Bearden: mathematician review (PhD mathematician)

Role: check derivations, notation, units and hidden assumptions. Inputs: facts.json, claims.md, and my own reading of the source text (pdftotext of Bearden - Final secret of free energy.pdf, and US6362718.pdf). Source tags: [read] = from the document, [web] = URL given, [memory] = recalled, confidence stated.

Source gap note: none for text. Both documents have a text layer. The patent is two-column OCR with garbled figures, so Figs. 6-8 data were not read as numbers; only the prose and the numerical sentences in the description were. Where a conclusion rests on [memory] alone, it is flagged.

A structural observation first. The 1993 paper contains no derivation in the mathematical sense. Its "pseudo-equations" [1]-[8] are bookkeeping identities (units: V_T x Coul_T = K_T, joules = volts x coulombs, which are dimensionally fine [read]). Everything that carries the free-energy claim is in prose definitions. The mathematician's job is therefore mostly to ask which statements are definitions, which are theorems, and which are assertions about the world dressed as definitions.

Corrections to Step 1

1. Date of the second paper. facts.json and claims.md call it "Update 15 February 1994". The PDF's list of contents says "Update - 5 February 1994" while the heading of the paper itself says "Update 15 February 1994" [read]. The source is internally inconsistent; neither is a Step 1 error but the date should be given as "5 or 15 Feb 1994 (the PDF disagrees with itself)".
2. Step 1 says the 1994 updates were "only sampled". That is true of Step 1, but they carry the only actual equations in the bundle, which I read (lines ~1700-1900 and the 13 May 1994 section). Key items are in claim 2 and 3 below. The Step 1 summary of the 15 Feb 1994 paper omits its central device idea: charging a capacitor "by massless displacement current" so the source is never "dissipated" [read]. That is the mathematical core of claim 3 and the place the error can be pinpointed.
3. Patent arithmetic. The patent's COP "3.44" does not follow from its own stated numbers: 48 W / 14 W = 3.43 (3.4286). Minor, but Step 1 reported 3.44 without noting this [read, with my arithmetic].
4. Patent internal consistency, not noted in Step 1: 450 turns / 40 turns = 11.25, so a plain transformer-like turns ratio on 100 V input would give about 1,125 V, not the 4,000 V quoted at 12 mA. Also reducing the output winding from 450 to 12 turns "drops output voltage to 106.7 V" (4000 x 12/450 = 106.7, consistent) "with increase in current to 0.5 A". Constant power would give 12 mA x 450/12 = 0.45 A, and 106.7 V x 0.5 A = 53 W, not 48 W [read plus my arithmetic]. The numbers are rough and unexplained, not a contradiction by themselves.
5. Step 1 says the quartz relaxation time of "about 10 days" is quoted against copper 1.5e-19 s. I checked: copper 1.53e-19 s is correct for tau = epsilon/sigma; fused quartz comes out near 51 days with sigma = 1e-17 S/m, so "about 10 days" is the right order of magnitude for an insulator, not an error [web: https://arxiv.org/pdf/0801.4182 and search result summary]. The numbers are not the problem; their use is (see claim 3).
6. facts.json describes the Whittaker/Ziolkowski results as "ordinary solutions of the wave equation". Refinement: Whittaker's 1903 result writes solutions of Laplace's equation as an integral of plane waves in COMPLEX directions, f(x cos v + y sin v + iz, v) integrated over v from 0 to 2 pi [web: https://arxiv.org/html/2205.08309v2]. These are not real time-reversed wave pairs. This matters to claim 1.

Claim 1: Energy = potential = an ordering in the virtual photon flux; a scalar potential contains a harmonic series of wave/antiwave pairs linking each charge to the universe

a. Internal consistency. Two parts. - Definitions. "Energy and potential have exactly the same definition" [read]. They do not have the same dimension: energy is joules, electric potential is joules per coulomb. Bearden concedes the issue himself ("it's used not really as energy, but rather as excess energy per coulomb", [read]) and then uses V_T x Coul_T. A definition that equates two quantities of different units is inconsistent unless "potential" is silently changed from phi to something else mid-argument; he does exactly that, switching between phi and "trapped energy". - Whittaker decomposition. Whittaker 1903 is a correct theorem: a harmonic function can be written as a superposition of plane waves with complex direction [web: https://arxiv.org/html/2205.08309v2]. For a static potential it is a mathematical identity (like a Fourier or Bessel expansion), it contains no time dependence for a static phi, and nothing in it assigns energy to the terms. A function being expandable in a series does not make each term a physical entity carrying energy. Bearden's step to "an infinite harmonic series of phase-locked time-forward EM waves going out ... and time-reversed waves coming back" and "infinite energy in each" [read] does not follow. The "time-reversed partner" is introduced from phase-conjugate optics (Yariv's distortion-correction theorem, which applies to a pumped nonlinear medium), not from Whittaker. I find no theorem in the cited papers [read: his own reference list; web check above on Whittaker only; Ziolkowski's papers I did not read] that yields it. - "Infinite energy". A sum of infinitely many terms each with finite positive energy density would diverge, but the actual field energy of a point charge is finite outside any radius R (q^2/(8 pi eps0 R)). No contradiction is shown, only asserted.

b. Experiment. The electrostatic energy U = (1/2) eps0 integral E^2 dV is verified in many experiments (capacitor energy, electrostatic generators, atomic energy levels). The "virtual photon flux ordering" has no measurable consequence distinct from textbook potential energy, so it neither agrees nor disagrees. Aharonov-Bohm (1959) is cited as "scalar interferometry proven" [read]; AB shows potentials affect phase in field-free regions, which is consistent with standard QED and does not require waves hidden in a static potential. Bearden's claim that AB becomes "distance independent" with Whittaker beams is an untested assertion [read].

c. Tested? No test exists. No one has measured a "bidirectional wave pair" in a static potential [read: paper cites none]. Ziolkowski's localized-wave experiments (cited in note 3) test real focused pulses in ordinary media; they are not tests of a decomposition of a static potential. I did not read those papers.

d. If true. Static electric energy would have to be reinterpreted as a flux through the vacuum from every point of the universe; this would conflict with the finiteness of total electrostatic energy and with locality in field theory. Mainstream QED would need new terms; none are written down.

Verdict: Split. (i) The Whittaker/Ziolkowski mathematics as pure math: Established (but it is a notation/expansion with no energy content). (ii) "Energy and potential are identical, an ordering of the virtual photon flux, and each potential carries unlimited energy from the universe": Internally inconsistent (units of energy vs. energy per charge) and Not testable as stated.

Claim 2: No EM force fields in vacuum; EM waves are longitudinal stress-potential waves; electron charge is not quantized but varies with ambient potential

a. Internal consistency. Several errors found by reading. - "E = -grad phi is incorrect for the vacuum"; the replacement is "PE = -grad phi" [read]. Then E is recovered "if and only if a charged mass is first introduced". But a test charge is exactly how E is operationally defined (E = F/q in the limit q goes to 0), so this is a restatement of an operational definition, not a refutation. The field exists as a solution of Maxwell's equations without a probe; radio waves propagate between a transmitter and a receiver through vacuum, carrying momentum and energy that can be measured (radiation pressure, light-sail forces) when they hit matter, and the Poynting theorem accounts for energy in transit. - Newton's third law argument: "forces occur in pairs, so there are no force fields in vacuum" [read]. Newton's third law holds for contact interaction; for fields, momentum conservation is restored by field momentum (Poynting vector / c^2). The pair partner of the force on a charge is the momentum change of the field. So the premise does not give the conclusion. - Longitudinal waves. A vacuum EM wave from Maxwell's equations has div E = 0 and div B = 0 in free space, so plane waves must be transverse (k dot E = 0). That derivation is a few lines of vector calculus. Bearden's "stress potential wave ... like a sound wave" would need a scalar field with a nonzero longitudinal component; Maxwell's equations plus gauge freedom show longitudinal and scalar potential modes are pure gauge in vacuum [memory, high]. Polarisation of light (Malus' law, polarisers) is only possible with transverse waves; that is an experimental fact he does not address. - Charge not quantized. His argument: otherwise "there could be no phi created on any charged particle q, and hence no E-field, and hence electrons would not move" [read, 13 May 1994]. This is a non sequitur: fields from fixed quantized charges exist in the standard theory. - Product rule error (a clear mathematical mistake). The 13 May 1994 paper writes charge as a coupled product (phi)(m) and then says "d/dt[(phi)(m)] = (dphi/dt)(dm/dt)" [read]. The derivative of a product is d(fg)/dt = f' g + f g', not f' g'. The result that current has "two components" coupled as a product of derivatives is therefore wrong as a derivation. It also has a units problem: i = dq/dt must be coulombs per second, but (dphi/dt)(dm/dt) has units of (V/s)(kg/s) = V kg / s^2, not A. - Units of D. The 1994 text says D (C/m^2) is "coulombs (charged mass rate of flow) per square meter" [read]. C/m^2 is a surface charge density (flux density), not a rate of flow; the rate is D times area per time, i.e. dD/dt (A/m^2). Also "div D = d(phi)/dl" is written "neglecting constants of proportionality" [read]; div D has units C/m^3, d(phi)/dl has units V/m; the constant that fixes it is eps0 and so div D = -eps0 laplacian phi in the static case. The pure replacement D to "d(phi)/dt" is dimensionally inconsistent as written (V/s versus C/m^2). - Displacement current. He treats displacement current as a "flow of pure potential" that carries energy without charge motion, and quotes Buchwald "no energy transformation into heat occurs for displacement currents" [read, note 8]. Displacement current is eps0 dE/dt, a term in Ampere-Maxwell; it carries no charge and so does not deliver charge to a plate. Charge on the plates arrives only via conduction current in the wire, which is why charge continuity (div J + d rho/dt = 0) holds at the plate. Plate charge Q = CV must come from the source.

b. Experiment. Charge quantization: Millikan, and modern tests of neutrality of matter (|q_p + q_e| below about 1e-21 e) [memory, high; Step 1 quoted the same]. Transverse polarisation and Maxwell's equations have been tested to extreme precision (radio, optics, LIGO-era timing for the speed of light; photon mass limit below 1e-18 eV [memory, medium]). Charge being independent of potential is tested: an electron has the same e in the 10^5 V field of a TV tube and on a ground electrode [memory, high].

c. Tested? Charge quantization and transversality are among the most heavily tested features of physics. Bearden proposes no experiment; nobody has replicated a potential-dependent charge. The claim is dismissed because it conflicts with existing data, not because a controlled test of Bearden's own prediction was run.

d. If true. Maxwell's equations as a model of radio, optics and the Standard Model's gauge structure (QED) would need replacement; polarisation, Millikan, and atom neutrality data would need explaining. These data currently falsify it.

Verdict: Split. Longitudinal vacuum EM waves replacing transverse ones: Tested, not supported (polarisation, Maxwell's theory) and Internally inconsistent in the "third-law" argument. "Charge varies with ambient potential": Tested, not supported (neutrality of matter, Millikan; I did not run a web check on the 1e-21 figure, it is [memory, high]). The product-rule step: Internally inconsistent.

Claim 3: A source used only to charge a long-relaxation-time collector, then disconnected while the collector discharges into a load, supplies energy from the vacuum at no cost; a car battery could run a car or a grid

a. Internal consistency. This is the claim where I can show the error with numbers. - Bearden's own equation [6]: K_T = V_T x Coul_T [read]. This is the energy "taken for free". Take a collector charged to V = 12 V with Q = C V = 12 microcoulomb (C = 1 microfarad). Bearden's K_T = V Q = 144 microjoule [my calculation from his equation]. But the energy actually stored in a capacitor is (1/2) C V^2 = 72 microjoule. Bearden's formula overstates the stored energy by a factor 2 because it uses the full potential V for every coulomb, while the potential builds up gradually from 0 to V as charge is added (integral of V dq = Q^2 / 2C). - Where K_T = V Q is correct: it is the energy that the source must give up to deliver charge Q at potential V, i.e. the work the battery does. So his "free" quantity equals the battery's expenditure. Half of it (72 microjoule) ends up in the capacitor and half is dissipated in the series resistance (this is true for any resistance, a standard result: integral of i^2 R dt = (1/2) C V^2 for charging from a constant source). The source is not "replenished by the vacuum": the battery's charge Q has moved through it, and its chemical energy is reduced by Q V. - The "no current during relaxation time" idea. Relaxation time tau = epsilon/sigma is the decay time of charge density in the BULK of a conducting material [web: https://arxiv.org/pdf/0801.4182]. For copper, tau = 1.5e-19 s is correct arithmetic: eps0 / (5.8e7 S/m) = 1.53e-19 s [web: same source as the search result above]. It is not a delay before current can flow into a circuit element. Connecting a capacitor to a battery draws current immediately, with time scale R C (for example 1 ohm x 1 microfarad = 1 microsecond), regardless of tau. Quartz at about 10 days or more is the time for injected charge to leak off an insulator; a "doped" conductor with tau of 1 ms in his example [read] would have conductivity sigma = eps0/tau = 8.85e-9 S/m, which is an insulator, not "a good conductor" as the paper says. This is a numerical inconsistency in his own prescription: tau = 1 ms cannot coexist with "still a good conductor" [read plus my calculation]. Even then, a collector with a 1 ms relaxation time has so little conductance (<1e-8 S/m) that very little current can flow to charge it. - "Amps are only dissipating coulombs" [read, definition of i]. Redefining current so that it excludes charging current is an arbitrary definition. dQ/dt flows whenever a capacitor charges, and the product V i is the power leaving the battery, by the Poynting theorem it flows through the fields around the wire. "We extract energy but no power" is contradictory: energy transferred in finite time is power x time. - Footnote 25 rewrites energy conservation as "the energy taken ... cannot exceed the sum of the trapped energy initially in the system and the unscattered energy that flowed in" [read]. This is a hypothesis that an unmeasured inflow exists; it is not a derivation and has no formula for the inflow. A conservation law with an unspecified source term is unfalsifiable as stated. - Heat pump analogy (note 20) [read]. Heat pump COP > 1 is Carnot-limited (COP up to T_h/(T_h - T_c)) and counts the electrical work input; the extra energy is heat taken from a cold reservoir whose energy fell. The analogy requires a depleting reservoir; Bearden's "vacuum" is claimed not to deplete. So the analogy does not carry over. - The "multiplication" argument (note 23, 10 wires x 100 V = 10 free sources) [read]. Connecting wires to a source puts them at the same potential, but each wire only carries charge as long as it holds some capacitance; no energy is created. Energy to charge N wires is N times the energy to charge one. This is also his note 14 argument (10,000 charges = 10,000 n joules), which is indeed ordinary arithmetic: total work scales with the number of charges, not a free multiplication.

b. Experiment. Charging a capacitor from a battery, with any resistance, and discharging into a load has been done in every electronics lab. Measured energy balance: stored (1/2) C V^2 plus resistive loss (1/2) C V^2 equals the battery's Q V. No excess energy is reported in any such standard measurement [memory, high; standard textbook result, e.g. Griffiths E&M, capacitor energy sections].

c. Tested? Bearden gave no experiment and no data. The February 1994 update says a patent application was filed [read]; the later US 6,362,718 does not use this scheme [read], so no test of this specific scheme appears in the documents. I found no independent test of the collector scheme [web search on MEG found nothing for it]. I did not find it dismissed untested by a body; it simply was never built with data published [web: https://www.zpenergy.com/modules.php?name=News&file=article&sid=3918 mentions only the MEG].

d. If true. Conservation of energy in circuit theory would fail for the simplest RC circuit; one could build a battery that never runs down by switching a capacitor. Circuit theory (Kirchhoff) and the Poynting theorem would be wrong in a regime that every switching power supply operates in daily.

Verdict: Refuted by calculation (the energy budget of charging a capacitor: battery gives Q V, capacitor holds half, shown above, with Bearden's K_T equal to the battery's outlay), plus Internally inconsistent (a 1 ms relaxation time implies an insulator, and "extract energy but not power"). It was never built with published data, so there is no experimental test to cite either way.

Claim 4: Closed classical EM cannot give overunity, so real overunity devices need a hidden open vacuum source; Maxwell's original quaternion theory, truncated by Heaviside, contained it and the gravity-EM unification

a. Internal consistency. - Logic. "If a device works totally within CEM it cannot be overunity; therefore overunity needs a hidden source" [read]. The first clause is true (energy conservation follows from Maxwell's equations via the Poynting theorem). The conclusion "so CEM must be flawed" is an inference that presupposes overunity exists, which is the claim at issue. It is a valid conditional (if overunity then not CEM) and not an argument for overunity. - Quaternions. Maxwell wrote some equations in quaternion notation and many in components [web: https://en.wikipedia.org/wiki/Oliver_Heaviside and the Bork/medium results; "quaternions were seldom used in derivations" per the search summary of the Treatise]. Heaviside (1884) reduced Maxwell's 20 equations to the 4 vector equations [web: https://medium.com/@zachbobby8/maxwells-quaternions-vs-heaviside-s-vector-calculus-699907b2843d, a secondary source]. The claim that the vector version "captured only a subset" is not supported: the Heaviside equations are logically equivalent to Maxwell's component equations except for the gauge/potential presentation. The quaternion "scalar part" is the divergence term; it is retained in Heaviside's form as div (E, B). Maxwell's 1873 gravity-EM unification does not exist in the Treatise; Maxwell tried gravity in his 1864 paper and concluded it did not work (stated in his Dynamical Theory) [memory, medium; I have the Maxwell Dynamical Theory text file in the scratch dir but did not read it here]. So that claim has no source in Maxwell [unverified; marked low-medium]. - Photons plus antiphotons coupling into "spin-2 gravitons" (Additional Comments, [read]): no equation is given; graviton spin 2 is standard, but a coupling of two spin-1 objects into spin 2 is not shown. - Barrett (note 22) argues higher-topology (non-abelian) EM can do things circuits cannot [read]; Barrett's actual papers I did not read, and I note the arXiv critique of the AIAS group "non sequitur" literature as a similar line [web: https://arxiv.org/pdf/physics/0302016, title only].

b. Experiment. No experiment contradicts Maxwell's equations at the scales of circuits. Quaternion and vector formulations give identical predictions in classical use because they are mathematically equivalent.

c. Tested? The historical claim (that Heaviside truncated Maxwell's theory) is a matter of history, checkable by reading the Treatise; it has been checked by historians (Bork and others) and the consensus is that Heaviside simplified notation and removed the potentials from the central role [web: http://kirkmcd.princeton.edu/examples/bork_76.pdf title only; I did not open it]. The physical claim of a hidden source is untested because no device that works is documented (see claim 5).

d. If true. Poynting's theorem would acquire a source term; the Standard Model's gauge sector would need to allow energy extraction from a ground state, which would undermine the vacuum's stability as the lowest energy state.

Verdict: Split. "Closed classical EM can't give overunity": Established (it is a theorem: Poynting). "Maxwell's quaternion theory had the gravity-EM unification that Heaviside lost": Historically superseded but instructive for the notation story (vector notation replaced quaternions, a real 1890s debate), and unsupported for the unification part (no source found; treat as Not testable as stated). "Real overunity devices need a hidden vacuum source": Plausible, untested only in the trivial sense that nothing is known to rule out an open system; as a statement of what Bearden must show it is Never properly tested.

Claim 5: The MEG (permanent magnet, Metglas core, pulsed input coils at 87.5 kHz) outputs more power than input (COP > 1; extrapolated 3.44 per output coil at 100 V), and can self-power

a. Internal consistency. - The patent's stated numbers: 100 V x 0.14 A = 14 W; 12 mA x 4000 V = 48 W per coil [read]; 48/14 = 3.43, not 3.44 [my arithmetic]. Two output coils would give 96 W, so total COP on the stated figures is 6.9, but the patent states 3.44 "for each" [read]. The 100 V figure is, in the patent's own words, extrapolated from the measured data [read], not measured. - Product of averages. The patent says "output current 12 mA average and output voltage 4000 V average", multiplied to 48 W [read]. For pulsed waveforms the average of v(t) i(t) is not the product of averages. It could be larger or smaller depending on phase. The patent also says measured real power "agrees" with the oscilloscope RMS power [read] but gives no waveform integral. - Transformer scaling. A 450-turn secondary driven by a 40-turn primary at 75-100 V would give about 840-1,125 V by turns ratio alone; 4,000 V implies either large transient spikes (the patent shows "transient spikes", Fig. 6E-6H [read]) or flux change greater than a simple switching ratio. Spikes are narrow and high frequency, exactly where measurement bandwidth errors occur. - Energy-conservation argument. In steady state, the magnetic energy in the core and magnet returns to the same state each cycle (the B-H trajectory is a closed loop). Then average electrical power out plus losses must equal average power in, plus any power from the permanent magnet. A permanent magnet in a closed cycle supplies none (it is a conservative store, not a source). I estimate its store: the magnet volume 40 x 25.4 x 38.1 mm = 38.7 cm^3 [read, dimensions]; with (BH)max about 300 kJ/m^3 for NdFeB [memory, medium; not web-checked], the maximum extractable energy is roughly 12 J; with B = 1.2 T the field energy density B^2/(2 mu0) = 573 kJ/m^3 gives about 22 J [my calculation]. If the stated excess power (96 W out minus 14 W in = 82 W) were drawn from the magnet, 12 J would be exhausted in 0.14 s. The patent reports no demagnetisation or run-down, so the energy must have come from the input, or the measurement is wrong. - Self-actuating form: claimed "apparent" from the measured COP (the patent says "Therefore, it is apparent that the generator can be built in a self-actuating form" [read]), but no self-running trial is described [read]. This is an inference from COP > 1, not a result.

b. Experiment. Transformers, flux-switching and saturable-core circuits are well understood; efficiency of a transformer is below 1 always. A COP above 1 in steady state is inconsistent with every power-conversion measurement ever made when measured correctly [memory, high].

c. Tested? Three points. - The patent itself reports measurements from the inventors (10-75 V) [read]. They are not independent, and the patent gives no instrument calibration, bandwidth or uncertainty [read]. - Naudin's amateur replication (JL Naudin, "MEG v2.0") reported COP 1.75 and had a coil failure from sparks above 3 kV [web: http://linas.org/mirrors/jnaudin.free.fr/2002.06.03/html/megdiag.htm and https://www.novakcorp.com/energy/experiments/megv2.htm]. These are amateur results of unknown measurement quality; I did not read his scope methodology. - A 2000 correspondence (article on ZPEnergy) quotes Bearden's team at COP 17-20 and claims of independent university/DoD-auspices testing "being arranged"; no results from a completed independent test are given in that source [web: https://www.zpenergy.com/modules.php?name=News&file=article&sid=3918]. I found no report of a completed, independent, neutral test and no published instrument-level energy balance (for example a calorimetric or battery-capacity-drain measurement with a long closed loop). So: no neutral controlled test on record that I could find; replications are amateur, unverified.

d. If true. Conservation of energy would fail for a device with no moving parts and no fuel; or the permanent magnet would have to be an energy supply, which contradicts its measured finite energy. The cheapest explanation (measurement artefact) would require no change in physics.

Verdict: Split. Strong form (steady-state COP > 1 with a permanent magnet as the only extra source, and self-running): Refuted by calculation (energy balance: the magnet holds roughly 12-22 J, the claimed excess would drain it in a fraction of a second; shown above). Weak form (the inventors measured COP > 1 on a bench): Never properly tested (no independent, instrumented test with a full energy balance on record; amateur replication reports are not neutral and their quality is unknown); the likely explanation is measurement artefact in pulsed, reactive, high-frequency waveforms, which I cannot confirm without the raw waveforms. The 3.44 figure is an extrapolation: Not testable as stated until a measured 100 V run is shown.

Summary table

Worked check for the reader (claim 3)

Charge C = 1 microfarad to V = 12 V through R = 1 ohm. Q = C V = 12 microcoulomb. Battery work = Q V = 144 microjoule. Energy stored = (1/2) C V^2 = 72 microjoule. Heat in R = integral of i^2 R dt = 72 microjoule. Total out of battery = 144 microjoule, which is Bearden's K_T. Discharging the capacitor into a load returns at most 72 microjoule, so the load receives half of what the battery gave up, and the battery has lost 144 microjoule. COP = 0.5, regardless of R.

Web checks done (budget 6, used 5)

1. MEG replication: Naudin COP 1.75 [web: http://linas.org/mirrors/jnaudin.free.fr/2002.06.03/html/megdiag.htm]; claims of independent tests pending only [web: https://www.zpenergy.com/modules.php?name=News&file=article&sid=3918].
2. Whittaker 1903 complex-direction plane-wave solution [web: https://arxiv.org/html/2205.08309v2].
3. Relaxation time copper 1.53e-19 s, quartz about 51 days [web: https://arxiv.org/pdf/0801.4182 (and search summary)].
4. Maxwell quaternions / Heaviside 20 to 4 equations [web: https://medium.com/@zachbobby8/maxwells-quaternions-vs-heaviside-s-vector-calculus-699907b2843d (secondary); Bork http://kirkmcd.princeton.edu/examples/bork_76.pdf not opened].
5. ZPEnergy article fetched for MEG test status. Still [memory] only: NdFeB (BH)max about 300 kJ/m^3 (medium); charge neutrality 1e-21 e (high); photon-mass limit (medium); Maxwell's 1864 gravity conclusion (medium). The Sweet vacuum triode claim was not checked.

Investigator findings (2026-10-05)

Scope. I read sites.md ("Leads for review") and all six research notes dated 2026-10-05 (archive-org-uspto, cheniere-mirror, company-records, epo-register, google-patents, jlnlabs). research/files/ is empty (nothing was downloaded). I did not re-fetch the web pages myself; everything tagged [web] below is the Investigator's reading, with the URL it gives, and I did not read the unread parts it lists (remarks pp. beyond 50,000 characters, Bearden's declaration, EPO annex pp.4-5, 2007/2008/2009 replies). Where I add arithmetic it is tagged [my calculation]. These findings bear almost entirely on claim 5, with some bearing on claims 1, 2 and 4 through Bearden's 2004 remarks. Claim 3 is untouched.

F1. Tektronix statement and the patent's power measurement (claim 5, questions a and c)

Finding [web: http://jnaudin.free.fr/meg/ths720p.htm, per research note]: Tektronix customer support, forwarded by builder Jon Flickinger and posted by Naudin in 2002, says the RMS measurement on a scope math waveform "is bogus and has no value"; Vrms x Irms x cos(theta) holds only for pure sinusoids, "THERE'S NO SUCH THING AS RMS WATTS"; true power is the mean of the point-by-point product v(t) i(t). The patent says its power figures "agree with RMS power measured using a Textronic THS730" [read, Step 1]. The instrument maker is not a neutral-in-the-dispute party in any obvious way (it sells scopes and has no stake in the MEG), but this is one support-desk reply relayed through a builders' list, so it is the maker's characterisation of a general rule, not a test of the MEG.

Mathematical weighing. The rule itself is correct and I can state it exactly. For periodic v(t), i(t) with period T, true average power is P = (1/T) integral of v(t) i(t) dt. The product Vrms x Irms is P only if v and i are proportional in shape (a resistor) or both are single-frequency sinusoids (then P = Vrms Irms cos(theta)). In general, by Cauchy-Schwarz, |P| <= Vrms x Irms, with equality only for proportional waveforms. So Vrms x Irms is an upper bound, never an estimate, of true power for a mismatched pair. This is the sharper form of what I wrote in claim 5a ("the average of v(t) i(t) is not the product of averages"): the error is one-signed for the "RMS power" product, it overstates.

Worked example (reader). Take v(t) = 10 sin(wt) volts and a current with a third-harmonic shape, i(t) = 2 sin(3wt) amperes. Vrms = 7.07 V, Irms = 1.41 A, so Vrms x Irms = 10 W. The true power is the mean of 20 sin(wt) sin(3wt), which is exactly 0 W (different harmonics are orthogonal). A "RMS power" reading of 10 W from a circuit that absorbs 0 W is the whole error, and it needs no energy to appear. Real MEG waveforms (spikes at 87.5 kHz, per the patent) are far from sinusoidal, so a large over-reading is arithmetically possible [my calculation]; whether it happened I cannot say without the raw waveforms.

Effect on my verdict. My earlier text said the artefact explanation was my inference and I could not confirm it. The finding does not confirm it, but it removes the possibility that the cited method is a valid measurement: the method the patent names cannot, as a matter of mathematics, establish COP > 1 on its own. I record this in the weak-form verdict (see table below). I make no claim that Tektronix measured a MEG.

F2. Naudin v2.1 and Flickinger: where the excess appears (claim 5, questions a-c)

Findings [web: http://jnaudin.free.fr/meg/megv21.htm and megnot01.htm, per research note]: - Naudin (status 12-06-00): control board alone 1.75 W; with a 9 W lamp, DC input 3.25 W; he credits 1.5 W of input to the lamp against "a measured OUTPUT = 6.76 Watts" (an apparent ratio of about 4.5); output signals "really measured" by several instruments "but unfortunately measurement artifacts remain possible"; a "conditioned" 100 kohm carbon resistor or a MOV load "is REQUIRED" for the measured output; "in most of cases the 'apparent' power measured seems greater than the heat dissipated by Joule's effect in the RLoad"; core saturation, hysteresis and calorimetry not checked; he is "pessimistic" about the calorimetric result; loop closure "has not yet been done". - Flickinger (independent builder; date not shown): with linear resistive loads "it is a waste of time ... if one expects to see any excess energy"; COP 1.75 to 5 appears only with MOVs (a Panasonic part number is named); he invites anyone to show COP > 1 with a linear load.

Mathematical weighing. 1. A MOV (metal-oxide varistor) has a strongly nonlinear current-voltage curve, roughly i = k v^alpha with alpha large. Voltage and current across it are then not proportional in shape, which is exactly the case where Vrms x Irms (F1) and the true mean of v i diverge. That the "excess" appears for a nonlinear load and disappears for a linear one is what the F1 arithmetic predicts if the excess is a measurement effect. It is also what one would see if the device really delivers surplus only to nonlinear loads, which has no stated mechanism in the patent. The data separate these only if true v(t) i(t) is integrated or the load heat is measured; neither was done. I treat this as consistent with the artefact explanation and not proof. 2. Naudin's "apparent output greater than Joule heat in the load" is itself a conservation test. Energy delivered to a resistive load reappears as heat (plus any radiation). If the heat is less than the claimed power, then the claimed power was not delivered. This is the same conclusion F1 gives, reached by a different route. 3. Naudin attributes the gap to EM radiation (he reports 2.8 mT near the load at 16 kHz and 1250 V/m at 50 cm). Two checks [my calculation]: (i) at 16 kHz the wavelength is about 18.7 km, and the radiation resistance of a small loop of area 0.03 m^2 is of order 1e-16 ohm (R_rad ~ 31,171 (A/lambda^2)^2), so radiated power is negligible; (ii) if 1250 V/m were a propagating wave its power density would be E^2/377 = about 4 kW/m^2, an absurd figure, so that field must be a reactive near field that stores energy and returns it, not power leaving the system. I therefore do not accept radiation as a plausible home for several watts. This is my arithmetic on his figures; I did not check how he measured them. 4. The 1.5 W credited to the lamp against 6.76 W output: the lamp's 9 W rating and 3.25 W total input are Naudin's own numbers and I cannot reconcile "1.5 W of input attributed to the lamp" with the rest without his method. Unknown; I note it as unresolved. 5. Single-source status. Each of Naudin's numbers is documented in one published source (his own page), and Flickinger's in one (Naudin's page). Naudin is an enthusiast with an interest in the MEG succeeding (it is his project), and he reports against his own interest where he writes that artefacts remain possible. Flickinger's statement is likewise a hobbyist's report. Neither is a controlled or neutral test. They are evidence on the question, and I weigh them as above. 6. My earlier text cited a Naudin COP of 1.75 for "MEG v2.0" and the Investigator records Flickinger's 1.75 to 5 for MOV loads. Both figures should be read with F1 and point 1 here, not as independent confirmation of COP > 1.

F3. EPO file (EP 1 446 862 B1) (claim 5, questions a, c, d)

Findings [web: https://patents.google.com/patent/EP1446862B1/en ; https://register.epo.org/application?number=EP01988155, per research notes]: the 2006 examination annex (pp.1-3 read by the Investigator) finds US 4,006,401 (deRivas, 1977) discloses "a motionless electromagnetic generator having the same structure as that claimed", the only difference being that the drive power there comes from a continuously applied alternating source and not from the permanent magnet; inventive step is argued over US 4,883,977 with FR 2,312,135 (Galley, 1975) or US 2,866,943. Granted claim 1 (2013) requires non-saturating switching and feedback of a portion of the output power to the switching circuit. The granted description keeps the extrapolated "3.44" sentence and the demagnetisation paragraph but drops the COP-versus-voltage figure and the sentences "output power was substantially higher than the input power", the THS730 agreement and "all of which were significantly greater than 1" (the Investigator compared page texts; when and why they were removed is in the unread file). Applicants of record were the five inventors; no opposition was filed.

Mathematical weighing. - The EPO's novelty reasoning is about structure and drive, not energy. Its finding that the structure is old and only the self-drive differs means the entire novelty of the European grant is the energy-balance claim of my claim 5: a loop that returns part of the output as drive. In my terms the claim is that closing the loop in a device whose open-loop COP > 1 gives net output. A grant is the office's finding that the claim is new and non-obvious; it is not a finding that COP > 1 was demonstrated, and the Investigator reports nothing in pp.1-3 on industrial applicability. I read nothing on how the EPO handled the energy question; unknown until the 2008 annex and the 2007/2009 replies are read. - The retained "3.44" sentence with the measured-COP plot and "all COP > 1" sentences removed: the retained number is, in the patent's own words, an extrapolation of a fitted curve (my claim 5a). In the European text it now stands without the data it was extrapolated from. That makes it less checkable, not more. The removal is a lead, not an inference of intent: a description may be trimmed for many reasons (clarity objections under Art. 84 were raised). I record it and draw no conclusion. - The self-powering condition can be written down. If open-loop COP = P_out/P_in = c, feeding back a fraction f of the output (f <= 1 including losses in the feedback path) sustains operation when f c >= 1, i.e. c >= 1/f. So a feedback efficiency of 0.5 needs c >= 2 just to run, and the measured c of 1.7-2.6 at the patent's own claimed values sits right at this edge. Because every unit of c above 1 is the claim to be tested, the loop adds no new physics: it is a restatement of c > 1 under a bookkeeping change. Closing the loop would be the decisive test: a running device with the supply removed is a measurement with no ambiguity about instrument method (it either runs or it does not). None is reported anywhere I have been given.

F4. USPTO continuation 10/081,847 and Bearden's 2004 remarks (claims 1, 2, 4, 5)

Findings [web: https://archive.org/details/gov.uspto.patents.application.10081847, per research note; only the first 50,000 of 85,447 characters of the remarks were read, and the Bearden declaration was not]: examiner Shawn Riley rejected all 14 claims under 35 USC 101 (no utility, violation of the first and second laws), 112 and double patenting (2004-01-23); required the applicants to "demonstrate how the magnetic flux stored within the permanent magnet is sufficient to contribute usable electrical energy ... and how the device then generates more power than what is available in the permanent magnet"; the applicants (2004-05-24) asked that the demonstration requirement "be held in abeyance"; the examiner refused (final rejection 2004-07-26); no response; abandoned 2005-05-16 (PlainSite: failure to respond). The examiner is an interested party only as the agency's voice; the applicants' remarks are legal advocacy for a patent.

Mathematical weighing. 1. The examiner's request is my own energy-budget calculation in legal form. It asks the applicants to show that the magnet holds enough energy to pay the claimed excess. My estimate was about 12 to 22 J (claim 5a), against an excess of about 82 W on the patent's extrapolated figures, which would drain the store in about 0.14 s [my calculation, earlier]. The applicants' remarks as summarised by the Investigator answer through the vacuum (Lee-Yang, Prigogine, Whittaker, Aharonov-Bohm), not through a magnet energy budget. The examiner's own windmill argument conflates COP with efficiency (as the Investigator notes), which is a real flaw, but the underlying request (an energy source large enough) is sound. 2. Aharonov-Bohm and E = -dA/dt (the central link the Investigator says the patent lacks). I can check this part directly [my calculation]. The remarks say that B is localised in the core and A is curl-free outside, and that sharp input pulses perturb A, giving "strong E-field energy pulses, by the well-known equation dA/dt = -E". The identity is the standard one (E = -grad(phi) - dA/dt). But where B = 0, curl A = 0, so locally A = grad(chi) and E = -grad(phi + d(chi)/dt), a pure gradient. A pure-gradient field has zero line integral round any closed loop that does not enclose changing flux; so it produces no EMF, and any EMF in a coil wound around the core is the ordinary Faraday value, the rate of change of the enclosed B flux (the same loop integral of A). Nothing in this identity creates energy: the Poynting theorem for the complete system (core, magnet, coils, source) still balances, and the power into a coil is the EMF times the current, an ordinary term already inside a transformer's energy account. The Aharonov-Bohm effect is a quantum-phase effect for charged particles; the quantity the remarks need (extra energy delivered to a classical circuit) does not follow from it. This bears on claim 4: the "open to a hidden source" step stays an assertion, not a derivation. I did not read the remainder of the remarks or the declaration, and there may be a derivation there I have not seen. 3. Heat-pump analogy (COP 3-4, "theoretical maximum about COP = 9.22"). A heat pump's COP above 1 is a true statement because the extra heat comes from a reservoir at temperature T_c; the Carnot limit is COP_max = T_h/(T_h - T_c), which gives 9.22 for example for T_h = 300 K and T_c about 267.5 K [my calculation; I do not know the inventors' temperatures]. The analogy supports "COP > 1 does not by itself violate conservation when there is an external reservoir". It does not locate a reservoir for the MEG. The only identified reservoir in the patent is the magnet, with a store of about 12-22 J (claim 5a), and the vacuum is not shown to be a source here. So this strengthens claim 4's conditional ("closed systems cannot give overunity") but not its antecedent (an open source exists). 4. The feedback example as summarised ("COP 5 open loop; 50% efficient feedback; COP = infinity"): I cannot reconstruct the arithmetic from the Investigator's one-line summary, and I did not read the original. In general COP = infinity is the definition (output / operator-paid input) with the input set to zero; it is not an extra energy source and the same f c >= 1 condition in F3 applies. Unknown beyond that. 5. "Dismissed without testing" or "never properly tested": the USPTO asked for a demonstration and the record shows none, and the applicants did not appeal. That is the strongest documentary evidence in the folder that a demonstration was asked for and not given. Competing explanations the Investigator lists (no unit could show net output; abandoning a redundant continuation was cheap; trade secrecy) are all consistent with the record, and the file says nothing about which is correct. I add one mathematical observation: a demonstration of the self-powered loop is cheap in principle (run, supply removed, watch the load) so the cost argument does not explain the absence of such a demonstration in some other forum; it explains only the lapse of this application. 6. Claims 1 and 2 (theory): the remarks are the fullest statement of how Bearden ties his vacuum/Whittaker theory to the MEG. My claim 1 and 2 verdicts rest on the 1993-94 papers' internal mathematics (units, product-rule error, charge quantisation), and they are unchanged by an application of the same words to the MEG in 2004. I note, per the Whittaker point already in Corrections 6, that the 2004 remarks as summarised call the magnet a "broken-symmetry dipole" absorbing energy from the vacuum; no calculation of the absorbed power is in the portion the Investigator read.

F5. Company, assignment and fee records (claim 5, absence of evidence; no mathematical content)

Findings [web: https://patents.google.com/patent/US6362718B1/en ; arc-sos.state.al.us, per research notes]: US 6,362,718 granted 2002-03-26; assigned by the inventors only in 2006 (nunc pro tunc, reel/frame 017706/0167, corrected 017746/0237) to "Magnetic Energy, Ltd., Alabama"; security agreement to Colmac Holdings Limited, Canada, recorded 2006-06-12 (017759/0537); lapse for the 4-year fee and reinstatement in 2006; 8-year fee paid late in 2010; 12-year fee paid 2013-09-17; expired. Alabama Secretary of State shows no entity named "Magnetic Energy" (searched once on the form; an earlier guessed-URL attempt returned no match, so the search is thin). Business directories list a "Magnetic Energy Ltd" in Huntsville (not opened).

Weighing. These are the historical-record facts bearing on why no later work is public; they do not bear on whether the mathematics is consistent. They are consistent with a small, thinly capitalised enterprise keeping a patent alive for 13 years and borrowing against it. The record does not say whether the loan was repaid or the lender took the patent. I give no verdict from them. I note only the contrast with Bearden's own 2002 letter (F6).

F6. Bearden letter of 15 August 2002 (claim 5; mathematics: the 2.5 kW and 10 kW plans)

Finding [web: https://billstclair.com/www.cheniere.org/correspondence/081502.htm, mirror of cheniere.org, per research note]: Bearden says Magnetic Energy Ltd has "a series of successful little experimental devices that prove the overunity and the concepts", no finished power units, a year of hard research away once funded; "bare bones" funding of $9 million per year for three years; demonstrations given "(with nothing hidden)" only to potential financial partners "whose technical representatives do their own measurements"; planned units of 2.5 kW and 10 kW; "every previous legitimate COP>1.0 system ... has been firmly defeated right at this same 'vertical cliff'" of scale-up funding. He is a co-inventor and spokesman, so these are his statements, and also the sole description of the investor demonstrations I have; no measurement report from the representatives has been found.

Weighing. (1) The patent's device delivers about 22 W (75 V input) [read]; a 2.5 kW unit is two orders of magnitude larger. Scaling a power transformer-like core is not a mathematical difficulty in itself (the same loss and saturation laws apply), so "a year of hard research" and "the vertical cliff" are claims about funding and engineering, not about a scaling law I can check. (2) His statement that investors' own technical staff "do their own measurements" is a claim of independent measurement. If such measurements exist they would be the neutral test my claim 5c found missing, and their absence from the public record is the anomaly the Investigator's competing explanations (commercial failure; secrecy; absorption; no net output) are meant to address. I cannot choose between them. This single source is documented in one letter, with no second source for any investor's measurement; I weigh it as the inventor's account of tests that may have happened, and I do not count it as a test on record. (3) The P.S. says that a charged capacitor or electret laid across a magnet, E perpendicular to H, is a "proven free energy generator" by Poynting's E x H. The Poynting vector E x H is a real energy flux, but in static fields with E and H perpendicular the vector is nonzero and its divergence is zero; the field circulates and delivers nothing to any load [my calculation: for static fields div(E x H) = H . (curl E) - E . (curl H) = 0, with no current source]. This is the textbook "hidden momentum / Feynman's disk" setup, not a power source. It bears on claim 4 and I judge it mathematically empty as a generator; the physicist may say more.

F7. Related work: flux-switching "static generators" (claim 5)

Finding [web: EPO annex; http://jnaudin.free.fr/meg/meg.htm, per research notes]: Starr 1957 (US 2,802,170), Toffolo 1963 (US 3,087,108), Garron 1968 (US 3,368,141), Galley 1975 (FR 2,312,135), deRivas 1977 (US 4,006,401), Richardson 1978 (US 4,077,001), Kunel (DE 3024814), Volkrodt (DE 3501076), Bramanti (US 4,675,615), Boday (CA 2172240), Keiichiro (US 5,926,083), Flynn (WO 00/07285, Parallel Path Technology / Magnetic Revolutions). Some are described by builders as claiming COP > 1 and some are plain permanent-magnet flux-switching generators; I did not read these patents, and I do not know which claim COP > 1. Mathematically the family is one idea: a magnetic circuit in which the flux seen by an output coil is switched between paths. The linear flux-linkage law e = -N dPhi/dt applies to all of them; energy delivered to the coils is bounded by the work done to switch the flux plus losses, as in my claim 5a energy balance (no new circuit element removes the balance). I keep the list in the record as related work; the EPO's own finding (F3) is that the MEG structure is not novel among them.

F8. Bearden's authorship of Foundations of Physics Letters 14(1) 2001 (a [memory, medium] item now [web])

Finding [web: http://jnaudin.free.fr/meg/meg.htm, per research note]: the author list on Naudin's index includes T. E. Bearden among 15 AIAS (Alpha Foundation's Institute for Advanced Study) authors. That retags the arbiter's Check 10. No mathematical weighing needed.

Verdict changes after the Investigator findings

Only claim 5 changes, and only in its reasons and weak form; no verdict label moves.

What would change a verdict: raw v(t) and i(t) from a MEG run, integrated to true mean power, with load heat (calorimetry); or a closed-loop run with the supply removed; or the investor representatives' measurement reports; or the unread EPO replies if they contain a derivation or data. Any of these would let me move the weak form to "Tested, not supported" or, if net output were shown, to something I would need to recompute.

Web checks for this section: none newly made by me (the 6-call budget was spent in the first pass); the Investigator's [web] findings are reported as the Investigator's reading. My own claims here are my calculation (radiation resistance and near-field, Cauchy-Schwarz bound, Poynting divergence, curl-free A), none a retag of a [memory] fact.