16$. The stabilization threshold and every exact value outside these cases were open, and the 2026 papers list exact values and lower bounds for $k\ge 5$ as open problems.

\begin{theorem}[main theorem]\label{thm:main}
For $k=5$ and every $n\ge 11$,
\[
\Z(\PP{n}{5})=
\begin{cases}
6 & n=11,\\
8 & n=12,\\
9 & n=13,\\
8 & n=14,15,16,\\
10 & n=17,18,19,20,21,\\
11 & n=22,\\
12 & n\ge 23.
\end{cases}
\]
In particular the least $n_0$ with $\Z(\PP{n}{5})=12$ for all $n\ge n_0$ is $n_0=23$.
\end{theorem}

\paragraph{Contributions.} Beyond the theorem itself, the paper offers the following.
\begin{enumerate}[label=(\roman*)]
\item A complete, self-contained adaptation of the first-force boundary method of \citet{arxiv2610_24059} to $k=5$ (Section~\ref{sec:lower}). The method needs a parameter $p$ (the number of first forces examined); for $k=5$ we prove that the parameter $p=16$ used for $k=4$ fails for every $n\ge 22$ (Proposition~\ref{prop:p16}), that $p=18$ is the smallest parameter that works on the tested range, and that no $p\le 24$ works at $n=25$ (Computational Result~\ref{cr:obstruction}). The finite window left by the deletion lemma, $26\le n\le 154$, is closed by externally checked DRAT certificates.
\item A certified table of exact values for $11\le n\le 48$ in which every entry has an upper-bound witness replayed by an independent program and a lower bound supported by at least one externally checkable or exhaustive argument, with the three-valued status \emph{fully certified / solver-supported / candidate} never collapsed (Section~\ref{sec:results}).
\item A systematic falsification campaign (Section~\ref{sec:robustness}): a pre-registered holdout, a second exhaustive enumeration by a different reduction principle and a different closure algorithm, exhaustive validation of the SAT encodings on small instances, negative controls for the proof checker, stress tests of the deletion lemma, and heuristic attacks on the theorem. Nothing was overturned.
\end{enumerate}

\paragraph{What is new and what is not.} We state this carefully, because several entries of Theorem~\ref{thm:main} were known. The bound $\Z(\PP{n}{5})\le 12$ is Theorem~2.2 of \citet{alameda2018families} as reported by \citet{rashidi2020petersen}, who prove it as their Theorem~2.2. The value at $n=11$ is Theorem~2.6 of \citet{rashidi2020petersen}, independently confirmed by exhaustive search in \citet{arxiv2607_19412}. The values at $n=14,16$ ($8$) and $n=19,21$ ($10$) are \emph{implied} by the isomorphisms $\PP{14}{5}\cong\PP{14}{3}$, $\PP{16}{5}\cong\PP{16}{3}$, $\PP{19}{5}\cong\PP{19}{4}$, $\PP{21}{5}\cong\PP{21}{4}$ (Proposition~\ref{prop:iso}) together with Theorem~1.1 of \citet{arxiv2610_24059}; they do not appear to have been stated for $k=5$ but we do not claim them. The values at $24\mid n$ are attributed to \citet{alameda2018families}; we could not access that paper's full text in our environment and rely on the statement reported in \citet[Remark~2.4]{rashidi2020petersen}, which our computation confirms at $n=24$ and $n=48$. To our knowledge, the values at $n\in\{12,13,15,17,18,20,22,23\}$, the statement $\Z(\PP{n}{5})=12$ for all $n\ge 23$, and the sharpness of the threshold are new. The proof technique for the lower bound is borrowed from \citet{arxiv2610_24059}; our contribution there is the $k=5$ analysis (the parameter, the obstruction at $n=25$, the finite window and its certificates), not the method.

\paragraph{Organization.} Section~\ref{sec:related} reviews prior work. Section~\ref{sec:prelim} fixes definitions and proves the foundational facts (order independence of the closure, the fort characterization, symmetry, graph facts, the isomorphism). Section~\ref{sec:upper} proves the upper bounds. Section~\ref{sec:lower} develops the boundary method and proves the correctness of the two SAT encodings used for certificates. Section~\ref{sec:framework} describes the exact-computation framework and its certification vocabulary. Section~\ref{sec:results} assembles the proof of Theorem~\ref{thm:main} and reports the certified table, the obstruction at $n=25$ and the negative results about the method. Section~\ref{sec:robustness} reports the falsification campaign, Section~\ref{sec:discussion} discusses competing explanations and open questions, Section~\ref{sec:limitations} states limitations, and Section~\ref{sec:conclusion} concludes. The appendices contain the remaining proofs, the computational protocol, the reproducibility notes and additional tables.

\begin{figure}[t]
\centering
\begin{subfigure}[t]{0.47\textwidth}\centering
\includegraphics[width=\linewidth]{fig1_anatomy_P13_5.pdf}
\caption{$\PP{13}{5}$: $\gcd(13,5)=1$, the inner edges form one $13$-cycle.}
\end{subfigure}\hfill
\begin{subfigure}[t]{0.47\textwidth}\centering
\includegraphics[width=\linewidth]{fig1_anatomy_P15_5.pdf}
\caption{$\PP{15}{5}$: $\gcd(15,5)=5$, the inner edges form five triangles.}
\end{subfigure}
\caption{Anatomy of $\PP{n}{5}$: outer cycle $u_iu_{i+1}$ (dark grey), spokes $u_iv_i$ (light grey) and inner chords $v_iv_{i+5}$ (dark grey). The inner subgraph is a union of $\gcd(n,5)$ cycles of length $n/\gcd(n,5)$ (Proposition~\ref{prop:graph}).}
\label{fig:anatomy}
\end{figure}