Beacon Principle II: Finite Closure for Rank Beacons on Finite Graphs

Many discrete dynamical systems on countable state spaces raise a basic
finite-closure question: do all trajectories eventually fall into a finite
closed region of state space? Although the underlying definitions are often
elementary, most existing approaches rely on probabilistic heuristics or ad hoc
case distinctions, and they do not isolate a general mechanism that forces such
finite-closure.

In this paper we introduce a finite-closure framework based on a discrete
\emph{rank Beacon} on a finite state graph. For each bit-length $m$ we encode
a discrete-time dynamics on the finite state space
\[
  S_m := \mathbb{Z} / 2^m \mathbb{Z},
\]
equip the induced transition graph $G_m$ with an integer-valued rank function
$r_m : S_m \to \mathbb{N}$, and interpret $r_m$ as a discrete energy measuring the
structural distance of a state from the expected terminal behaviour. The key
requirement is a forced-decay inequality: outside a finite core region $C_m$, the
rank decreases by at least a fixed amount along every edge of $G_m$.

Once such a rank function and core region exist, the dynamics becomes purely
combinatorial. Every orbit on $G_m$ can only decrease the rank finitely many times
before it enters $C_m$, and forward invariance of $C_m$ then forces the orbit to
remain inside $C_m$ forever. We call the data $(G_m, r_m, C_m)$ satisfying these
conditions a \emph{rank Beacon}, and we formulate \emph{Beacon Principle II} as a
finite-closure theorem for such rank Beacons on finite directed graphs.

For concrete applications, this framework separates the structural and
computational tasks. On the structural side, one constructs $r_m$ and $C_m$ so
that the forced-decay and forward-invariance conditions hold, typically using
problem-specific encodings of the underlying dynamics. On the computational
side, one analyses the limiting shape of $C_m$ as $m \to \infty$ and verifies
that no new terminal behaviours appear; this part is naturally expressed in
terms of $\Sigma_1$-style certificates on finite ledgers.

Before stating our main results it is helpful to recall how the present setup
relates to classical discrete Lyapunov theory.  On a finite directed graph one
usually combines three ingredients: (a) a Lyapunov function or rank function
that decreases along transitions; (b) an absorbing set in which the function is
not forced to decrease; and (c) a decomposition into terminal strongly
connected components.  Beacon Principle~II repackages these ingredients into a
triple of \emph{window}, \emph{target} and \emph{positivity}.  The forced rank
decay outside a finite core is encoded by the positivity of a windowed
rank–difference target, and the core itself plays the role of an absorbing set
that is forward invariant.  A key benefit of this repackaging is that the data
and inequalities involved admit natural $\Sigma_1$ certificates.

We intend this paper for researchers in nonlinear analysis, discrete dynamics
and numerical analysis.  Our aim is to provide a unified Lyapunov-type
finite-closure framework on finite graphs that may be useful across these
communities.