The Beacon Principle for Finite Closure

We formulate a \emph{beacon principle} for finite-closure phenomena in dissipative
dynamical systems: once we fix a finite beacon $(\text{window},\ \text{target},\ \text{positivity bound})$,
any dynamics that is dissipative with respect to a beacon-compatible energy admits an
explicit finite-closure radius $R_{\mathrm{fc}}(\mathcal{B},V,W_\infty)$, expressed in terms of
rational data and suitable for $\SigOne$ verification. The analytic core is a family
of \emph{beacon kernels} $K_{\Xi,\lambda}^{(\tau)}$, obtained by convolving a compactly supported
window $w_\Xi$ with a Yukawa kernel $G_\lambda$ and a Poisson smoother $P_\tau$. We prove a
quantitative \emph{uniform window positivity} theorem: for each choice of parameters
$(\Xi,\lambda,\tau,\Delta)$ the smoothed beacon kernel admits a strictly positive lower bound
$\delta_{\mathrm{pos}}(\Xi,\lambda,\tau,\Delta)$ on $[-\Delta,\Delta]$, and we construct outward-rounded
rational lower bounds.

On top of this kernel analysis we build a general \emph{finite closure theorem}. Given
a state space $\mathcal{X}$, an energy functional $V$, and a beacon observation
$b(x)=K_{\Xi,\lambda}^{(\tau)}*\Phi(x)$, we show that if $V$ dissipates whenever $b(x)$ is large,
and $b(x)$ cannot be small when $V(x)$ is large, then all trajectories are confined to a
finite sublevel set $\{V\le R\}$. In this sense $V$ plays the role of a \emph{beacon energy}
that is actively drained through the observation channel $b$, and the choice of beacon
triple encodes the \emph{agency} of the designer: where one listens, what one listens to, and
what level of energy one commits to detect.

The \emph{beacon principle} states that the entire mechanism is determined by a triple
\[
  \text{beacon} = (\text{window},\ \text{target},\ \text{positivity bound}),
\]
where the target is of one of three structural types: state, deviation, or gradient.
We illustrate the framework with minimal examples from bridge dynamics, a semantic
``meaning OS'', and security kernels; in each case the finite-closure behavior is an
instance of the same abstract theorem.
\textbf{Keywords:} finite closure; nonlinear systems; Lyapunov stability; 
ultimate boundedness; kernel-based verification.