Hopper-WJ (William J. Hooper): Mathematician review (Step 2)

Reviewer role: PhD mathematician. I check derivations, notation, units and hidden assumptions. Physics verdicts belong chiefly to the physicist, but verdicts below are on the Step 2 scale because the arithmetic decides several of them.

Source tags: [read] from a document (page given where known), [web] from an Internet source with URL, [memory] recalled with confidence. "My calculation" means I computed it with numpy in .venv. Hooper's name is spelled "Hooper" in his documents; the folder says "Hopper".

No source gap for the two folder documents. Note that neither the patent nor the essay contains measured data. I therefore read the Investigator's downloaded copies of "Similarities of the Motional Electric & Gravitational Forces" (files/hooper-similarities.pdf) and the New Horizons manuscript (files/hooper-NewHorizons.pdf, chapter 3), plus Gibson, Millis and Williamson (NASA TM-106963) and Klicker. Where a conclusion rests on a document I did not open fully, I say so.

0. Plain-language setup (for the reader)

• Motional electric field. If a conductor moves with velocity v through a magnetic field B (B is the magnetic flux density, tesla), each charge feels a force per unit charge v x B (the Lorentz force, per charge). Textbook name: motional EMF (EMF = electromotive force). Griffiths ch. 7 [memory, high]. Hooper writes it E = B x V, with the cross product reversed in sign convention (V x B vs B x V); his sign handling is loose, but only magnitudes matter below.
• Hooper's twist. He says the magnetic field loops around a current-carrying wire physically travel along the wire at the electron drift velocity V (drift velocity = the slow average speed of conduction electrons, v = I/(n e A), where I is current, n is conduction-electron density, e the electron charge, A the wire cross-section). A "moving" field would, he says, make an electric field B x V in the empty space around the wire.
• The generator. Over 4000 wires in series, half carrying current up and half down. Outside, the two magnetic fields cancel, but both B and V flip sign between the two halves, so B x V keeps its sign and the two B x V fields add (E = 2 B1 x V1). I checked this sign algebra: it is correct as algebra [read: Similarities p.3].

1. Corrections to Step 1

1. The "immune to shielding" experiments are described, and they are not WWII aircraft work. claims.md (claim 2) says the experiments behind the immunity claim "(first from WWII aircraft work) are not described in either document". That is true of the two folder documents. But the Investigator's file of New Horizons chapter 3 ("Experimental Confirmations by Electrostatic Shielding") describes three experiments in detail: the "Ironing Board" (credited to Egbert Jones and W. J. Hooper, Physics 401 project, Principia College, 26 March 1954), the "Trapeze" and the "Aluminum Box" [read: New Horizons ch. 3, PDF text, about printed pp. 26-31]. The Investigator's note that the manuscript "holds the WWII aircraft drift-meter shielding experiments" should be corrected: the WWII drift-meter project is the story of how Hooper arrived at the question (printed p.1-2 of New Horizons ch. 1 and the essay p.1-2); the quantitative shielding table I found is dated 1954. Whether other, WWII-era, shielding measurements exist I could not find.
2. All three chapter-3 experiments are closed-circuit motional-EMF tests. Standard theory predicts their outcome (details under claim 2). Step 1 treated "immunity to shielding" as one claim; it is two (see claim 2).
3. Table 3.1 (Ironing Board) has small arithmetic slips (my recomputation of v, and of 100 x B x l x v with B = 0.55e-4 Wb/m2, l = 0.30 m, n = 100): trial 8 lists v = 0.590 cm/s but 20.0/29.0 = 0.690 (the printed 11.4 microvolt theory value matches 0.690); trial 6 lists theory 11.6 microvolts where 0.665 cm/s gives 10.97; trial 7 lists 10.9 where 0.680 gives 11.2. The printed theory column sums to 204.5 but the numbers listed sum to 205.0, so the mean theory value is 10.79, not 10.76, and the observed-versus-theory gap is about 1.4%, not "1.11%". These do not change the conclusion (agreement within a few percent with ordinary flux cutting) [read: New Horizons ch. 3 table; my calculation].
4. Similarities paper: notation and typography errors the Investigator did not list. - Eq. (3) is printed as V = [2 pi P.D. / (mu0 n I)] x (ln r2 - ln r1). The ln factor must divide, not multiply. Integrating Eq. (2), P.D. = (mu0 n I V / 2 pi) ln(r2/r1), gives V = 2 pi P.D. / (mu0 n I ln(r2/r1)). With his inputs (n = 4020, I = 10 A, r1 = 0.1 m, r2 = 0.1035 m, P.D. = 5e-6 V) I get 1.81e-2 m/s (his 1.78e-2 m/s is consistent); the formula as printed gives 2.1e-5 m/s. - "j = 0.04172 amp/cm2" cannot be a current density. The next line uses 239.69 = 10/0.04172 as the numerator, so 0.04172 cm2 is the wire cross-section (about AWG 11, 4.17 mm2) and 239.69 A/cm2 is the current density. Notation slip. - "d = 9.96 g/cm3" should be 8.96 (copper), and the denominator prints 8.96e22 where the n he used is 8.49e22 cm-3 (d N / M = 8.96 x 6.02252e23 / 63.546). My recomputation: 239.69/(8.49e22 x 1.6021e-19) = 1.762e-2 cm/s, which matches his printed result [read: Similarities pp.3-5; my calculation].
5. Gibson's "parabola" column. The dry-ice table (Gibson 1983, Fig. 6) has a column headed "Parabola @ 30 amps" holding 1.2, 5.9, 18.8, 45.8, 95. Those are 95 x (I/30)^4, not 95 x (I/30)^2 (which would be 10.6, 23.8, 42, 66, 95). A fit of the five dry-ice readings 1.5, 6, 17, 45, 95 microvolts at 10-30 A gives exponent 3.78 (my calculation). The Investigator's I^4 is right; the column label is Gibson's mislabel [read: Gibson, rexresearch-gibson.htm].
6. Additions to the Investigator's leads (the Investigator did not open Klicker's body). Klicker's 1986 thesis (Tesla Book Co. printing) says Hooper's shielding experiments "have been duplicated and verified at Montana State University", citing a student report by T. Lamb, "Unique Properties of Motional Electric Fields", EE 470 Special Topics, Montana State University, 1985, work "preformed by J. C. Stover and K. A. Klicker". It also says a project to measure the motional field around an electron beam with a shielded capacitor and electrometer "was initiated" but "limitations in size of the electron beam ... and the sensitivity of equipment (specifically an electrometer) prevented the completion" [read: Klicker, files/archive-Klicker..._djvu.txt, pp.10, 52, 54]. So there is one reported (student-level, unpublished, source not seen) duplication of the shielding experiments and one attempted but unfinished test of the moving-flux hypothesis. Klicker states he "has not been able to disprove Hooper's claims" and shows that the moving-flux hypothesis gives a correction equal in size to the second-order (v squared over c squared) term in special relativity, which agrees with Moon and Spencer's version of Ampere's law "when the magnetic flux drift velocity is assumed to be one half the charge carrier drift velocity". That bears directly on claim 1 (below). Klicker is a sympathetic reader writing a thesis, not an independent test.
7. Fine points confirmed. I checked the patent's corrected drift speed: J/(n e) = 480 A/cm2 / (8.49e22 cm-3 x 1.602e-19 C) = 3.53e-2 cm/s, so the Certificate of Correction value 3.6e-2 is right and the printed 3.6e12 cm/s (about 100 times the speed of light, 3e10 cm/s) was a misprint [my calculation; read: patent p.3, p.6]. "10^10" is really 1e8/3.6e-2 = 2.8e9.

2. Claim-by-claim review

Claim 1. Magnetic flux around a current moves along the conductor with the electron drift velocity V, so a current produces a motional electric field E = B x V.

a. Internally consistent? As algebra, yes. As physics, it has a hidden assumption: that "flux lines" are objects that can be assigned a velocity. For a steady current in a straight wire, B(r) depends only on distance from the wire and does not change with time or position along the wire. A pattern that is unchanged by shifting along the wire has no definable speed: the B field at every fixed point is constant, so dB/dt = 0 and Faraday's law (curl E = -dB/dt) gives no induced E. Hooper's "V" is therefore not a measurable property of the field; it is assigned by postulate ("my experimental discovery that this magnetic flux actually moves", patent p.3, with no experiment described there). It also matters that E = B x V, applied to a single wire in the lab frame, predicts a nonzero field outside a wire that is electrically neutral; Gauss's law and charge neutrality give E = 0 there (apart from surface charges from the resistive voltage drop). The equation Hooper needs is also internally awkward in Eq. (1) of the Similarities paper: B x V per set of wires is treated as a field B1 of "2010 conductors" concentrated on the axis, which is fine outside the bundle by Ampere's law, but the patent never says which B (one set, or net) enters P.D. = B V l (patent Eq. 2). - Size check that Klicker's result supports: a moving-flux term for a single moving charge equals the second-order relativistic correction, of relative size (v/c)^2. With v = 1.8e-4 m/s that is about 4e-25 [my calculation from Klicker's stated equivalence]. That is the mainstream's own prediction for a "motional" effect of drift motion, and it is far below any measurement. - Unipolar induction (FIG. 11 of US 3,656,013, Cravens' "no critical experiment ... whether a magnetic flux rotates with its source"): for an axially symmetric magnet, rotating it changes nothing about B at any point, so "does the flux rotate?" has no operational difference in the symmetric case; the standard result is that the EMF depends on the motion of the conductor relative to the lab, whichever way one pictures the lines [memory, high]. So I would call that sub-question "not testable as stated" in the symmetric geometry, and Hooper's yoke experiment as an interpretation of a measurement that standard theory already explains.

b. Consistent with experiment? The textbook motional EMF E = v x B for conductors cut by flux: yes, many experiments. Moving flux from steady currents causing external E: no experiment in the record that discriminates. The only test I know of is Hooper's own (claim 2).

c. Tested? Hooper (own capacitor measurements, with an unnamed "disinterested expert" who confirmed the electrometer readings with an oscilloscope, per Hooper); Gibson (a participant, later company president and publisher, so an interested party); Millis and Williamson, NASA Lewis 1995 (surface-charge readings "too erratic to reach definite conclusions"); Klicker (analytical thesis; electron-beam experiment not finished). No independent replication of the claim-specific observable.

d. If true: Maxwell's equations plus Gauss's law would need an additional field source: a steady neutral current would have an external E of order v B, not v B (v/c)^2 times anything. Klicker notes that the requirement conflicts with classical theory but matches special relativity at second order; so the mainstream already has the (tiny) term.

Verdict, split. - Weak form (motional EMF E = v x B when a conductor moves through a field): Established. - Strong form (flux of a steady current travels with the electrons and creates a measurable, unshieldable external E): Never properly tested (no neutral replication of the capacitor measurement; NASA's attempt inconclusive). It sits at odds with Maxwell plus neutrality by a calculation (zero predicted external field apart from surface charges), and the sub-question "does flux move/rotate" is Not testable as stated for symmetric geometry.

Claim 2. A non-inductive winding has no net magnetic field, yet the B x V fields add, giving an isolated field that charges an external capacitor and is "immune to shielding", including a grounded steel shield.

a. Mathematically consistent? - Sign algebra: correct (B2 x V2 = (-B1) x (-V1) = B1 x V1). - Own data against own model. Hooper's model gives P.D. = (mu0 n I V / 2 pi) ln(r2/r1), where V = I/(n_e e A) is the drift speed, so P.D. is proportional to I squared (a parabola) for a constant temperature. He reports a parabola for DC, which fits; but for AC he reports a straight line [read: Similarities p.5, Fig. 3]. With AC, B and V both follow sin(wt), so B x V keeps one sign and averages to a value proportional to I_rms squared, again a parabola. A straight line contradicts his own model. A straight line is what coupling to the current would give (stray magnetic or capacitive pickup), so the AC result looks to me like a signal of that kind, though I cannot prove it. - Dry ice: Gibson's readings fit I to the fourth power, not I squared (exponent 3.78, above). Gibson reads this as a change in conduction-electron count; but n in a metal is fixed by the number of atoms (about 8.5e22 cm-3 in copper, set by one conduction electron per atom, which Hall-effect measurements confirm [memory, high]); drift speed is J/(n e) and does not depend on temperature or resistance at fixed current density. Millis and Williamson state the same [read: NASA TM-106963, drift-velocity equation discussion]. So the I^4 law is not predicted by the theory. - Magnitude. Hooper's own number: his classical drift speed is 1.762e-2 cm/s and the speed he extracts from 5 microvolts is 1.78e-2 m/s, which is about 101 times larger. He says "a decrease of 100-fold in n, as indicated by Fermi-Dirac statistics" would remove the gap, and elsewhere that his values are "in good agreement with accepted values" [read: Similarities pp.5-6]. Both statements fail: Fermi-Dirac statistics determine the Fermi energy from n, they do not reduce n by 100; and a factor of 100 is not "good agreement". Put the other way, with the accepted n his model predicts P.D. = 5e-6/101 = 4.9e-8 V (50 nanovolts) for 10 A, and he reports 5 microvolts. The data therefore disagree with his model by two orders of magnitude, unless n is declared adjustable. One measurement cannot give both V and n, so the "measure n" proposal is circular as presented. - Field implied. E = 2 B1 V (B1 = 402 gauss equivalent at r = 0.1 m for 2010 conductors x 10 A) = 1.43e-3 V/m. He measured a maximum stray B of about half a gauss, about 1/800 of one set's field, so the cancellation is good [read: Similarities p.6; my calculation]. - Field type. The implied field E(r) = 1.43e-4/r (V/m, r in metres) is radial, curl-free and steady. A steady curl-free field is E = -grad(phi) with phi proportional to ln r, which is exactly the field of a line charge. Gauss's law then needs a charge per unit length of n I V / c^2 = 8e-15 C/m on the axis (my calculation, using Hooper's V). No such charge exists in a neutral winding. A field that is "not electrostatic" but is steady, curl-free and radial is therefore not a Maxwell field; Hooper wants a third kind of field (his 1969 "three electric fields" thesis), which is not an error of algebra but is a claim that departs from Maxwell's equations, and the mathematics of his own derivation does not supply the extra equations such a field would obey. - Faraday's flux rule makes the shielding results trivial. In the Ironing Board, Trapeze and Aluminum Box experiments, the measured quantity is the EMF around a closed circuit, the line integral of E around the loop. An electrostatic shield contributes an E_s with zero circulation (the integral of E_s around a closed loop is zero because E_s = -grad phi), so it cannot reduce a closed-loop EMF, with or without a shield; Hooper's Table 3.1 (observed 10.64 against theory about 10.8 microvolts, using v B l with n = 100, B = 0.55e-4 Wb/m2, l = 0.30 m) is exactly the ordinary flux-cutting result and I reproduced the theory values [my calculation; read: New Horizons ch. 3]. The same logic applies to Klicker's reading of Montana State. "Immune to shielding" in this sense is correct and standard physics, not a new field. - Open-field case. A capacitor reading in a region enclosed by a grounded conductor is different: an electrostatic field is excluded by a closed grounded conductor. Hooper's setup puts the capacitor inside the grounded cabinet and pail, and the electrometer head is inside it too; the pail therefore does not test the claim that the generator field passes through a shield and appears outside it, because the detector is on the same side as the source, and the shield is the capacitor's inner plate. The claim that the field "passes through" a grounded steel jacket depends on that geometry and is not independently shown.

b. Consistent with experiment? The closed-circuit shielding results: yes (flux rule). The generator readings (5 microvolts at 10 A; up to 95 microvolts dry-ice at 30 A): no discriminating comparison exists. A thermoelectric explanation is plausible in size: copper is about +2 microvolts per kelvin [web: https://en.wikipedia.org/wiki/Seebeck_coefficient via the search summary]; I could not confirm a value for copper against Hooper's stainless steel plates (plausibly 1-10 microvolts/K, low confidence [memory]), so a few kelvin between the inner and outer capacitor junction would give microvolts. The generator dissipates about 3300 W at 30 A [read: Similarities p.6] and the thermal EMF would also scale with I squared (heating), so the parabola does not distinguish the two. Hooper's quick-stepping control (readings a few seconds each, 30-25-20-15-10 then back) addresses part of this and found "substantially identical" values, but he adds that when the device warms up "each of the observed potential differences becomes enhanced" [read: Similarities p.6], i.e. a heating-dependent component exists. Gibson also reports 8 to 10 degrees of room-temperature variation changed readings, "the higher the temperature, the higher the readings", and a power-supply imbalance that changed readings from 28 to 17 (with a 11 microvolt negative excursion) before balancing [read: Gibson]. Those are the signatures of artefacts the author had to chase at the microvolt level, though Hooper and Gibson present them as controlled.

c. Tested? Hooper and Gibson (interested parties, 1968-71, power supply built for the purpose; controls: quick stepping, outer capacitor to remove contact-potential change, gaussmeter and five-wire pickup loop for stray flux, balanced supply for lead-wire charging). The "disinterested expert" is unnamed [read: Similarities p.9], reported as Hooper's characterization. Millis and Williamson (NASA Lewis, 1995; propulsion-screening agency, so an interest in a large effect) tried surface-charge readings with a circuit-board capacitor and 0.001 mV voltmeter and found them "too erratic to reach definite conclusions"; they did not pursue it. Their absence of a result is not a refutation of the capacitor claim. The generator has not been rebuilt and measured by a neutral party as far as I found. The Montana State report (1985) duplicated shielding demonstrations (not the generator).

d. If true: a steady, curl-free E field not derivable from charges: Gauss's law and the Maxwell-Faraday equation would need a new term; E would not be a gauge-invariant function of charge and current distributions alone.

Verdict, split. - Closed-circuit motional EMF is not removed by electrostatic shielding: Established (the flux rule; Hooper's own 1954 data agree with it). - The generator produces an isolated external field that charges a capacitor outside a shield: Never properly tested (no neutral replication; NASA's attempt inconclusive). The reported numbers disagree with Hooper's own model by a factor of about 100, the AC and dry-ice data do not follow the I-squared law the model predicts, and plausible artefacts (thermal EMF, lead charging) are of the same size. I would not call it dismissed without testing: Hooper's own controls and a NASA attempt exist, but none is conclusive.

What would settle it (calculation or experiment). Hooper's field must be present at microsecond time scales after a current step (thermal EMF takes seconds to minutes), must scale as ln(r2/r1) with capacitor radii, must fall when n changes (aluminium has about twice copper's n, so V = J/(n e) halves for the same J), and must keep its sign when the current reverses. Thermal and contact EMFs fail some of these tests. This is a bench job within a student lab's reach.

Claim 3. The capacitor voltage (P.D. = B V l) directly measures electron drift velocity and conduction-electron density; drift velocity at large current density is higher at low temperature, by up to about 10^10 in superconductors.

a. Mathematically consistent? No, in four ways. 1. Drift speed v = J/(n e) has no temperature dependence at fixed current density J, because n is fixed by atom count. Resistivity and voltage cancel (Millis and Williamson say this). My calculation reproduces the patent's own corrected number: 480 A/cm2 gives 3.5e-2 cm/s in copper. 2. The patent says the superconducting drift speed "approaches 10^8 cm/s" (the Fermi speed, about 1.6e8 cm/s in copper: I get 1.57e6 m/s). A superconducting current is carried by paired electrons, and the maximum speed of the pair flow is the Landau critical (depairing) velocity v = Delta / p_F, where Delta is the energy-gap parameter and p_F the Fermi momentum [web: https://arxiv.org/pdf/2008.12732 and the search summary of the Landau depairing velocity]. With Delta about 1.5 meV and k_F = 1.36e10 m-1 for copper-like density (my inputs; Delta of a typical low-Tc metal, [memory, medium]), v_c is about 170 m/s = 1.7e4 cm/s, corresponding to J of about 2e8 A/cm2. So 1e8 cm/s is nearly four orders of magnitude above the physical ceiling, and the largest enhancement over 3.6e-2 cm/s is about 4.7e5, not 1e10 (my calculation). The 1e10 figure is therefore refuted by an accepted law applied correctly, independent of any experiment. 3. Extracting both V and n from one P.D. reading is circular (see claim 2). 4. The "factor 100" gap in Hooper's own number (claim 2).

b. Consistent with experiment? Hall-effect measurements fix n at about one electron per atom for copper [memory, high]; no experiment shows n or v varying as Gibson reports ("almost ten times as many conduction electrons" in dry ice). Gibson's resistance fall 3.7 to 2.3 ohm on cooling is ordinary resistivity behaviour.

c. Tested? Only Hooper and Gibson; the superconducting wire version was never reported (Hooper died 14 February 1971, per Investigator's single source). NASA tested coils in ice (no weight change) but not the drift-velocity claim.

d. If true: Fermi-Dirac statistics and the Drude/Sommerfeld picture of conduction would be wrong by orders of magnitude, as would Hall-effect and Landau-criterion results.

Verdict, split. The 1e10 superconducting enhancement and the temperature dependence of drift speed at fixed J: Refuted by calculation. The proposal that the capacitor voltage is a measurement of drift speed and n: Internally inconsistent as presented (circular, and off by 100 from its own model).

Claim 4. The field polarizes neutral atoms and attracts matter like gravity, so Earth's gravity is an electric (polarization) effect.

a. Mathematically consistent? The elementary step is standard: an induced dipole p = alpha E feels the force F = alpha (E . grad) E = (alpha/2) grad(E^2), attractive toward stronger |E| (Jeans, Griffiths ch. 4) [memory, high]. The step from there to gravity is not derived; there is no equation linking it to mass. My checks: - Size. For Hooper's measured E = 1.4e-3 V/m at r = 0.1 m and a typical atomic polarizability alpha = 4 pi eps0 x 1e-30 m3 (about 1.1e-40 C m2/V [memory, high for order of magnitude]), F = alpha E^2 / r = 2.3e-45 N per atom, against the weight of a copper atom of 1.0e-24 N. The ratio is 2e-21. To reach one gravity, E would have to be about 3e7 V/m (about ten times air's breakdown field) (my calculation). - Distance law. For his radial 1/r field (cylindrical), grad(E^2) falls as 1/r^3; for a point-like source E goes as 1/r^2 and the force as 1/r^5. Gravity falls as 1/r^2 for a point mass. The polarization mechanism has a different distance law. - Mass dependence. Polarization force goes as alpha, not mass m. Gravity gives the same acceleration to all materials. A neutron is nearly non-polarizable but falls like any mass. Modern tests find no composition dependence: the MICROSCOPE satellite gives eta(Ti, Pt) = [-1.5 ± 2.3 (stat) ± 1.5 (syst)] x 10^-15 [web: https://arxiv.org/abs/2209.15487]. A polarization origin of gravity would give differences many orders larger. - "Earth's gravity polarizes matter" is stated without a defining field E_g or any value.

b. Consistent with experiment? Dielectrophoresis (neutral dielectric pulled into a nonuniform field) is established; Hooper's identification with gravity conflicts with equivalence-principle tests above.

c. Tested? The essay says the attraction experiment was planned, not run [read: essay p.7]. The patent's pendulum and gravity-meter tests are predictions. Hooper's own 1957-59 gravity-meter work was withdrawn (claim 6). Never run by anyone else in a controlled way.

d. If true: Newtonian and general-relativistic gravity (universal coupling to energy) would be replaced by an electromagnetic polarization force; the equivalence principle would fail at the tested 1e-15 level.

Verdict, split. Strong form (gravity is the polarization force of a B x V field): Refuted by calculation (magnitude about 1e-21 of weight at measured fields; wrong distance law; coupling to alpha, not m). Weak form (a nonuniform E field attracts neutral polarizable matter): Established as dielectrophoresis. Whether the generator's field at the claimed strength exists is claim 2's question.

Claim 5. A strong enough B x V field could cancel or reverse the polarization gravity induces, so objects lose weight or are repelled (anti-gravity), with possible spacecraft lift.

a. Mathematically consistent? The patent's FIG. 3 and FIG. 4 devices give a uniform vertical field and Hooper says that the pull and push are "equal" so there is no attraction, only a change in "the state of polarization already existing ... produced by the gravitational field of the earth". In his own dipole law a uniform field gives zero force on any induced dipole (E . grad E = 0 for uniform E), so a uniform B x V field changes the force only through the cross term alpha E_b . grad E_g with the (undefined) gravitational field E_g. The patent gives no equation connecting polarization to weight and no field strength needed. The internal logic is thus incomplete: the effect depends on a quantity never defined. FIG. 3 is also an ordinary (inductive) coil: a collapsing field in the coil induces a conventional axial E by Faraday's law, and a conductive shell around the leg carries eddy currents with the usual Lenz-law repulsion, a conventional force that would mimic a weight change; the patent does not address it [read: patent p.4; my reading]. - Size under the most favorable superconducting assumptions: B = 10 T, V at the Landau critical speed 170 m/s gives E = 1.7e3 V/m; with the force law above, the force on an atom is 3e-9 of its weight, so about nine orders of magnitude short even with unrealistic values (my calculation; uses the same alpha). - The patent's claims are for the hardware only; the five claims say nothing about gravity.

b. Consistent with experiment? The weight null at the NASA test, below.

c. Tested? Yes, once, independently: Millis and Williamson (NASA Lewis, 1995, AIAA-95-2601 / TM-106963). They built self-cancelling coils as described, weighed a 555 g acrylic disk on a balance (threshold 0.02 g, i.e. about 4e-5 of the weight), read a servo accelerometer (threshold about 40 micro-g), and ran the coils in ice. Result: "No weight changes or confirmed alterations of the gravitational field were observed with any of the coils at any of the tested current levels. Freezing the coils made no difference." They attribute Hooper's reported changes (as described in Gibson) to possible thermal convection, saying he "may have misinterpreted thermal effects". Caveats: NASA is a government agency screening for propulsion, interested in a large effect; they took quick on/off readings without warm-up, while Hooper let coils warm up; they did not test a superconducting coil. They state that a "minuscule" effect below threshold cannot be excluded [read: NASA TM-106963; Investigator note]. Hartman (1996) says Hooper "and others" witnessed gravitational and antigravitational forces; this is documented in one secondary review, no witness is named, and I found no second source. Hooper's own essay says the attraction experiment was planned and the patent says "should"; Gibson's 1983 monograph, as summarized by the Investigator, reports capacitor data but no weight change. I keep Hartman's statement in the record beside those.

d. If true: conservation of momentum with no reaction mass, plus the equivalence-principle and field-strength problems in claim 4.

Verdict, split. Weight change from room-temperature or ice-cooled self-cancelling coils at the tested currents: Tested, not supported. Superconducting version and spacecraft lift: Never properly tested, and Refuted by calculation at attainable fields on the polarization-force estimate above.

Claim 6. Earlier evidence linking this field to gravity: the 1957-59 Fisher/Hooper spinning-magnet rotor gave gravity-meter deflections, but a sound absorber removed them and sound or a buzzer reproduced them; the project collapsed; Hooper kept the belief.

a. Mathematically consistent? Not a derivation; no numbers given for rotor speed, magnet strength, meter type or deflection [read: essay pp.2-3]. One logical point: Hooper reports that a sound absorber removed the signal. A field "immune to shielding" would not be stopped by an acoustic barrier, so the absorber result itself shows the measured effect was not that field. Hooper does not draw this inference; his continued belief rests on the WWII-era shielding claim (claim 2).

b. Consistent with experiment? The acoustic artefact is the simplest reading of what he reports.

c. Tested? By Hooper himself, who recorded the result against his own earlier claim ("Then came the disappointment ... Thus our joint project collapsed"). The APS presentations (1958-59) are confirmed by an independent historian, Helge Kragh (Varying Gravity, Springer 2016, pp. 68-69), who identifies Fisher (1891-1966) and says the work "resulted in two US patents to Hooper" [web: dokumen.pub mirror via Investigator; cite the Springer edition]. The original deflection data are not in any document I found.

d. If true: n/a (withdrawn by Hooper).

Verdict: Tested, not supported (by the inventor's own controls). Documented fully as the inventor's own negative; the later claim that gravity-meter readings came from the field has no support in the record.

Claim 7. The field can penetrate barriers, enabling communication, power and detonation signals through concrete or armor, and electrical energy "available everywhere on the earth" could be tapped.

a. Mathematically consistent? No equations or magnitudes are given. The communication proposal is a capacitor-electrometer receiver for a pulsed coil; the magnitudes involved (5 microvolts at 10 A and 4000 conductors close to the source) say the signal is tiny beyond a few coil radii. For the power claim: a static field that behaves like an electrostatic field is conservative (the line integral of E around any closed path is zero), so no net energy can be drawn from it cyclically without depleting a source. The claim does not say what the source is, so it is not a quantity that can be checked.

b. Consistent with experiment? Through-barrier signalling by low-frequency magnetic induction is real, but it is ordinary magnetic coupling through non-magnetic walls [memory, high], not Hooper's field.

c. Tested? Not reported. The patent calls it "I have demonstrated that AC current can be used in my device" only for the generator [read: patent p.5].

d. If true: new energy source and signal channel outside Maxwell theory.

Verdict: through-barrier low-frequency signalling by magnetic induction: Established (a different mechanism). Hooper's version and the free-energy claim: Not testable as stated (no source, no magnitude), with the conservative-field argument as a further objection. No test on record.

3. Investigator's findings I am keeping in the record and how I weigh them

• US 3,656,013 (second patent, filed 19 April 1968): FIG. 11 unipolar yoke experiment (deflection follows the iron yoke rather than the magnets, which Hooper reads as flux rotating with the magnetic system) and a counter-rotating rotor in a grounded high-permeability box. By the symmetric-geometry argument in claim 1, interpretive rather than discriminating [web: https://patents.google.com/patent/US3656013A/en; I did not re-derive FIG. 11 from the drawings].
• Similarities paper, New Horizons manuscript, Gibson monograph: the only quantitative accounts by the inventor and his collaborator. Reported controls (quick stepping, outer capacitor, gaussmeter and five-wire loop, balanced supply) are kept; my conclusions on them are in claim 2. The 15 V AC from the rotor at 10,000 rpm is ordinary induction in a pickup coil beside spinning magnets and is not a test of the claim [read: Similarities p.8].
• Gibson 1983 is a participant and later company president, so an interested party; her data are the only ones for dry ice. Her I^4 fit conflicts with Hooper's own model.
• Millis and Williamson 1995 (NASA): the only independent controlled test; the weight null and the inconclusive capacitor are both stated above. Its description of Hooper's measurements comes from Gibson's monograph, so it tests the weight prediction as the patent states it, not Hooper's own capacitor apparatus.
• Klicker 1986: moving flux equals the relativistic second-order term (v^2/c^2); sympathetic author; Montana State student duplication of shielding demos (report not seen).
• Hartman 1996: one secondary review reporting that Hooper "and others" witnessed gravitational and antigravitational forces; no witness named; no second source found yet. Not dismissed here; it is unverified against Hooper's own essay, which says the attraction test was only planned.
• Cravens 1990 (USAF): "There has been no critical experiment which can disprove whether a magnetic flux rotates with its source": an excerpt only; the full text was not available to the Investigator. That sentence fits my point that the question has no measurable difference in symmetric geometry.
• Historical record (Investigator): Electrodynamic Gravity, Inc. dissolved 9 December 1991; Hooper died 14 February 1971 (single source: a Burridge snippet on Scribd, per the Investigator) before the grants of both patents; Gibson says no one continued the work and in 1983 "all the original equipment is still available". From the mathematics side, the unfinished business is a rebuilt generator and a time-resolved capacitor test; nothing in my review contradicts the Investigator's reading that the silence fits the death of the sole investigator and the end of private funding, but I did not examine corporate records.
• Later patents citing Hooper (Banduric, PST Associates, WWK Trust, Lockheed Martin Energy Research): citation is not endorsement; I did not review their claims. Kept on the record as related work.
• Moray cross-link (Gibson, New Horizons): Hooper believed T. Henry Moray had tapped Earth's gravitational field. I did not assess Moray's apparatus; see the TernionAG material.

4. Summary table