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Mathematician review: Alexander V. Frolov (folder "Frollov-Alexander")

Reviewer role: PhD mathematician (derivations, notation, units, hidden assumptions). Documents read by me: "single wire power transmission.pdf" (3 pp., full text) and "FreeEnergy.pdf" (18 pp., full text layer; patent appendices skimmed) [read]. Source gap: none for these two documents. Everything else (Brown, Sweet, Searle, Avramenko, Strebkov, Kozyrev, Bearden, Whittaker papers) is reported second-hand by Frolov; I did not have those primary works. Conclusions about those rest on Frolov's report plus the web checks tagged below.

Corrections to Step 1

  1. FreeEnergy.pdf how_read: Step 1 wrote "text layer sampled" but its own summary says the whole body text was read. Accurate description: full text layer extracted; body read in full, patent/organization appendices skimmed. I re-read the body text myself [read].
  2. claims.md claim 5 overstates Frolov. He does not simply call the others' devices real examples of free energy. In FreeEnergy.pdf he (i) says "We have to exclude ideas on electrokinetic apparatus since it is just a reactive method and propulsion force is result of ionization flow" and keeps only Brown's patent 3,187,206 surface-asymmetry idea; (ii) says Searle's disk "main effect is based on well known Lorentz forces" with ionization currents [read]; (iii) says the unbalanced-wheel and Archimedes-screw paradoxes disappear if treated as "two different systems" and "classical theory can explain" them [read]. So for wheels and screws he partly concedes the conventional reading. Step 1 key_claims for Brown "produce propulsion without ionization" is right for patent 3,187,206 but he elsewhere excludes the ionization-flow devices.
  3. Step 1 omits a concession that matters mathematically: in the Tesla section Frolov writes "in each receiver the power can't be more than power of this generator" [read]. That is in tension with claim 3 (efficiency above 100%) in the same article.
  4. Step 1 sources tagged the Avramenko plug "medium, not web-checked". I checked: see Web checks below. Tag upgraded to [web].
  5. Arithmetic in Step 1 (0.02 J per spark, 0.1 W) is correct; I reproduced it (0.5 x 1e-8 x 2000^2 = 0.02 J; x 5 Hz = 0.1 W). I add below the charging-current check, which Step 1 did not do.

Web checks (budget used: 3 searches)

Mathematical tools used in the review (for the reader)

  • Work by a force field F along a path: W = integral of F . dr. For a conservative field (F = -grad U, with U the potential energy), W around any closed loop is zero because W = U(start) - U(end).
  • Displacement current: Maxwell's term eps0 dE/dt, the current-like quantity through a capacitor gap. Reactive ("wattless") power: the part of AC power that flows back and forth; its time average is zero.
  • Capacitor energy: E = (1/2) C V^2 (C capacitance in farads, V volts).

Claim 1: The potential (scalar) field can do net work around a closed cycle, without fuel, if the negative-work half is removed by pulsing, gradients, polarity change or screening.

a. Mathematical consistency. The claim is a half-truth. Frolov states correctly that for a static conservative field, W around a loop is zero. He then proposes to "exclude" the negative-work part by pulsing, polarity change or screening [read, FreeEnergy.pdf p.12]. Here is the hidden assumption: once the field is switched in time, E = E(r,t) and F = qE(r,t) is no longer conservative along the actual trajectory. Net work per cycle can then be nonzero, which is standard (a particle accelerator, a motor). But the energy then equals the work done by whoever switches the field, plus the field energy exchanged with the source. His text never accounts for the energy cost of the switching, screening or polarity change. "The power source is not required to keep the potential field in force" is true for a static field (holding a charge in a field costs nothing) but irrelevant to a cycle that repeats, because the repeating requires the field to be changed, and changing it costs work. Worked toy example: charge q = 1 uC in a uniform field E = 1000 V/m over d = 1 m. Forward leg: W = qEd = 1 mJ. Frolov's trick: "screen" the return leg so E = 0 there. Net work per cycle = 1 mJ. But inserting and removing a conducting screen in a field is not free: the screen carries induced charge, and moving it around the charge system costs the energy that was gained (electrostatic energy changes by exactly that amount when the configuration is changed back to the start). Total energy over a full cycle of the whole system, including the screen, is zero for a static configuration. The article omits this bookkeeping; it treats the screen as a costless boundary condition. Also: the vacuum tube example (grid under cathode, pulse switched off before electrons reach anode) [read] describes a transient where the source does not deliver "conduction current" in the interval. Electrons that arrive carry kinetic energy that was supplied by the source charging the electrode capacitance; there is no conservation violation, and the pulsed-source energy is not shown to be below the output. No numbers given. Notation: the article's "gradient of the gradient is the second derivative of the potential" (capacitor with unequal plates) is correct as a statement (force on a polarizable dielectric is proportional to grad of E^2, a well-known result, F = (1/2) alpha grad(E^2)) but the force is not a source of continuous energy; it moves the ball to the high-field region and stops there.

b. Consistency with experiment. Consistent with every ordinary experiment (non-static fields do net work, motors, accelerators). Inconsistent with the specific "free" reading: no experiment known to me shows a cycle returning to its initial state with net energy gain [memory, high]. Frolov provides no data tables or error analysis for any device in the article [read].

c. Tested? The strong form (net work per cycle without any energy input) is the perpetual-motion-of-first-kind claim; countless patent-office and laboratory examinations have not found a device; no controlled test of Frolov's own proposal is documented in the files. His own device claim (the 1995 flat-spiral coil with "asymmetric mutual induction", load "free from input power") has no measurements, instruments, or numbers [read]. Never performed in a documented controlled way, so not "dismissed without testing" or "tested and failed" on the record available to me.

d. If true. Energy conservation (Noether's theorem for time-translation symmetry) would fail, or the Hamiltonian of a closed cycle would not be single-valued. Faraday/Maxwell electrodynamics itself would be unchanged for the time-varying case; what would change is the claim that field energy is accounted by the Poynting theorem. That is a deep change.

Verdict, split: - Weak form (time-varying fields do net work per cycle when someone supplies the switching energy): Established (textbook). - Strong form (net work per cycle with no net energy supplied by the system or operator): Refuted by calculation (conservation of energy and the Poynting theorem; shown above by the screen bookkeeping) and, as written, lacking any accounting of the switching cost, thus Never properly tested for any specific Frolov device.

Claim 2: A single-wire line with an Avramenko plug carries only wattless displacement current yet delivers real power to a load "for free" (home test: 2 kV, 10 nF spark gap, 5 Hz, 0.1 W out vs 3 W in).

a. Mathematical consistency. The arithmetic is correct and internally consistent. Check [read + own calculation]: - Energy per spark: A = (1/2) C V^2 = 0.5 x 1e-8 F x (2000 V)^2 = 0.02 J. - Power: P = A f = 0.02 J x 5 Hz = 0.1 W. - Efficiency: 0.1 / 3 = 3.3%, matching his "about 3%". - Required average charging current: I = Q f = C V f = 1e-8 x 2000 x 5 = 1.0e-4 A = 0.1 mA. That is tiny. - Required stray capacitance to the surroundings: a conventional displacement current at 10 kHz and 2 kV is I = 2 pi f C_s V. For I = 0.1 mA, C_s = I / (2 pi f V) = 1e-4 / (2 pi x 1e4 x 2000) = 8e-13 F, about 0.8 picofarad (my estimate; I computed 7.96e-13 F). Any wire, secondary winding and nearby objects have stray capacitance of that order or larger. So the effect is fully explained by an ordinary capacitive return path through stray capacitance to ground/air, with the diodes rectifying. This is the conventional model (Avramenko plug = rectifier with capacitive coupling) [web, sources above]. - Spark gap check: breakdown of air is about 3 kV/mm for uniform fields [memory, high], so 0.5-1 mm gap breaks down near 1.5-3 kV, matching his 2 kV. Consistent. Definitional problems: "wattless current" (his word) means reactive current in power engineering, which carries zero average power [memory, high]. But he says the load receives real (wattful) power through the diodes. Both statements can hold only if the diodes convert reactive AC displacement current to DC through nonlinearity, which is standard rectification, no new category of current. His "closed circuit consumes the source" statement conflates two things: the circuit through the generator side is closed through the stray capacitance and the ground, so it is also a closed circuit in the displacement-current sense. Maxwell: total current (conduction plus displacement) is divergence-free, so there is no truly open circuit. The "single wire" has a closed loop of total current via capacitance. Logical gap: the 3 W is the DC power in at the low-voltage generator; no measurement is given of how the generator input changes with and without the load, nor of its no-load consumption. A step-up transformer at 10 kHz and a generator will dissipate several watts with no load. The claim "free since not connected with conduction current in source circuit" is a non sequitur: the output 0.1 W is below the input 3 W, so even taken at face value nothing exceeds unity.

b. Consistency with experiment. Fully consistent with ordinary electrical engineering: capacitive coupling, Goubau/single-wire lines, Tesla-style high-frequency single-terminal lamps, rectifier-multiplier circuits [web]. No measurement in the article contradicts energy conservation. The 60 mA x 50 V = 3 W for Avramenko's lamp [read] is a product of RMS or peak values without phase; for a reactive circuit, P = V I cos(phi) <= V I, so the 3 W is an upper bound unless the phase and waveform are given (they are not).

c. Tested? Frolov's own test is a home experiment with no controlled measurement of input power under load vs no load, no waveforms [read]. Avramenko, Strebkov and colleagues report 20 kW-class single-wire systems [web]; these are proponent reports of power delivered, not of over-unity. I found no source showing a measured output exceeding input. No independent replication of "free" power is on record in what I have read or searched.

d. If true. The Poynting theorem and Maxwell's equations would need a second power-carrying channel ("longitudinal displacement-current wave" with free power). Ordinary Maxwell electrodynamics has transverse waves and quasi-static fields only in vacuum; longitudinal electromagnetic waves exist in plasmas and as near-field components, but carry no free energy.

Verdict, split: - Weak form (a single-wire capacitively coupled line with diodes can deliver small real power to a load): Established (conventional rectified capacitive coupling; my 0.8 pF estimate shows stray capacitance suffices). - Strong form ("free", cost to the source independent of load, because the current is displacement/wattless): Never properly tested (no input-power measurements in the article) and, for his own numbers, Refuted by his own data: output 0.1 W < input 3 W shows no gain. As a conceptual statement, "wattless current becomes wattful free" is Internally inconsistent as defined (reactive current has zero mean power; the conversion requires a nonlinear element fed by the source).

Claim 3: Resonance (Tesla transformers, standing waves, quarter-wave lines) can raise the output until efficiency exceeds 100%.

a. Mathematical consistency. Resonance raises voltage and current amplitudes: a series RLC circuit has Q = (1/R) sqrt(L/C), and the voltage across C is Q times the drive voltage. Example: L = 1 mH, C = 1 nF, R = 10 ohm: sqrt(L/C) = 1000 ohm, Q = 100, so 1 V drive gives 100 V across C [own calculation]. But the average input power is V_s^2 / (2R) at resonance, the real power is dissipated in R, and stored energy is replenished each cycle only by the source. "Efficiency" defined as output real power over input real power is always at most 1 for a passive linear circuit. Frolov's remark "the efficiency can be more than 100%" (Tesla's resonance transformer) [read, FreeEnergy.pdf p.15] confuses voltage or amplitude gain with power gain. He also says "the output power of the generator is not important... the question of power in load is the question of amplitude" (1994 article) [read], which is wrong dimensionally: power is amplitude squared over impedance and is bounded by what the source supplies. In the 2003 article he states "in each receiver the power can't be more than power of this generator" [read], contradicting the 100% claim. Quarter-wave check: at 100 kHz, lambda = c / f = 3e8 / 1e5 = 3000 m, lambda/4 = 750 m [own calculation], so the "quarter-wave" tuning he mentions at kHz-range frequencies implies lines of hundreds of metres or a lumped-inductor loading; no such numbers are given.

b. Consistency with experiment. Resonant transformers do give large voltage gain (Tesla coils reach MV); no measured power gain above unity is documented [memory, high]. Frolov gives none in these files.

c. Tested? Not tested by him in the documents (no data). Tesla's historical claims are second-hand. Resonant circuit power balance is the most routinely tested in electrical engineering (every radio); it never gave over-unity [memory, high].

d. If true. Linear passive circuit theory (impedance, Poynting theorem for LTI circuits) would fail. Reactive stored energy would have to be created.

Verdict: Voltage/amplitude gain by resonance: Established. Power efficiency above 100%: Refuted by calculation (the Q example above: power dissipated in R equals source power; the article itself concedes "power can't be more than power of this generator"), and the article is Internally inconsistent on this point.

Claim 4: Energy conservation is "generalized" to a four-dimensional balance (entropy process in forward time balanced by an anti-entropy process in reversed time); the vacuum/aether is an unlimited source whose density sets the rate of time (Kozyrev).

a. Mathematical consistency. No equations are given. The statements are: "total energy of four dimensional system is amount of energy of processes in time and energy of processes in reversed time, it is constant and equal to zero"; and "A + B = 0, 0 = A + B ... creation of energy is possible in theory" [read, FreeEnergy.pdf pp.13-14]. This is an algebraic triviality: any quantity can be assigned a compensating opposite term, A + (-A) = 0. That does not demonstrate that A can be extracted into a load; it shows nothing without a physical definition of the anti-process, its energy, and a coupling to the load. Definitions are inconsistent: "Energy is a quantitative characteristic for different forms of motion" but then "work... as transformation of energy forms", then "free power in load" paired with "anti-entropy in reversed time". "Density of aether" is substituted for Kozyrev's "density of time" with no definition, units, or measuring procedure. A "rate of time" that varies by season and time of day (said to explain Sweet's output varying) is not defined against a clock: if all clocks and processes slow equally there is no observable. As stated this is not a mathematical theory. Further: the statement that the aether density "determines hardness of cause-effect connections" has no mathematical content. The text also asserts conservation is exactly preserved while requiring that "some change of energy density of space should be detected" and the "curvature of space-time" change; this is a general-relativity phrase used without the Einstein equations (G = 8 pi T), where energy-momentum conservation is already built in locally (the divergence of T vanishes).

b. Consistency with experiment. Time-reversed anti-processes balancing entropy have no experimental support; entropy increase is observed in all macroscopic processes. Kozyrev: no conclusive reproducible results in independent replications [web, link above]. Zero-point energy extraction at macroscopic power: no demonstration [memory, high] (Casimir force exists but is not an energy source in a cycle).

c. Tested? Frolov proposes seasonal and daily variation of free-energy device outputs as a test [read]; no data are offered. Kozyrev's torsion-balance and gyroscope work has attempts at replication that did not confirm [web]. Not tested by Frolov.

d. If true. The second law of thermodynamics and causality would have to give way (backwards-in-time processes performing work now); energy conservation would need replacing by a new "balance" law with an observable hidden partner; local energy conservation, a theorem from time-translation symmetry (Noether), would no longer hold in the observed sector.

Verdict: Not testable as stated (no definition of the anti-process, the aether density or a measurable difference), with the energy-extraction consequence Never properly tested and the generalization of conservation as written Internally inconsistent (conservation said to be preserved while "creation of energy" in the observed system is asserted). Kozyrev's part: Historically superseded but instructive is too generous; I would use "Never properly tested; independent replications reported negative [web]".

Claim 5: Other people's anomalous devices (Brown asymmetric capacitors, Sweet VTA, Searle disks, Cooper, Hubbard, unbalanced wheels) are real examples of the effect, including reactionless thrust and gravity reduction.

a. Mathematical consistency. Frolov offers no calculations for any of them. Quantitative checks I can do from his text: - Brown "efficiency million to one" [read]: if thrust F and power P are quoted, thrust-to-power F/P in N/W for an ion wind device is bounded by about 1/v_exit; for a lifter in air, typical measured values are of order 1e-2 N/W (a few grams force per tens of watts) [memory, medium; not web-checked]. "Million to one" is undefined (a ratio of what to what?); a capacitor holding charge draws near-zero DC power, so thrust per watt diverges only if the steady thrust were real without any leakage current; but a steady force without leakage current would violate momentum conservation with the field momentum unless something is expelled. Not a calculation Frolov provides. - Sweet VTA: "weight decrease proportional to output power", "self-cooling about 20 degrees" [read]. Weight loss accompanying power extraction would require momentum exchange; self-cooling of magnets 20 K below ambient while delivering electric power would require a heat pump (type-two perpetual-motion machine, in his own classification). No power, mass or temperature measurements with error bars are given. - Searle disk: Frolov himself explains it with ionization currents and the Lorentz force [read], i.e. a conventional (possible) mechanism. - Unbalanced wheels, Archimedes screws: he concedes a two-systems resolution consistent with classical theory [read]. Statics check: for a wheel with weights moved radially, the torque from the raised side at larger radius is offset by more weights on the descending leg; a free-body analysis gives equilibrium (the centre of mass cannot be shifted permanently by an internal mechanism of a closed loop returning to the same configuration, because the work around a loop in uniform gravity is zero: W = mg times the net height change = 0). - Cooper's patent claim: depolarizing atoms to "release them from gravitation" [read] has no equation.

b. Consistency with experiment. Brown-type thrust in air is conventionally explained by ion wind [web, sources above]; vacuum tests reportedly show no effect, with proponents disputing [web]. Sweet VTA: not independently confirmed in anything I found or can cite; I have no source either way beyond [memory, medium]. Perpetual wheels: Frolov lists historical claims (Bhaskar, Worcester 1620 with an alleged royal test) that have no accessible neutral source here; all prior historical wheels failed on later scrutiny [memory, medium; not web-checked].

c. Tested? Frolov reports others' claims; none documented here with data. Brown: tested by others, with reported results largely attributing to ion wind [web] (some low-pressure claims contested). Sweet, Searle, Hubbard, Godin-Roshin (7 kW, 100 kg axial force claimed [read]): I found no independent replication in what I read; I did not search for them (budget), so I say "no replication known to me", not "failed".

d. If true. Conservation of momentum (reactionless thrust) and of energy (net output from a static field), plus the equivalence principle for gravity-shielding claims; a coupling of electric field to gravity would require a modification of general relativity.

Verdict: Heterogeneous; I give it per device. - Brown thrust in air: Established as a real force; conventionally explained by ion wind (the "no ionization" strong form: Tested, not supported per the vacuum-test reports I could only read in secondary summaries [web]). - Sweet VTA (over-unity, weight loss, self-cooling): Never properly tested on the record I have (reports by proponents, no data in the article). - Searle disk: Plausible, untested as a conventional ionization/Lorentz force machine, as Frolov himself explains it; the free-energy and anti-gravity part Never properly tested. - Unbalanced wheels and screws: Refuted by calculation (zero net work around a closed loop in uniform gravity, shown above), and Frolov himself concedes the classical resolution. - Cooper, Hubbard, Hendershot, Godin-Roshin: Never properly tested (claims only; no data in these files).

Overall assessment

What is mathematically sound in Frolov: capacitor energy and spark power arithmetic; the observation that a time-varying field does net work per cycle; the existence of resonance gain in voltage; a nonuniform field force on a dielectric (grad E^2). What fails: the step from "net work is possible in a time-dependent field" to "free energy", by omitting the energy needed to vary the field; confusing amplitude gain with power gain in resonance; defining "wattless" current and then using it as a power carrier; replacing the energy budget by an undefined "anti-process". The most instructive point for a learner: in Frolov's own numbers (3 W in, 0.1 W out), nothing exceeds unity, and the effect is explained by stray capacitance (about 0.8 pF suffices for the 0.1 mA charging current).