Passive Signal as Passive Controller: A Systems-Theoretic Feasibility Analysis of the Signal‑Worker Ontology

#1. Introduction

Engineers who must stabilize fragile quantum or mesoscopic states—such as superconducting devices, driven materials, or biological energy‑transfer structures—currently face a trade‑off between two control paradigms. Active control continuously injects energy, achieving stabilization at high thermodynamic cost and losing stability when the drive is removed. Passive control encodes stabilizing information in the environment’s structure, offering cheap and persistent stabilization but lacking a general theory to guarantee that a proposed passive structure indeed behaves as a controller. The cost of this gap is concrete: without a certification criterion, a designer cannot decide whether a candidate passive scaffold will stabilize the target state, cannot compare candidate scaffolds quantitatively, and cannot falsify claims made for them. The Signal‑Worker ontology addresses the trade‑off by interpreting active quantum control as reliance on an external, energy‑intensive Signal and passive structural control as relying on an intrinsic, architectural Signal, hypothesizing a Complexity‑Efficiency relationship linking scaffold complexity to stabilization performance. The missing bridge is a precise control‑theoretic statement: does the Signal satisfy a dissipation inequality with a storage function? Without such a bridge, the ontology’s central engineering question cannot be certified, designed against, or falsified.

A prior feasibility analysis [9] restated the Signal as a discrete‑time controller block, derived output transformations for integrator‑type dynamics, and bounded passivity degradation caused by output quantization. That work produced arithmetic‑checkable quantities but stopped at the block‑diagram level, omitted connections to the corpus’s thermodynamic arguments, and left the engineering pathway unspecified. This paper completes the arc and, in doing so, corrects an interpretive error in the predecessor’s quantization analysis: the published Section 4.3 stated that the quantization budget is tolerable only for states small enough that $\delta\sum_k x_k^{2}$ stays above $5.0$; the inequality in fact requires states large enough, because the dissipation term must exceed the degradation budget. The corrected interpretation is derived in Section 4.3 and confirmed by the structure of the inequality itself: the net margin is $\delta\sum_k x_k^{2} - \Delta_{\text{quant}}$, which is non‑positive when the dissipation is too small, i.e., when states are too small.

Contributions are:

1. A block‑diagram formalization of the Signal‑Worker decomposition as a discrete‑time passivity problem (Section 3).
2. Exact derivations, with every arithmetic step shown, of the output maps under which pure and leaky integrator Signals are passive, including a counterexample for the naive output (Section 4).
3. A worst‑case quantization degradation bound ($5.0$ energy units over $100$ steps at $q = 0.1$) with the corrected state‑magnitude threshold $x_\ast \ge 0.726$, and an illustrative example quantifying delay margin, quantization energy cost, attractor radius, and robustness budget (Sections 4.3–4.4, summarized in Section 5).
4. A mapping between the derived dissipation rate $\delta$ and the corpus’s passive‑versus‑driven distinction (Section 6).
5. A practical pathway: identification of users, TRL assessment, the smallest next test, and the primary engineering constraints (Section 7).

#2. Background and Related Work

[1] Passive iFIR Filters for Data‑Driven Control. The work designs passive controllers as the parallel action of an integrator and a finite‑impulse‑response filter, with transfer function $C(z)=\frac{\gamma T_s}{1-z^{-1}}+\sum_{k=0}^{m-1}g_k z^{-k}$. It shows that for passive plants the closed‑loop stability is guaranteed by construction and that the design requires no plant model, directly supporting the notion that an integrator‑based Signal block can be rendered passive by construction.

[2] Passivity‑based PI control of first‑order systems with I/O communication delays. This paper treats a delayed communication channel as a transport PDE and proves that the overall loop remains passive when a scattering transformation is applied, despite the Nyquist plot of a pure delay lying outside the right‑half plane. The result motivates the delay‑margin analysis in Section 4.4, where a transport delay models the spatial separation between Signal and Worker.

[3] Sampled‑data control design for systems with quantized actuators. The authors design sampled‑data state feedback for continuous‑time linear systems with uniform input quantization; this design is used as a basis for the quantization analysis in Section 4.3. Their framework underpins the quantization analysis of Section 4.3, treating the quantized Signal output as a bounded perturbation of the ideal passive inequality.

[4] Composite Adaptive Control Barrier Functions for Safety‑Critical Systems with Parametric Uncertainty. This work shows that safety guarantees based on control barrier functions can fail under parametric uncertainty, and introduces adaptive mechanisms to retain safety. It illustrates the broader theme that model uncertainty can invalidate guarantees, which is reflected in the limitations discussion of Section 6.

[5] Robust Nonlinear Optimal Control via System Level Synthesis. The authors decompose an uncertain nonlinear system into a nominal trajectory and a norm‑bounded error, jointly optimizing both. Their nominal‑plus‑error decomposition informs the robustness‑budget calculation in Section 4.4, where the Signal’s passivity margin is eroded by a bounded deviation.

[6] Safe Control Synthesis via Input Constrained Control Barrier Functions. This paper introduces Input‑Constrained CBFs that construct inner safe sets when full input authority is unavailable. The attractor‑radius result of Section 4.4 mirrors the inner‑safe‑set construction, highlighting how quantized, bounded outputs can be interpreted as input constraints.

[7] On $\ell_2$-performance of weakly‑hard real‑time control systems. The authors develop LMI conditions for $\ell_2$ performance under weakly‑hard packet‑loss constraints, showing how loss sequences can be modeled and analyzed. Their framework motivates the discussion of packet‑loss constraints as a future extension in Section 6.

[8] On the relationship between control barrier functions and projected dynamical systems. This work proves that CBF‑controlled systems are Krasovskii‑like perturbations of projected dynamical systems, providing a quantitative bound on the perturbation. It supports the treatment of quantization as a bounded additive perturbation in Section 4.3.

[9] QNFO: Passive Signal as Passive Controller (prior feasibility analysis). The predecessor formalized the Signal as a discrete‑time block, derived exact passivity for a pure integrator, strict passivity for a leaky integrator with dissipation rate $\delta = \alpha - \tfrac{\alpha^{2}}{2}$, and a worst‑case quantization degradation of $5.0$ energy units. The present paper re‑derives those results with full arithmetic, fixes the interpretation of the state‑magnitude condition, extends the treatment with an illustrative example, and adds the engineering pathway the predecessor lacked.

[10] QNFO: Structural vs Driven Quantum Coherence. The corpus paper proposes the Signal‑Worker ontology, built on the premise that information is physical and that maintaining the order of a quantum state has an irreducible energy cost as quantified by Landauer’s principle, making thermodynamic efficiency the ultimate metric for any stabilization mechanism. It introduces two complexity metrics—Krylov complexity for the dynamical complexity of the Worker state and the Lossless Complexity Index (LCI) for the structural complexity of the Signal scaffold—and applies the framework to classify actively driven systems as thermodynamically inefficient and passive structural systems as efficient. The present paper imports no numerical values from that work; the connection is structural and is developed in Section 6.

#3. Methods

#3.1 Definitions

Discrete‑time passivity. A discrete‑time system with state $x_k\in\mathbb{R}^n$, input $u_k\in\mathbb{R}$, and output $y_k\in\mathbb{R}$ is passive if there exists a storage function $V(x)\ge 0$ with $V(0)=0$ such that for all $k$,

The right‑hand side $u_k y_k$ is the supply rate.

Strict passivity with dissipation rate. If additionally

the system is strictly passive with dissipation rate $\delta$.

Block interpretation of the S‑W ontology. The mapping from the corpus [10] to a control block diagram is

u_k (instruction) ──► [ SIGNAL ] ──y_k──► [ WORKER ] ──► work output
                        (delocalized)   (localized state)

The research question becomes: does a storage function $V_S$ exist such that the Signal block alone satisfies the dissipation inequality?

#3.2 Candidate Signal dynamics

Because the corpus provides no dynamical equations, we analyze the two minimal candidate dynamics:

Pure integrator: $x_{k+1}=x_k+u_k$.

Leaky integrator (leakage $\alpha\in[0,1]$): $x_{k+1}=(1-\alpha)x_k+u_k$.

For both, the quadratic storage function $V(x)=\tfrac{1}{2}x^{2}$ is natural. The analysis seeks output maps $y_k=\beta x_k+\gamma u_k$ that satisfy the dissipation inequality.

#3.3 Perturbation analysis under quantization and delay

Following the quantized‑actuator framework of [3], we bound how uniform input quantization of resolution $q$ degrades the passivity inequality over a finite horizon. In the illustrative example (Section 4.4) we also evaluate delay‑phase loss using the transport‑delay model of [2] and a robustness budget inspired by the nominal‑plus‑error decomposition of [5].

#4. Analysis

All numerical inputs are either structural constants of the candidate dynamics or explicitly stated illustrative parameters. Every arithmetic step is shown.

#4.1 Exact passivity of the pure integrator

Setup. Dynamics $x_{k+1}=x_k+u_k$; storage $V(x)=\tfrac{1}{2}x^{2}$; output $y_k=\beta x_k+\gamma u_k$.

Step 1: storage change.

Step 2: passivity requirement. Require

i.e.

Step 3: coefficient choice. The mixed term changes sign unless its coefficient vanishes, so $\beta=1$. Then $\frac{1}{2}-\gamma\le 0$ forces $\gamma\ge\frac{1}{2}$. The minimal exactly‑passive choice is $\gamma=\frac{1}{2}$, giving

Numerical check. With $x_0=0$, $u_0=0.1$: $x_1=0.1$, $V(x_1)=\tfrac{1}{2}(0.1)^{2}=0.005$, $\Delta V=0.005$; $y_0=0+\tfrac{1}{2}(0.1)=0.05$, $u_0 y_0=0.1\times0.05=0.005$. Equality holds, confirming exact passivity.

#4.2 Passivity of the leaky integrator

Setup. Dynamics $x_{k+1}=(1-\alpha)x_k+u_k$ with $\alpha\in[0,1]$; storage $V(x)=\tfrac{1}{2}x^{2}$; output $y_k=\beta x_k+\gamma u_k$.

Step 1: storage change.

Step 2: subtract supply.

Step 3: coefficient choice. To eliminate the mixed term we set $\beta = 1-\alpha$. Setting $\gamma=\frac{1}{2}$ removes the $u_k^{2}$ term. The remaining coefficient of $x_k^{2}$ is $-\alpha+\frac{\alpha^{2}}{2}= -\delta$ with

Thus the passive output is

Numerical verification for $\alpha=0.1$. $\beta = 0.9$, $\gamma = 0.5$, $\delta = 0.1 - 0.005 = 0.095$.

Zero supply: $x_0=2$, $u_0=0$ → $x_1=0.9\times2=1.8$. $V(x_0)=2$, $V(x_1)=1.62$, $\Delta V=-0.38$. $-\delta x_0^{2}= -0.095\times4 = -0.38$. Equality holds.

With supply: $x_0=2$, $u_0=0.5$ → $x_1=0.9\times2+0.5=2.3$. $V(x_1)=2.645$, $\Delta V=0.645$. $y_0=0.9\times2+0.5\times0.5=2.05$, $u_0 y_0=1.025$. $\Delta V - u_0 y_0 = -0.38 = -\delta x_0^{2}$. Inequality holds with margin $0.38$.

Counterexample for the naive output. With $y_k = x_k$ ($\beta=1,\gamma=0$) and $\alpha=0.1$,

At $x_k=1$, $u_k=-1$ the expression equals $-0.095 + 0.1 + 0.5 = 0.505 \gt 0$; passivity fails.

#4.3 Quantization degradation bound

Assumptions. (i) Uniform output quantization with resolution $q=0.1$ ⇒ $|y_k - y_k^{q}| \le q/2 = 0.05$. (ii) The Worker receives the quantized output, so the actual supply is $u_k y_k^{q}$. (iii) Input magnitude bounded: $|u_k| \le U_{\max}=1$. (iv) Horizon $N=100$ steps.

Derivation. Per‑step supply error:

Cumulative worst‑case degradation:

Interpretation (corrected). The leaky integrator dissipates $\delta \sum_{k} x_k^{2}$. The net passivity margin over the horizon is

If state magnitudes are of order unity, the total dissipation over 100 steps is $\delta \times 100 \times 1 = 0.095\times100 = 9.5$ (for $\delta=0.095$), exceeding the quantization budget of $5.0$; the net margin is then $9.5-5.0=4.5\gt 0$. The budget is therefore tolerable only for states large enough that $\delta\sum_k x_k^{2} \ge 5.0$. Assuming a constant state magnitude $x_\ast$,

Under the stated assumptions, the bound $\Delta_{\text{quant}} \le N U_{\max} q/2$ is exact and cannot be tightened without reducing $q$, $N$, or $U_{\max}$.

#4.4 Illustrative example: delay margin, energy cost, attractor radius, robustness budget

All parameters in this subsection are illustrative choices, stated explicitly; every derived number follows from them by shown arithmetic. No number below is an empirical measurement.

Input list. $k_S = 2.0$ (static gain), $\epsilon_S = 0.1$ (Signal margin), $\epsilon_W = 0.2$ (Worker margin), $T_d = 0.5\ \text{s}$ (transport delay), $\omega_c = 1.0\ \text{rad s}^{-1}$ (crossover frequency), $\Delta = 0.05$ (quantization step), $u_{\max}=1.0$ (norm bound), $\lambda_W = 0.5\ \text{s}^{-1}$ (Worker contraction constant).

Delay margin. Phase lag at $\omega_c$:

Combined dissipative margin proxy:

Normalized delay loss:

Normalized margin:

Safety ratio:

Quantization energy cost.

Nominal per‑packet supply bound $E_0 = u_{\max}^{2}=1.0\ \text{J}$. Relative overhead:

Attractor radius.

Robustness budget. Combined margin $M=0.6$. The Signal’s margin erosion tolerance is $\delta \lt \epsilon_S = 0.1$. Fraction of combined margin:

Thus the Signal can tolerate up to a $16.7\%$ erosion of the total loop margin before its own passivity fails.

#5. Results

All results are direct outputs of the derivations in Section 4.

• R1 (integrator passivity). The pure integrator is exactly passive under $y_k = x_k + \tfrac{1}{2}u_k$; supply equals storage change identically (Section 4.1).

• R2 (leaky integrator passivity). For any $\alpha\in(0,1]$ the leaky integrator is strictly passive under $y_k = (1-\alpha)x_k + \tfrac{1}{2}u_k$ with dissipation rate $\delta = \alpha - \tfrac{\alpha^{2}}{2}$. At $\alpha=0.1$ the margin is $-0.095\,x_k^{2}$ (Section 4.2). The naive output $y_k = x_k$ fails, with counterexample margin $+0.505$ (Section 4.2).

• R3 (quantization budget, corrected interpretation). With $q=0.1$, $|u_k|\le1$, and $N=100$, the worst‑case cumulative passivity degradation is $\Delta_{\text{quant}} = 5.0$ energy units (Section 4.3). The budget is tolerable only when the accumulated dissipation $\delta\sum_k x_k^{2}$ reaches $5.0$; under a constant‑magnitude assumption this requires states of magnitude at least $0.726$. States smaller than this leave a positive net deficit.

• R4 (delay margin, illustrative). Delay phase lag $0.5\ \text{rad}\approx28.65^{\circ}$; safety ratio $\rho \approx 4.71$ (Section 4.4).

• R5 (quantization energy cost, illustrative). $E_q = 0.025\ \text{J}$ per packet, i.e. $2.5\%$ of the nominal supply bound (Section 4.4).

• R6 (attractor radius, illustrative). $r_q = 0.05$ in normalized Worker units (Section 4.4).

• R7 (robustness budget, illustrative). Margin erosion tolerance $16.7\%$ of the combined loop margin (Section 4.4).

Projection. For any future specification of Signal dynamics, the formulas of Sections 4.3–4.4 apply unchanged. The qualitative conclusions—exact integrator passivity under the corrected output, failure under the naive output, positive safety ratio, percent‑level quantization cost, nonzero attractor radius—hold for any parameter set satisfying $\epsilon_S,\epsilon_W\gt 0$ and $\rho\gt 1$. Quantitative values are example‑specific and follow exactly from the stated inputs.

#6. Discussion

What the framing buys. By recasting the Signal as a passive controller, the ontological claim becomes a checkable dissipation inequality. This connects to design templates where passivity is enforced by construction ([1]), preserved across transport delays ([2]), and compatible with quantized sampled delivery ([3]). The derivations demonstrate that the framing yields concrete, arithmetic‑checkable quantities (R1–R7).

The corpus connection. The dissipation rate $\delta = \alpha - \tfrac{\alpha^{2}}{2}$ operationalizes the corpus’s passive‑versus‑driven distinction. In the corpus’s terms [10], an actively driven system requires continuous external energy (effectively $\delta\approx0$), whereas a leaky Signal with $\alpha\gt 0$ dissipates stored instruction even without supply, matching the passive structural control described in the ontology, where information encoded in structure achieves stability with minimal energy cost. The mapping is structural; the corpus supplies no numerical LCI values, so calibration remains open.

The corrected interpretation and why the error mattered. The predecessor’s Section 4.3 claimed that the quantization budget is tolerable only for states small enough that $\delta\sum_k x_k^{2}$ stays above $5.0$. The inequality $\delta\sum_k x_k^{2} \ge \Delta_{\text{quant}}$ admits the opposite reading: dissipation must reach the budget, and dissipation grows with state magnitude. A second reading of the same derivation confirms the correction: at constant magnitude $x_\ast$, the threshold $x_\ast \ge 0.726$ is a lower bound, so states below it fail and states above it pass. The practical consequence reverses: small‑signal operation is the fragile regime, not the safe one, and a designer must either increase signal amplitude, reduce $q$, shorten $N$, or reduce $U_{\max}$ to protect small states.

Limitations and failure modes.

1. Postulated dynamics. The integrator models are assumptions; any different Signal dynamics require a new storage‑function derivation.
2. Quadratic storage. The analysis relies on $V(x)=\tfrac{1}{2}x^{2}$; alternative storage functions could alter passivity conclusions.
3. Scalar system. Extensions to multi‑input, multi‑state systems are non‑trivial and not covered.
4. Illustrative parameters. Numbers in Section 4.4 are chosen for illustration; real systems will have different values.
5. Bounded input and finite horizon. The quantization budget scales linearly with $N$ and $U_{\max}$; longer horizons or larger inputs increase degradation.
6. Delay model. The margin proxy $\rho$ is heuristic; a full $\sigma$-stability proof of the kind performed in [2] would be required for rigorous guarantees.
7. Model uncertainty. As shown in [4], parametric uncertainty can invalidate safety guarantees, and adaptive mechanisms are needed to retain them; the robustness budget R7 quantifies only a first‑order erosion.
8. Packet loss. The analysis does not incorporate weakly‑hard loss sequences; the LMI framework of [7] would be needed for such extensions.
9. Discontinuity treatment. Quantization is treated as an additive bounded perturbation, following the perturbation view of [8]; regularity conditions required by that theorem are not verified here.

What would falsify the claims.

• A concrete Signal dynamics specification for which no storage function $V_S$ satisfies the dissipation inequality for any output map of the form $y_k=\beta x_k+\gamma u_k$.
• An arithmetic error in the counterexample margin (e.g., if the computed margin were not $+0.505$ under the naive output).
• A demonstration that the corrected threshold direction is wrong: a construction in which states smaller than $0.726$ retain a non‑negative net margin under the stated assumptions would falsify the corrected interpretation.
• Empirical evidence that the corpus’s passive structural control requires the Signal to inject energy without prior supply, contradicting the passive‑controller interpretation.

Limitations list (assumptions and sensitivity). 1. Postulated integrator dynamics. The analysis assumes the Signal follows either a pure or leaky discrete‑time integrator; deviations would require re‑derivation of the storage function. Sensitivity: results scale with the actual dynamics; small model mismatch can invalidate the passivity inequality. 2. Quadratic storage function. Using $V(x)=\tfrac12 x^2$ may not capture energy for nonlinear or higher‑order Signal models; alternative storage functions could change the dissipation rate. Sensitivity: if the true storage is larger, the derived $\delta$ may be conservative. 3. Scalar system. The derivations treat a single‑input, single‑state block; multi‑dimensional extensions are non‑trivial and may introduce coupling terms that break the coefficient‑vanishing strategy. Sensitivity: additional states could reduce the achievable margin. 4. Illustrative parameters. Numbers in Section 4.4 (gains, delays, quantization step) are chosen for illustration; real systems may have different values, affecting the quantitative margins. Sensitivity: margins scale linearly with gains and inversely with delay. 5. Bounded input and finite horizon. The quantization budget assumes $|u_k|\le U_{\max}=1$ and $N=100$ steps; longer horizons or larger inputs increase degradation proportionally. Sensitivity: $\Delta_{\text{quant}}$ grows as $N U_{\max} q/2$. 6. Heuristic delay proxy. The safety ratio $\rho$ uses a simplified phase‑loss estimate; a full $\sigma$‑stability analysis could yield a different bound. Sensitivity: larger delays reduce $\rho$ roughly proportionally to $\phi_d$. 7. Uncalibrated corpus mapping. The connection between the dissipation rate $\delta$ and the corpus’s passive‑versus‑driven distinction is conceptual; without empirical LCI values the mapping remains qualitative. Sensitivity: calibration would be needed to translate $\delta$ into thermodynamic efficiency.

#7. Practical Implications and Engineering Pathways

(a) Who would use the result and which decision, design parameter, process or policy changes if it holds. The immediate users are control engineers and theorists tasked with evaluating or formalizing claims from the Signal‑Worker research programme. The result gives them a concrete certification target: a storage‑function inequality with the explicit output map $y_k=(1-\alpha)x_k+\tfrac{1}{2}u_k$ (for a leaky Signal) or $y_k=x_k+\tfrac{1}{2}u_k$ (for a pure integrator). If the result holds, the design parameter that must be implemented is the corrected output map; any implementation that reports the naive state output $y_k=x_k$ would violate passivity. A second user group is designers of passivity‑based controller structures; the iFIR pattern of [1] provides a template for constructing a Signal block that satisfies the derived output map by design rather than by postulate. Beyond these groups the work is foundational: it establishes the form of a certification question that any future physical or simulated Signal must satisfy before a deployment decision can be made.

(b) Current readiness, stated against the NASA Technology Readiness Level definitions. The work is purely analytical, has not been peer‑reviewed, and contains no experimental or simulation validation. Accordingly it resides at Technology Readiness Level 1–2 (basic principles observed and reported; formulation of a concept). Analytical work of this kind never exceeds TRL 3, and no manufacturing or deployment readiness can be claimed.

(c) The smallest next test, prototype, pilot or calculation that would de‑risk or falsify the result: what is measured and which outcome refutes it. The smallest de‑risking step is a calculation rather than a physical experiment. Given any concrete specification of Signal dynamics (e.g., a discrete‑time model supplied by the corpus or a candidate physical scaffold), one would attempt to fit a storage function $V_S(x)$—starting with the quadratic ansatz $V_S=\tfrac{1}{2}x^2$—such that the dissipation inequality

holds for all admissible $(x_k,u_k)$. Failure to find any output map of the form $y_k=\beta x_k+\gamma u_k$ that satisfies the inequality for the given dynamics would falsify the central claim. A second, equally minimal step is a numerical simulation of the leaky integrator with quantized output ($q=0.1$) over $N=100$ steps under the bound $|u_k|\le1$; observing a cumulative supply error exceeding the derived bound $\Delta_{\text{quant}}=5.0$ energy units would falsify the quantization‑budget claim.

(d) The engineering, manufacturing or deployment constraints that would bind first (materials, tolerances, components, data, compute, energy, cost drivers, scale‑up, regulation), each stated only when the grounding supports it or Section 4 derives it; otherwise list it as an open question.

1. Specification data. No engineering step is possible until an explicit Signal dynamics model is provided; this is the primary binding constraint today (derived from the lack of corpus equations).
2. Quantization resolution. The degradation bound $\Delta_{\text{quant}}=N\,U_{\max}\,q/2$ shows that the actuator/output resolution $q$ and the horizon $N$ are the first numerical design constraints once a dynamics model exists.
3. Channel delay and loss. The safety‑ratio $\rho$ depends on transport delay $T_d$ and crossover frequency $\omega_c$; any increase in $T_d$ erodes $\rho$ and must be bounded to keep $\rho\gt 1$ (heuristic from Section 4.4).
4. Model uncertainty. The robustness‑budget analysis (Section 4.4) limits allowable erosion of the combined margin $M$ to $16.7\%$; parameter‑identification error must therefore be kept below this fraction.
5. Packet loss. Weakly‑hard loss constraints described in [7] are not yet integrated with the dissipation inequality; handling loss sequences remains an open question.
6. Physical implementation. No material, tolerance, or regulatory constraints can be listed until a concrete Signal hardware or scaffold is proposed; these remain open questions.

(e) Order‑of‑magnitude cost or performance bounds only when every input is derived in Section 4 or cited, labeled as projections with their assumptions.

• Dissipation‑rate range. From $\delta=\alpha-\alpha^{2}/2$ with $\alpha\in[0,1]$ we obtain $0\le\delta\le0.5$ (maximum at $\alpha=1$). This is a projection derived directly from the formula in Section 4.2.
• Quantization‑budget scaling. The worst‑case cumulative degradation is $\Delta_{\text{quant}}=N\,U_{\max}\,q/2$. At the operating point $N=100$, $U_{\max}=1$, $q=0.1$ this yields $\Delta_{\text{quant}}=5.0$ energy units (Section 4.3). Doubling $q$ to $0.2$ would double the budget to $10.0$, assuming $N$ and $U_{\max}$ unchanged. Halving $q$ to $0.05$ would halve it to $2.5$, under the same assumptions.
• Per‑packet quantization energy cost. With $\Delta=0.05$ and $u_{\max}=1$, the excess energy per packet is $E_q\le(\Delta/2)u_{\max}=0.025$ J, i.e. $2.5\%$ of the nominal supply bound $E_0=u_{\max}^{2}=1.0$ J (Section 4.4).
• Attractor radius. Using the contraction constant $\lambda_W=0.5\ \text{s}^{-1}$, the guaranteed attractor radius is $r_q=(\Delta/2)/\lambda_W=0.05$ (Section 4.4).

All these numbers are algebraic consequences of the formulas derived in Section 4 and the illustrative parameters explicitly listed there; they are therefore projections with the stated assumptions.

(f) Direct risks or misuse paths, where one exists.

• The principal risk is rhetorical misuse: citing the passivity results as empirical validation of the Signal‑Worker ontology would be misleading, because the analysis only shows internal consistency of a block‑diagram abstraction on postulated dynamics.
• A second misuse would be transferring the illustrative quantitative values (e.g., $\rho\approx4.71$, $2.5\%$ overhead) to real systems without re‑deriving them for the actual parameters; such transfer would overstate performance.
• Finally, reliance on the quadratic storage function may conceal cases where alternative storage functions are needed, leading designers to overlook potential instability if the true Signal dynamics deviate from the assumed integrator form.

These risks highlight the need for careful model validation before applying the certification criteria to physical systems.

#9. Conclusion

We have completed the feasibility arc for the Signal‑Worker ontology's central engineering question. The Signal, modeled as a discrete‑time controller block, is exactly passive as a pure integrator under the output map $y_k = x_k + \frac{1}{2}u_k$ and strictly passive as a leaky integrator under $y_k = (1-\alpha)x_k + \frac{1}{2}u_k$ with dissipation rate $\delta = \alpha - \frac{\alpha^{2}}{2}$ — a result that fails under the naive state output, with counterexample margin $+0.505$. Output quantization degrades the passivity budget by at most $N\,U_{\max}\,q/2$, which is $5.0$ energy units at the stated operating point. A labeled illustrative example quantifies the delay margin ($\rho \approx 4.71$), quantization energy cost ($2.5\%$ per packet), attractor radius ($r_q = 0.05$), and robustness budget ($16.7\%$ margin erosion tolerance). The dissipation rate $\delta$ provides the corpus's passive‑versus‑driven distinction a tunable control‑theoretic analogue, though the mapping remains uncalibrated because the corpus supplies no dynamical equations or numerical complexity values. The hypothesis is well‑posed and internally consistent at the block‑diagram level; it resides at Technology Readiness Level 1–2 as pure analysis; and the smallest next test is the fitting of a storage function to any concrete Signal dynamics, with failure defined as the nonexistence of any output map rendering the dissipation inequality satisfied.

#Appendix A. Divergence Report

No divergent claims were identified among the independent drafts. All substantive statements in the manuscript are convergent across the source versions.

#Appendix B. Claim Attribution

#Provenance and AI Involvement

• This paper was drafted, reconciled, reviewed, revised and checked by the QNFO automated research pipeline (qnfo-research-exec 0.32.0-no-registrar). Language models were tools in that process; they are not authors. The named author is accountable for the content.
• Models: drafting by openai/gpt-oss-120b, zai-org/glm-5.3-flash; reconciliation, completion, review, revision and verification by openai/gpt-oss-120b, zai-org/glm-5.3-flash, moonshotai/kimi-k2.6, with the QNFO model router as a fallback.
• Sources: every reference is drawn from a bibliography retrieved from arXiv and the QNFO corpus before drafting; every cited work's full text was retrieved (arXiv HTML, or the QNFO paper's body) and the writers were given excerpts of its own sections; statements about a cited work are restricted to its retrieved title, abstract and those excerpts.
• Checks passed before publication: required sections present and finished, cross-references resolve, reference list rendered from the retrieved bibliography, no assistant residue or placeholders, adversarial model review with its blocking findings resolved, and a citation existence check at publication.
• Independent recomputation: a separately generated script recomputed 16 quantitative claims from the paper's stated inputs; 0 differed from the stated value at the stated precision.
• Not checked: whether each cited source supports the sentence that cites it beyond the retrieved full-text excerpts; derivation steps other than the recomputed claims. This paper has not been peer reviewed, and this version was published by the pipeline without a prior manual review by the author.

#References

[1] Passive iFIR Filters for Data-Driven Control. arXiv:2403.06640v2. https://arxiv.org/abs/2403.06640v2 [2] Passivity-based PI control of first-order systems with I/O communication delays: A complete sigma-stability analysis. arXiv:1507.01146v1. https://arxiv.org/abs/1507.01146v1 [3] Sampled-data control design for systems with quantized actuators. arXiv:2208.05694v3. https://arxiv.org/abs/2208.05694v3 [4] Composite Adaptive Control Barrier Functions for Safety-Critical Systems with Parametric Uncertainty. arXiv:2601.17683v3. https://arxiv.org/abs/2601.17683v3 [5] Robust Nonlinear Optimal Control via System Level Synthesis. arXiv:2301.04943v3. https://arxiv.org/abs/2301.04943v3 [6] Safe Control Synthesis via Input Constrained Control Barrier Functions. arXiv:2104.01704v1. https://arxiv.org/abs/2104.01704v1 [7] On $\ell_2$-performance of weakly-hard real-time control systems. arXiv:2305.07875v3. https://arxiv.org/abs/2305.07875v3 [8] On the relationship between control barrier functions and projected dynamical systems. arXiv:2309.04378v1. https://arxiv.org/abs/2309.04378v1 [9] https://papers.qnfo.org/papers/passive-signal-as-passive-controller-a-systems-theoretic-feasibility-analysis-of. QNFO: Passive Signal as Passive Controller: A Systems-Theoretic Feasibility Analysis of the Signal‑Worker Ontology. [10] https://papers.qnfo.org/papers/structural-vs-driven-quantum-coherence. QNFO: Structural vs Driven Quantum Coherence.

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