Smith-Don: Mathematician review (Step 2)

Reviewer: PhD mathematician. Date: 2026-10-08. Project: Don (Donald L.) Smith, resonant air-core coil and capacitor "free energy" devices.

Tags: [read: p.N] from the project documents (page images where noted); [web] with URL; [memory] with confidence; [computed] my own arithmetic (scripts run in a scratch directory, not deliverables).

Source gap (repeated from claims.md)

The folder holds no measurement, test report, instrument reading or patent grant. The binder (Resonance-Energy-Methods-Don-Smith.pdf, 96 pp.) holds Smith's claims and illustrative calculations; don-smith.pdf (Joel Lagace) is a secondary reconstruction, not Smith's words. This review therefore checks mathematics, units and energy budgets, not hardware. Where a conclusion about a device would need data, I say so. Pp.10-35 of the binder and pp.39-41, 46-50 of Lagace were digest-only in Step 1; I re-read Lagace pp.39-50 myself (text layer) and the binder pp.44, 47-49 and magazine p.37 as page images.

Method in one paragraph (for the reader)

Every claim below is tested the same way. (1) Write down what is stored, what is delivered and what is supplied, with units: energy in joules (J), power in watts (W = J/s), charge in coulombs (C), capacitance in farads (F), inductance in henrys (H), frequency f in hertz (Hz = cycles per second). (2) Check the formulas dimensionally: both sides must have the same units. (3) Check the arithmetic. (4) Do the energy budget: energy leaving a closed system cannot exceed what entered plus what is drawn from an identified reservoir. A stored energy is not a delivered power: a capacitor holding 50 kJ holds 50 kJ however fast it is switched.

Key quantities used throughout

• Energy stored in a capacitor C charged to V: W = (1/2) C V^2 (J). Energy stored in an inductor L carrying current I: W = (1/2) L I^2 (J).
• Series resonance: an inductor L and capacitor C in series resonate at f0 = 1/(2 pi sqrt(L C)). Impedance Z (the AC generalisation of resistance, in ohms) is Z = R + j(omega L - 1/(omega C)), omega = 2 pi f. At resonance the two reactive terms cancel and Z = R. It does NOT become zero. The quality factor Q = omega0 L / R measures how many radians of oscillation it takes to lose the stored energy; Q is infinite only if R = 0. [memory, high; textbook circuit theory]
• Reactive power: the volt-amperes that slosh in and out of L and C each cycle and do no net work. Sustaining a tank that stores energy W at frequency f against a loss resistance requires real power P = omega W / Q. [memory, high]

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Claim 1. Energy transferred by resonant induction scales with the square of frequency: E = (1/2) C V^2 x f^2

a. Is the mathematics internally consistent? No.

I read pp.44, 47-49 as page images and the 1996 magazine printing (ExtraOrdinary Science, Apr/May/Jun 1996, p.37) [read: page image p.44, p.47-49; Investigator file p.7 image].

1. Two incompatible formulas. p.44 prints the capacitor formula "W = .5 X CE X C.P.S." with E "volts squared": (1/2) C E^2 times f to the first power. p.47 prints "Joules = [0.5 C x V's squared] x C.P.S. squared": (1/2) C V^2 f^2. p.48 says transfer is "a direct function of the squaring of the cycles per second" [read].
2. The worked examples use different formulas. Example A: (1/2)(10 uF)(120 V)^2 = 0.072 J; x 60^2 = 259.2 [read: p.48]. That is f^2. Example B: (1/2)(10 uF)(100,000 V)^2 = 50,000 J; the page leaves the last line "[50,000 x One Million squared = ]" blank and states "greater than 50 Mega K-Joules" [read: page image p.49]. The formula gives 50,000 x 10^12 = 5 x 10^16. The stated 50 Mega K-Joules = 5 x 10^10 is 50,000 x 10^6, which is f to the first power. So A uses f^2 and B's stated answer uses f. [computed, confirmed]
3. A third version. The magazine printing (1996) reads "Joules = 0.5 x .000,010 x 100,000^2 x CPS^2 = 500 x One Million squared" and "more than 500 MegaJoules" [read: page image]. The intermediate "500" is wrong: 0.5 x 0.00001 x 10^10 = 50,000, not 500 (the binder has 50,000). And 500 x (10^6)^2 = 5 x 10^14, not 500 MJ (5 x 10^8). The three printed answers for the same example are 5 x 10^8 J (magazine), 5 x 10^10 (binder) and 5 x 10^16 (the formula). Step 1's note recorded the differing magazine answer but not that its intermediate line is also wrong. [computed]
4. The frequency ratio on p.56 contradicts the squared law. "The resonate induction system, cycling at 60 million times per second produces one million times the energy produced by the present energy systems" [read: p.56]. 60 MHz / 60 Hz = 10^6. Under f^2 the ratio is 10^12; "one million times" is the first-power ratio. [computed]
5. Units. (1/2) C V^2 is joules. Multiplied by f it is J/s = watts. Multiplied by f^2 it is J/s^2, which is neither energy nor power. Smith labels the f^2 result "Joules [Watt Seconds]" (p.47-48) and example B "[Watts]" and "Mega K-Joules" (mega x kilo = giga). The p.38 statement "E = (Volts x Amperes) x Cycles Per Second squared" has units W/s^2 = J/s^3, a different unit again, and it is offered as "the electrical equivalent of E = mc^2", which is an energy [read: p.38]. p.47 defines C "in microfarades" yet both examples insert 0.000010, i.e. farads. p.44 "Both one henry and one farad equal one volt" is dimensionally false (1 H = 1 V s/A; 1 F = 1 A s/V). p.36 "Coulombs (Volt-Amperes)" equates charge (A s) with power (W). [read; computed]
6. Where f^2 comes from in the text: analogy, not derivation. p.43: in the kinetic energy formula (1/2) m v^2, "intensity of voltage and amperes times cycles per second replace velocity". If "velocity" is V x f then (1/2) C (V f)^2 = (1/2) C V^2 f^2. But V f has units of V/s, not a velocity, and there is no reason a capacitor's energy should depend on the speed of an unrelated cycle. p.39's "second round for the law of squares" is assertion [read].

Worked example (reader): what the formula could legitimately mean

Take example B as printed: C = 10 uF, V = 100 kV, f = 1 MHz. - Stored energy (1/2) C V^2 = 50 kJ. [computed] - (1/2) C V^2 f = 5 x 10^10 W = 50 GW. This is "the stored energy counted a million times per second". For a capacitor on an AC supply, the reactive power is Q_r = 2 pi f C V_rms^2 = 4 pi times (1/2) C V^2 f (if V is the rms voltage) = 6.3 x 10^11 VA for example B and 54.3 VA for example A. [computed] Step 1's note says the same; the equality assumes the quoted voltages are rms. Reactive power returns to the source every half cycle and does no net work. - To keep 50 kJ ringing at 1 MHz you must replace what the loss resistance dissipates: P = omega W / Q = 2 pi (10^6)(5 x 10^4)/Q = 3.1 x 10^11 / Q W. For Q = 100 that is 3.1 GW of input; for Q = 1000, 0.31 GW. [computed] A higher stored energy at higher frequency costs more, not less, to maintain. - The capacitor's reactance at 1 MHz is 1/(omega C) = 0.016 ohm; at 100 kV it would carry about 6.3 x 10^6 A, and resonating it at 1 MHz needs L = 2.5 nH, a few millimetres of wire. No such device is described. [computed]

For the 1996 suitcase claim (Smith's own account, below), "12 V at 0.3 A in; 120 V at 80 A out" is 3.6 W in and 9,600 VA out (ratio about 2,700) [web: Investigator, energyevo.org, 2015/01/18 compilation]. The output is volts times amperes read by an engineer; for AC or pulsed reactive loads that product is apparent power, not real power (real power = V I cos(phase angle)). The statement does not say which was measured. A continuous 9.6 kW is also 9.6 kWh per hour, far beyond a 12 V gel cell (a 7 Ah cell stores 84 Wh [computed from a typical rating, memory medium]) unless the "system" recharges it. This is checkable only on the 1996 videotape (Investigator lead).

b. Consistent with established experiments? No.

Standard circuit theory and every measured resonant circuit give energy stored in L and C and loss set by R and Q; no measurement shows delivered energy rising as f^2 at fixed stored energy. Legitimate places where powers of f appear: the EMF induced in a pickup coil in a field B0 sin(omega t) is proportional to omega, so power into a fixed resistor goes as f^2; power radiated by a short dipole at fixed current goes as f^2 and at fixed dipole moment as f^4 [memory, high]. These are rates (watts) under stated conditions, and they do not multiply a stored energy by f^2. Smith does not invoke them.

c. Tested? By whom? No test of the f^2 law is on record. The binder contains no measurement. The only quantitative performance statement by Smith is the suitcase claim (above); the recording exists (Investigator: 1996 workshop video, 2 h 39 min) and I have not seen it.

d. If true, what would have to change? Energy conservation as applied to linear circuits (Poynting's theorem; energy stored vs power delivered) and the definition of the joule. A system whose output energy grows as f^2 would deliver unbounded energy by raising f.

Verdict: split.

• Strong form (energy delivered = (1/2) C V^2 f^2): Internally inconsistent (formula versions disagree, unit mismatch, p.56 ratio contradicts it) and Refuted by calculation as an energy gain (stored energy is not delivered energy; the loss budget above).
• Weak form ((1/2) C V^2 f as a circulating, reactive rate): a true expression for something real (reactive volt-amperes, to a factor 4 pi), but it is not energy delivered to a load. Established as circuit theory; it supports no gain.

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Claim 2. At resonance impedance (resistance) becomes zero: room-temperature superconductor; Ohm's law does not apply; "volts and amperes are equal until work is introduced"

a. Internally consistent? No.

• "Volts and amperes are equal" [read: p.55-56, 63, 74]. Volts and amperes have different units and cannot be equal; read numerically it holds only if |Z| = 1 ohm (V = I Z). If Z were zero with finite V, I = V/Z would be unbounded, not equal to V. The text asserts both. [computed]
• "Corrected" Ohm's law [read: p.71]: "Volts = Energy Available", "Ampere = the rate of dissipation". A volt is joules per coulomb, not energy; the rate of dissipation is watts (V x A), not amperes. "Until wasting occurs there are no amperes" [read: p.45] confuses current with loss; current flows without dissipation in a lossless LC tank.
• p.44 "Both one henry and one farad equal one volt": false by units (see Claim 1).

Worked example (reader): a real series resonant circuit

L = 2.53 uH, C = 10 nF (f0 = 1 MHz), R = 1 ohm, driven by a 10 V amplitude source at resonance. Then |Z| = R = 1 ohm (not zero), I = 10 A, Q = omega L / R = 15.9, capacitor voltage = Q x 10 V = 159 V, peak energy stored = (1/2)(10 nF)(159 V)^2 = 0.127 mJ, and the source supplies (1/2)(10 A)^2 (1 ohm) = 50 W continuously to make up the dissipation. [computed] Voltage is magnified by Q; energy is not created; the source pays the loss every cycle. Reducing R raises Q and the current for a given source, and the input power with it.

The "superconductor" and skin-effect statement

p.48: at radio frequency "the electrons do not pass through the conductor ... encircle the conductor and are free of the conductor's resistance" [read]. Mainstream: this is the skin effect. Current crowds into a surface layer of depth delta = sqrt(2 rho/(omega mu)), 65 micrometres in copper at 1 MHz [web: https://en.wikipedia.org/wiki/Skin_effect; my computation gives 65.2 um]; the AC resistance of a thick wire rises roughly as sqrt(f), "approaches one half of the ratio of the radius to the skin depth" times the DC value [web: same]. For a 1 mm radius wire at 1 MHz the AC resistance is about 7-8 times the DC value [computed]. That is the opposite of freedom from resistance. A superconductor has exactly zero DC resistance and expels magnetic field (Meissner effect); a resonant circuit has finite Q (tens to a few thousand for air coils) [memory, high].

b. Experiments? Every measured LC resonator has finite Q. Superconducting resonators exist but need cryogenic temperature or special materials.

c. Tested? Not by Smith in the record. Any Q meter or network analyser would measure it; none reported.

d. If true: superconductivity theory (BCS and later), and the Q factor itself, would have to be rewritten.

Verdict: Internally inconsistent (units; V = I Z) and Refuted by calculation (Z = R at series resonance; Q finite; skin-effect resistance rises with f).

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Claim 3. The extra energy comes from the Earth's and air's ambient electrical and magnetic background, drawn in by resonance and grounding

a. Is the mathematics internally consistent? The numbers offered do not follow from the stated inputs, and the units conversions are not dimensionally meaningful.

1. Earth's "17.6 x 10^18 exajoules" [read: p.54, 67]. From the text's own inputs, 3.9 EJ/yr x 4.5 x 10^9 yr = 1.76 x 10^10 EJ, a factor 10^9 below the printed figure. [computed; confirms the deep read] Both numbers lack a source. Scale check: 1 EJ = 10^18 J. The printed total is 1.76 x 10^37 J. Sunlight intercepted by Earth is about 1.2 x 10^17 W, so over 4.5 Gyr it delivered about 1.7 x 10^34 J [computed; solar intercept memory, high]. The printed figure exceeds all the solar energy Earth ever received by about 1000 times, while the text says the electrons "accumulated from the solar plasma". Even the corrected 1.76 x 10^28 J is not an identified, extractable store.
2. Unit equivalences [read: p.55, 68]: "Gauss (one unit = 100,000 volts), Oersted (one unit = 50,000 volts), ... One gamma is the magnetic flux equivalent of 10 active volts." The gauss and gamma (1 gamma = 1 nanotesla = 10^-9 T) are units of magnetic flux density B; the oersted is a unit of field strength H. Volts measure electric potential. By Faraday's law a changing flux gives an EMF: emf = d(Phi)/dt with Phi = B x area. So "volts per gamma" has no meaning without an area and a rate of change. Example: 1 nT through 1 m^2 oscillating at 1 MHz gives an EMF amplitude of 2 pi x 10^6 x 10^-9 = 6.3 mV. [computed] A fixed 10 V per gamma is not dimensionally defined.
3. The aeromagnetic "correction" example [read: p.55, 69]: remote reading 1,600 gammas at 1,000 ft: 1,000 x 1,000 x 1,600 = 1.6 x 10^9, which the text calls "1.6 trillion gammas" (a trillion is 10^12). x 10 V = 1.6 x 10^10 V, printed "16 trillion volts" (1.6 x 10^13). So the printed result is 1000 times the product of the printed factors. [computed] The method itself (multiplying a reading by distance squared) has unit problems (gamma x ft^2 is not gamma) and does not match magnetostatics: a dipole field falls as 1/r^3 and a monopole-like term as 1/r^2, and surveys are corrected by upward or downward continuation, not by one factor. [memory, high] The claim "gamma readings exceed one trillion gammas" is 10^12 nT = 1,000 tesla; Earth's field is 25,000-65,000 nT (25-65 microtesla) [memory, high], and the strongest laboratory steady fields are tens of tesla.
4. "Lightning ... ten trillion joules" per flash [read: p.54, 67]: one source gives an average flash of about 500 MJ (5 x 10^8 J) [web: https://en.wikipedia.org/wiki/Atmospheric_electricity]. The printed figure is about 2 x 10^4 times larger. Smith's own "200,000 emissions ... 2,000 quadrillion watt seconds" is consistent only with his 10^13 J.
5. Ambient voltage "200,000 volts ... 500,000 at the poles" [read: p.39]: no instrument or place given. The global atmospheric electric circuit has an ionosphere-to-ground potential of order 400,000 V ("about 400,000 volts with respect to the surface") [web: same Wikipedia page]. (Step 1's note quoted 200-300 kV from memory; the web source says about 400 kV; same order.) That is a potential difference between the ionosphere, tens of kilometres up, and the ground. A conductor near the ground at one point does not "see" it as a battery between two terminals.

Worked example (reader): how much power is in the atmospheric electric field?

• Fair-weather surface field about 100-120 V/m and conduction current density about 2 pA/m^2 [web: same page]. Power per square metre, J x E = 2 x 10^-12 A/m^2 x 130 V/m = 2.6 x 10^-10 W/m^2. [computed]
• Whole Earth: 2 pA/m^2 x 5.1 x 10^14 m^2 = about 10^3 A of global fair-weather current (my product of web density and surface area) times 4 x 10^5 V = about 0.4 GW for the entire planet, about 10^-6 W/m^2 averaged over the surface. [computed; area memory, high] This is the whole flow of the global circuit, equal to one mid-size power station, and it is already what drives the weather electricity.
• Smith's 1996 suitcase claim, 9.6 kW from a briefcase, would be a few ten-thousandths of that whole planetary flow, collected by one coil. A 12 V input at 0.3 A has been put against the 3.6 W it supplies.

The "ground" step

An earth connection ties the circuit to a single large conductor at one potential. Energy needs a potential difference across the load. The binder never specifies two points (the unequal potentials it speaks of, p.71 batteries of "unequal potentials", are an ordinary battery-balancing argument and carry no measured values). "Electrons pulled into the system" from earth conserves charge, not energy.

b. Experiments? Atmospheric-electricity collectors (Franklin, Kelvin water dropper, antennas) are established but yield microwatts. Energy conservation with open systems: Smith, p.31, 34 says conservation laws "are excluded" for open non-linear systems, with a quotation attributed to Einstein, no citation [read]. In physics an open system exchanges energy with an identified surroundings, and the exchange is counted; conservation is not suspended.

c. Tested? Smith's own claims: the suitcase demonstration (his account, 2001; the audience-picked engineer unnamed); the 1994 lecture demonstration as described by Rick Friedrich (an interested party: he sold a Smith-derived kit): the flat-coil part ran "to the point of the neon bulbs", part failed, with Friedrich attributing the failure to a 12 V battery on a 24 V module [web: Investigator, YouTube BHr3eDELyHk]. A 2016 bench reproduction on a forum ("tswift") reports about 1 W with a self-charging battery and notes his own possible meter artefact; a 2009 forum thread opened by a poster asked whether Smith was "heavily miscalculating" (capacitor banks of 8000 uF at about 400 V = 640 J each [computed]) [web: Investigator, energeticforum.com]. These are bench accounts, not controlled tests. None includes an input and output energy measurement over a defined period.

d. If true: the atmosphere or ground would be a gigawatt-scale accessible reservoir; or conservation of energy would fail in open circuits. The first would be a large measurable change in atmospheric electricity budgets.

Verdict: split.

• Strong form (kilowatts drawn from ambient electrical background by resonance and grounding): Refuted by calculation for the atmospheric-electric budget quoted above; the numbers in its support (exajoules, gamma-volts, lightning, 16 trillion volts) are Internally inconsistent.
• The suitcase demonstration as an event: Never properly tested (a recording exists; no neutral instrumented measurement is on record; not dismissed by any test).
• Weak form (a small amount of energy can be collected from the atmospheric field): Established, at nano- to microwatt levels.

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Claim 4. An air-core two-coil system gains energy through the turn ratio and by "squaring" flux

a. Internally consistent? No.

• "Each turn at L-1 acquires 2,000 volts" (8,000 V over 4 turns) [read: p.45]: arithmetic correct. The step "each L-1 turn is replicated in every L-2 turn, so more L-2 turns give a net gain" [read: p.61, 63, 72]: this is the ordinary transformer relation V2 = V1 (N2/N1) for tight coupling, where N1 and N2 are the numbers of turns. Voltage rises with turns; current falls by the same ratio; power is unchanged: V2 I2 = V1 I1. "Net gain in the Energy present" confuses voltage gain with energy gain. The speech version divides "energy" over turns, the paper divides "potential" (volts) [read: p.72 vs p.61]; these are different quantities.
• "The flux lines are squared and are additive" [read: p.43, 45]. Magnetic flux Phi = (1/L) times the integral of volts over time... more simply, Faraday: V = d(Phi)/dt, so flux is proportional to applied volts at fixed frequency (linear). In electrostatics the field between plates is proportional to V (linear); the energy density is proportional to V^2. So the p.43 grid ("two volts = four, four volts = sixteen") is a correct picture of energy, mislabelled as flux. Then volts add linearly turn by turn in a secondary (V2 = N2 x volts per turn); there is no turn-by-turn squaring. [memory, high]
• The "dominos" and "ping pong" analogies [read: p.72, 74] are not quantitative. Dominos store potential energy in each piece, so the total is the sum of the stored energies, and the energy to flip the first is a small trigger. That models a trigger releasing stored energy, not creation of energy; the stored energy has to be identified, and for coils it is not.

Worked example (reader): the printed numbers, closed

Use Smith's p.45 numbers. Input: 12 V at 0.3 A (his suitcase figures) = 3.6 W. The HV module makes 8,000 V; L-1 has 4 turns, 2,000 V per turn. Suppose L-2 has 400 turns and perfect coupling. Then the secondary voltage is 2,000 x 400 = 800 kV, but the output current can be at most 3.6 W / 800 kV = 4.5 microamperes. [computed] Any larger current would require a larger input. Real air-core (Tesla) coils have loose coupling (coupling coefficient k about 0.1-0.2) and the voltage gain is set by resonance: V2/V1 is about sqrt(L2/L1) = sqrt(C1/C2) for lossless transfer [memory, high], again with energy conserved; breakdown of air (about 3 MV/m) limits the voltage in practice.

b. Experiments? Transformer and Tesla-coil behaviour is thoroughly measured; efficiencies below 100%.

c. Tested? Smith names no measurement of L-2 output. The 1997 schematic (binder p.36) carries the note "will have to experiment for correct number" for the capacitor bank, which indicates an untested design at that date [read: p.36, page image].

d. If true: transformer theory (Faraday, Lenz) and Poynting's theorem.

Verdict: Refuted by calculation (power in a transformer with the printed numbers cannot exceed input power) and Internally inconsistent (flux/energy squaring; "energy" vs "potential" divided over turns).

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Claim 5. Dipole / capacitor-plate collection (the "Dipole Transformer Generator", MX application)

a. Mathematics consistent? Cannot be checked: no equations.

pp.11-19 are patent-style text and four drawings; no frequency, dimension, voltage, current or output is printed [read: digest p.11-19, MX application text]. The terms "Heaviside current component" and "magnetic waves ... change to electrical energy" are not defined with a formula. (In Poynting's theorem the energy flow is S = E x H, with E the electric field and H the magnetic field strength; flux that is not intercepted by a conductor is not "wasted" but circulates or radiates. [memory, high]) The text says prototypes were "fully tested for utility by the Inventor" but prints no results [read: p.12].

Worked bound (reader): how much is "radiated and wasted"?

A short dipole of length l has radiation resistance R_rad = 80 pi^2 (l/lambda)^2 ohms [memory, high]. For l = 1 m at 1 MHz (lambda = 300 m): R_rad = 0.0088 ohm; a 10 A current radiates (1/2)(10^2)(0.0088) = 0.44 W. [computed] The rest of the energy near such a small antenna is reactive near field that returns to the source. A capacitor plate placed in that field can draw real power only by loading the source (coupled-circuit action); the power received cannot exceed what the source delivers. There is no unloaded reservoir of "radiated energy" to collect beyond the radiated watt-level share. Whether Smith's plates act as ordinary coupled plates is a measurement question, not settled in the documents.

b. Experiments? Near-field capacitive and inductive coupling for power transfer is established with efficiency below 100% and the source loaded.

c. Tested? Not in the record. The Mexican application (MX NL02000035 A, filed 15 Nov 2002, published 20 May 2004, "Application Discontinuation" on Google Patents) is an application, not a grant [web: Investigator, patents.google.com/patent/MXNL02000035A/en]. The IMPI file history would show the outcome.

d. If true: little would need to change if the plates just couple; Poynting theorem would need to change if more power came out than went in.

Verdict: Not testable as stated in its mathematics (no quantities), and Never properly tested as a device claim (no neutral measurement; not dismissed by a test).

The "mirror plate" effect (Lagace p.2) is also unquantified; a charged plate near another induces charge on it (ordinary electrostatics), and grounding the second lets induced charge flow once, giving a transient current limited by the energy of the first plate's field.

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Claim 6. Capacitor displacement-current scheme (Lagace's reconstruction of Smith's "1 F / 1 kW")

This is Lagace's mathematics, not Smith's [read: don-smith.pdf p.2-3]. I checked the deep read and read pp.39-50 myself.

a. Internally consistent? Chapters 3-4 nearly; chapters 6-7 no.

Confirmed from the deep read and my recomputation: - Initial energy: (1/2)(10 uF)(10 V)^2 = 0.5 mJ, not the printed 5 mJ; figures C/D label the capacitors 10 mF (0.5 J) while the prose says 10 uF [read: p.5-6; page image]. All later figures are on the document's own x10 scale. Ratios are unaffected. - Redistribution to 5 V each and "half the energy is lost" is correct and independent of the resistance [read: p.6-7]. Charge conservation fixes the 5 V; the energy lost is (1/2)(C/2)(delta V)^2 = half. - Polarity-flip step: the final 0 V / 0 V is right, but "A now holds all the energy" at 0 V contradicts (1/2) C V^2 = 0 [read: p.9-10]. - Ch.4 step per-pulse 1.25 mJ vs the p.7 loss 2.5 mJ vs p.15 total "2.5 + 2.5 = 5 mJ": a factor-2 slip in the printed per-pulse figure [read: p.14-15]. - Ch.6: printed state after step 1 already totals 6.25 mJ for a 5 mJ input; the printed 10 mJ counts Cap A's recharged 5 mJ as output [read: p.26-29].

New from my checking: the 100% capture of chapter 4 is not over-unity. Two transfers each dissipate 50% of the starting energy (first the 10 V -> 5 V redistribution, then the flipped +/-5 V -> 0 V), so recovering all of it with an ideal transformer and rectifier would be 100%: conservation is respected in the ideal limit (the budget "5 mJ in, up to 5 mJ out" the deep read noted). The over-unity claims start in chapter 6 (200%) and chapter 7 (950%).

Simulation (reader): does the primary "feel" the load?

I simulated two equal capacitors C = 10 uF (V0 = 10 V on one, 0 on the other) joined through a series inductor L = 1 mH (the transformer primary), loss resistance R, with the secondary load reflected into the primary as a series resistance R_L (an ideal transformer with a resistive load looks like this to the primary). Fraction of the starting energy that ends up in the load, in the loss resistance, and left in the capacitors: [computed]

(Rounding: the last column is about 0.50 in each row; the small deviations are the simulation stopping while current is not exactly zero.) So when the load resistance dominates, the secondary load takes essentially the whole 50% that would otherwise heat the resistor, exactly Lagace's step 1 ("captures the energy normally lost"). But the load takes it by acting as series resistance in the primary loop: the same current flows, and the primary does feel it. The statement "the primary does not feel the secondary load because there is no closed current loop" [read: p.35] is contradicted by Lagace's own Figure D, which draws T1's primary in series in the closed loop C1-T1-C2-SW2 [read: page image p.13], and by Ampere-Maxwell: the displacement current in the capacitor gap equals the conduction current in the leads (current continuity), so the "displacement current" through the capacitor is the same current the primary carries, not an extra source. [memory, high]

Chapter 7 budget (pp.40-50, which I read in full)

Printed: C = 1 F, V = 100 V: 5 kJ in A [correct: (1/2)(1)(100^2) = 5,000 J]; redistributes to 50 V each, 2,500 J left, 2,500 J released [correct]; Cap C captures 1,250 J per redistribution, 2,500 J over two; auxiliary load receives 2,250 J "assuming 90% efficiency in the return-path recovery"; total output 2,500 + 2,250 = 4,750 J; "The system recovers 90% of the initial energy in Cap A via the return path" = 4,500 J; the battery supplies the remaining 500 J; efficiency 4,750/500 = 950%; at a 1 s cycle, 2,500 W + 2,250 W = 4.75 kW [read: Lagace chapter 7, pages in the 40s]. The arithmetic (products, Wh conversions: 500 J = 0.139 Wh, 4,750 J = 1.32 Wh) is correct. The budget is not: - Pool of energy per cycle: A starts with 5,000 J and is topped up by 500 J. Dispositions printed: 4,500 J returned to A, 2,500 J to C, 2,250 J to the auxiliary load = 9,250 J, against 5,000 J (5,500 J with the battery). [computed] - The auxiliary 2,250 J is described as the energy "remaining in Cap B", but B holds 1,250 J after redistribution (formula) [read: Lagace chapter 7]; 2,250 = 90% of 2,500, the released energy, which was already counted in C. - If 90% of A's energy comes back to A each cycle, the only energy that leaves the loop for loads is the loss, which the battery replaces: at most the 500 J supplied (500 W at 1 s per cycle), less the fraction lost as heat in the loop. The printed output (4,750 J) exceeds that by a factor 9.5. [computed] - Output counted as "efficiency" is a ratio of loop throughput to battery top-up, not of output to total energy supplied (the first 5,000 J must be put in once). - The conclusion says "Don Smith's work peaked over 35 years ago" [read: Lagace closing chapter]; the dated record puts it at 1994-2005 (Investigator): the secondary source misstates Smith's chronology. - "A 1 F capacitor at 10 V stores the same energy as a 0.01 F capacitor at 100 V" [read: Lagace closing chapter]: correct, both 50 J.

The original Smith claim "a single 1 F capacitor could produce 1 kW" has no source, date or occasion [read: p.2]. By itself it is arithmetically possible only as a rate: 1 F at 44.7 V holds 1 kJ, so 1 kW is that whole energy replaced each second; the question is where the replacement comes from.

Other published form

Vladimir Utkin (about 2012) reports Smith's arrangement "the capacitor is charged by sparks and powerful displacement current is produced. The transformer with the ferromagnetic core is collecting this current" [orchestrator fix 2026-10-08: the source reads "is produced", not "is around"; the sentence is Utkin's caption for a 2006 rendering titled "Don Smith Device" (credited to Zero Point Films), not Smith's words; see deep/further-investigator-files.md] [web: Investigator, free-energy-info.tuks.nl/VladimirUtkin.pdf]. The idea predates Lagace's write-up; Utkin gives no numbers I could check.

b. Experiments? Two-capacitor energy loss, LC transfer and transformer recovery are classic and measured; recovery up to the 100% ideal is limited by resistance and core/switch losses. No measurement shows more than 100%.

c. Tested? Lagace's 2026 videos claim a working "modified Don Smith replication" with a cold input inverter and output melting power supplies [web: Investigator, YouTube Tiji0JqY90o]; I have not seen them, they are a first-hand claim by the author of the secondary document (an interested party), without published energy accounting in the project files. Capacitor figures C and D appear to be simulator screenshots with a single 5.0 V reading [read: deep read].

d. If true: conservation of energy as applied to circuits would fail; or an external energy source would have to be identified and metered (Lagace names none with a number).

Verdict: split.

• Chapters 3-4 (recover up to the 100% of otherwise-dissipated energy with a transformer and rectifier): Plausible, untested as an energy-recovery scheme, with arithmetic slips (factor 10 in the initial energy; factor 2 in the per-pulse figure; A "holds" energy at 0 V). It is ordinary circuit physics and respects conservation.
• "The primary does not feel the secondary load": Refuted by calculation (simulation above; Lagace's own figure).
• Chapters 6-7 (200% and 950%, 4.75 kW): Internally inconsistent (budget does not close) and Refuted by calculation (output cannot exceed input in a closed energy budget; no external source entered with a number).

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Claim 7. Smith's account of suppression (Morgan/Edison/Tesla; Patent Office "metering group")

a. Mathematics consistent? Not mathematics; I checked the logic.

The Patent Office item [read: p.50-51] chains four steps: (1) over-unity applications are referred to the "metering group"; (2) so the Office accepts logarithmic measuring devices; (3) so it accepts the square law for kinetic energy; (4) so it knows over-unity exists ("direct and absolute evidence"). Each step is a non sequitur. A referral to a measurement-related examining group (if it occurred) shows an examiner classified the subject matter, not an opinion about physics. Logarithmic scales (decibels) are a convenience for wide ranges, not evidence of non-linearity in AC circuits: for fixed R, L, C the AC equations are linear [memory, high]. The Office's published position on devices claiming more energy out than in is inoperability, which is a position that over-unity is not known to exist (the physicist should treat this fully). Step (1) gives no art unit, examiner, application number or date; Smith had a stake as the holder of a pending application (08/100,074, 4 February 1992) [read: p.50-51, 56]. Smith's claims are his account, stated with its stake; absence of a record in the folder is not contradiction. The USPTO file wrapper of 08/100,074 (not obtained; Investigator lead) would show which group handled it. The Morgan/Edison/Tesla meeting [read: p.37]: no date, place or source; mathematics not involved.

b-d. b: not applicable to experiments. c: file wrapper and the MX file history (IMPI NL/a/2002/000035) not examined. The Investigator found Perreault (2002) relaying Smith's statement that his Mexican patents "have been approved" while the published record is an application marked discontinued [web: Investigator, energyevo.org 2016/08/05 and Google Patents]. The 2001 tape was edited by the organiser, who says he removed about five minutes where an audience member questioned Smith (interested party: Perreault, event organiser and video seller). d: not applicable.

Verdict: Not testable as stated (an assertion about institutional motive and an internal practice); the inference chain is invalid, and the factual question (file wrapper, Mexican outcome) is Never properly tested by records not yet obtained.

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Investigator's findings, weighed (kept in the record whatever the verdict)

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Corrections to Step 1 (earlier documents)

1. claims.md, claim 1, and the deep read (pp.36-53): they say the binder's two worked examples use different versions of the formula. Confirmed from page images [read: p.44, 47-49]. Add: p.56 ("60 million cycles per second ... one million times the energy") is a third place where the first power of f is used, contradicting the "squared" claim in the same document. Evidence: 60 MHz / 60 Hz = 10^6 (f) vs 10^12 (f^2).
2. Investigator's correction and claims.md on example B: the magazine printing's intermediate line "500 x One Million squared" is itself wrong (0.5 x 10^-5 x 10^10 = 50,000, not 500), and 500 x (10^6)^2 = 5 x 10^14, not the printed 500 MJ. Three answers, not two, none matching the printed formula (5 x 10^16). Evidence: magazine p.37 page image; computation.
3. Deep read (pp.54-75), "For context [memory, high]... ionosphere ... about 200-300 kV": the web source gives about 400,000 V (https://en.wikipedia.org/wiki/Atmospheric_electricity). The same order of magnitude; state 200-400 kV or "about 400 kV" and keep the source.
4. Deep read, p.55 gamma example: the deep read notes the arithmetic (1.6 x 10^9 vs "1.6 trillion"; 1.6 x 10^10 V vs "16 trillion volts"). Add that the premise "gamma readings exceed one trillion gammas" equals 1,000 tesla (Earth's field is about 0.025-0.065 mT), and that gamma and gauss are magnetic units that have no fixed conversion to volts (an EMF needs an area and a rate of change).
5. Deep read, reactive-power remark "Q = 4 pi x (1/2) C V^2 f": true when V is rms (checked: 54.3 VA for example A). If V is the peak the factor is 2 pi. Clarify.
6. claims.md, claim 3: "the Earth holds 17.6 x 10^18 exajoules". Add that the figure also exceeds the solar energy Earth has received in 4.5 Gyr (about 1.7 x 10^34 J) by about 1000 times, and differs by 10^9 from Smith's own rate x age. Also: Smith's "ten trillion joules" per lightning flash is about 2 x 10^4 times the 500 MJ average given by [web: Wikipedia, Atmospheric electricity].
7. claims.md, claim 6: "The printed balances do not close" is right for chapters 6-7, but chapters 3-4 do close in the ideal limit (two 50% transients, 100% recovery; no over-unity). The printed 4 kW figure comes from chapter 7: "4.75 kW" arises from adding the aux load (2,250 J, 90% of the released 2,500 J, already counted in Cap C) to Cap C (2,500 J); recovered A (4,500 J) is a third count of the same pool (9,250 J against 5,000-5,500 J). Chapter 7 is a worked example; its arithmetic products are right.
8. claims.md and the deep read, claim 6 "Printed outputs: 10 mJ from 5 mJ (200%)": the deep read's ceiling analysis stands. Add the simulation result: a series-inductor transfer lets the load take about 50% (the whole transient loss) per transfer if R_L is much larger than R, so the primary does feel it.
9. Lagace, p.47 ("Don Smith's work peaked over 35 years ago"): the Investigator's correction stands (dated record 1994-2005); no further change.
10. facts.json / Step 1 describes p.38 "E = (V x A) x CPS^2" without units. Units are J/s^3 (W/s^2), a third unit (p.47's formula is J/s^2).

Web checks done (budget about 6)

1. https://en.wikipedia.org/wiki/Atmospheric_electricity: potential about 400 kV, current density about 2 pA/m^2, surface field about 100-120 V/m, average lightning flash about 500 MJ (retagged [web]; my [memory] figures of 200-300 kV and about 1 GJ were low-confidence and differ). I did not find a global total current in that page; I derived about 1 kA from density x area and say so.
2. https://en.wikipedia.org/wiki/Skin_effect: copper skin depth 65.2 um at 1 MHz; AC/DC resistance ratio about half the radius to skin depth; resistance rising as the square root of frequency (matches my computation). Not checked on the web (not load-bearing): d'Arsonval dates; Ruhmkorff, Oudin and similar dates in the deep read (kept as [memory]); gel-cell capacity (medium); solar intercept 1.2 x 10^17 W (memory, high); the radiation-resistance formula (memory, high).

Summary of verdicts

1. f^2 law: Internally inconsistent; Refuted by calculation as an energy gain (weak f-form is reactive power, Established).
2. Zero impedance, room-temperature superconductor: Internally inconsistent; Refuted by calculation (Z = R at resonance; skin effect raises R).
3. Ambient background energy: Refuted by calculation (strong form); supporting numbers Internally inconsistent; suitcase demonstration Never properly tested; weak form Established at microwatt level.
4. Turn ratio and flux squaring: Refuted by calculation; Internally inconsistent.
5. Dipole / plate collector: Not testable as stated; Never properly tested.
6. Lagace capacitor scheme: ch.3-4 Plausible, untested (arithmetic slips); "primary does not feel load" and ch.6-7 (200%, 950%) Refuted by calculation and Internally inconsistent.
7. Suppression account: Not testable as stated; inference chain invalid; records Never properly tested.