Many discrete dynamical systems on countable state spaces raise a basicfinite-closure question: do all trajectories eventually fall into a finiteclosed region of state space? Although the underlying definitions are oftenelementary, most existing approaches rely on probabilistic heuristics or ad hoccase distinctions, and they do not isolate a general mechanism that forces suchfinite-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 encodea 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 thestructural distance of a state from the expected terminal behaviour. The keyrequirement is a forced-decay inequality: outside a finite core region $C_m$, therank 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 purelycombinatorial. Every orbit on $G_m$ can only decrease the rank finitely many timesbefore it enters $C_m$, and forward invariance of $C_m$ then forces the orbit toremain inside $C_m$ forever. We call the data $(G_m, r_m, C_m)$ satisfying theseconditions a \emph{rank Beacon}, and we formulate \emph{Beacon Principle II} as afinite-closure theorem for such rank Beacons on finite directed graphs. For concrete applications, this framework separates the structural andcomputational tasks. On the structural side, one constructs $r_m$ and $C_m$ sothat the forced-decay and forward-invariance conditions hold, typically usingproblem-specific encodings of the underlying dynamics. On the computationalside, one analyses the limiting shape of $C_m$ as $m \to \infty$ and verifiesthat no new terminal behaviours appear; this part is naturally expressed interms of $\Sigma_1$-style certificates on finite ledgers. Before stating our main results it is helpful to recall how the present setuprelates to classical discrete Lyapunov theory. On a finite directed graph oneusually combines three ingredients: (a) a Lyapunov function or rank functionthat decreases along transitions; (b) an absorbing set in which the function isnot forced to decrease; and (c) a decomposition into terminal stronglyconnected components. Beacon Principle~II repackages these ingredients into atriple of \emph{window}, \emph{target} and \emph{positivity}. The forced rankdecay outside a finite core is encoded by the positivity of a windowedrank–difference target, and the core itself plays the role of an absorbing setthat is forward invariant. A key benefit of this repackaging is that the dataand inequalities involved admit natural $\Sigma_1$ certificates. We intend this paper for researchers in nonlinear analysis, discrete dynamicsand numerical analysis. Our aim is to provide a unified Lyapunov-typefinite-closure framework on finite graphs that may be useful across thesecommunities.▼GhostDriftMathmaticalInstitue HPhttps://www.ghostdriftresearch.com/%E8%A4%87%E8%A3%BD-adic