The Beacon Principle for Finite Closure
A Mathematical Foundation for Agency-Driven Stability

Introduction

Many controlled systems exhibit a common qualitative behavior: despite potentially unbounded state spaces and ongoing disturbances, trajectories remain confined to a finite region determined by design parameters and disturbance levels. Examples include the bounded oscillation of a damped bridge under wind and traffic, the stabilization of battery state-of-charge in energy systems, the suppression of anomalies in security kernels, and the resolution of semantic inconsistency in meaning-oriented operating layers. We term this phenomenon finite closure.

The beacon principle isolates a common structural choice underlying these phenomena: an agent (designer, operator, or semantic participant) decides where to place a finite window, what to monitor through it, and how much signal is considered significant. This choice constitutes what we call agency: the finite-closure certificate is never purely geometric, but is inextricably tied to a particular beacon window and target selected by an involved party. The mathematical object that reflects this choice is the beacon energy $V$, whose dissipation is enforced whenever the beacon observation is large.

The aim of this paper is to isolate a simple analytic mechanism behind such finite closures and to express it in a form that is uniform across applications. The central object is a beacon: an observable built from a finite window and a regularizing kernel that "monitors" a chosen aspect of the state. Informally, the beacon principle states:

Once we fix how far we look (the window), what we look at (the target), and how strongly the window responds (a positivity bound), the finite-closure radius $R_{\mathrm{fc}}(\mathcal{B},V,W_\infty)$ of the dynamics is determined.

Formally, this is captured by the beacon finite-closure principle (Theorem 5.4). Analytically, the beacon is realized as a convolution operator \[ b(x) = K_{\Xi,\lambda}^{(\tau)} * \Phi(x), \] where $K_{\Xi,\lambda}^{(\tau)}$ is a smoothed beacon kernel depending on a span $\Xi>0$, a Yukawa decay parameter $\lambda>0$, and a smoothing scale $\tau>0$, while $\Phi$ is a target constructed from the state. We distinguish three structural types of targets:

• State type (S): the beacon window is placed directly on the state.
• Deviation type (D): the beacon window measures deviations from a reference or prediction.
• Gradient type (G): the beacon window probes an energy gradient or driving force.

These three types correspond to common sensing paradigms in applications.

The technical backbone of the paper is a uniform window positivity (UWP) result. For each choice of parameters $(\Xi,\lambda,\tau,\Delta)$, we prove that the smoothed beacon kernel $K_{\Xi,\lambda}^{(\tau)}$ is strictly positive on $[-\Delta,\Delta]$ and derive an explicit lower bound $\delta_{\mathrm{pos}}(\Xi,\lambda,\tau,\Delta) > 0$.

Analytically, the paper establishes three main results:

1. Quantitative uniform window positivity: Theorem 3.3 provides an explicit lower bound.
2. $\SigOne$-friendly rational positivity certificates: Proposition 3.5 demonstrates how to convert parameters into a rational lower bound $\widehat{\delta}_{\mathrm{pos}}$ suitable for formal verification.
3. Beacon finite-closure principle and separation of design choices: Theorem 5.4 asserts that the finite closure radius depends solely on the beacon design, the energy, and the disturbance level, thereby separating agency from dynamical details.

Beacon Kernel: Finite Windows and Yukawa Convolution

In this section, we fix a one-dimensional domain $\RR$ (time or space). All kernels are assumed to be real-valued and integrable.

Finite Window

Yukawa Kernel

Beacon Kernel and Beacon Transform

Beacon Targets: Three Structural Types

Uniform Window Positivity and Quantitative Lower Bounds

Poisson Smoothing and Smoothed Beacon Kernel

Uniform Window Positivity on Compact Intervals

For $\Delta>0$, let $\delta_{\mathrm{pos}}(\Xi,\lambda,\tau,\Delta) := \inf_{|t|\le\Delta} K_{\Xi,\lambda}^{(\tau)}(t)$.

Proof.
First, we estimate the lower bound of the beacon kernel $K_{\Xi,\lambda} = w_\Xi * G_\lambda$. By definition, $\mathrm{supp}\,w_\Xi\subset[-\Xi,\Xi]$, $w_\Xi\ge0$, and $\int_\RR w_\Xi(s)\,ds = 1$. The Yukawa kernel is \[ G_\lambda(t) = \frac{1}{2\lambda} e^{-\lambda|t|}, \] which is even and monotonically decreasing in $|t|$. For any $|t|\le\Delta$ and $s\in[-\Xi,\Xi]$, we have $|t-s|\le \Xi+\Delta$, so \[ G_\lambda(t-s) \;\ge\; \frac{1}{2\lambda}\,e^{-\lambda(\Xi+\Delta)}. \] Therefore, \[ K_{\Xi,\lambda}(t) = \int_\RR w_\Xi(s)\,G_\lambda(t-s)\,ds = \int_{-\Xi}^{\Xi} w_\Xi(s)\,G_\lambda(t-s)\,ds \;\ge\; \frac{1}{2\lambda}e^{-\lambda(\Xi+\Delta)} \int_{-\Xi}^{\Xi} w_\Xi(s)\,ds = \frac{1}{2\lambda}e^{-\lambda(\Xi+\Delta)} \] holds for all $|t|\le\Delta$. Next, we evaluate the Poisson smoothing \[ K_{\Xi,\lambda}^{(\tau)} = P_\tau * K_{\Xi,\lambda}. \] The Poisson kernel $P_\tau(t) = \frac{1}{\pi}\frac{\tau}{t^2+\tau^2}$ is also even and monotonically decreasing in $|t|$. For any $|t|\le\Delta$, \[ K_{\Xi,\lambda}^{(\tau)}(t) = \int_\RR P_\tau(t-u)\,K_{\Xi,\lambda}(u)\,du \;\ge\; \int_{-\Delta}^{\Delta} P_\tau(t-u)\,K_{\Xi,\lambda}(u)\,du. \] If $|t|\le\Delta$ and $|u|\le\Delta$, then $|t-u|\le 2\Delta$, so \[ P_\tau(t-u) \;\ge\; \min_{|v|\le 2\Delta} P_\tau(v) = \frac{1}{\pi}\frac{\tau}{4\Delta^2+\tau^2}. \] Using the lower bound for $K_{\Xi,\lambda}(u)$ obtained above for $|u|\le\Delta$, we get \[ K_{\Xi,\lambda}^{(\tau)}(t) \;\ge\; \frac{1}{\pi}\frac{\tau}{4\Delta^2+\tau^2} \int_{-\Delta}^{\Delta} K_{\Xi,\lambda}(u)\,du \;\ge\; \frac{1}{\pi}\frac{\tau}{4\Delta^2+\tau^2} \cdot \frac{1}{2\lambda}e^{-\lambda(\Xi+\Delta)} \cdot 2\Delta, \] which simplifies to \[ K_{\Xi,\lambda}^{(\tau)}(t) \;\ge\; \frac{\tau\,\Delta}{\pi\,\lambda\,(4\Delta^2+\tau^2)} e^{-\lambda(\Xi+\Delta)}. \] The right-hand side is clearly positive for $\Xi,\lambda,\tau,\Delta>0$, implying $\delta_{\mathrm{pos}}(\Xi,\lambda,\tau,\Delta)>0$. ∎

$\SigOne$-Friendly Outward Rounding

Construction Outline.
Given rational inputs $(\Xi^\uparrow,\lambda^\uparrow,\tau^\uparrow,\Delta^\uparrow) \in \QQ_{>0}^4$ (assuming outward rounding such that $\Xi\le\Xi^\uparrow$, etc.), we use the lower bound formula from Theorem 3.3: \[ \delta_\star(\Xi,\lambda,\tau,\Delta) = \frac{\tau\,\Delta}{\pi\,\lambda\,(4\Delta^2+\tau^2)} \exp\bigl(-\lambda(\Xi+\Delta)\bigr). \] We construct the rational number $\widehat{\delta}_{\mathrm{pos}}$ as follows:

1. Calculate $q := \lambda^\uparrow(\Xi^\uparrow+\Delta^\uparrow)\in\QQ_{>0}$.
2. Take a bounded decreasing sequence $(r_n)_{n\ge1}\subset\QQ_{>0}$ such that $r_n \downarrow e^{-q}$ (e.g., using Taylor polynomials with directed rounding).
3. Select some $n_\star$ and define $\mathrm{ExpLow}(q) := r_{n_\star}$. Then $0<\mathrm{ExpLow}(q) \le e^{-q}$ holds.
4. Finally, define \[ \widehat{\delta}_{\mathrm{pos}} := \frac{\tau^\uparrow\,\Delta^\uparrow}{\pi\,\lambda^\uparrow\,(4(\Delta^\uparrow)^2+(\tau^\uparrow)^2)} \,\mathrm{ExpLow}\bigl(\lambda^\uparrow(\Xi^\uparrow+\Delta^\uparrow)\bigr). \] This is clearly rational and satisfies \[ 0 < \widehat{\delta}_{\mathrm{pos}} \le \delta_\star(\Xi,\lambda,\tau,\Delta). \]

Proof of Proposition 3.5.
The above construction relies only on rational arithmetic and a lower approximation of the exponential function by a decreasing rational sequence. The calculation procedure is entirely describable in a $\Sigma_1$ manner (finite steps of construction + "there exists an $n_\star$"). The monotonicity from Theorem 3.3 and the choice of rounding direction ensure \[ 0 < \widehat{\delta}_{\mathrm{pos}} \le \delta_\star(\Xi,\lambda,\tau,\Delta) \le \delta_{\mathrm{pos}}(\Xi,\lambda,\tau,\Delta). \] ∎

Finite Closure via Beacon Dissipation

Beacon Dissipation Inequality and Coercivity

Finite Closure and Forward Invariance

Proof of Theorem 4.4.
From Assumptions 4.2 and 4.3, for times when $V(x(t))\ge R_0$, we have \[ \frac{d}{dt}V(x(t)) \;\le\; -\kappa\,\|b(x(t))\|^2 + \gamma\,\|w(t)\|^2 \;\le\; -\kappa m\,V(x(t)) + \gamma\,\|w(t)\|^2. \] Let $a := \kappa m>0$ and $d(t):=\gamma\|w(t)\|^2$. Then \[ \frac{d}{dt}V(x(t)) + a\,V(x(t)) \le d(t). \] By Grönwall's inequality, \[ V(x(t)) \le V(x(0))e^{-at} + \int_0^t e^{-a(t-s)} d(s)\,ds. \] From Assumption 4.1 regarding the disturbance, $d\in L^1([0,\infty))$, and \[ \int_0^t e^{-a(t-s)} d(s)\,ds \le \left(\sup_{u\ge0} e^{-au}\right) \int_0^\infty d(s)\,ds \le \int_0^\infty d(s)\,ds = \gamma \int_0^\infty \|w(s)\|^2 ds = \gamma W_\infty^2. \] Therefore, for all $t\ge0$, \[ V(x(t)) \le V(x(0))e^{-at} + \gamma W_\infty^2. \] We now establish the two assertions:

• Ultimate Boundedness: For any initial value, for sufficiently large $t$, $V(x(0))e^{-at} \le (\gamma/(\kappa m))W_\infty^2$, so there exists a time after which \[ V(x(t)) \le \frac{\gamma}{\kappa m}W_\infty^2. \] By definition, \[ V(x(t)) \le R_{\mathrm{fc}} := \max\Bigl\{R_0, \frac{\gamma}{\kappa m}W_\infty^2\Bigr\}. \] Thus, the trajectory is ultimately captured by $\Omega_{R_{\mathrm{fc}}}=\{V\le R_{\mathrm{fc}}\}$.
• Forward Invariance: Assume the initial state satisfies $V(x(0))\le R_{\mathrm{fc}}$. Suppose for the sake of contradiction that there exists a time $t_1>0$ where $V(x(t_1))>R_{\mathrm{fc}}$. By continuity, immediately before $t_1$, $V(x(t))\ge R_0$ holds, so the above differential inequality applies. However, the estimate for $t\in[0,t_1]$ gives \[ V(x(t_1)) \le V(x(0))e^{-at_1} + \gamma W_\infty^2 \le R_{\mathrm{fc}}, \] which contradicts $V(x(t_1))>R_{\mathrm{fc}}$. Thus, $\Omega_{R_{\mathrm{fc}}}$ is a forward invariant set.

The Beacon Principle: Window Placement as a Design Choice

Window, Target, and Positivity Bound

Abstract Beacon Principle

Proof of Theorem 5.4 (Reduction from Chapter 4).
From the definition in Section 5.2, for a beacon triple $\mathcal{B}=(W,\Phi,\delta)$:

• The choice of window $W$, Yukawa kernel, and Poisson kernel ensures uniform window positivity via Theorem 3.3 in Chapter 3: \[ K_{\Xi,\lambda}^{(\tau)}(t) \ge \delta > 0 \quad (|t|\le\Delta). \]
• The structural type (S, D, G) of target $\Phi$ makes the beacon observation $b(x) := K_{\Xi,\lambda}^{(\tau)} * \Phi(x)$ coercive in the sense that "large energy $V$ forces large $b(x)$." This provides the beacon compatibility condition (Assumption 4.3 type): \[ V(x)\ge R_0 \;\Rightarrow\; \|b(x)\|^2 \ge m\,V(x). \]
• At the dynamical level, the system is designed to satisfy the dissipation inequality (Assumption 4.2 type) for the beacon-compatible energy $V$: \[ \frac{d}{dt}V(x(t)) \le -\kappa\|b(x(t))\|^2 + \gamma\|w(t)\|^2. \] The $L^2$ level $W_\infty$ of the disturbance $w$ is given by Assumption 4.1.

Therefore, the definition of beacon compatibility precisely satisfies Assumptions 4.1–4.3 of Chapter 4. Applying Theorem 4.4, it follows that for the finite radius determined as \[ R_{\mathrm{fc}} = R_{\mathrm{fc}}(\mathcal{B},V,W_\infty), \] all trajectories are ultimately captured by $\Omega_{R_{\mathrm{fc}}}$, and $\Omega_{R_{\mathrm{fc}}}$ is forward invariant. This coincides with the statement of Theorem 5.4. ∎

Examples: Bridge, Meaning OS, and Security Kernel

Prime Beacon: Outlook toward Analytic Number Theory

(Note: Detailed development is left to a separate paper.) We consider the error term $x_{\mathrm{pr}}$ associated with the zeros of the Riemann zeta function as the state, and the weighted square integral as the "prime energy $E_{\mathrm{pr}}$". This can be formulated as a gradient-type (G) beacon, suggesting the finite closure property of the error term in the prime number theorem.

Bridge Dynamics (State-Type Beacon)

The state consists of displacement and velocity, and vibrations in a specific interval (window) are monitored. If energy dissipation is guaranteed by structural damping and feedback, an explicit finite bound for the bridge oscillation amplitude is obtained.

Meaning OS (Gradient-Type Beacon)

The state space is a space of semantic configurations, and the semantic energy $E_{\mathrm{sem}}$ penalizes inconsistencies or conflicts. The beacon target $\Phi$ is placed on the energy gradient (semantic tension). The finite closure theorem indicates that "persistent high tension cannot survive without triggering dissipation through the beacon window."

Security Kernel (Deviation-Type Beacon)

The system monitors the deviation between the actual behavior $S(x)$ and a reference (normal) behavior $S_{\mathrm{ref}}(x)$. The anomaly energy $V$ is defined as the norm of the deviation. If the response by the security kernel (throttling or isolation) causes dissipation, the anomaly energy is suppressed to a finite level.

Appendix A: Sufficient Conditions for G-Type Beacons

Proof.
Assume $V(x)\ge R_0$. By Assumption A.1, \[ \|\nabla V(x)\|^2 \;\ge\; 2\mu\,V(x) \] and \[ \|b(x)\| = \bigl\|B(\nabla V(x))\bigr\| \;\ge\; c\,\|\nabla V(x)\|. \] Therefore, \[ \|b(x)\|^2 \;\ge\; c^2\,\|\nabla V(x)\|^2 \;\ge\; 2\mu c^2\,V(x). \] Setting $m := 2\mu c^2$, we obtain the form of Assumption 4.3: \[ V(x)\ge R_0 \;\Longrightarrow\; \|b(x)\|^2 \ge m\,V(x). \] ∎

References