Dollard-Eric: review by a PhD mathematician (Step 2)

Reviewer role: checks derivations, notation, units and hidden assumptions. Source tags: [read] from the documents, [web] with URL, [memory] with confidence.

Source gap (repeated from Step 1). The two scans of the "Free-Energy Research of Eric Dollard" compilation are image-only; I read the key pages as images (Transmission of Electricity Part I file pp.13-15, Part II file pp.17-19, interview pp.8 and 11). EPDPaper.pdf (telluric waves) has only a very poor OCR layer; I read the OCR text of the key passages (antenna, complex wave, applications, reciprocity) but not the drawings. It contains no numbers, no measurements and no test records that I could find. Dollard's books (Theory of Wireless Power, Symbolic Representation of the Generalized Electric Wave) are not in the folder. Mathematical verdicts below on the telluric-wave system therefore rest on a verbal description only, not on equations.

Corrections to Step 1

1. The Free-Energy facts entry lists as a key claim "The velocity of light changes continually, changing the capacity of all capacitors." I did not locate that statement in the pages I read, so I cannot confirm it. Not used below.
2. claims.md item 1 says conductors "reflect the field". More exactly, Dollard writes (Part I, p.6, [read: page image]) that the conducting material "does not so much conduct as it does repel or reflect magnetism, or electro-magnetic energy in general", arguing from the Meissner expulsion of field by superconductors. That is a generalisation of a true fact (see claim 1).
3. claims.md item 4: the "hysteresis of the aether" idea is Dollard quoting C.P. Steinmetz quoting a discussion of Kennelly (AIEE 1891) [read: page image, JBR May-Jun 1988 p.12]. Step 1 had this right but did not note that the quoted passage itself describes retardation of the field, i.e. radiation loss, not a separate loss (see claim 4). This is a substantive point, not a correction of fact.
4. The EPDPaper entry is typed "patent" and year "unknown". It has no patent number or filing data in what I read; it is a typescript in patent-claim style. Treat as an unpublished design proposal. [read: OCR]
5. Part I and Part II of "Transmission of Electricity" disagree with each other on the role of C = 1/c^2; Step 1 did not notice this (see claims 1 and 2).

Claim 1. Energy is carried by the space between conductors, not electrons in the wires; conductors reflect the field and insulators are the true conductors.

a. Mathematics. Split the claim in two. (i) "Energy flows in the field outside the wire": this is the Poynting vector S = E x H (E electric field, H magnetic field, x is the vector cross product). For a wire carrying current, the textbook result is that S points into the wire's side and equals the Joule heating, and energy reaches a load through the surrounding fields. [web] Feynman Lectures II ch.27, https://www.feynmanlectures.caltech.edu/II_27.html. Dollard's picture is correct here, but he presents it as a refutation of the "electron-flow" picture; in Maxwell theory both are true (electrons carry charge and are pushed by the field; the field carries the energy). (ii) "Insulators are the true conductors": this is a definitional inversion; nothing is derived. The formulas he gives are unit-fragile. Part I defines C = 1/c^2 ("the capacitance of the electric circuit", c the speed of light) and L = l0^2, where l0^2 = l1 l2 is a length times a spacing (cm^2) [read: page image p.5]. Then: - Period: t0^2 = -LC = -(l0/c)^2 has dimension s^2 (consistent up to the sign, which comes from his j convention). OK. - Impedance: Z0^2 = -L/C = -l0^2 c^2 has dimension cm^4/s^2, so Z0 is cm^2/s. In the emu (electromagnetic CGS) system a resistance is cm/s, so the dimension is off by one length. The formulas are only coherent if C and L are per-unit-length coefficients with a hidden length factor, which the text does not state. Worked check: in emu, the vacuum permittivity is 1/c^2 (s^2/cm^2) and the permeability is 1, so "C = 1/c^2, L = l^2" is the textbook vacuum permittivity and permeability relabelled as circuit elements, with a geometric factor assumed. - Internal inconsistency: Part I says the inverse square of the light velocity "is called the capacitance". Part II (p.10, [read: page image]) says "the notion has occurred that these dimensions establish the propagation velocity of electric transmission ... This concept may have been the most significant obstacle to the understanding of electric transmission." So Part II rejects what Part I's definition is used for. Both cannot be a foundation.

Reflection by conductors: true and quantified. Skin depth delta = sqrt(2/(omega mu sigma)); for copper (sigma = 5.8e7 S/m) at 1 MHz, delta = sqrt(2/(2 pi 1e6 x 4 pi 1e-7 x 5.8e7)) = 6.6e-5 m (66 micrometres), so a few skin depths of copper reflect almost all of an incident wave. Dollard's "conductors reflect" is the standard high-conductivity limit of Maxwell theory, not a refutation of it. [memory: high; computation done here]

b. Experiments. Poynting energy flow is consistent with every power-transmission measurement; transmission lines carry TEM waves in the dielectric. Insulators do not carry steady currents: resistivity of glass is ~1e10 to 1e14 ohm m against 1.7e-8 for copper (a 1e18 to 1e22 ratio) [memory: high]. "True conductor" holds only in the sense that the energy resides in the dielectric region.

c. Tested. No test of the claim as distinct from Maxwell theory exists; it makes no different prediction in the part that is correct and a false prediction (conduction in insulators) in the part that is not.

d. If true. One would have to drop Ohm's law and the carrier picture of solid-state physics (Hall effect, drift velocity measurements).

Verdict: split. The "energy flows in the space around the wires" part: Established (as standard Poynting theory, not a departure). The "electrons do not carry the current / insulators are the true conductors" part: Internally inconsistent (Part I vs Part II on C = 1/c^2, plus unit mismatch in Z0) and Not testable as stated as a distinct theory.

Claim 2. The dielectric part of induction propagates independently, faster than light (pi/2 times c), with large "dielectric currents" in free space without electrons.

a. Mathematics. In free space the Maxwell curl equations, curl E = -dB/dt and curl B = mu0 eps0 dE/dt, give wave equations for both E and B with speed c = 1/sqrt(mu0 eps0). Neither field can propagate by itself: each is the source of the other. In the near field (distances much less than a wavelength) one field dominates and quasi-static calculations are valid, which is why a ferrite loop antenna responds to the magnetic part and a whip to the electric part, as Dollard says (p.12 of JBR 1988, [read: page image]); that observation is correct. It does not imply separate propagation speeds. The instantaneous Coulomb force (Coulomb gauge) is a calculational convention; physical fields still obey retardation. Dollard's own quoted Steinmetz passage in Part II relies on a finite propagation time, which contradicts instantaneous or superluminal dielectric propagation within the same article.

Origin of pi/2: Tesla's 1899 Colorado Springs figure of 471,240 km/s is exactly (pi/2) x 299,792 km/s. [web] https://teslauniverse.com/nikola-tesla/articles/faster-light. The ratio equals half the earth's circumference over its diameter: a signal travelling at c along a diameter and a signal along the surface half-circle reach the antipode at the same time at speed (pi/2)c over the surface path. I.e. pi/2 is geometry (arc divided by chord) combined with a resonance measurement, not a medium property. Dollard's statement that pi/2 "will appear in wave calculations" is not derived anywhere I read. [read: page image] Wheatstone's 1834 value of about 288,000 miles/s (about 1.5 c) came from a rotating-mirror measurement of spark timing in wire, widely regarded as a measurement error. [web] https://harmoniaphilosophica.com/2013/04/20/electricity-faster-than-light/ (search result summary; I did not retrieve the primary paper). Dollard cites Wheatstone as claiming this "dielectric induction" speed, so the cited experiment is a flawed timing result, not evidence for a new mode.

"Dielectric current of many amperes through free space": displacement current eps0 dE/dt is real and is in Maxwell's equations; it is the term that makes capacitors pass AC. So "current without electrons" is standard. What would be new is its being independent of the magnetic field it generates; Ampere-Maxwell law forbids that.

b. Experiments. No measured signal has exceeded c in vacuum; phase velocities above c occur (waveguides, evanescent regions) but carry no energy or information faster than c. Radio-wave speed measurements agree with c to parts per 1e9. [memory: high]

c. Tested. Tesla's own earth-resonance timing; Wheatstone's measurement. No independent replication of superluminal electrical signalling is on record in the material provided. Not asserted beyond that.

d. If true. Special relativity's invariant speed for signals, and Maxwell's equations, would need a second propagating field with its own constants.

Verdict: split. Near-field E-versus-B antennas and displacement current: Established. Independent superluminal (pi/2 c) propagation: Refuted by calculation for the pi/2 origin (geometry, shown above) and Internally inconsistent with Dollard's own Part I/Part II treatment of C = 1/c^2; as an experimental claim of superluminal signalling it is Never properly tested in the sources here (only historical timing measurements, one known erroneous).

Claim 3. Capacitance is dielectric energy stored inward, dual to inductance's outward storage; sudden release can exceed light speed and yield infinite current.

a. Mathematics. The qualitative duality is real: energy U = (1/2) C V^2 = (1/2) eps0 E^2 per volume in the electric field, and U = (1/2) L I^2 = B^2/(2 mu0) per volume in the magnetic field, with E and B swapping roles under the duality of Maxwell's equations. That a capacitor is a field store rather than an electron "bucket" is correct (plate charges are the field's boundary condition). Problems: - "The smaller the space, the more energy": at fixed voltage, C = eps A/d rises as plate gap d falls, so U = (1/2) C V^2 rises. At fixed charge Q, U = Q^2 d/(2 eps A) falls. Dollard's statement is true only under the fixed-voltage assumption and is stated as unconditional ("exact opposite of magnetism"). A hidden assumption. "Series versus parallel" is unexplained. - Infinite current: his argument shorts a charged ideal capacitor (zero resistance) and says current "jumps to infinity". The circuit equation L q'' + R q' + q/C = 0 shows the current is bounded: with L > 0, I_peak = V sqrt(C/L) at R = 0. Worked example: C = 1 microfarad charged to V = 1000 V (stored energy (1/2)(1e-6)(1e6) = 0.5 J) shorted through a loop of L = 1 microhenry gives I_peak = 1000 x sqrt(1e-6/1e-6) = 1000 A. If L is made smaller the current is larger but never infinite; a real short has L > 0 and R > 0. His own text concedes this one paragraph later ("The inertia of the magnetic field limits the rise of current to a realistic value") [read]. The "infinity" is a limit of an unphysical idealisation (L = 0), as for the inductor case where the voltage L dI/dt is infinite for an instantaneous break (real: arcing, stray capacitance). - "Velocity that may exceed light" for the field: no equation is given; the "velocity of the field" is a metaphor for dB/dt (flux rate), and "EMF equals field velocity" is a loose paraphrase of Faraday's law EMF = -d(flux)/dt. - Free-space capacitance "is the sum of mutual capacities to all conductors of the universe": field lines end on charges, not only on conductors, and an isolated sphere of radius R has a finite capacitance C = 4 pi eps0 R, with the field lines ending at infinity (the usual convention). For the earth R = 6.37e6 m: C = 4 pi (8.854e-12)(6.37e6) = 7.1e-4 F (about 709 microfarads). [memory: high; computation done here] So an isolated conductor stores energy without a second conductor; Dollard's "no free space capacity" is false as a statement about C but a fair intuition about Maxwell capacitance-coefficient matrices. - Free inductance of an isolated filament being infinite (Steinmetz): correct mathematically for an infinite straight wire (external-field inductance per length (mu0/2 pi) ln(R_out/a) diverges as R_out grows). Fine. - "Supercapacitance" converting energy into matter: no equation or threshold given; mass-energy conversion requires pair-production energies of 1.022 MeV per electron-positron pair, which is not approachable by capacitor impedance. Not a mathematical claim.

b. Experiments. RLC discharge currents of capacitor banks are routinely measured and match I = V sqrt(C/L) and ringing; no superluminal current is observed. [memory: high]

c. Tested. Standard capacitor-discharge physics is tested daily; Dollard's extra claims (exceeding c, infinity, matter) are not tested by anyone in the material.

d. If true. Circuit theory with L > 0 would fail; relativistic causality would fail.

Verdict: split. Capacitance as a field store, LC resonance at omega = 1/sqrt(LC), Z0 = sqrt(L/C): Established. Infinite current and superluminal collapse: Refuted by calculation (finite I_peak = V sqrt(C/L), shown). "Energy into matter / supercapacitance": Not testable as stated.

Claim 4. Energy a tower does not return to the circuit or dissipate as heat is not radiated but lost to "hysteresis of the aether".

a. Mathematics. The quoted Steinmetz passage (via Kennelly) argues that if field propagation were instantaneous no energy would be dissipated; because propagation takes a finite time, "the retardation of the magnetic disturbance behind the magneto-motive force" produces a loss, which Steinmetz calls hysteresis of the aether. [read: page image JBR 1988 p.12] That retardation loss is exactly what is now called radiation resistance (energy permanently carried off by outgoing waves). Dollard presents the quote as an alternative to radiation ("Many believe this lost energy is radiated ... there exists evidence that it may not be the correct interpretation"). Read in context the quote supports radiation, so the argument contradicts its own citation. The claim "only a small fraction is returned and heat does not account for the rest" is a statement about radiation resistance: for a quarter-wave tower the radiation resistance (about 36 ohms) exceeds the ground and wire loss resistance, so most input power is radiated. [memory: high that a quarter-wave monopole has ~36 ohm radiation resistance; Dollard's "large spacing, little heat" matches this.]

A quantitative check: an ideal quarter-wave monopole over perfect ground has gain G = 3.28 (5.15 dBi) [memory: medium-high], so the field at distance r with P radiated is E_rms = sqrt(30 P G)/r. For P = 1 kW, r = 1 km: E = sqrt(30 x 1000 x 3.28)/1000 = 0.314 V/m, i.e. about 313 mV/m (the value commonly quoted as the FCC "unattenuated field at 1 km for 1 kW"). [memory: medium; the computation is mine]. Field strengths of that order are routinely measured for AM stations at ground level, so the missing energy is observed in the radiated field. [memory: high that AM field-strength surveys exist; not sourced to a specific paper]

b. Experiments. AM radiated field strengths are measured and used for coverage maps; ground-wave and sky-wave propagation are well characterised. [memory: high]

c. Tested. Not tested as a rival to radiation. Dollard promises an exploration "in part III" which is not in the folder. Nothing in the sources shows a measured energy deficit.

d. If true. Energy conservation would need a new aether dissipation channel that nobody can detect; antenna theory would be unnecessary.

Verdict: Refuted by calculation (radiation resistance accounts for the "missing" energy and the measured far field is of the expected size; his own Steinmetz quote describes radiation by another name). The idea that the loss is not radiation: Internally inconsistent with its own citation.

Claim 5. Telluric (earth) electric waves with a velocity and a counter-velocity; a loaded aerial-ground antenna, filter network and parametric amplifier receive them, reject EM interference, and give advance seismic warning.

a. Mathematics. What I could read [read: OCR text of EPDPaper.pdf, with errors]: the antenna carries "a complex wave ... one quantity in the dimension of velocity (the real part) and another quantity in the dimension of counter-velocity (the imaginary part) ... analogous to impedance as a complex quantity consisting of resistance, the real part, and reactance, the imaginary part." No equation, no definition of counter-velocity, no units, no dispersion relation, no numbers anywhere in the pages I read. In standard theory a propagation constant gamma = alpha + j beta (alpha attenuation per metre, beta phase per metre) is complex, and a phase velocity omega/beta can be read off from it. If "counter-velocity" were the attenuation analogue it would be a well-defined, mainstream idea (damped waves in a lossy earth); but Dollard calls it a velocity of a second, counter-propagating wave and says the antenna is "non electro-magnetic". That cannot be assessed because the mathematics is not given.

Reciprocity: the Lorentz reciprocity theorem holds for linear media with symmetric permittivity, permeability and conductivity. It fails for non-reciprocal media (magnetised ferrites, magnetised plasma such as the ionosphere) and for active elements. Dollard's system contains parametric amplifiers and negative-resistance elements, which are active, so a non-reciprocal system is possible; but his assertion is about the earth-antenna transfer of energy, with no derivation. The "boundary condition or circuit law is not applicable" statement (item 2, the same page) is not a mathematical statement I can check.

The rejection aerial: cancelling EM interference by subtracting a signal from a second aerial is ordinary common-mode / noise-cancellation practice. That part is plausible.

b. Experiments. Earth-current (telluric) signals are measured in geophysics (magnetotellurics) at ELF/ULF; their propagation is electromagnetic diffusion in conducting rock, not a separate non-EM wave. [memory: high]. Earthquake precursors: Fraser-Smith et al. reported 0.01 Hz ULF magnetic noise before the 1989 Loma Prieta earthquake at Corralitos [web] https://ui.adsabs.harvard.edu/abs/2007AGUFM.S41D..03C/abstract (and summary of the 2009 critique at https://www.researchgate.net/publication/222682123_On_the_reported_magnetic_precursor_of_the_1989_Loma_Prieta_earthquake); Thomas, Love and Johnston (2009) re-analysed 21 months of the same sensor's data and concluded the anomaly was an instrument artefact; Campbell (2009) attributed it to natural geomagnetic disturbance [web] https://agupubs.onlinelibrary.wiley.com/doi/10.1029/2008JA013932. The original authors dispute this. So the one best-known precursor claim is contested and not accepted as a working warning method. I found no source for any test of Dollard's system, and no measurement record in the folder.

c. Tested. Not by anyone on record in the provided material. No construction, data or replication of the telluric-wave apparatus is documented. [read: absence in the folder; I did not search the literature beyond the above for Dollard-specific tests.]

d. If true. Maxwell's equations in conducting media would need a second wave type; a solid-earth non-EM electrical wave would need new physics; reliable earthquake prediction would overturn the present consensus that short-term prediction is not currently possible [memory: high].

Verdict: Not testable as stated (the "velocity/counter-velocity" wave is not defined mathematically; the document gives no numbers) for the physical wave claim; Never properly tested for the engineering claim of seismic warning (no test on record in the sources; general ULF precursors are contested, see above). Whether it was dismissed without testing is not documented in the materials.

Claim 6. Dielectric fields are the "productive" side of electricity, a source of free energy and Reich-style orgone effects; magnetism only consumes.

a. Mathematics. In the interview, Dollard says "Free energy will come from the dielectric field where energy grows rather than decays" and "the magnetic wave is the one that is consumptive and retarded, whereas the dielectric wave is productive and advanced" [read: page image, JBR Mar-Apr 1987 p.28]. In Maxwell theory the electric and magnetic energy densities, (1/2) eps0 E^2 and B^2/(2 mu0), are equally positive and, in a lossless LC system, trade energy back and forth with total energy constant. "Growth rather than decay" is not an equation but an assertion; a growing-energy solution of the LC circuit requires negative resistance (an energy source) and the sum of energies obeys dW/dt = -I^2 R, which is non-positive for passive circuits. The "advanced versus retarded" phrase echoes phasor lead/lag (current leads voltage in a capacitor, lags in an inductor by 90 degrees), but a phase lead does not store or create energy: the instantaneous power V I averages to zero for pure reactances. This is the likely source of the idea; it is a misreading of phase as energy gain.

b. Experiments. No closed system has produced more energy than supplied in a controlled measurement; Reich's orgone claims have no accepted controlled confirmation [memory: high].

c. Tested. Dollard says in the same interview that every free-energy claimant he has examined is a fraud except one unnamed person [read: page image p.28]; no device, data or test is given for the exception. So, in his own words, nothing public was tested positive.

d. If true. Conservation of energy (first law of thermodynamics; Noether's theorem for time translation) would need an exception.

Verdict: Refuted by calculation for the form that a passive dielectric field supplies growing energy (energy balance dW/dt = -I^2 R <= 0 shown above); the "orgone" link: Not testable as stated.

Summary

• Claim 1: split. "Energy flows in the field" is Established (Poynting); insulators-as-conductors and electrons-not-involved are Internally inconsistent / Not testable as stated.
• Claim 2: split. Near-field E/B antennas and displacement current Established; independent pi/2 c propagation Refuted by calculation (pi/2 is arc/chord geometry) and internally inconsistent with Part I; no test on record for superluminal signalling.
• Claim 3: split. Capacitor as a field store and LC resonance Established; infinite current and superluminal collapse Refuted by calculation (I_peak = V sqrt(C/L)); supercapacitance Not testable as stated.
• Claim 4: Refuted by calculation / Internally inconsistent (own citation describes radiation).
• Claim 5: Not testable as stated (wave) and Never properly tested (seismic warning); no mathematics provided to check.
• Claim 6: Refuted by calculation (energy balance); orgone Not testable as stated.

Medium-confidence [memory] items not web-verified: 36-ohm radiation resistance and 3.28 gain of a quarter-wave monopole; the 313 mV/m FCC reference figure; Earth capacitance (computed from the formula, so independent of memory). The Wheatstone and Tesla pi/2 facts are web-sourced from secondary sites only (primary papers not retrieved).