The Exponent Is the Price: Quantum Tunneling and the Architecture of Wireless Electric Grids

The request that motivates this paper — build a wireless electric grid on quantum tunneling — is worth stating precisely enough to be wrong:

H₀. There exists a device architecture in which quantum-mechanical tunneling of charge carriers serves as the primary power-transport channel across macroscopic distances, at efficiency and power density competitive with conducted transmission.

We reject H₀ decisively in §2, and then argue that the rejection is more informative than an acceptance would have been, because the form of the rejection identifies exactly which currency the physics is demanding. This is the analytical move used throughout: a no-go theorem is a price list written in the imperative mood.

Three successor hypotheses survive:

• —H₁ (evanescent). The 1D barrier equation governing tunneling also governs classical evanescent electromagnetic coupling; substituting a massless carrier converts an intractable exponent into a geometric one. Accepted, with quantified limits.
• —H₂ (terminal). Tunneling enters the wireless grid as rectifier, switch, and sensor rather than as transport. Accepted.
• —H₃ (layered). No single mechanism spans the wireless grid; the currency changes character at each length scale, so the wireless grid is necessarily heterogeneous. Accepted as the paper's synthetic claim.

2.1 The exponent

For a particle of effective mass $m^{\ast }$ and energy $E$ incident on a rectangular barrier of height $V$ and width $d$, the WKB transmission is

For a free electron, evaluating the constants:

[T] For a metal–vacuum–metal junction with $\mathrm{\Phi }\approx 4.5$ eV, $\kappa =10.87{\mathrm{n}\mathrm{m}}^{-1}$.

Gap       2κd        T
0.5 nm    10.9       1.9 × 10^-5
1 nm      21.7       3.6 × 10^-10
10 nm     217        10^-94
1 µm      2.17×10^4  10^-9442
1 m       2.17×10^10 10^(-9.4×10^9)

2.2 The invoice

[T] To deliver one coulomb ( $6.24\times {10}^{18}$ electrons) across a 1 m vacuum gap by direct tunneling requires $\sim {10}^{9,440,000,000}$ incidence attempts. For calibration: the observable universe contains $\sim {10}^{80}$ baryons and has existed for $\sim {10}^{61}$ Planck times, giving an absolute ceiling of $\sim {10}^{141}$ elementary events in all of cosmic history. The requirement exceeds the budget by a factor of ${10}^{9,439,999,859}$.

This is worth stating in this vulgar form because it forecloses a specific rhetorical move. There is no clever geometry, no array multiplicity, no resonance, and no material that recovers ${10}^{9.4\times {10}^9}$ . Arrays buy linear factors. Resonances buy polynomial factors. The deficit is exponential in a quantity of order ${10}^{10}$.

[T] Setting a marginal detection floor of $T={10}^{-10}$, the maximum single-barrier electron tunneling gap is:

$d_{\max }=1.06$ nm at $\mathrm{\Phi }=4.5$ eV (metal–vacuum) $d_{\max }=7.1$ nm at $\mathrm{\Phi }=0.1$ eV (engineered low-barrier heterostructure)

Conclusion: H₀ is rejected. Charge tunneling is a nanometre-scale phenomenon and will remain one under any materials engineering consistent with known physics. [T]

2.3 Reading the invoice: three levers

The rejection came from a single number, and that number is a product of three factors:

$d$: bounded below by atomic spacing ($\sim$0.2 nm) and above by nothing — but the whole problem is that we need it large. $\mathrm{\Phi }$: reducible from ~5 eV to ~0.1 eV by heterostructure design; buys $50\approx 7\times$ in $d$. Real, but a rounding error against ${10}^{10}$. $m^{\ast }$: here is the interesting term. It is the only factor in the exponent that is a property of the carrier rather than the barrier. Effective-mass engineering in semiconductors reaches $m^{\ast }\sim 0.02{\mathrm{m}}_e$

(InSb), buying $50\approx 7\times$.

But $m^{\ast }$ has a limit the others do not: it can be exactly zero.

This is the permission structure. The barrier problem does not say "no transport at range." It says: *the exponent is priced in $m^{\ast }$

, and you may choose your carrier.*

3.1 The formal identity

Separate the Helmholtz equation for a monochromatic field in a source-free region, with transverse wavenumber $k\parallel$

:

=(k ∥ 2

− c 2 ω 2

)ψ

Compare the time-independent Schrödinger equation in the classically forbidden region:

𝜓 ∂z 2 ∂ 2 ψ

= ℏ 2 2m ∗ (V−E)

ψ

[T] These are the same equation. The correspondence is exact:

(3.1) κ e 2

= ℏ 2 2m ∗ Φ

⟷κ γ 2

=k ∥ 2

− c 2 ω 2

(3.1)

This is not a metaphor. Frustrated total internal reflection is the optical realisation, and is correctly and standardly called photon tunneling. Evanescent decay across a sub-wavelength gap and electron decay across a barrier are the same boundary-value problem.

3.2 The epistemic firewall

This is the point at which the literature on "quantum wireless power" typically goes wrong, and it is worth being explicit.

The correspondence is exact at the level of the differential equation and empty at the level of the mechanism. Magnetoquasistatic coupling between two 85 kHz coils is a solution of Maxwell's equations with no quantum content whatsoever; the field is a coherent state with photon occupation numbers of order ${10}^{30}$, and every observable is classical. Calling it "tunneling" is legitimate as a statement about the mathematical structure of the field decay, and illegitimate as a claim about quantum mechanics doing work in the device.

[T] The correspondence licenses the scaling analysis below. It licenses nothing about quantum advantage, entanglement, coherence, or any other quantum resource in the power link.

Any claim that a macroscopic WPT link derives performance from quantum mechanics — as opposed to from classical resonance — should be treated as false until the claimant exhibits a measurable that distinguishes it from the classical solution. [T]

3.3 Sizing the discount

For a sub-wavelength resonator of characteristic size $a$, the near field is dominated by $k\parallel \sim 1\mathrm{/}a$, and since $a\ll \lambda$ we have $k\parallel \gg \omega \mathrm{/}c$:

• —𝜅
• —𝛾
• —≈
• —1
• —𝑎

(3.2) κ γ

≈ a 1

(3.2)

[T] The evanescent decay length equals the source size. A 30 cm coil has a 30 cm decay length. This single fact is the whole of resonant WPT.

The discount ratio:

m − 1 × 0.3 m ≈ 3.3 × 10 9 κ γ

κ e

= ℏ 2m e

Φ

a=1.09×10 10 m −1 ×0.3 m≈3.3×10 9

[T] Choosing a massless carrier reduces the decay coefficient by $3.3\times {10}^9$ for a 30 cm resonator. The exponent goes from $2{\mathrm{m}}_e\mathrm{\Phi }\mathrm{/}\mathrm{\hslash }$ — a compound of the electron rest mass and the vacuum work function, neither of which is negotiable — to $1\mathrm{/}a$, which is a line item in a bill of materials.

This is the central result of Part I. The impossible thing becomes an engineering choice the moment the correct currency is identified. The currency was carrier mass.

The mass discount is real but not unconditional. Having bought a geometric exponent, we now discover what geometry charges.

4.1 Coupled-mode theory

Two resonators with coupling rate ${\kappa }_c$ and intrinsic loss rates ${\mathrm{\Gamma }}_1,{\mathrm{\Gamma }}_2$. Define the strong-coupling figure of merit:

where $k=M\mathrm{/}\sqrt{L_1{L}_2}$ is the magnetic coupling coefficient. [T] At optimal loading, the maximum end-to-end efficiency is

• —$U=1$: ${\eta }_{\max }=17.2\mathrm{\%}$
• —$U=10$: ${\eta }_{\max }=81.9\mathrm{\%}$
• —$U=19$: ${\eta }_{\max }=90.0\mathrm{\%}$
• —$U=100$: ${\eta }_{\max }=98.0\mathrm{\%}$
• —$U=200$: ${\eta }_{\max }=99.0\mathrm{\%}$
• —$U=2000$: ${\eta }_{\max }=99.9\mathrm{\%}$

[T] Efficiency is a function of $U$ alone. Distance enters only through $k$.

4.2 The Q¹⁄³ range law

For coaxial coils of radius $a$ at separation $d\gg a$, with $M={\mu }_0\pi a^4{N}_1{N}_2\mathrm{/}(2d^3)$ and $L={\mu }_0{N}^{2}aG$ (form factor $G\approx 7$):

Substituting into (4.1) with $Q_1=Q_2=Q$ and solving for range at fixed $U$:

[T] Range scales as the cube root of quality factor.

With $a=0.3\mathrm{m}$, $U_{\mathrm{t}\mathrm{a}\mathrm{r}\mathrm{g}\mathrm{e}\mathrm{t}}=19$ ($\eta =90\mathrm{\%}$):

• —- $Q={10}^3$: $d_{\max }=0.68\mathrm{m}$ — Litz copper, routine [E]
• —- $Q={10}^4$: $d_{\max }=1.47\mathrm{m}$ — Optimised copper / low-loss dielectric [E]
• —- $Q={10}^6$: $d_{\max }=6.8\mathrm{m}$ — HTS coil, 77 K [C]
• —- $Q={10}^9$: $d_{\max }=68\mathrm{m}$ — Beyond any demonstrated low-frequency coil [O]
• —- $Q={10}^{12}$: $d_{\max }=683\mathrm{m}$ — Exceeds best SRF cavities by $\sim 10\times$ [O]

[T] The cube root is the wall. Extending range from 1 m to 1 km at fixed aperture demands $Q\times {10}^9$. Distribution feeders run 1–30 km. There is no $Q$ that reaches them.

The alternative is to scale $a$, which enters linearly. A 10 km link at $Q={10}^6$ requires $a=215\mathrm{m}$ — a superconducting resonator the footprint of a large generating station. This is where the second wall appears.

4.3 The reactive-power wall

Power crosses the gap through the mutual term; the primary must circulate the rest. To order of magnitude,

[T] (The $O(1)$ prefactor depends on loading and is not pinned here; the $(d\mathrm{/}a{)}^3$ scaling is robust.)

• —EV pad: $P=11\mathrm{k}\mathrm{W}$, $d=0.15\mathrm{m}$, $a=0.25\mathrm{m}$, $k=1.04$, $P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}=11\mathrm{k}\mathrm{V}\mathrm{A}$
• —Warehouse node: $P=100\mathrm{k}\mathrm{W}$, $d=20\mathrm{m}$, $a=5\mathrm{m}$, $k=3.5\times {10}^{-3}$, $P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}=29\mathrm{M}\mathrm{V}\mathrm{A}$
• —Distribution feeder: $P=10\mathrm{M}\mathrm{W}$, $d=1\mathrm{k}\mathrm{m}$, $a=50\mathrm{m}$, $k=2.8\times {10}^{-5}$, $P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}=357\mathrm{G}\mathrm{V}\mathrm{A}$
• —Transmission line: $P=1\mathrm{G}\mathrm{W}$, $d=100\mathrm{k}\mathrm{m}$, $a=1\mathrm{k}\mathrm{m}$, $k=2.2\times {10}^{-7}$, $P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}=4.5\mathrm{P}\mathrm{V}\mathrm{A}$

[T] A single 1 GW / 100 km near-field link would circulate $4.5\times {10}^{15}$ VA of reactive power. Global installed electrical generating capacity is $\sim 8\times {10}^{12}$ W. The circulating power exceeds it by a factor of $\sim 560$.

This is the decisive result for H₀-adjacent proposals. Near-field WPT does not forbid grid-scale transmission; it prices it in circulating volt-amperes at $P(d\mathrm{/}a{)}^3$, and no amount of $Q$ touches that number — $Q$ governs what fraction of the circulating power is lost, not how much must circulate. [T]

4.4 The exposure wall — a derived scaling

The circulating current produces a field, and the field is regulated. We derive the compliance geometry.

A coil of radius $a$, $N$ turns, current $I$ has moment $m=\pi a^{2}NI$. From $P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}=\frac{1}{2}\omega L{I}^2$ with $L={\mu }_0{N}^{2}aG$:

(Note $m$ is independent of $N$, as it must be.) The compliance radius where on-axis $B={\mu }_{0}m\mathrm{/}(2\pi r^3)$ falls to the regulatory limit:

Substituting $P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}=(2G\mathrm{/}\pi )P{d}^3\mathrm{/}{a}^3$ from (4.5), the coil radius cancels exactly:

[T] The public exclusion radius of a magnetoquasistatic power link depends only on delivered power and link distance — not on aperture size.

This deserves emphasis because it removes the designer's principal degree of freedom. One can trade coil size against $Q$, against current, against turns count — and the exclusion zone does not move. It is a property of the service being sold, not of the hardware selling it. $R_c\propto P^{1\mathrm{/}6}{d}^{1\mathrm{/}2}$: weakly sensitive to power, strongly sensitive to range.

Regulatory inputs. ICNIRP (2020) supersedes the 100 kHz–10 MHz range of ICNIRP (2010). General-public whole-body reference levels, 0.1–30 MHz, averaged over 30 min: $H_{\mathrm{i}\mathrm{n}\mathrm{c}}\le 2.2\mathrm{/}{f}_{\mathrm{M}\mathrm{H}\mathrm{z}}$ A m⁻¹ and $E_{\mathrm{i}\mathrm{n}\mathrm{c}}\le 300\mathrm{/}{f}_{\mathrm{M}\mathrm{H}\mathrm{z}}^{0.7}$ V m⁻¹. Local exposure (6 min): $H_{\mathrm{i}\mathrm{n}\mathrm{c}}\le 4.9\mathrm{/}{f}_{\mathrm{M}\mathrm{H}\mathrm{z}}$ A m⁻¹. Peak instantaneous, 100 kHz–10 MHz: $H_{\mathrm{i}\mathrm{n}\mathrm{c}}\le 21$ A m⁻¹. [E]

• —6.78 MHz (ISM): $H_{\lim }=0.325\mathrm{A}{\mathrm{m}}^{-1}$, $B_{\lim }=0.41\mu \mathrm{T}$
• —13.56 MHz: $H_{\lim }=0.162\mathrm{A}{\mathrm{m}}^{-1}$, $B_{\lim }=0.20\mu \mathrm{T}$
• —85 kHz (SAE J2954): below ICNIRP 2020 scope, $B_{\lim }\approx 27\mu \mathrm{T}$ [E, verify]

[T] Below 100 kHz, ICNIRP (2010) governs, with a general-public flux-density reference level roughly 66× more permissive than the 2020 value at 6.78 MHz. This is why deployed automotive WPT operates at 85 kHz. The frequency was not chosen by coupling physics — coupling is frequency-independent in $k$ — but by the shape of the regulatory curve immediately below its discontinuity. [C, strong] The industry standard sits where it does because of a boundary in a document.

Worked exclusion radii at 85 kHz, $B_{\lim }=27\mu \mathrm{T}$:

• —- Warehouse node: $P=100\mathrm{k}\mathrm{W}$, $d=20\mathrm{m}$, $R_c=9.7\mathrm{m}$, keep-out area 300 m²
• —- Feeder: $P=10\mathrm{M}\mathrm{W}$, $d=1\mathrm{k}\mathrm{m}$, $R_c=147\mathrm{m}$, keep-out area 6.8 ha
• —- Transmission: $P=1\mathrm{G}\mathrm{W}$, $d=100\mathrm{k}\mathrm{m}$, $R_c=3.2\mathrm{k}\mathrm{m}$, keep-out area 32 km²

[T] A 1 GW wireless transmission link would impose a 3.2 km public exclusion radius — 32 km² of uninhabitable right-of-way, against roughly 0.3 km² for a conventional 100 km overhead line at 60 m width. Wireless transmission is land-hungrier than wire by two orders of magnitude, which inverts its principal claimed advantage. [T]

4.5 The cryogenic break-even theorem

Superconductivity is the obvious route to the $Q$ that buys range. It has a price with a closed form.

At fixed distance, resonator loss is $P_{\mathrm{l}\mathrm{o}\mathrm{s}\mathrm{s}}=P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}\mathrm{/}Q$. Raising $Q$ by $\rho$ reduces loss by $\rho$. But the refrigerator consumes $1\mathrm{/}\mathrm{C}\mathrm{O}\mathrm{P}$ watts of wall power per watt lifted. Break-even requires:

[T] With Carnot $\mathrm{C}\mathrm{O}\mathrm{P}=T_c\mathrm{/}(T_h-T_c)$ and practical machines at 10–20% of Carnot:

• —- 77 K (LN₂/HTS): Carnot COP 0.345, practical COP ~0.07, required $Q_{\mathrm{S}\mathrm{C}}\mathrm{/}{Q}_{\mathrm{C}\mathrm{u}}>15$, range gain 2.5×
• —- 20 K: Carnot COP 0.071, practical COP ~0.010, required $Q_{\mathrm{S}\mathrm{C}}\mathrm{/}{Q}_{\mathrm{C}\mathrm{u}}>100$, range gain 4.6×
• —- 4 K (LHe/LTS): Carnot COP 0.014, practical COP ~0.002, required $Q_{\mathrm{S}\mathrm{C}}\mathrm{/}{Q}_{\mathrm{C}\mathrm{u}}>500$, range gain 7.9×

[T] At the break-even point, the range gain is exactly $(1\mathrm{/}\mathrm{C}\mathrm{O}\mathrm{P}{)}^{1\mathrm{/}3}$ — because range goes as $Q^{1\mathrm{/}3}$ while the energy penalty goes as $Q^1$. A system operating at 20 K must find a $100\times$ $Q$ advantage merely to break even energetically, and that $100\times$ purchases only $4.6\times$ in range.

[C] This is the honest verdict on superconducting near-field WPT: the cryogenic overhead scales faster than the benefit, by a power of three. HTS at 77 K is the only regime where the arithmetic is close, and it buys 2.5×.

4.6 Where the near-field niche actually ends

Compare aperture requirements against radiative beam-forming (developed in §6). Setting the near-field coil radius from (4.4) equal to the Fresnel-zone aperture radius $a_{\mathrm{f}\mathrm{f}}=\sqrt{\tau \lambda d\mathrm{/}\pi }$ and solving:

[T] At 2.45 GHz ($\lambda =12.24\mathrm{c}\mathrm{m}$), $\tau =2$, $U=19$:

• —- $Q={10}^3$ (copper): crossover distance 0.40 m
• —- $Q={10}^4$: crossover distance 1.9 m
• —- $Q={10}^6$: crossover distance 40 m

Beyond ~0.4 m, microwave beam-forming needs a smaller aperture than resonant coupling to move the same power — with normal-conductor coils, which is what exists. [T]

This is a genuine retrodiction, not a fit. The model was constructed from first principles with no reference to deployment data, and it places the near-field/far-field crossover at ~0.4 m. Every commercially deployed WPT system — Qi phone charging (~5 mm), medical implant links (~1–3 cm), SAE J2954 automotive pads (~10–25 cm), industrial AGV charging (~5–15 cm) — operates below that line. [E] The single class of demonstrated systems approaching it is heavy-duty vehicle charging under SAE TIR J2954/2, specified to 500 kW with larger ground assemblies. [E]

The theory predicted the envelope. The envelope is what exists. This is the strongest available evidence that the pricing model is correct, and correspondingly the strongest evidence that proposals to extend near-field WPT to distribution ranges are not incremental but categorical.

Having excluded tunneling from transport, we now find it is load-bearing at both terminals — and in one place, structurally unavoidable.

5.1 The rectifier: MIM tunnel junctions

This is the strongest genuine claim in the paper. Above roughly 1 THz, conventional semiconductor rectification fails: minority-carrier transit times and junction capacitance impose an RC ceiling no doping profile evades. Metal–insulator–metal (MIM) tunnel diodes rectify by direct barrier tunneling, a process with femtosecond transport time, and are therefore the only known rectifying element viable for infrared and terahertz rectennas. [E]

Asymmetry arises from dissimilar electrode work functions producing a trapezoidal barrier; multi-insulator stacks (MIIM, MI³M) engineer step or resonant barriers for larger zero-bias nonlinearity. Reported figures of merit: asymmetry ~6 at 0.1 V and responsivity 4.3 A/W at 0.35 V for ALD-grown MI³M stacks; zero-bias responsivity $-0.6$ A/W for Au/Al₂O₃/Ti. [E] Notably, one careful study found that resonant tunneling in triple-insulator stacks does not deliver the low-bias enhancement previously reported, and that dissimilar-metal asymmetry is the more promising route — a negative result worth more than the positive ones it corrects. [E]

The trilemma. Three requirements are mutually antagonistic, and the constraint is derivable:

• —Impedance match to the nano-antenna:
• —𝑅 𝑑 ≈ 50 – 200

Ω R d

≈50–200 Ω. RC cutoff: $f_c=1\mathrm{/}(2\pi R_{d}C)>f_{\mathrm{s}\mathrm{i}\mathrm{g}\mathrm{n}\mathrm{a}\mathrm{l}}$

• —.
• —Responsivity $\beta =I^{\mathrm{\prime }\mathrm{\prime }}\mathrm{/}2I^{\mathrm{\prime }}$: large nonlinearity requires thick or high barriers, which raise $R_d$

exponentially.

With $C={ϵ}_0{ϵ}_{r}A\mathrm{/}t$, ${ϵ}_r=9$ (Al₂O₃), $t=2$ nm:

Junction area $C$ Max $R_d$

at 28.3 THz (10.6 µm)

( 100 nm ) 2 (100 nm) 2 398 aF 14 Ω — unmatched, unbuildable

( 50 nm ) 2 (50 nm) 2 100 aF 56 Ω

( 20 nm ) 2 (20 nm) 2 16 aF 354 Ω — matchable

[T] Long-wave infrared rectification requires junction areas below ~ ( 30 nm ) 2 (30 nm) 2 . This caps power per element, forcing array multiplicity: a $10\mu m^2$ antenna under 1 kW m⁻² collects ~10 nW, so a 1 W panel requires $\sim {10}^8$ elements, and 1 m² of collector requires $\sim {10}^{10}$ junctions.

[C] The binding constraint on optical rectennas is therefore not the diode physics — which is understood — but interconnect and yield across ${10}^{10}$ devices per square metre, each delivering nanowatts at tens of millivolts. Series-parallel network losses and the statistics of junction-parameter dispersion will dominate system efficiency long before any single-device figure of merit does. This is a falsifiable prediction about where the field's effort will be misallocated. (See P3, §7.)

5.2 The switch: tunnel FETs and the Boltzmann floor

Every thermionic switch — every MOSFET ever built — is bounded by

$S$ min = 𝑘 𝑇 𝑞 ln 10 ≈ 60 mV/decade at 300 K SS min

= q kT

ln10≈60 mV/decade at 300 K

[T] This is a statistical-mechanical floor, not a fabrication limit. It follows from the Boltzmann tail of the source carrier distribution, and it is the reason supply-voltage scaling stalled in the mid-2000s: cutting $V_T$

by 60 mV multiplies off-state leakage by ten.

TFETs switch by gate-modulated band-to-band tunneling rather than thermionic emission over a barrier. Because the tunneling window is bounded by band edges rather than by a thermal tail, the Boltzmann floor does not apply. [T] Sub-60 mV/decade operation has been demonstrated experimentally since 2007 (52.8 mV/dec, 70 nm Si n-channel); Ge-source S-TFETs report ~45 mV/dec with $I_{\mathrm{o}\mathrm{n}}\mathrm{/}{I}_{\mathrm{o}\mathrm{f}\mathrm{f}}>{10}^5$ at $V_{DD}=0.5$ V. [E] The persistent liability is low drive current and ambipolar conduction. [E]

Grid relevance — and this is the pivotal reframe of §5: tunneling is the only known mechanism that breaks a thermodynamic bound on switching energy. A wireless grid is not a link; it is a control problem. Dynamic road charging requires coil-segment switching at vehicle speeds; a resonant node requires continuous impedance retuning against a moving load; exclusion-zone interlocks require always-on presence sensing. All of it is switching energy, all of it is bounded by 60 mV/decade, and TFETs are the only announced escape.

[C] The efficiency ceiling of a distributed wireless grid will be set by its control electronics, not its links, once link efficiency exceeds ~95% — because control is continuous and always-on while transfer is intermittent.

5.3 The sensor and the clock: Josephson junctions

Cooper-pair tunneling through a weak link obeys $I=I_c\sin \varphi$ and $V=(\mathrm{\hslash }\mathrm{/}2e)\dot{\varphi }$

, giving the exact frequency-voltage relation $2e\mathrm{/}h=483.5979$ GHz/mV. [T] Consequences for grid instrumentation:

• —Voltage metrology. The Josephson effect realises the SI volt. Every calibration chain in every grid meter terminates in a tunneling phenomenon. [E] Tunneling already defines the unit in which the grid is billed.
• —Current sensing. SQUIDs offer non-contact, galvanically isolated current measurement at sensitivities inaccessible to Hall or Rogowski devices — directly relevant to fault location and to metering a link that has no conductor to clamp.
• —Cryogenic control. Single-flux-quantum logic switches at ${10}^{-19}$ J/gate, some four orders below CMOS, and is the natural control layer for any system already carrying a cryostat for HTS resonators.

[C] The structural connection. The $Q$ that buys range in §4.2 comes from superconductivity. Superconductivity is macroscopic phase coherence of a Cooper-pair condensate. Josephson tunneling is that coherence made measurable. The tunneling and the range are not two technologies bolted together — they are the same physics observed at two scales. This is the one place where H₀'s intuition was pointing at something real, arriving by the wrong route.

5.4 Tunneling as the adversary

Symmetry demands the negative case. In today's grid, tunneling appears overwhelmingly as loss:

• —Gate-oxide direct tunneling sets static leakage in every power-converter controller. [E]
• —Fowler–Nordheim injection into gate dielectrics is a dominant wear-out mechanism in SiC and GaN power devices, and therefore a limit on converter lifetime. [T/E]
• —Field emission from asperities initiates vacuum-interrupter breakdown in medium-voltage switchgear. [E]

[C] Integrated over installed base, tunneling currently costs the grid far more energy than it delivers, and will continue to for at least a decade. Any honest roadmap must show the crossover date at which tunneling becomes net-positive infrastructure. We do not believe that date is calculable with present data. [O]

If the wireless grid is to reach beyond metres, it must radiate. The pricing changes character completely — from circulating VA to aperture area.

6.1 Aperture pricing

The Fresnel-zone beam-collection efficiency between two apertures is governed by the Goubau parameter:

(6.1) τ= λd A t

A r

,η beam

≈1−e −τ 2 (6.1)

[T] $\tau =2$ gives 98.2% capture. Equal apertures then require radius $a=\tau \lambda d\mathrm{/}\pi$

:

• —Link $d$ $\lambda$ Aperture radius (each end)
• —Campus 1 km 12.2 cm (2.45 GHz) 8.8 m
• —Regional 100 km 12.2 cm 88 m
• —Regional 100 km 1.06 µm (laser) 0.26 m GEO→ground 35,786 km 12.2 cm 1.67 km

[E, consistency check] The GEO figure of 1.67 km radius is independently corroborated: Space Solar's proposed Cassiopeia satellite is designed at 1.8 km span for GEO microwave delivery. The formalism reproduces the industry's own sizing without adjustment.

[T] Note the wavelength leverage. Aperture scales as $\lambda$

, so moving from microwave to optical buys 12.2 cm / 1.06

12.2 cm/1.06 μm

=339× in aperture radius. This — not efficiency — is why laser beaming is winning the demonstration race.

6.2 State of the art DARPA POWER / PRAD (May 2025): >800 W optical delivered over 8.6 km in a 30 s transmission, >1 MJ across the campaign, ~20% end-to-end conversion, through the dense lower atmosphere. Prior record: 230 W at 1.7 km. [E] Phase 2 targets relay-to-target transmission; Phase 3 targets 5 kW across 200 km via three airborne relays at ~60,000 ft. [E] Star Catcher (Nov 2025): >1.1 kW optical to commercial off-the-shelf solar panels, >10 MJ across the campaign, at NASA Kennedy. [E] Caltech SSPD-1 / MAPLE (2023): first space-to-Earth directed microwave power transfer; power detected at Pasadena; 10-month mission deliberately pushed to failure to expose weaknesses. [E] SAE J2954 (2024): light-duty WPT standardised to 11 kW at up to 93% efficiency, with differential inductive positioning for alignment; J2954/2 specifies heavy-duty to 500 kW static and dynamic; J2954/3 for dynamic transfer in preparation. [E]

[C] The gap between the near-field and far-field state of the art is instructive: deployed near-field systems move 10⁴ W at 10⁻¹ m; demonstrated far-field systems move 10³ W at 10⁴ m. Five orders of magnitude in distance for one order in power. The trade is exactly what (4.6) and (6.1) predict.

6.3 The far-field's own wall

Beaming replaces the reactive-power wall with a flux-density wall. Delivering 1 MW through the 244 m² aperture computed above implies 4.1 kW m⁻² at the aperture, against an ICNIRP 2020 general-public reference level of 10 W m⁻² at 2.45 GHz. [E] The main beam is 410× over limit and is intended to be — the entire safety architecture lives in sidelobe suppression, keep-out volumes, and interlock latency, not in the beam itself.

[C] Far-field power beaming is not regulated as a field-exposure problem but as an aiming problem. Its risk profile is therefore governed by control-system reliability and failure modes, not by exposure physics — which places it in a different regulatory category than anything the electricity sector currently administers, and probably in a different one than the one it will initially be assigned. [O]

The synthetic claim (H₃). Each length scale is denominated in a different currency, and the currencies are not convertible:

Layer Range Mechanism Price paid in Status L0 0–10 nm Electron/Cooper-pair tunneling Barrier exponent $2{\mathrm{m}}^{\ast }\mathrm{\Phi }d\mathrm{/}\mathrm{\hslash }$ Deployed as rectifier/switch/sensor [E] L1 0–0.4 m Magnetoquasistatic coupling Alignment tolerance Commercial [E] L2 0.4–40 m Resonant coupling, high $Q$ Circulating VA + cryogenics + exclusion radius Research [C] L3 40 m–10⁵ m Microwave/optical beam Aperture area + pointing reliability Demonstration [E] L4 >10⁵ m Space solar Aperture × launch mass Proof of concept [E]

[T/C] There is no wireless technology that spans the grid. Conducted transmission is uniquely privileged in that a single mechanism — Ohmic conduction in a guided TEM mode — covers 10⁻³ m to 10⁶ m with efficiency varying only logarithmically. No wireless mechanism does this, because every wireless mechanism prices range in a quantity that diverges: exponentially (L0), cubically (L2), or quadratically in aperture (L3/L4).

This is the paper's central architectural claim, and it is a reversal of the usual framing. Wireless power is normally presented as the successor to wire. The pricing analysis says it is a complement occupying the niches wire handles badly — moving interfaces, sealed volumes, hostile environments, and orbits — and that a "wireless grid" is not a grid rendered wireless but a wired grid with wireless terminations.

7.1 The metering problem

One consequence deserves separate statement, because it is not a physics problem and is therefore easy to miss.

A conductor is an excludable good: power flows where copper goes, and a meter can be clamped around it. A field is not excludable. Any receiver within $R_c$

of a resonant node can extract power from the shared mode, and — because it loads the same resonance — its extraction is observable to the transmitter only as an aggregate change in $Q_{\mathrm{l}\mathrm{o}\mathrm{a}\mathrm{d}\mathrm{e}\mathrm{d}}$

, not as an identified draw.

[C] Wireless power reintroduces a commons problem into a system that solved it physically in the nineteenth century. Every wireless node therefore requires cryptographic power accounting: authenticated load signatures, challenge-response modulation of receiver impedance, or per-receiver spread-spectrum keying. The metering layer of a wireless grid is a computing problem of comparable difficulty to the link, and it has no analogue in the wired grid. [O]

We flag this as the most under-analysed component of every wireless-grid proposal we are aware of, including the ones that are otherwise technically careful.

Dated, with the observation that would refute each.

Prediction Horizon Refuted by Confidence

P1 No demonstrated electron-tunneling power transfer across any gap > 100 nm at efficiency > 10⁻⁶ 2031 Any peer-reviewed demonstration >0.999 P2 No commercial near-field WPT product delivering > 1 kW at $d\mathrm{/}a>6$ 2031 Product datasheet 0.95 P3 Full-system optical/IR rectenna DC conversion efficiency remains < 5% under broadband illumination 2031 Certified measurement 0.85 P4 The highest-power deployed wireless grid asset remains vehicle charging at $d<0.3$ m — not any long-range system 2031 Deployment record 0.90 P5 At least one optical beaming demonstration exceeds 10 kW at > 10 km 2031 (confirmed by) DARPA POWER Phase 3 or successor 0.70 P6 No superconducting near-field WPT system in public distribution service 2035 Utility deployment 0.93 P7 No space-solar system delivers > 1 MW continuous to a terrestrial grid 2035 Grid interconnection record 0.80

P3 is the load-bearing prediction. It follows from §5.1: if the binding constraint is interconnect statistics across ${10}^{10}$ nanowatt junctions rather than diode figures of merit, then continued improvement in reported asymmetry and responsivity will not translate into system efficiency. Should P3 fail while single-junction figures of merit stagnate, the trilemma analysis is wrong. Should P3 hold while single-junction figures of merit continue improving, it is confirmed in the informative direction.

Honest statement of the conditions under which this paper is wrong. [O]

Room-temperature superconductivity at low AC loss. Equation (4.7) collapses if $COP\to 1$. Range still scales only as $Q^{1\mathrm{/}3}$, but the energy penalty vanishes and L2 extends to perhaps 100 m. This is the single development that most changes the analysis. Note the specific requirement: not merely a high $T_c$

, but low hysteretic and coupling AC loss at 10⁴–10⁷ Hz, which is a distinct and less-studied property. Revision of exposure limits. $R_c\propto B_{{\lim }^{-1\mathrm{/}3}}$

, so a 10× relaxation shrinks exclusion zones by 2.15×. ICNIRP (2020) already relaxed 100 kHz–30 MHz reference levels relative to ICNIRP (1998) on improved dosimetry [E]; further revision is plausible but the cube root blunts it. Metamaterial near-field enhancement. Negative-permeability slabs can amplify evanescent components, effectively reducing ${\kappa }_{\gamma }$

below $1\mathrm{/}a$. Whether this survives realistic loss is unresolved; the general expectation is that loss in any passive negative-index medium eats the enhancement, but we are not aware of a proof for the magnetoquasistatic case. [O] A rectifier that is fast without being small. The trilemma of §5.1 assumes $f_c=1\mathrm{/}2\pi R_{d}C$ with parallel-plate $C$. A geometry decoupling capacitance from junction area — travelling-wave rectification, distributed tunnel junctions, or a mechanism with no lumped junction at all — would remove the ${10}^{10}$ elements/m² requirement and with it P3. [O] The de Sitter caveat. We have used flat-space Maxwell throughout. Nothing here depends on cosmological structure, and we make no claim that it should.

T = theorem/derivable · E = empirical/cited · C = conjecture · O = open

Claim Tag Basis

nm⁻¹ for free electrons T Eq. 2.1, constants C2 Metal–vacuum tunneling gap ceiling ≈ 1 nm at $T={10}^{-10}$ T Eq. 2.1 C3 1 C across 1 m requires ~ ${10}^{9.4\times {10}^9}$ attempts T Direct C4 H₀ (tunneling as macroscopic transport) is rejected T C2, C3 C5 Schrödinger–Helmholtz correspondence is exact T Eq. 3.1, standard C6 Correspondence licenses scaling, not quantum advantage T §3.2 C7 Evanescent decay length = source size for sub-λ resonators T Eq. 3.2 C8 Mass discount ≈ $3.3\times {10}^9$ for $a=0.3$ m T §3.3 C9 ${\eta }_{\max }=U^2\mathrm{/}(1+1+U^2{)}^2$ T CMT, Eq. 4.2 C10 $k\approx 0.224(a\mathrm{/}d{)}^3$ for coaxial coils, $d\gg a$ T Eq. 4.3, $G\approx 7$ C11 Range $\propto Q^{1\mathrm{/}3}$ T Eq. 4.4 C12 $P_{\mathrm{c}\mathrm{i}\mathrm{r}\mathrm{c}}\gtrsim P(d\mathrm{/}a{)}^3\mathrm{/}0.224$; $O(1)$ prefactor unpinned T Eq. 4.5 C13 1 GW/100 km link circulates ~4.5 PVA ≈ 560× global capacity T C12 C14 $R_c\propto P^{1\mathrm{/}6}{\mathrm{d}}^{1\mathrm{/}2}$, independent of coil radius T Eq. 4.6, derived here C15 ICNIRP 2020 general public 0.1–30 MHz: $H\le 2.2\mathrm{/}{f}_{MHz}$

A/m E ICNIRP 2020 Table 5 C16 85 kHz chosen for regulatory discontinuity, not coupling physics C C15 + SAE J2954 C17 1 GW wireless transmission needs 3.2 km exclusion radius (32 km²) T C14 C18 Cryo break-even requires $Q_{SC}\mathrm{/}{Q}_{Cu}>1\mathrm{/}COP$ T Eq. 4.7 C19 At break-even, range gain is only $(1\mathrm{/}COP{)}^{1\mathrm{/}3}$ T C11, C18 C20 Near-field/far-field aperture crossover ≈ 0.4 m at $Q={10}^3$ T Eq. 4.8 C21 All deployed WPT lies below the C20 crossover E Qi, J2954, implants C22 C21 is an unfitted retrodiction of C20 C §4.6 C23 MIM tunnel diodes are the only viable IR/THz rectifiers E Femtosecond transport C24 MI³M asymmetry ~6 @0.1 V, responsivity 4.3 A/W @0.35 V E ALD literature C25 Resonant tunneling does not enhance low-bias rectification E Negative result, cited C26 28.3 THz rectification requires junctions < ~(30 nm)² T RC + match, §5.1 C27 ~ ${10}^{10}$ junctions/m² required for IR rectenna collection T C26 + flux C28 Rectenna binding constraint is interconnect/yield, not diode FoM C C27 → P3 C29 60 mV/dec is a Boltzmann floor, not a fabrication limit T Statistical mechanics C30 TFETs demonstrated ~45–53 mV/dec; low $I_{\mathrm{o}\mathrm{n}}$

persists E 2007–2025 literature C31 Wireless grid efficiency ceiling set by control, not links, above ~95% C §5.2 C32 $2e\mathrm{/}h=483.5979$ GHz/mV; Josephson realises the SI volt E/T Metrology C33 Superconducting $Q$ and Josephson tunneling are the same physics C §5.3 C34 Tunneling is currently net-negative for grid energy balance C §5.4 C35 Crossover date for C34 not calculable with present data O §5.4 C36 ${\eta }_{\mathrm{b}\mathrm{e}\mathrm{a}\mathrm{m}}=1-e-{\tau }^2$, $\tau =A_t{A}_r\mathrm{/}\lambda d$ T Goubau C37 GEO link needs ~1.67 km aperture; matches Cassiopeia's 1.8 km T/E §6.1 C38 DARPA PRAD: >800 W over 8.6 km, ~20% conversion, May 2025 E DARPA C39 Star Catcher: >1.1 kW optical, >10 MJ, Nov 2025 E Press release [verify] C40 SAE J2954: 11 kW light-duty at up to 93%; /2 to 500 kW E SAE 2024 C41 Beaming main beam runs ~410× over ICNIRP by design T/E §6.3 C42 Beaming risk is an aiming problem, not an exposure problem C §6.3 C43 No wireless mechanism spans the grid; conducted transmission is unique T/C §7 C44 A "wireless grid" is a wired grid with wireless terminations C C43 C45 Fields are non-excludable; wireless power reopens the commons problem C §7.1 C46 Cryptographic metering is a first-order requirement, not an add-on O §7.1 C47 Metamaterial evanescent enhancement under loss: unresolved O §9.3 C48 A capacitance-decoupled fast rectifier would void P3 O §9.4

Ledger summary: 22 T, 12 E, 14 C/O. The paper's load is carried by derivable results; the conjectures are concentrated in architecture (C31, C43–C46) and in the rectenna bottleneck (C28), which is where the falsification effort should go.

The request was a wireless grid built on quantum tunneling. The literal version fails by nine orders of magnitude in an exponent, and the failure is not close.

But reading the failure correctly returns something better than a refusal. The exponent $2{\mathrm{m}}^{\ast }\mathrm{\Phi }d\mathrm{/}\mathrm{\hslash }$ is not a wall; it is an itemised bill, and one of its line items is the carrier's rest mass. Set that to zero and the same equation describes resonant near-field power transfer, which is real, standardised, and in production. The physics never said no. It said: *this costs $m^{\ast }$

, and you did not have to bring an electron.*

Then it presented three more invoices — circulating volt-amperes, exclusion radius, cryogenic overhead — and those turn out to be the ones that actually bound the technology. They price the near-field grid into a 40 cm envelope, which is exactly the envelope in which it has been deployed. The model was not fitted to that fact.

And tunneling, having been excluded from the link, turns out to be indispensable at the terminals: the only rectifier fast enough for infrared, the only switch that escapes the Boltzmann floor, the definition of the volt in which the whole grid is billed.

The cartoon closes. The toy models are internally consistent and they identify the structure of the problem: five layers, five incommensurable currencies, and one unsolved accounting problem that no one is working on. They do not resolve the real engineering. The question of what a genuinely non-excludable power commons does to a grid that has never had one remains entirely open.