Mathematician review: Ainslie-Rosemary-Ann

Reviewer role: PhD mathematician (derivations, units, notation, hidden assumptions). Inputs checked: facts.json, claims.md, and my own reading of electric_heater_experiment.pdf (full text layer), quantum_october_2002.pdf (all 5 pages rendered and viewed, [read: page image p.1-5]), magnetic_field_model.pdf (abstract and the proton section, text layer), Ainslie_heater_circuit.pdf (grep of Kelly's operating notes).

Source gap (repeated from claims.md). The patent body (WO 99/38247), the ABB/BP/Cape Hope reports and the raw oscilloscope data are not in the folder. Nothing below depends on them except where marked. Conclusions about the experiment rest on the authors' own tables ([read]) plus my arithmetic; statements about replications rest on a [web] search only, and I did not open those threads.

Symbols used

V = 24 V supply; R = 10 ohm load; L = 8.64 microhenry load inductance; d = fraction of time the MOSFET conducts (duty); Rshunt = 0.25 ohm; P = average power; f = switching frequency.

Corrections to Step 1

1. The "17.5 W average, 1.22 MJ" pair is internally inconsistent in the source, and Step 1 repeated both without noticing. 17.53 W x 997 min x 60 s = 1.049 MJ, not 1.22 MJ. The 1.22 MJ equals 20.4 W average. The cause is visible in Table 2 / Table 1 (the Joules column, [read: page image p.7] and 2009 text): I integrated the table's own Watts column (trapezoid) and compared with the Joules column. Through 960 min the two agree within about 3% (983 kJ vs 957 kJ). Then the Joules column jumps 135.6 kJ over 23 min (98 W) and 129.8 kJ over 14 min (155 W), while the Watts column in those rows reads 18.5 and 20.4 W. Those last two rows also show the load temperature rising sharply (ΔT 54.3 then 59.9 K) so the test ended in a different state. Using the authors' own watts, the energy ratio is 1.049 MJ / 67.6 kJ = 15.5, not 18. The "greater than 16" figure in the conclusions therefore does not follow from their own numbers either (15.5 before any error margin).
2. Claim 4 mis-sourced. Step 1 says the extra energy comes from an unseen medium "(Kelly's reading)". Ainslie's own text says the opposite: "nowhere is there a claim to have a closed system"; the experiment "relies on input of energy from a supply source"; and in the appendix "the energy delivered by the battery goes back to the battery. And the energy dissipated at the load comes from the load" (energy released from magnetic fields bound in the resistor, which "peel off as photons", causing "fatigue"). [read: 2009 pdf pp.13-18]. The environmental-energy story is Kelly's [read: Ainslie_heater_circuit.pdf].
3. Claim 3 contrast is overstated. The 2009 paper does not simply "still quote 2.4 kHz": it says adjusting the gate resistor "results in random oscillation that overrides the predetermined frequency and duty cycle" [read: 2009 p.5-6]. What 2009 drops is the 143-200 kHz / 1.3% figures and the separate 12 V gate battery (confirmed: no mention anywhere in the 2009 text). Step 1 is right about the omitted battery.
4. Quantum arithmetic note confirmed and extended. 12.45 V x 0.039 A = 0.49 W is right. It is also physically plausible: the IRFPG50 has Qg up to 190 nC and Ciss 2800 pF [web: https://www.digikey.com/en/products/detail/vishay-siliconix/IRFPG50/171751 ; datasheet figures returned by search], so gate drive at 200 kHz is about Qg x f = 190 nC x 200 kHz = 0.038 A, close to the reported 0.039 A. That corroborates the oscillation frequency and shows the gate battery supplies roughly 0.5 W that the 1.13 W figure omits (my calculation). Note the authors' 0.039 A is inferred from the voltage drop of a 4 Ah battery, a very coarse measurement.
5. Step 1 says figures "not inspected as images" for the 2009 reprint; fine. The Quantum article's Figure 4 (probe positions) shows "common" at the battery negative end of the shunt and channel b at the other end, i.e. a single-ended measurement across a 0.25 ohm shunt that is four 1 ohm carbon resistors in parallel. This detail matters below and was not in Step 1.
6. Step 1's note that the patent/Kelly COP values are unverified is correct; I did not verify them either.

Central calculations (used by several claims)

A. What ordinary theory predicts for this circuit (my calculation). With the switch on, steady current is V/R = 2.4 A. Time constant L/R = 8.64 uH / 10 ohm = 0.86 microseconds (corner frequency R/(2 pi L) = 184 kHz). Average battery power with duty d is P = d V^2/R = d x 57.6 W. The load is in series with the supply, so the load's average I^2R heating equals the same d x 57.6 W to first order; the energy stored in L, (1/2) L I^2 = 25 microjoules per pulse, is recycled through the flyback diode and is already counted because it came from the battery. Even at f = 200 kHz this would be at most 5 W if it were extra, but it is not extra. - Stated setting, d = 3.7%: P = 2.1 W battery and about 2.1 W in the load (plus about 0.06 W from the diode loop). - Self-oscillating state, d = 1.3% (2002 text): 0.75 W. - Reported battery power 1.13 W corresponds to d_eff = 1.13/57.6 = 2.0%. - Reported load heating 17.5 W corresponds to d_eff = 17.5/57.6 = 30% (about 44% if the MOSFET on-resistance is about 2 ohm in series with the 10 ohm; Rds(on) for the IRFPG50 is [memory, medium confidence], not checked). Kelly reports about 55% mark-space in the oscillating state [read: Kelly pdf lines about 123-129]. So the heating is what ordinary battery-fed operation at a 30-50% effective duty would give. The "anomaly" is that the shunt-based battery power corresponds to only 2%.

B. Energy-conservation constraint (my derivation). Over a periodic steady state the net energy stored in L is zero, so average power from the battery (net, signed) plus gate-supply power equals average power dissipated in load + MOSFET + diode + shunt + radiation. Hence the claim P_load = 17.5 W with P_battery = 1.13 W requires the instantaneous battery current to be nearly symmetric about zero: rms load current is sqrt(17.5/10) = 1.32 A, while the mean battery current implied by 1.13 W is 1.13/24 = 0.047 A. A battery that alternately gives and receives about 1.3 A rms, with the return phase driven through the MOSFET body diode, is a very specific waveform, and the authors show none of the numbers (positive-part power, negative-part power).

C. Why the shunt reading is fragile (my estimate, assumption stated). The shunt drop for 2.4 A is 0.25 x 2.4 = 0.6 V. A loop or shunt inductance of only 20 nH with a 10 ns current edge gives L di/dt = 20e-9 x 2.4 / 10e-9 = 4.8 V, eight times the resistive signal. Four paralleled carbon resistors and a single-ended scope probe at hundreds of kHz with ringing are exactly the setup where inductive pickup swamps I x R. The authors' own method (treat positive samples as battery output, negative as return, take the difference) makes the answer depend on any zero offset: the sign split is performed on a signal whose zero is uncertain. Search results report that skeptics' analysis of Ainslie's shunt waveform found the ringing sitting asymmetrically around zero, and that replicators saw the effect only with sufficiently inductive shunts [web: https://overunity.com/13743/rosemary-ainslie-quantum-magazine-circuit-cop-17-claims/30/ ; search-result snippet only, thread not opened, so low reliance].

D. The sampling window. The article says each analysis used "a sample range of 1.2 microseconds" holding about 1200 samples [read: page image p.4, p.3]. At the stated 2.4 kHz the period is 417 microseconds, so a 1.2 microsecond window is 0.3% of a cycle; at 143 kHz (period 7 microseconds) it is 17%. An average over a fraction of a cycle is not a cycle average unless the waveform is stationary within it. The text says the waveforms "vary greatly from one cycle to another". The sample count and number of windows used for 1.13 W are not given.

E. The error analysis is mis-specified. "Even if all negative voltage sample measurements are discounted by these values [10%, 1.5%]" [read: 2009 p.12] treats the sign split as the only uncertainty. The relevant bound is a positive-part-only power (all negative samples set to zero), and the authors do not give it. Taking 10% of the negative samples off a quantity that is a difference of two large numbers cannot bound a result near zero. A net 1.13 W from a difference of two terms each of order tens of watts (if P_load is truly 17.5 W from the battery) has relative error far above 10%.

F. Battery voltage is not a discriminating check. Nominal capacity is 24 V x 20 Ah = 480 Wh = 1.73 MJ. The reported battery energy (67.6 kJ) is 3.9% of that, so no voltage fall would be expected under their own number. The load energy (1.05-1.22 MJ) would be 60-70% of nominal capacity if it did come from the battery, which would be a substantial discharge. The readings are 24.8 V at the start and 24.3 to 25.1 V later, taken under switching load with a scope display to 0.1 V resolution. The authors "omitted" the capacity comparison, calling it "too many vagaries" [read: 2009 p.10]. The voltage record is consistent with a modest discharge and is not decisive either way; I did not attempt a state-of-charge conversion (lead-acid curves are [memory], low confidence and not used for any verdict).

Claim-by-claim

Claim 1: 17.5 W in the load vs 1.13 W from the battery (gain 16-18), steady for 16.5 h, battery voltage steady

a. Mathematics: Partly. The arithmetic of V^2/R = 13.32^2/10 = 17.74 W is correct, and 17.74/52 = 0.341 W/K is correct. The Control Watts column is exactly 17.74 x (ΔT/52) (checked: 48.1 K gives 16.41 W), i.e. an assumed linear law. The energy totals are inconsistent (see Correction 1: 1.049 MJ vs 1.22 MJ). The gain figure of 16-18 therefore has no single consistent value. The 1.13 W figure is not reproducible from the data shown (no waveform, no number of samples). The 2002 and 2009 tables differ in the final rows' voltage (24.76 average vs 24.3 in the 2009 table row) [read]. b. Experimental consistency: Heating by thermal calibration is a legitimate method in principle. Energy conservation (calculation B) is violated if both numbers are right, so one must be wrong; calculation A shows the heating is ordinary for a 30-50% effective duty, pointing at the electrical input measurement and the true duty, not at new physics. The 0.5 W gate battery is omitted from the input. c. Tested: Authors only. Independent testing claimed (ABB, Cape Hope, BP) but no report in the folder; Cape Hope's role is only calibration of the instruments, per the article itself [read: p.8]. A web search turned up many forum replication threads and video demonstrations, with some enthusiast groups saying they reproduced the heating and a skeptic (Poynt99) reporting measurement artifacts and COP of about 1 [web: https://www.youtube.com/watch?v=-c1py6CrgOY ; title only, not watched]. No peer-reviewed or neutral controlled replication was located; I did not verify any of those claims. d. If true: energy conservation, not just "unity efficiency", would be violated for an isolated circuit. No change to Maxwell's equations is needed for the claim as measured, because the likelier explanation is instrumentation. Verdict: Not supported as measured; the numerical result is internally inconsistent (1.22 MJ vs 1.049 MJ from their own watts), and the electrical-input measurement is a method that cannot support the claim (calculations C, D, E). In the scale: Internally inconsistent (energy column vs power column and error analysis) for the numbers as published; strong form (genuine net energy gain) Never properly tested by a neutral party on this record (only author-run tests and skeptic bench replications in unverified forum material). Not "Refuted by calculation" because no calculation can rule out the specific waveform without the raw data; calculation B shows what that waveform must look like.

Claim 2: Back EMF from the inductor produces the gain, partly recharging the battery; gain rises with frequency

a. Mathematics: Back EMF is v = L di/dt, standard. The maximum energy returnable per cycle is (1/2) L I^2 = 25 microjoules at 2.4 A, which was itself drawn from the battery, so it cannot exceed what was supplied (calculation A, B). No equation in the papers connects frequency to gain; the dependence is asserted ("the faster the frequency the better the return") without a model. In the ideal circuit load power is d V^2/R, independent of f except for edge effects. Appendix text asserts the reverse (dissipation proportional to the "number of transitions") with no mathematics, and states that "no part of that model has been developed mathematically" [read: 2009 p.15]. b. Experiment: Flyback and freewheel diodes are standard and well tested; they recirculate energy, they do not create it. Battery recharge by a returning current is possible in principle only up to the energy taken from it. c. Tested: Only inside the same circuit. The claim "repeatable across many circuit variants" is unsourced; no data on the variants are given. d. If true: the energy-conservation constraint in B and the circuit-theory result that a series source plus passive network cannot deliver net negative power would fail. Verdict: Back-EMF recycling itself: Established (and gives no gain). "Back EMF produces gain, increasing with frequency": Never properly tested, no controlled test or model on record; as a mathematical statement about a passive network it is Refuted by calculation (A and B), provided the circuit is as drawn.

Claim 3: Tuned gate resistor makes the 555 self-oscillate far above its setting (143-200 kHz in 2002; over 500 kHz per Kelly) and this causes the gain

a. Mathematics: No derivation. The two frequency figures differ by a factor over 2.5 and Kelly's 55% duty differs from the 1.3% of the article, which says the circuit regime changed between reports or was not measured the same way. A gate drive of Qg x f = 0.038 A at 200 kHz agrees with the measured gate-battery current, which supports the 2002 oscillation frequency. b. Experiment: Parasitic oscillation of a MOSFET driven through a series gate resistor with high Ciss is well known engineering behaviour. Kelly's observation that only some 555 makes do it is plausible as a device-dependence. c. Tested: Described by the author and by Kelly, a compilation author, not measured by a neutral party in this folder. d. If true: nothing in mainstream physics changes for the oscillation itself. That oscillation could produce a gain is a separate claim, answered under 1 and 2: it also changes the effective duty (calculation A), which would raise heating legitimately. Verdict: Oscillation: Plausible, untested by neutral measurement but ordinary physics. "This is what makes gain appear": Not testable as stated until the effective duty and cycle-averaged battery power are measured on a stable waveform; at present the changed duty alone explains the heating.

Claim 4: Unity is breached; second law and conventional energy transfer do not constrain it; extra energy from an unseen medium or the resistor's own bound fields

a. Mathematics: Not a mathematical claim. No equation is offered. Dropping the second law "because magnetic fields comprise particles" [read: 2009 p.2] is a non sequitur: the second law applies to any particle system. "Unity" is never defined as a ratio of defined quantities (supplied energy includes gate battery; Step 1 point 4). The paper also concedes it is not a closed system, undermining Kelly's reading. b. Experiment: Contradicts the entire record of energy-accounting in circuits at the instrumentation level used; no experiment cited that the conservation law has failed in a circuit. c. Tested: Not by anyone with a neutral protocol. The authors "invite disproof" ("It now remains to be disproved"), which reverses the burden of proof, but does produce a falsifiable protocol (below). d. If true: first law (energy conservation) plus the second law for circuits would need rewriting; Poynting's theorem and Kirchhoff's laws would need an extra source term. Verdict: As stated, Internally inconsistent for the Kelly environmental-energy form versus Ainslie's own "no closed system" statement and the resistor-supplied form; the "second law not constraining EM interactions" argument is Not testable as stated (no measurable prediction from it other than the claimed ratio). The measured ratio is addressed under Claim 1.

Claim 5: Current is movement of magnetic dipole "zipons" (tachyons at about 2c); magnetism primary; proton = three electrons; model predicted the over-unity result and resolves non-locality, dark matter, mass ratio

a. Mathematics: Checked the proton mass-ratio calculation [read: magnetic_field_model.pdf, section "The Proton"]. Steps as written: electron = 3 truants = 0.5 x 3 = 1.5; second truant "four times bigger" 1.5 x 4 = 6; third 6 x 4 = 24; then 3 x 6 x 6 = 108 and 3 x 24 x 24 = 1728; "1837.5 minus 1.5 = 1836". Findings: (i) 108 + 1728 = 1836 exactly; the extra +1.5 and -1.5 cancel and are decoration; (ii) the numbers 4, 4 and the squaring are chosen with no stated principle, so the result is a fit to a known target, not a prediction; (iii) it is inconsistent with the paper's own units: the electron's mass is given as 1.5 units, so a proton of 1836 units gives a ratio of 1224, not 1836; and a proton built from three electrons would have 4.5 units; (iv) the measured proton/electron mass ratio is 1836.15 [memory, high confidence], and the paper itself says "some variation ... is, no doubt, required", so even the fit is not exact; (v) three electrons carry charge -3e, the proton +e, and no charge accounting is attempted; (vi) "mass" of a photon set to 1 contradicts the zero photon mass the paper concedes. The velocity-mass inverse proportion (v x m constant) is an assumption, not derived; relativistic tachyon formulas give imaginary rest mass, not a "mass below that of a photon". "Predicted over-unity" is asserted but no derivation from the model to the circuit is given; the paper says it is "not developed mathematically". b. Experiment: Deep inelastic scattering shows quark substructure of the proton and the electron shows no structure to current limits [memory, high confidence; standard physics]. Beta decay and charge conservation also conflict with a proton made of electrons [memory, high]. c. Tested: Not tested; no measurable prediction is stated except the circuit result, which does not discriminate it. d. If true: relativity (superluminal speed limit), the Standard Model, charge conservation, and quark evidence would all need replacing. Verdict: Mass-ratio derivation: Internally inconsistent (units give 1224, ratio fitted by hand-picked factors). Zipon/tachyon cosmology and "current is moving magnetic fields": Not testable as stated (no equations, no measurable differences). The claim that the over-unity result confirms the model: Not testable as stated, and it fails even by the paper's own concession that it does not prove the model.

Claim 6: Institutions refused to test (Stellenbosch refused even a demonstration) while companies (ABB, BP, Sasol, Fluke distributors) tested or supported it

a. Mathematics: Not applicable. b. Experiment: Not applicable. c. Evidence: The refusal is the author's report in the Pretoria News (14 Nov 2002) [read: page image]. The same article says the exhibiting centre could not obtain the oscilloscope and the computer used for the displayed models. The Quantum article's own wording is narrower than the headline: Cape Hope Metrology Laboratory confirmed instruments were calibrated; Spescom "attended demonstrations"; ABB "conducted independent tests ... confined to the evaluation of instantaneous power"; BP's Green "applied tests to measure the effect ... on battery duration"; Sasol representatives "are offering a bursary"; none certifies the over-unity result [read: Quantum p.8]. No test report is in the folder, and the BP battery-duration result is not given. d. If true: that would be a sociology of science matter, not a physics change. Verdict: Never properly tested by a neutral party on this record. Whether it was "dismissed without testing" by a university rests on one self-reported refusal; the sponsor-company statements do not amount to independent confirmation. I cannot split this further without the ABB/BP reports. Not scored on the physics scale beyond "Never properly tested".

Single decisive check for the whole project

Log the raw waveforms with a differential probe across a low-inductance coaxial shunt, plus the gate-supply current, and compute cycle-averaged P_battery(total) = sum of V x I over an integer number of cycles. At the same time measure P_load calorimetrically against the DC calibration. Compare with calculation A: a gain over 1 would then need to be reproduced with the sign-split and duty reported. A drop-in alternative needing no scope: run the same circuit from a large capacitor bank (charge in, charge out) and measure the energy change, which is immune to shunt pickup.

Summary

Heating by calibration is arithmetically fine; the input measurement is where the whole result hangs and it is fragile, under-documented and internally inconsistent in the energy column. The theory pieces (Claim 5) contain a mass-ratio computation that fails its own units.