Conditional Regime Feasibility in VENDOR.Max
A conditional mathematical model defining sufficient coefficient conditions for a stable operating regime $\Omega$. Numerical power levels, boundary closure, and validation metrology are outside the scope of this article.
Scope. The framework does not assert that the required coefficients are achieved in the present prototype. It states only that, if such coefficients are achieved, the proposed model admits a mathematically self-consistent operating regime. Whether these conditions are realized physically is left to experimental validation.
Engineering Principle 2. Existence of the regime is governed by the ability of the system to maintain a bounded, self-consistent pre-breakdown field state. Power extraction, if any, is a separate condition imposed on that state. Formally, $\Omega = \Omega_{LC}\cap\Omega_{\mathrm{prebreakdown}}$ under a self-consistency (fixed-point) condition linking the two projections (§3, §5).
Central thesis. A stable operating regime exists when, and is identified with, an attracting self-consistent state of the pre-breakdown field:
Existence of this state does not, by itself, entail power delivery ($\exists\,\Omega\not\Rightarrow P_3>0$).
Regime as a dynamical object
A regime is a persistent operating process: a trajectory $\mathbf{x}(t)$ that converges to a bounded invariant set $\Omega$ while satisfying phase-synchronization and bounded-energy conditions. Existence of $\Omega$ is the object of this article; load delivery is treated as a separate condition (§6) and does not follow from regime existence alone.
State and the regime-forming operator $\mathcal{N}(\mathbf{x};\alpha)$
State (energy variables plus medium state, §3):
The regime-forming element is a general operator $\mathcal{N}(\mathbf{x};\alpha)$, parameters $\alpha$ identified experimentally. Minimal properties required for a bounded regime: threshold (onset above $E_{\mathrm{on}}$); nonlinear conductivity / discharge response (modulates $\sigma_g$); amplitude limitation / saturation (fixes a limit amplitude $A^\star$); phase-dependent response (participates in synchronization); optional memory / hysteresis (augments the state).
Candidate classes (not assumed a priori; discriminated by measurement): NDR-like; parametric-modulation; switching/hybrid; hysteretic-discharge; delayed-feedback. The model defines the minimal operator properties required for regime existence; validation determines which class the prototype realizes and whether the corresponding coefficients exceed threshold.
Assumptions. A0 deterministic reduced-order model; A1 bounded state variables; A2 lumped-parameter approximation; A3 bounded pre-breakdown medium; A4 stable synchronization window; A5 finite losses; A6 measurable state / proxy variables.
Field-formation layer (central element)
The present model treats the architecture as not fully reducible to a conventional transformer representation $I_A\to\Phi_A\to V_2,V_3$, because the coupling flux is assumed to depend on the state of the pre-breakdown medium. The flux is a regime-dependent field state produced by the interaction of (i) the controlled pre-breakdown medium, (ii) the Tesla-type resonant excitation stage, and (iii) the capacitive-node dynamics:
Medium state (general form):
(One admissible realization is the Townsend-with-recombination form $\dot n_e=\alpha_{\mathrm{ion}}(E_g)n_e-\beta n_e^2+S$; it is an example, not an assumption.)
Field flux $\Phi_A(t)=\mathcal{F}_\Phi(E_g,n_e,I_A,\kappa_T)$; induced voltages $V_k=-N_k\dot\Phi_A$, $V_{k,\mathrm{rms}}\approx\omega_0 M_k I_{A,\mathrm{rms}}$.
Pre-breakdown window (existence constraint). The regime is modeled as a controlled pre-breakdown conductive state, not an arc. Define the normalized window coordinate
Self-consistency. $\Omega_{LC}$ constrains the energy coordinates $(q_C,\varphi_A,\varphi_2,\varphi_3)$ (a bounded LC limit cycle); $\Omega_{\mathrm{prebreakdown}}$ constrains the medium coordinates $(E_g,n_e,\sigma_g)$. These are distinct projections of $\mathbf{x}$, coupled by the field map $\mathcal{F}_\Phi$. A regime exists only at their self-consistent intersection — a fixed point of the medium $\to$ field $\to$ current loop. Writing the loop as an amplitude map,
a regime corresponds to a fixed point $A^\star$ of $\mathcal{T}$ lying in the window. Then
Non-emptiness of $\Omega_{LC}$ and $\Omega_{\mathrm{prebreakdown}}$ separately is necessary but not sufficient; existence of the fixed point $A^\star=\mathcal{T}(A^\star)$ is the nontrivial content.
Tesla-type stage. Tesla-type resonant excitation stage means a high-voltage resonant field-forming stage; the term does not imply any non-classical energy source. With $W_{\mathrm{field}}=\tfrac12\int_V\epsilon|E_g|^2dV+\tfrac12\int_V\mu|H_A|^2dV$: field-formation gain $K_T$ and $K_{\mathrm{field}}=W_{\mathrm{field}}/W_A$.
Model property. $K_T$, $K_{\mathrm{field}}$, and $Q_A$ scale field magnitude and reactive (circulating) energy; field magnitude and active-power transfer are treated as distinct quantities throughout. (This is the definition of reactive power, not a device-specific claim.)
Coupling and sustainment coefficients
Field sufficiency (geometry/induction): $K_{\Phi,2}=\dfrac{\omega_0 M_2 I_{A,\mathrm{rms}}}{V_{2,\mathrm{crit}}}$, $K_{\Phi,3}=\dfrac{\omega_0 M_3 I_{A,\mathrm{rms}}}{V_{3,\mathrm{crit}}}$. Carrier density enters through the window $n_e\in[n_{\min},n_{\max}]$ of $\Omega_{\mathrm{prebreakdown}}$ (no separate ionization threshold, to avoid double-describing the same constraint). Regime damping: $K_{\mathrm{damp}}=P_{\mathrm{loss,regime}}/(\omega_0 W_A)
Regime sustainment (local, Contour A only):
$K_{\mathrm{fb}}\ge 1$ means only that the feedback path is sufficient to compensate local regime losses in Contour A under the model assumptions. It is a coefficient of local regime sustainment, not a proof of total energy closure and not a proof of net power delivery. The feedback path $\text{Secondary}\to\text{Rectifier}\to\text{BMS}\to C_{2.1\text{–}2.3}$ is the only subsystem input to Contour A considered in the present model; its $P_{\mathrm{loss},A}$-only denominator deliberately excludes $P_3$ (see §6).
Regime existence (sufficient conditions)
Conjecture H1 (sufficient conditions for regime existence). The following conditions are conjectured sufficient for regime existence within the present model. Under assumptions A0–A6, if
then the realization admits a bounded invariant set $\Omega$ at amplitude $A^\star$ with convergence from a neighborhood. A merely existing but non-attracting fixed point ($|\mathcal{T}'(A^\star)|\ge 1$) does not establish an attracting operating regime within this model. Necessity of these conditions is not claimed. A proof is outside the scope of the present article. The coefficients are functions of $L_A,C_\Sigma,M_2,M_3,Q_A,\kappa_T,\mathcal{N}$; their numerical values are not claimed here.
Toward a proof: converting the amplitude-map argument $A^\star=\mathcal{T}(A^\star)$ into a theorem (existence and stability of the fixed point via averaging / describing-function) requires the qualitative form of $\mathcal{N}$ and $\sigma_g$ and order-of-magnitude parameters — admissible within a TRL-gated disclosure boundary.
Tertiary extraction as a separate condition
Load delivery does not follow from regime existence:
Extraction requires, in addition to $\exists\,\Omega$, the independent field-sufficiency condition $K_{\Phi,3}\ge K_{\Phi,3}^{\mathrm{crit}}$ and a separate extraction condition
whose explicit form and numerical resolution — together with the quantitative boundary-energy interpretation, including the identification of any sustaining term, if required — are deferred to the companion validation work. This article records only that extraction imposes a condition distinct from, and not implied by, regime existence; the two results are not combined.
Discrimination by measurement (outside the present scope, stated for completeness)
Future validation would determine which candidate class $\mathcal{N}$ realizes; whether $E_g$ stays in $\Omega_{\mathrm{prebreakdown}}$ ($0
What this article does not claim
- It does not prove autonomous operation or net power delivery.
- It does not assign a numerical value to any coefficient or power.
- It does not claim necessity of the §5 conditions, only sufficiency.
- It does not assert that the feedback path reaches the required sustainment threshold in hardware — that is a condition to be tested.
- $K_T,K_{\mathrm{field}},Q_A$ scale field/reactive quantities, distinct from active-power transfer.
- Regime existence (§5) does not imply tertiary extraction (§6); $K_{\Phi,3}$ is not part of the regime-existence set — it belongs to the separate extraction condition.
The article reduces regime feasibility to a sufficient-condition set $\{K_{\mathrm{window}},K_T,K_{\Phi,2},K_{\mathrm{fb}},K_{\mathrm{damp}}\}$ for a stable attracting fixed point $A^\star=\mathcal{T}(A^\star)$ of the pre-breakdown field state, with $\Omega=\Omega_{LC}\cap\Omega_{\mathrm{prebreakdown}}$ (Principle 2). Numerical coefficients, power delivery performance, and boundary closure remain outside its scope.
Frequently asked questions
Does this article establish net or autonomous power?
No. It establishes neither net nor autonomous power. It defines sufficient model conditions for a stable operating regime $\Omega$. Power delivery is a separate condition (§6) and is not implied by regime existence: $\exists\,\Omega\not\Rightarrow P_3>0$.
Is regime existence the same as device-level operation?
No. Existence of an attracting fixed point $A^\star=\mathcal{T}(A^\star)$ means the modeled dynamics can settle into a bounded pre-breakdown state. Whether the physical prototype realizes that state, and whether it can deliver load, are empirical questions for future validation.
Is $K_{\mathrm{fb}}\ge 1$ a claim that feedback sustains full device operation?
No. $K_{\mathrm{fb}}\ge 1$ is a condition on local regime losses in Contour A only; its denominator excludes $P_3$ by construction. It is a sustainment coefficient, not a proof of total energy closure and not a proof of net delivery. Whether it is reached in hardware is to be tested.
Does the high field of the Tesla-type stage mean energy gain?
No. $K_T$, $K_{\mathrm{field}}$, and $Q_A$ scale field magnitude and reactive (circulating) energy. A high-$Q$ stage can support large circulating reactive quantities; active-power transfer is a separate measurement question. Field magnitude and active-power transfer are distinct quantities.
Is the conjecture (§5) a theorem?
No. H1 is a conjecture; a proof (existence and stability of $A^\star$ via averaging / describing-function) is outside this article's scope and requires the qualitative form of $\mathcal{N}$ and $\sigma_g$.
Where do numerical power levels, extraction balance, and boundary-energy questions go?
Outside this article — into companion boundary-accounting and validation work, together with the explicit extraction inequality $\mathcal{C}_{\mathrm{extract}}\ge 0$.
What does "pre-breakdown" mean here, and is this an arc?
It is a controlled conductive state below the arc-transition threshold, $0
References
Reference scope: the citations support the mathematical methods (port-Hamiltonian formulation, nonlinear-dynamics existence/stability, describing-function analysis) and the high-voltage / pre-breakdown discharge physics used in this model. They do not assert net or autonomous power generation.