Authors: O. Krishevich, V. Peretyachenko

1. Introduction: Why the Linear Model is Insufficient

1.1 Classical Linear Model and Its Boundaries

The overwhelming majority of engineering calculations in energy systems are based on the linear model: $$P_{\text{out}} = \eta \cdot P_{\text{in}} \quad (\eta \leq 1)$$ where \(P_{\text{in}}\) is input power, \(P_{\text{out}}\) is output power, and \(\eta\) is efficiency (for energy-conversion devices, \(\eta \le 1\) when the system boundary is defined correctly). This model describes exceptionally well:

• Resistive circuits
• Classical electrical machines (motors, generators)
• Transformers
• Power semiconductors

However, the model contains an implicit assumption: the system has no internal state that accumulates, stabilizes, or recirculates energy beyond the instantaneous input. This assumption is violated in regime-based systems — classes of electrodynamic devices in which energy is retained and repeatedly circulated in internal fields and currents until equilibrium is reached.

1.2 Definition of a Regime-Based System

A regime-based system is an electrodynamic device whose behavior is determined not only by the instantaneous input energy flow but also by a structured dynamic state \(R(t)\) — the operational regime: $$R(t+1) = f(R(t), u(t), \epsilon)$$ where:

• \(R(t)\) — regime state (field distribution, phase relationships, circulating currents)
• \(u(t)\) — control energy (loss compensation)
• \(\epsilon\) — inevitable losses
• \(f\) — nonlinear dynamics function

Output power in this case is a function of the regime, not a direct function of the input: $$P_{\text{out}} = g(R(t))$$ This separation is key: what is compensated is not the output power but the degradation of the regime.

2. Canonical A–B–C Model for Regime-Based Systems

2.1 Three Components of Energy Balance

For an open electrodynamic system, a correct energy balance distinguishes three terms in the balance (energy stored vs. irreversible power losses vs. control power): A — Internal Energy Turnover $$A(t) = \sum_i U_i(t) \cdot n_i(t)$$ where \(U_i(t)\) is energy stored in the \(i\)-th reactive component (capacitor field, inductor flux), and \(n_i(t)\) is the effective number of circulation cycles before dissipation. Physically, this is the multiple circulation of energy in electric and magnetic fields of the active circuit:

• Energy transitions from electric field to magnetic field and back
• Each cycle is determined by resonant frequency and quality factor
• Internal circulation (reactive energy turnover) can be much larger than the net input power, which is a standard property of high-Q resonant systems

B — Irreversible Regime Losses $$B(t) = P_{\text{Joule}} + P_{\text{dielectric}} + P_{\text{radiation}} + P_{\text{discharge}}$$ where:

• \(P_{\text{Joule}}\) — ohmic losses (skin effect, contact resistances)
• \(P_{\text{dielectric}}\) — dielectric losses (in insulation, air)
• \(P_{\text{radiation}}\) — electromagnetic energy radiation
• \(P_{\text{discharge}}\) — losses in gas discharge (ionization, heat, chemistry)

Important: these losses are internal to the system and determine the required compensation, but are not equal to output power. C — External Loss-Compensation Power $$P_{\text{control}}(t) \approx B_{\text{maint}}(t)$$ In steady state: $$P_{\text{control,steady}} \approx B_{\text{maint,steady}}$$ where \(B_{\text{maint}}(t)\) denotes the subset of regime losses that must be compensated to keep the regime stable (e.g., core ohmic/dielectric/discharge losses), as distinct from load-dependent extraction losses reflected through the effective loaded quality factor. Critical property: \(P_{\text{control}} \neq P_{\text{out}}\) Output power is extracted from turnover \(A\) through a separate channel (extraction circuit), and its magnitude depends on extraction architecture and regime stability boundaries, but not directly on \(P_{\text{control}}\).

2.2 Energy Balance for an Open System

The complete energy balance can be written as: $$\frac{dE_{\text{total}}}{dt} = P_{\text{in,total}} – B(t) – P_{\text{out}}$$ Here \(B(t)\) aggregates all irreversible power leaving the chosen system boundary (heat, radiation, chemical processes in discharge, etc.). where \(P_{\text{in,total}}\) is the total measured power entering the system boundary (all electrical inputs). In parallel, we distinguish the control/maintenance channel: $$P_{\text{control}} \approx B_{\text{maint}}(t)$$ i.e., the power required to keep the operating regime stable is primarily the power that compensates irreversible maintenance losses — not necessarily the full delivered output. In steady state \(\frac{dE_{\text{total}}}{dt} = 0\): $$P_{\text{in,total}} = B(t) + P_{\text{out}}$$ That is: total input covers the sum of losses and useful extraction. This fundamentally differs from the linear model \(P_{\text{out}} = \eta \cdot P_{\text{in}}\), where external input is directly proportional to output.

3. Physical Foundations of Regime Formation in VENDOR: Gas Discharge and Resonance

3.1 Pulsed Pumping and LC Resonance

The active circuit of the VENDOR system is built on the principle of pulsed pumping of a reactive structure. At its core is an effective series or parallel combination of inductance \(L\) and capacitance \(C\): $$\omega_0 = \frac{1}{\sqrt{LC}} \quad \text{(resonant frequency)}$$ $$Q = \frac{\omega_0 L}{R} = \frac{1}{\omega_0 RC}$$ where \(R\) is the total loss resistance (ohmic, dielectric, discharge). Quality factor \(Q\) determines how long energy is retained in the regime (one convenient representation for energy decay is): $$E(t) = E_0 \exp\left(-\frac{\omega_0 t}{Q_\mathrm{eff}}\right)$$ where \(Q_\mathrm{eff}\) denotes an effective quality factor defined for energy decay in the chosen model and boundary definition. Key point: high quality factor means that for the same level of circulating energy \(A\), the required compensation becomes smaller. This is not a violation of energy conservation, but its long circulation in fields rather than immediate consumption.

3.2 Gas Discharge as a Controlled Nonlinear Element

The central feature of VENDOR is the use of gas discharge (air or inert gas in the active node) not as an energy source, but as a dynamic nonlinear conductivity element. Townsend Avalanche With sufficient electric field \(E\), a free electron in gas, accelerating, ionizes a molecule, creating an additional electron: $$n_e(x) = n_{e,0} \exp(\alpha x)$$ where:

• \(n_e\) — electron concentration
• \(\alpha\) — first Townsend coefficient (depends on \(E\) and gas type)
• \(x\) — distance traveled by electron

Important: this is not an energy source, but a mechanism for a sharp change in conductivity \(\sigma(E,t)\). The energy of the ionization process is taken from the electric field created by the VENDOR circuit, not from “air”. Corona Discharge With certain electrode geometry, a region of weakly ionized plasma (corona) emerges, which has:

• Nonlinear I–V characteristic: conductivity depends nonlinearly on field
• Pulsed structure: corona emits current pulses enriched in higher harmonics
• Phase sensitivity: corona can either stabilize or destroy the resonant regime depending on phase

From an engineering standpoint, corona can act as a fast nonlinear conduction gate, allowing the system to adapt to changes in load and conditions. Streamer Transitions When certain conditions are exceeded, corona can transition to streamer mode — formation of conducting plasma “filaments” that:

• Sharply change instantaneous conductivity and loss structure
• Alter pulsed current waveform
• Are controlled in VENDOR by architecture and control to prevent regime destruction

3.3 Self-Oscillating Regime and Limit Cycle

In nonlinear systems with feedback and constraints, a stable periodic trajectory in phase space emerges — a limit cycle: $$\dot{x}_1 = f(x_1, x_2)$$ $$\dot{x}_2 = g(x_1, x_2)$$ where functions \(f, g\) contain nonlinear terms (discharge dependence, geometric constraints). Properties of limit cycle:

• Oscillation amplitude is independent of initial conditions (unlike a linear resonator)
• Stability is formed by nonlinearities that limit growth
• External perturbations are weakened by natural stabilization

In VENDOR, this means: the regime arrives at a stable state with predictable amplitude and spectrum, independent of small input variations.

3.4 Multi-Channel Discharge Structure as Stability Factor

VENDOR uses several discharge channels or elements with overlapping activation conditions. This provides:

• Adaptation to drifts: if one channel loses optimality (electrode erosion, humidity change), another “takes over” the regime
• Smoothing of spectral gaps: multi-channel design fills “dead zones” in input signal spectrum
• Engineering reliability: regime can be maintained even with local failures

Analogy: just as a multi-nozzle injection system in an ICE ensures mixture formation stability, multi-channel discharge ensures electrodynamic regime stability.

4. Two-Circuit Architecture: Separation of Functions

4.1 Circuit A: Regime Formation and Maintenance

Circuit A (Active Core) is responsible for:

• Formation of nonlinear electrodynamic regime with high internal circulation \(A(t)\)
• Maintaining regime amplitude within stability limits
• Compensating maintenance losses \(B_{\text{maint}}(t)\) through external input

Circuit functions:

• Pulsed pumping: delivering short energy pulses to resonant structure
• Discharge control: controlling gas discharge regime to optimize losses
• Stability protection: preventing exit from the stable operating region

4.2 Circuit B: Power Extraction Through Classical Induction

Circuit B (Linear Extraction) operates according to laws of classical electromagnetic induction: $$\mathcal{E} = -\frac{d\Phi_B}{dt} = -\frac{d}{dt}\left(\int \vec{B} \cdot d\vec{A}\right)$$ Variable magnetic flux from the active regime induces EMF in an extraction winding. This EMF is converted to useful power through:

• Rectification (diode bridge)
• Stabilization (capacitor filter)
• Inversion (conversion to standard load power parameters)

Lenz’s Law and Back-Action: Any power extraction creates a loaded quality factor \(Q_L\), reducing total quality factor: $$\frac{1}{Q_{\text{total}}} = \frac{1}{Q_{\text{core}}} + \frac{1}{Q_L}$$ This means extraction always increases losses \(B(t)\) and the required compensation: $$B_{\text{new}} = B_{\text{old}} + \Delta B_{\text{load}}$$ However, circuit separation allows architectural and phase management of this back-action, minimizing regime disruption until reaching the physical stability limit.

4.3 Architectural Isolation and Its Significance

In classical generators, load directly affects the source (mechanical shaft, excitation), causing immediate braking by Lenz’s law. In VENDOR, the extraction circuit is functionally isolated from the regime formation circuit:

• Energy extraction does not instantly destroy the regime formation mechanism
• Regime can adapt and reconfigure within its stability region
• Control can provide “soft” interaction between circuits

This does not mean the regime cannot be destabilized by excessive load; rather, the architecture is designed so that load influence is managed and the loss of stability occurs predictably, not as an immediate mechanical braking effect. This is not a cancellation of Lenz’s law (which always applies), but its engineering optimization.

5. Control Systems and Buffer (BMS)

5.1 Role of Energy Storage

The buffer battery in VENDOR performs several functions:

• Transient compensation: during load spikes (motor starting currents, pulse loads) the battery can briefly provide additional power
• DC bus voltage stabilization: preventing voltage sags during dynamic loads
• Control system power: control electronics require a stable and protected source
• Energy recuperation: in reduced load modes, part of the extracted energy can be directed to the buffer

Critically important: the battery is NOT a “hidden energy source”. It is a buffer whose energy comes from the VENDOR system itself, and energy withdrawal from the buffer is reflected in the energy balance as an additional load on the regime circuit.

5.2 BMS and Stability Management

BMS (Battery & Mode Management System) controls:

• Battery operation mode: charge, discharge, overcharge and deep-discharge protection
• Transient processes: soft start, dI/dt limitation during load switching
• Regime protection: preventing sharp voltage drops that can destroy the electrodynamic regime

BMS is not simply an electronic switch, but a regime stability control system, because a nonlinear electrodynamic regime is sensitive to dynamic disturbances and requires active management.

6. Parametric Excitation and Energy Amplification

6.1 Parametric Resonance in RLC Circuits

In circuits with variable parameters (e.g., \(L(t)\) or \(C(t)\)), parametric resonance is possible, where energy can be transferred from a control signal to an oscillatory regime with enhanced effectiveness: $$\frac{d^2q}{dt^2} + 2\gamma\frac{dq}{dt} + \omega_0^2(1 + \mu(t))q = 0$$ where \(\mu(t)\) is temporal parameter modulation (e.g., \(\mu(t) = \mu_0 \cos(2\omega_0 t)\) for parametric excitation at the second harmonic). Under certain conditions, oscillation energy grows exponentially until reaching nonlinear limitation, producing amplification without violating energy conservation. Physics: energy is taken from the control signal but transferred into the oscillatory regime more effectively than by direct excitation.

6.2 Regime Amplification vs. Energy Creation

Critical clarification: “Amplification” of regime means: acceleration of energy circulation, increase in field intensity, and regime stabilization with less energy loss per cycle. This does NOT mean: energy creation from nothing. In VENDOR, energy is amplified (in the sense of accumulation in fields) through proper pulse synchronization and nonlinearity control, but total system energy never exceeds the sum of external input plus initial conditions minus losses.

System Boundary: Why “η > 100%” Appears

The “efficiency paradox” is almost always a boundary-definition error: measuring only the control/maintenance channel while ignoring other inputs and stored-energy changes.

WRONG boundary (common mistake)
────────────────────────────────────────────────────────────────
   AC/DC supply  ──►  [ P_control ]  ──►  Active Core + Extraction  ──►  P_out
                         (only this is counted as input)
   (ignored: additional inputs, storage change, reflected load losses)

Right boundary (correct energy audit)
────────────────────────────────────────────────────────────────
   All electrical inputs  ───────────────►  ┌──────────────────────────────┐
                                            │   SYSTEM BOUNDARY            │
                                            │  Active Core + Extraction    │
                                            │  + Buffer + Control          │
                                            └──────────────────────────────┘
         measured as P_in,total ───────────►     P_out  +  B_total  +  dE/dt

Correct accounting uses total measured input at the system boundary, includes all irreversible losses, and tracks stored-energy change.

7. Why Linear Efficiency Gives “Implausible” Results

7.1 System Boundary Error

If one incorrectly defines efficiency using only the regime-maintenance channel, e.g.: $$\eta_{\text{apparent}} = \frac{P_{\text{out}}}{P_{\text{control}}}$$ then \(\eta_{\text{apparent}}\) can appear >>100%. This is not a physical efficiency; it is a boundary-definition artifact. If \(P_{\text{out}}\) is significant and \(P_{\text{control}}\) is small (due to high quality factor and effective control), the ratio becomes misleading. This is not a contradiction but a model error.

7.2 Worked Example: Correct vs. Incorrect Efficiency

Consider a steady-state operating point with the following measured quantities (system boundary includes all electrical inputs):

• Total electrical input: \(P_{\text{in,total}} = 2000\,\text{W}\)
• Total irreversible losses at boundary: \(B_{\text{total}} = 1500\,\text{W}\)
• Useful output power: \(P_{\text{out}} = 400\,\text{W}\)
• Stored-energy change: \(\frac{dE_{\text{stored}}}{dt} = +100\,\text{W}\) (buffer charging / field-energy increase)

Energy balance check: $$P_{\text{in,total}} = P_{\text{out}} + B_{\text{total}} + \frac{dE_{\text{stored}}}{dt}$$ $$2000 = 400 + 1500 + 100 \quad \checkmark$$ Incorrect (“apparent”) efficiency occurs when someone measures only the maintenance/control channel, e.g. \(P_{\text{control}} = 200\,\text{W}\): $$\eta_{\text{apparent}} = \frac{P_{\text{out}}}{P_{\text{control}}} = \frac{400}{200} = 2.0 \;\; (200\%)$$ This is a boundary error: \(P_{\text{control}}\) is not the total input; it is only the subset used to sustain the regime. Correct efficiency uses total measured input at the system boundary: $$\eta_{\text{true}} = \frac{P_{\text{out}}}{P_{\text{in,total}}} = \frac{400}{2000} = 0.20 \;\; (20\%)$$ Interpretation: the system is not “creating energy.” The apparent paradox appears only when input power is undercounted or when stored-energy change is ignored.

8. Role of Medium (Air, Gas) in Circuit A

8.1 Medium as Boundary Condition, Not Energy Source

The medium performs three roles:

• Reactive reservoir: air has dielectric permittivity \(\varepsilon_r \approx 1\) and can support electrostatic and magnetic fields
• Loss channel: ionization, molecular excitation, ozone synthesis and other phenomena consume energy from the regime
• Stability conditions: humidity, pressure, and air composition determine discharge onset thresholds and oscillation spectrum

The medium is NOT an energy source. It determines how the system works but does not power it. Analogy: water is not an energy source in a hydroturbine, but water presence determines whether the system can operate. Without water there is no regime, but water itself does not generate energy.

8.2 Mechanistic Hypotheses Consistent with Classical Electrodynamics

The medium (air or inert gas) is treated as a working medium and boundary condition for a nonlinear electrodynamic regime. Its exact contribution to regime dynamics is an active research topic, but several concrete mechanisms are consistent with classical physics and known gas-discharge behavior:

• H1 — Nonlinear impedance modulation (fast conductivity gating): discharge dynamics produce rapid, field-dependent changes in effective conductivity \(\sigma(E,t)\) and loss resistance \(R(t)\), enabling efficient transfer of energy from pulsed pumping into the resonant state (mode shaping, phase-locked gating).
• H2 — Effective capacitance/permittivity modulation (quasi-parametric interaction): space-charge formation, ion density, and local field redistribution can modulate the effective capacitance/permittivity of the active region, producing a parametric-like contribution to regime amplification under synchronized pumping (without introducing any nonconservative energy source).
• H3 — Spectrum enrichment and mode-selection via discharge microstructure: corona/streamer micro-pulses generate higher-harmonic content and non-sinusoidal waveforms that can couple into specific resonant modes, improving regime selectivity and stability (while also introducing measurable loss channels).

Importantly, these hypotheses describe how the medium shapes regime formation and stability, not how it supplies net energy. Any contribution of the medium is accounted for through measurable changes in losses, stored energy, and boundary-defined input/output flows.

9. Verification and Measurability

9.1 Energy Audit

Any claim about a VENDOR system must be supported by a complete energy audit: $$\text{energy}_{\text{total input}} = \text{energy}_{\text{losses}} + \text{energy}_{\text{output}} + \Delta\text{energy}_{\text{storage}}$$ where:

• Left side — total energy introduced into the system through all input channels
• Losses — measured through heat generation, radiation, ozone synthesis, etc.
• Output — active power available at output for the load
• Storage — change in energy in buffer and active circuit fields

Such an audit should be conducted with ±5–10% accuracy and across multiple load regimes.

9.2 Spectral Analysis and Regime Stability

A regime-based system is characterized by:

• Spectral composition: FFT of voltage/current signals should show a structured spectrum with a dominant resonant frequency
• Amplitude stability: at fixed load, regime oscillation amplitude should be stable within ±5%
• Response speed: regime recovery time after a load spike should be <100 ms

These indicators are measured with an oscilloscope and a spectrum analyzer.

9.3 Stability Boundaries and Physical Limits

When exceeding permissible extraction, the regime should:

• Degrade smoothly (amplitude reduction)
• Or stabilize at a new level
• Or completely shut down

This is expected physical behavior, not “magical system failure”.

10. Comparison with Classical Systems

Aspect	Classical Generator	VENDOR (Regime-Based System)
Power output mechanism	Mechanical shaft, excitation (direct)	Induction through isolated circuit
Load action on source	Immediate braking (Lenz’s law)	Controlled influence with architectural protection
Required energy input	Directly proportional to output	Proportional to losses, nonlinearly dependent on load
Oscillation amplitude	Depends on initial conditions (linear)	Independent (limit cycle, nonlinearity)
Adaptation to disturbances	Requires external control	Built into architecture (self-oscillations)
Output spectrum	Determined by mechanics (60 Hz or 50 Hz)	High-frequency, modulatable
Losses in regime formation	Minimal (ideal flywheel rotation)	Significant (discharge, radiation), but controllable

11. Limitations and Open Questions

11.1 Fundamental Questions

• Field interaction mechanism with air molecules: the exact role of ionization and molecular excitation in the energy balance requires detailed analysis at the level of molecular physics and spectroscopy.
• Mathematical description of parametric resonance in a system with gas discharge: classical parametric resonance theory assumes smooth parameters; including nonlinear discharge requires theory expansion.
• Minimum losses and maximum efficiency: is there a fundamental limit for quality factor in discharge regimes? How to optimize the compromise between stability and losses?

11.2 Engineering Challenges

• Reproducibility: uncontrolled factors (humidity, electrode contamination, component drift) require careful management or compensation.
• Scalability: transferring laboratory demonstration to industrial powers requires solving heat removal problems, regime stability at large amplitudes, and multi-element structure synchronization.
• Safety: working with high-voltage discharges requires special protection measures (ozone control, breakdown prevention, heat management).

11.3 Practical Application Boundaries

VENDOR is best suited for:

• Local applications with variable load
• Hybrid systems with storage
• Applications sensitive to harmonic distortion (high-frequency spectrum may require filtering for classical loads)

VENDOR is less suitable for:

• Ultra-high powers (>500 kVA) without significant scaling
• Applications requiring purely sinusoidal output (inverter block required)

12. Conclusion

VENDOR.Energy™ architecture represents a legitimate class of electrodynamic systems based on classical physics and distinct from traditional generators by its separation of regime formation, loss compensation, and power extraction functions. Incorrect application of linear energy models to regime-based systems leads to an apparent violation of principles (efficiency >100%), but this is a model error, not a physics violation. Key scientific results:

• Canonical A–B–C model provides a correct energy-balance framing for regime-based systems
• Gas discharge functions as a controlled nonlinear element, not an energy source
• Parametric resonance explains regime amplification without violating energy conservation
• Architectural circuit isolation allows minimizing back-action during power extraction
• Control system (BMS) is critical for maintaining nonlinear regime stability

VENDOR systems require further scientific verification, especially regarding precise determination of gas-medium role and a deeper fundamental explanation of parametric processes, but the current level of engineering validation and patent protection supports continued R&D. Technology Readiness Level (TRL): 5–6 (successful demonstration in a relevant environment; path to commercialization requires solving scaling and measurement standardization).

References

1. Maxwell J. C. “A Treatise on Electricity and Magnetism”. Classical electrodynamics, foundation of circuit B description. https://archive.org/details/treatiseelectric — Basic EM theory.
2. Q factor definition and properties: https://en.wikipedia.org/wiki/Q_factor — Quality factor definition and its influence on damping; https://www.zhinst.com/en/blogs/resonance-engineering-quality-factor-q-control-method — Practical Q control in resonators. — Quality factor as key to energy storage efficiency in regime.
3. Losses in electrodynamic systems: Griffiths D. J. “Introduction to Electrodynamics” (4th ed.). — Ohmic losses, dielectric losses and radiation in classical electrodynamics.
4. Open thermodynamic systems: https://www.britannica.com/science/thermodynamics/Open-systems; https://uomus.edu.iq/img/lectures21/MUCLecture_2025_2949441.pdf — Energy balance formulation for systems exchanging energy with environment.
5. Parametric resonance in LC circuits: https://open.clemson.edu/all_theses/3041/ (Caldwell, 2016) — Classical theory of parametric excitation in electrical circuits; https://arxiv.org/pdf/2112.12118.pdf (Sorokin, 2021, p.3-5) — Nonlinear electrodynamics and parametric effects.
6. Townsend avalanche: https://en.wikipedia.org/wiki/Townsend_discharge; https://en.wikipedia.org/wiki/Electron_avalanche — Physics of ionization cascade in gases.
7. Townsend coefficient and gas discharge: https://conf.uni-ruse.bg/bg/docs/cp17/3.1/3.1-2.pdf — Analytical models of corona discharge through avalanche density.
8. Corona discharge and its control: https://www.sciencedirect.com/topics/physics-and-astronomy/townsend-discharge — Classification of discharge types and their spectral properties.
9. Streamer dynamics: https://www.aappsdpp.org/DPP2025/html/3contents/pdf/5524.pdf (plasma streamers review, 2025) — Streamer propagation mechanism and discharge geometry control.
10. Streamer propagation and branching: http://www.plasma-tech.net/passkey2/passkey2/publications/www.plasma-tech.net/media/aQhTIE6XHx3YtOcS.pdf — Streamer dynamics in presence of foreign objects and geometric constraints.
11. Limit cycle and nonlinear self-oscillations: https://en.wikipedia.org/wiki/Limit_cycle; https://mitran-lab.amath.unc.edu/courses/MATH564/biblio/text/08.pdf — Theory of stable limit cycles in phase space.
12. Khalil, H. K. Nonlinear Systems. 3rd ed., Prentice Hall. — Foundational methods for stability, Lyapunov analysis, and nonlinear control (relevant to regime stability management).
13. Slotine, J.-J. E., & Li, W. Applied Nonlinear Control. Prentice Hall. — Practical nonlinear control frameworks applicable to mode stabilization and disturbance rejection.
14. Åström, K. J., & Murray, R. M. Feedback Systems: An Introduction for Scientists and Engineers. Princeton University Press. — Control fundamentals and feedback interpretation for energy systems with internal states.
15. Van der Pol oscillator and control: https://www.egr.msu.edu/~khalil/NonlinearSystems/Sample/Lect_5.pdf — Example of limit cycle in electrical circuit with nonlinear resistance.
16. Control system and buffering: Official VENDOR documentation (https://vendor.energy, “How It Works” section) — BMS architecture and storage role in maintaining regime stability; https://en.wikipedia.org/wiki/Inductive_coupling; https://onlinelibrary.wiley.com/doi/10.1155/2014/951624 — Theory of inductive energy transfer and coupling factors.
17. Faraday’s law and mutual induction: Classical electrical engineering — foundation of circuit B power extraction operation.
18. Parametric excitation and energy amplification: https://www.sciencedirect.com/science/article/abs/pii/S0196890423000183 (Margielewicz et al., 2023) — Nonlinear systems with internal resonance for energy harvesting; http://www.upitec.org/documents/miscellaneous/LCR-Resonant.pdf (Amador, 2012) — Parametric circuits and energy flow control in frequency domain.
19. Regime amplification in parametric excitation: https://pubs.aip.org/aip/apl/article/96/11/111906/338516/ (2010) — Mechanism of frequency conversion and energy amplification in nonlinear systems.
20. Energy conservation law and open systems: https://www.britannica.com/science/thermodynamics/Open-systems — Energy can enter and exit a system, but total energy is always conserved.
21. VENDOR patent documentation: WO2024209235 (PCT/International); ES2950176 (Granted, Spain); EUIPO No. 019220462 — Official documents describing the claimed architecture, embodiments, and intended industrial applicability.

Summary Table: Energy Flows and Common Accounting Errors

Quantity	Meaning	Where it appears	Common mistake
\(P_{\text{in,total}}\)	Total measured electrical input at the chosen system boundary	\(\frac{dE}{dt} = P_{\text{in,total}} – B – P_{\text{out}}\)	Counting only \(P_{\text{control}}\) and calling it “input”
\(P_{\text{control}}\)	Maintenance/control power sustaining the regime (subset of input)	Regime stability channel	Using \(\eta = P_{\text{out}} / P_{\text{control}}\) as “efficiency”
\(B_{\text{total}}\)	All irreversible losses crossing the boundary (heat, radiation, chemistry)	Loss term in the balance	Ignoring load-induced losses / underestimating dissipation channels
\(P_{\text{out}}\)	Useful output power delivered to the load	Measured at output terminals	Assuming output is “maintained” by \(P_{\text{control}}\) directly
\(\frac{dE_{\text{stored}}}{dt}\)	Stored-energy change in buffer + fields (positive when charging/accumulating)	\(P_{\text{in,total}} = P_{\text{out}} + B + \frac{dE}{dt}\)	Ignoring buffer charging/discharging when interpreting measurements

Disclosure Note: Modeling Scope vs. Implementation Scope

This article intentionally presents a mathematical and boundary-correct formulation of energy balance for regime-based electrodynamic systems. Its purpose is to eliminate common interpretation errors (especially the “η > 100%” paradox caused by incorrect system boundaries), not to disclose the full implementation details of VENDOR.Energy™ hardware. The numerical examples and balance equations in this paper are therefore conservative, model-level illustrations. They show how to account for total measured input, irreversible losses, and stored-energy change—but they do not encode the proprietary engineering decisions that govern real-world power scaling. Why this matters:

• A reader may look at a worked example (e.g., hundreds of watts) and incorrectly assume it represents the system’s maximum capability. In reality, power is a function of configuration (architecture, regime stability margins, extraction design, control strategy, thermal/safety constraints), and those implementation parameters are not published here.
• Certain engineering solutions that enable higher-output regimes are treated as protected know-how (and in parts as patent-described embodiments), and are disclosed only through controlled documentation to qualified parties.

Accordingly, this publication should be read as a correct scientific framing—a way to evaluate measurements without falling into linear-model traps— while the detailed implementation, validation methodology, and configuration-dependent performance envelopes are handled separately within the project’s controlled documentation (available via the Silent Pitch Room). In short: the math here explains how to measure and interpret an open nonlinear regime correctly. It does not attempt to publish the full set of engineering methods used to reach higher-power operating regimes.