Physicist review (Step 2): Charles Proteus Steinmetz

Reviewer: PhD physicist. Inputs read: claims.md, facts.json, all three deep/ reads, digests as needed, sites.md "Leads for review" and "Corrections", research/ notes (summaries). Tags: [read] from the project documents or the deep reads, [web] with URL, [memory] with confidence, [calc] my own arithmetic (SymPy/NumPy in the project venv).

Source gap: none for text. Only parts of the three books were deep-read (see claims.md); I rely on the deep readers' page-image readings for equations and have re-checked the load-bearing numbers myself. I did not open the Investigator's downloaded PDFs (1898, 1912 papers, US 1,230,615, finding aid); I use them as the Investigator reports them and say so. [read]

Stance: Steinmetz was a mainstream, institutionally embedded engineer (GE, Union College, AIEE president). Most of what these books contain is standard, correct and still taught. The interesting physics for a learner is where his explanations differ from what the mainstream later settled on, and where his arithmetic or models fail. I judge each claim separately, and keep the Investigator's finds beside the verdicts.

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Claim 1. Sustained (continual or cumulative) oscillation needs a continuous energy supply, which comes from the power source through an arc, not from the line's stored energy

a. Mathematics: There is no equation in the 1914 pages, only an energy-balance argument (dW/dt = P_supply - P_dissipated; deep read paraphrase, flagged by the deep reader as not printed). It is trivially consistent. [read: Discharges pp.142, 149]

b. Experiments: Fully consistent with the Duddell singing arc (1899-1900), the Poulsen arc transmitter, and with operating experience of arcing grounds on isolated-neutral systems. The J. AIEE 1923 discussion of arcing-ground suppressors shows that the overvoltage problem was real (Investigator, web: WorldRadioHistory J. AIEE 1923-05/06; not re-read by me). A line's stored energy is finite, C V^2/2 plus L I^2/2; a train of oscillations that lasts seconds to hours cannot come from it. The 22 kV, 40 Hz oscillograms (Figs. 59-65, Cunningham artificial line, Trans. AIEE 1911) are described by Steinmetz, not tested by me. [read: p.143-145]

c. Tested: Yes, and widely. Duddell shunted an LC circuit across a DC carbon arc and obtained audio oscillation; Poulsen and Pedersen took it to radio frequency. [web: https://en.wikipedia.org/wiki/William_Duddell, summary of Duddell/Poulsen; priority also credited to H. T. Simon 1898 per the same search] Arcing grounds were controlled later by neutral grounding and (1916, Petersen coil) neutral reactors [memory: high]. Replicated many times.

d. Nothing. This is mainstream physics, and it is Steinmetz's own: "the engine or turbine power" feeds the oscillation (p.149). His statement that it "is not a resonance phenomenon" (p.150) is the correct point that the frequency is chosen by the circuit and the energy comes from the source (a self-excited oscillator, not a driven resonance).

Verdict: Established.

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Claim 2. The arc supplies the energy through a lag of voltage behind current, a "negative energy cycle or reversed hysteresis loop" (Fig. 66); no equation

a. Mathematics: Steinmetz writes no equation in 1914, so nothing can be inconsistent. I can test the physics of the claim with a one-line model. Take a falling static characteristic with negative differential resistance -R_n (R_n > 0, Fig. 66 stationary branch: about 80 V at 1 A to 61 V at 3 A, slope near -10 ohm [read: page image p.148, deep reader's estimate]) and let the arc voltage follow it through a first-order thermal lag with time constant tau. Then for current I sin(wt) the arc impedance is Z = -R_n/(1 + j w tau). The in-phase part, which sets average power, is Re Z = -R_n/(1 + w^2 tau^2), still negative (the arc delivers power to the oscillating circuit). The quadrature part gives voltage = -R_n I [sin wt - w tau cos wt]/(1 + w^2 tau^2), which is higher while the current is rising, exactly the sense of the loop in Fig. 66 [calc]. So the loop Steinmetz photographed is what a negative resistance with thermal lag looks like; the loop area is a reactive (inductive-like) quantity and does not by itself measure the energy delivered. The lag reduces the available negative resistance by 1/(1 + w^2 tau^2) and so sets the high-frequency limit and amplitude. The energy comes from the falling static characteristic, i.e. negative differential resistance. This is the order of explanation Steinmetz himself adopts in 1917 and 1923 (see Investigator below), so his later work contradicts the 1914 wording only in emphasis. [calc + read]

A note on the loop: area enclosed by a v-i curve is the integral of v di, which is not the average power (the integral of v i dt). A "reversed hysteresis loop" read as an energy cycle in v-i space is therefore a loose analogy for a magnetic B-H loop. This is why the 1914 wording is misleading, not wrong in spirit.

b. Experiments: The real part of the picture (arc in an LC shunt oscillates, frequency set by LC) is confirmed (claim 1). The titanium-carbon loop of Fig. 66 is one oscillogram, 0 to 4.1 A in about 1/120 s; the data are not tabulated; no repeat or second source in the folder. I have no reason to doubt it. [read: p.147-148]

c. Tested: The phenomenon extensively; the specific lag-loop-as-energy-source wording was not tested separately. Steinmetz's own 1913-14 treatment is of hunting in synchronous machines (Proc. AIEE Jan 1914), where a phase lag in the restoring force does the job (this is a different and correct mechanism: negative damping). Hunting itself is well established. [read: p.147; web (IEEE) via Investigator]

d. Nothing changes. The mainstream account is "negative differential resistance plus power source gives a self-excited oscillator" (Duddell, Barkhausen-Kurz, later tunnel-diode and Gunn oscillators [memory: high]).

Investigator findings that bear (kept whatever the verdict): - Theory and Calculation of Electric Circuits (1917), Ch. X, secs. 96-99: "the arc, when shunted by an oscillating circuit, has an effective negative resistance ... and thereby generates alternating power, from the consumed direct-current power", worked example 18.5 / 20.3 / 27.5 ohm; lag limits amplitude (p.190-192) [web: https://archive.org/details/theorycalculatio00stei, Investigator]. This is the physically correct account and it makes the 1914 statement that the mechanism "is still practically unknown" a statement about the 1914 book, not Steinmetz's final position. Full text not read by me; the Investigator recommended an Opus deep read of printed pp.165-200. - Trans. AIEE 42 (1923) 470: "a negative resistance in combination with a source of power" converting machine frequency to oscillation frequency [web: https://ieeexplore.ieee.org/document/5060887, abstract only]. Full text behind a paywall. - Notebook items (Schenectady finding aid: "Frequency transformation by third class conductor", March 1922; "Nernst glower as oscillator"; 1912 "tests on titanium and carbind") would show Steinmetz's own measurements. Not examined. [web via Investigator]

Verdict, split: - The fact that the arc converts DC/low-frequency power into oscillation power (negative resistance form): Established. - The 1914 explanation, energy supplied by a reversed v-i hysteresis loop arising from lag: Historically superseded but instructive (the loop is real, but it is a symptom; energy supply is the negative slope of the characteristic, which Steinmetz himself adopted by 1917). - "Mechanism practically unknown for circuits": was accurate for his 1914 book; superseded by his 1917 text and by Duddell's 1900 account that preceded it (so "unknown" was, in 1914, an overstatement for arcs).

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Claim 3. General circuit equations (r, L, g, C per unit length), decrements u, m, s; surge impedance sqrt(L/C); distortionless r/L = g/C; compound circuits pass energy from low-loss to high-loss sections

a. Mathematics: Internally consistent. The deep reader verified the full derivation with SymPy (a^2 = rg - (rC+gL)b + LCb^2 = (bL - r)(bC - g); b = u -/+ sqrt(a^2/LC + m^2); one misprint "h^2 > LC s^2" which must read "<"; a wrong cross-reference). I add that I recomputed the Lecture IX numerical example (u0 = 3.6/4.5e-3 = 800; s = -100, +700, -100, -800) and the Cunningham line z = sqrt(0.3944/1.135e-6) = 589.5 ohm; both agree with the deep reader. [read: deep; calc]

The "energy-transfer constant" s is Steinmetz's bookkeeping device: it splits a transient into two components, one that gives up energy to the line as it travels and one that picks it up. In modern terms this is the decomposition of a distributed RLCG system into forward and backward travelling waves with different attenuation, and power flow is the Poynting vector (E x H) integrated over the cross-section. The pairing of "dissipation" and "transfer" decrements is a re-labelling of the standard solution, not new physics.

b. Experiments: The telegrapher's equation is Heaviside (1880s) and Kelvin (1855) [memory: high], tested for transmission lines, cables, wave guides. The distortionless condition r/L = g/C is Heaviside's [memory: high]. The oscillograms of Figs. 41 and 51-53 (28-mile line; 130-mile artificial line, matched end gives no reflection, open end gives repeated reflections) are exactly what the theory predicts. [read: p.113, 128-130]

c. Tested: Yes. Standard engineering; Cunningham's artificial line (Proc. AIEE Jan 1911) and Steinmetz's own tests supplied data. The Investigator found a patent (US 1,230,615, granted 1917, GE) using the surge-resistance idea: a series inductance shunted by a resistor equal to the transformer's sqrt(L/C), so high-frequency waves pass out of the transformer but reflect from the line (Investigator; I did not open the PDF). The principle (resistive termination equal to surge impedance prevents reflection) is standard. Whether the device performed as claimed is unknown to me. [web: https://patents.google.com/patent/US1230615A/en]

d. Nothing.

Limitation, stated fairly: the model assumes constant r, L, g, C with no separate radiation term, so it is a quasi-TEM theory, correct while the conductor spacing is small against the wavelength. The books fold radiation into an effective resistance (preface to the 3rd edition, quoted in the deep read). [read]

Verdict: Established.

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Claim 4. Line resonance and surge effects: quarter-wave line (constant potential to constant current, very high open-circuit voltage); f1 = 7.5e9 / l0; transformer coils with lowest damping at 10,000-100,000 cycles are the most dangerous place for arc-fed oscillation

a. Mathematics: Internally consistent, apart from printed slips. - f = S/4l: with S = 186,300 miles/s, S/4 = 46,600; the book's 47,000 is rounded. [calc] - f1 = 7.5e9/l0 with l0 in cm equals c/(4 l0), c = 3e10 cm/s. Dependence on length only, independent of wire size and height, follows because L C = l0^2/c^2 for any lossless TEM line in a uniform medium; the log terms cancel in his eq. 24-25. [read: page image p.370-371; calc] Table check: 10 miles = 1.609e6 cm gives 4,661 cycles vs 4,680 in the Investigator's 1898 table; 100 miles gives 466 vs 475 [calc]. A 1-2 per cent difference in constant (mile conversion or c), not a physical discrepancy. - Transformer-coil example: I recomputed the 283 independently from the line-attenuation form. R = 0.1 ohm x 100 = 10 ohm; Z0 = sqrt(0.02/1e-8) = 1,414 ohm; alpha l = R/(2 Z0) = 0.00354; open-end gain = 1/sinh(alpha l), about 1/(alpha l) = 283. [calc] 50,000 V supply of which 10 per cent third harmonic (5,000 V at 177 Hz) gives 1.4 MV. [read: p.355-356] (One wrinkle in the deep read: it quotes E_open = E1/u and then 2/(pi u); the number 283 comes out of the second form and out of my independent calculation, so the arithmetic is right whichever form is printed.) The quarter-wave frequency: 1/(4 sqrt(LC)) with L total 2 H, C total 1e-6 uF-scale gives 177 Hz [calc]. This is a voltage, not a power, gain (the output current is small, as he notes: constant-current behaviour). - Printed error: "e = i0 sqrt(L/C) = 300,000 V" with i0 = 750 A and sqrt(L/C) = 200 ohm is 150,000 V, and the series coefficient pi/2 should be pi/4 (deep reader's correction). [calc: 750 x 200 = 150,000] Not a conceptual flaw but it doubles the stated overvoltage in the short-circuit interruption example.

b. Experiments: Quarter-wave transformation is the standard result of transmission-line theory and is used in impedance transformers (lambda/4 transformer: Zin = Z0^2/ZL) [memory: high]. Resonant overvoltage on long lines (Ferranti effect) is textbook. High-frequency winding resonances of transformers (kHz to MHz) and their role in insulation failure under fast transients are a modern research topic [memory: medium]. His claim that "danger frequencies" lie at 10,000-100,000 cycles is asserted with no measurement in the pages read; the 1917 book repeats it as "frequently been observed" (Investigator). Arcing-ground overvoltages were an accepted operating hazard in 1923 (J. AIEE discussion, Investigator).

c. Tested: The existence of line resonance and of arcing-ground overvoltage: yes. His specific numerical claims for transformer coils (frequencies, "lowest damping"): not tested in anything in the folder. He gave a proposed cure, a resistance in series with a capacity across the line, which passes the high frequency and blocks machine frequency (p.149-150). The cure's effectiveness is not demonstrated in the pages. The design study (60,000 kW over 700 miles with a quarter-wave line) was never built so far as the pages state. [read]

d. Nothing. A caution for the reader: the 283x gain needs an essentially pure sinusoidal source at exactly the quarter-wave frequency and a lossless insulation; real corona and dielectric loss cap the gain, as he acknowledges in the "infinity as far as insulation is concerned" language.

Dating (Investigator): the f1 = c/4l frequency-length table already appears in Steinmetz, Electrical World, 27 Aug 1898 [web: tuks.nl, research/files/steinmetz-1898-natural-period-transmission-line-lightning.pdf; I did not read the PDF]. This corrects "first appears in the Transient book".

Verdict, split: - Line resonance, quarter-wave behaviour, f1 = c/4l, and the existence of arc-fed cumulative surge: Established. - The 10,000-100,000 cycle "danger frequency" of transformer coils as stated, and the proposed cure: Never properly tested in this record (asserted; later literature on winding resonance supports the general idea, [memory: medium], not his numbers). Not dismissed; simply not examined in the folder.

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Claim 5. Lightning flashes in clouds are oscillatory discharges stepping between rain drops, modelled by a multi-gap-arrester chain with distributed series capacity; gap voltages add to 24 times the end voltage, so a mile-long flash needs only a few hundred million volts

a. Mathematics: The qualitative point (a series chain of gaps can have gap voltages summing to many times the end-to-end voltage because they are out of phase along the chain) is correct for any wave-carrying chain: it is the same as the statement that the sum of magnitudes of voltage drops along a transmission line exceeds the net voltage. The worked example does not reproduce. Using his own printed constants (L = 2e-8 H, C0 = 1e-13 F, C = 4e-11 F, r = 1 ohm, g = 4e-6 mho, f = 1e8 Hz, n = 300, e0 = 30 kV) I get x = 12.57 ohm, xc = 39.8 ohm, Z = 1 - 27.2 j ohm, Y = 4e-6 + 62.8e-6 j mho, propagation constant a = sqrt(ZY) = 0.0414 - 0.00056 j per cylinder. The series arm is net capacitive at 100 MHz (series resonance at 178 MHz [calc]), ZY is nearly real positive, and there is no propagating wave: the gap voltage decays exponentially from 1.8 kV at the line end to about 15 V at the ground end, and the sum of the gap voltages is about 44.7 kV, a ratio of 1.5 to the 30 kV applied, not 24. [calc, finite chain with boundary conditions e = e0 at l = 0, e = 0 at l = n; agrees with the deep reader's E5.] His printed alpha = 0.0021, beta = 0.0412 look like the real and imaginary parts interchanged; with those values the gap voltage would be roughly constant near 2.4 kV, and 300 x 2.4 kV / 30 kV gives the 24:1. The "7.86 wavelengths" is also off (beta = 0.0412 per cylinder gives 1.97 wavelengths in 300 cylinders; 7.86 equals 24.72/pi). A propagating regime does exist: if the series arm is net inductive (my trial values L = 2e-7 H gives a gap-sum ratio near 9; 1e-6 H gives 1.8), so the effect is parameter-dependent, not impossible. [calc] The conclusion "a few hundred million volts" is an extrapolation of a 24:1 ratio to a cloud with no cloud parameters used. [read: p.397]

b. Experiments: Modern lightning physics does not support drop-to-drop oscillatory discharge as the propagation mechanism. The accepted picture is a stepped leader of hot, conducting channel whose tip enhances the local field to breakdown strength, with streamer zones ahead of the tip, the whole channel carrying 100-1000 A at an average speed of 2e5 m/s [web: https://arxiv.org/pdf/0906.0429, and search summary]. Ambient cloud fields of about 200 kV/m are well below the 2 MV/m air-breakdown figure; the puzzle of initiation is the modern open problem, and the field is enhanced locally by hydrometeors [web: search summary, informal source quoting Physics Today; treat as secondary]. Estimated potential differences in intracloud discharge modelling: tens to hundreds of MV, one study about 300 MV [web: https://pure.psu.edu/en/publications/production-of-very-high-potential-differences-by-intracloud-light/]. So Steinmetz's "few hundred million volts" lands in the range of modern estimates, though by a route that is not independently supported. His premise that no cloud can sustain breakdown field across a mile of straight air is, in its essence, the modern leader-propagation argument.

c. Tested: The cloud model was not tested. Laboratory consequences were pursued instead: GE's 1922 lightning generator (10,000 A at over 100,000 V for 10 microseconds; AP, NY Herald 3 Mar 1922, [web: https://www.loc.gov/resource/sn83045774/1922-03-03/ed-1/?sp=6], Investigator). My check of those figures: 10,000 A x 100,000 V = 1e9 W peak, 1e-5 s gives at most 10 kJ [calc]. They are self-consistent; they test equipment, not the cloud hypothesis. Steinmetz cites no cloud measurement.

d. Nothing in mainstream physics would need to change if true; it would add a mechanism for intracloud propagation. At present the stepped-leader/streamer picture is preferred on observation.

Verdict, split: - The numerical demonstration (24:1 for his constants): Internally inconsistent (printed alpha, beta do not follow from printed Z, Y; with the constants given the ratio is about 1.5). Flagged to the Mathematician; I have not re-derived his eq. 17-19. - The cloud mechanism (oscillatory discharge between rain drops): Never properly tested (offered as "probable", no measurement; modern observation favours the leader picture, which is a reason to doubt it but not a test of it).

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Claim 6. Dielectric loss and corona: constant power factor in solid insulation; true dielectric hysteresis doubtful; a layered dielectric with a leaky layer imitates it; air breaks down at about 30 kV/cm peak; corona starts at an "energy distance"; P = a(f + 25)(e - e0)^2 (Peek)

a. Mathematics: Consistent. Constant power factor means P = e i cos(phi), i proportional to f e, so P proportional to f e^2: the stated form. [read: p.180] Air: 30 kV/cm peak = 21.2 kV/cm rms [calc: 30/sqrt 2]. Two-layer (mica and varnished cloth) analysis is a correct series-RC calculation; its two limits give a power factor falling as 1/f (low leakage) or rising as f (high leakage), with a broad plateau between, and this is the physical content of the "constant power factor" imitation. [read: p.183-186] The glass-plate example: with two 0.3 cm plates (k = 4) and a 0.4 cm air gap at 25 kV, uniform D gives E_glass = 25/(0.6 + 0.4 x 4) = 11.4 kV/cm and E_air = 45.5 kV/cm [calc]. The printed 8.4 and 35.7 kV/cm do not reproduce with the dimensions as the deep reader reports them (the geometry on page image p.192 may be different from my reading; unverified). Either set exceeds the 30 kV/cm limit, so the stated conclusion (the air gap sparks over with the glass inserted at 25 kV) holds regardless. Peek's formula: [web: https://eepower.com/technical-articles/understanding-corona-discharge-in-high-voltage-transmission-lines/] gives P = 241e-5 (f + 25)/delta x sqrt(r/D) x (V - V0)^2 kW/km/phase, with different texts using 241, 244 or 390 per mile, so the "scale factor" the deep reader queried is a known textbook variation; the structure (f + 25) and (V - V0)^2 agrees with Steinmetz's quoted form. Peek's formula is described as unreliable at low loss (V0/V < 1.8) per the same search.

b. Experiments: Air breakdown near 30 kV/cm peak at standard conditions, proportional to air density: correct [memory: high; also web summary above gives 21.1 kV/cm rms]. Constant loss tangent over wide frequency range for many solids is observed and is now characterised as "constant-phase" or universal dielectric response (Jonscher) [memory: medium]. The claim that "true dielectric hysteresis is doubtful" is where later physics differs: dielectric relaxation (Debye, Cole-Cole) gives a real polarization lag and real molecular loss independent of leakage; the interfacial (layered) polarization model Steinmetz uses is the Maxwell-Wagner mechanism (Wagner, 1913) [memory: high]. So his layered model is a correct member of the family, but he under-rates the non-leakage molecular loss.

c. Tested: Peek's corona work (1910s) was experimental; Steinmetz quotes it. Dielectric loss measurements were routine (Schering bridge, 1919 [memory: high]). The data in the pages are quoted, not Steinmetz's own: "a loss is commonly observed" with no measurement given. [read: p.179-180]

d. Nothing.

Verdict: Established (observational core, air strength, Peek-type corona law, layered-dielectric explanation). His comment that dielectric hysteresis is doubtful is Historically superseded but instructive.

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Claim 7. A conductor radiates because the field travels at finite speed; this appears as an effective resistance (0.2 ohm per metre of No. 00 wire at 100,000 cycles, proportional to frequency), computed with sil/col functions

a. Mathematics: The function-theory is right. sil x is minus the cosine integral (-Ci) and col x is pi/2 - Si x; the deep reader confirmed the series and the tables to 6 decimals against the Glaisher tables, and flagged seven misprints. His numbers follow from his formula: r = 2 pi^2 f l0 1e-9 ohm gives 0.197 ohm for 100 cm at 1e5 Hz; I recomputed L and x (2.19 uH, 1.38 ohm), and 100 A through 0.195 ohm gives 1.95 kW. [calc] The deep reader also finds that the single-delay model agrees with the cylindrical-wave solution at the wire surface (2/pi col 0 = 1 = J0(0)) but not in the far field. [read: deep]

The physics is another matter. A wire carrying current with no return conductor is not a closed current (charge must accumulate at the ends), so the "no-return" result describes an unphysical configuration. The real radiation resistance of a short, uniformly carrying wire of length l is 80 pi^2 (l/lambda)^2 [memory: high]. At 100 kHz, lambda = 3,000 m, l = 1 m: R = 8.8e-5 ohm, and 100 A radiates about 0.9 W, compared with Steinmetz's 0.195 ohm and 1.95 kW. [calc] Even for a quarter-wave monopole over ground at this frequency (h = 1 m), R is of order 4e-5 ohm [calc]. So the printed figure is too large by a factor of about 2,000 for a one-metre length. With a return conductor (his case B), r proportional to f^2 l' (spacing l') is the right frequency trend for the electric-dipole part, but standard theory gives a dependence on the square of the spacing for a close pair (antiphase currents), whereas his single-delay model gives it linearly in l' [memory: medium; not checked on the page]. His table of values for 6 ft spacing and 30 ft height (about 0.48 ohm per km at some frequency; the figure is garbled in the deep read) therefore also cannot be trusted without checking against dipole radiation theory.

b. Experiments: Radiation resistances of antennas are measured routinely and are small at low frequency (VLF stations need very large antennas to reach ohms), in line with (l/lambda)^2 and not with 0.2 ohm per metre. Steinmetz himself says the radiated power for a spaced pair is far smaller than for the single wire. No measurement of the effective resistance of a wire at 100,000 cycles is given in the pages. [read: p.440-441, 446-451]

c. Tested: Not by Steinmetz in these pages. The concept (radiation resistance from retarded fields) is standard, developed in the same period (Poynting 1884, Hertz 1888, Abraham 1898-1902 [memory: high]); Steinmetz's sil/col formulation is a calculation, not an experiment.

d. Nothing. The finite speed of the field is already the basis of mainstream radiation theory. Steinmetz's statement is conceptually correct; his number for a single wire is not.

Investigator leads for this claim (kept): Steinmetz's 1912 GE address treats the ether as both gas and solid, a contradiction (Investigator); Four Lectures on Relativity and Space (1923) gives his mature "field of energy" view; "Steinmetz Decoded" (modern secondary site) reads the shift from ether to field. Eric Dollard builds on Steinmetz's symmetric treatment of magnetic and dielectric fields (Discharges pp.13-14) and on finite-velocity radiation as an energy component of inductance; Dollard's wider claims (Journal of Borderland Research 1990, "Theory of Antirelativity", 2013) are not accepted by mainstream engineering [memory: high]. That label is the mainstream's characterization, recorded as such; I have not read Dollard's papers and make no judgment of them here. Steinmetz's own complex-number (symbolic) method for AC is standard phasor analysis today. [web via Investigator; memory: high]

Verdict, split: - The principle (finite field velocity causes radiation, appearing as an effective resistance): Established. - The quantitative figure for a single wire (0.2 ohm/m at 100 kHz, 1.95 kW/m at 100 A) and the no-return model: Refuted by calculation (standard short-antenna radiation resistance is about 9e-5 ohm for 1 m at 100 kHz; radiated power about 0.9 W at 100 A, about 2,000 times smaller).

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Corrections to Step 1

1. claims.md claim 2 and facts.json (Discharges entry): The statements "Steinmetz gives no equation", "later steps must not credit him with a derivation" and "does not use the words 'negative resistance'" are accurate for the 1914 Lectures VIII-X but incomplete as a picture of Steinmetz. Theory and Calculation of Electric Circuits (1917) Ch. X and Trans. AIEE 42 (1923) give an explicit negative-resistance account with numbers (Investigator, web, not read by me in full). I agree with the Investigator's correction. I add a physics correction: the lag/hysteresis loop of Fig. 66 does not carry the energy supply; it is the signature of a negative resistance with thermal lag, and the lag reduces the supply (see Claim 2, [calc]). claims.md should say the energy comes from the falling characteristic and that the 1914 wording is loose.
2. claims.md claim 4 / facts.json: the f1 = c/4l table is first printed in Electrical World, 27 Aug 1898 (Investigator). Agree. The "danger frequencies" remain uncorroborated by any measurement in the folder; I treat them as "never properly tested", not as wrong.
3. claims.md claim 5: the deep reader's flag is confirmed by my independent calculation: with the printed constants the chain does not propagate and the gap-voltage sum is about 1.5 times the end voltage, not 24. Claims.md should state this as a result, not only "flagged for the Mathematician". Also, the modern estimate of intracloud potential (tens to hundreds of MV) is of the same order as Steinmetz's "few hundred million volts" claim, so the weakness is in the demonstration and mechanism, not in the magnitude.
4. claims.md claim 7 / facts.json: the 0.2 ohm per metre figure is quoted as Steinmetz's number without a physical check; it is about 2,000 times the standard radiation resistance of a 1 m wire at 100 kHz, and the "no return" geometry is unphysical. The claim should be reported as principle correct, number not supported.
5. Deep read of the Transient book, E_open = E1/u and 283: the text gives both E1/u and 2/(pi u); only the second reproduces 283 (and my independent distributed-line formula gives 283). Minor internal inconsistency in the deep read, no effect on the verdict.
6. Glass-plate example (Alternating-Current, Ch. XIV): with the geometry as the deep read states it, I get 11.4 and 45.5 kV/cm, not 8.4 and 35.7. The conclusion is unchanged, but the printed figures should not be used in the primer before the page image p.192 is rechecked.
7. Citation: "Instability of Electric Circuits", Proc. AIEE 33(1), Jan 1914, pp.13-29, per IEEE (Investigator). Agree.

Web checks done (budget about 6)

• Peek corona formula constants and structure: https://eepower.com/technical-articles/understanding-corona-discharge-in-high-voltage-transmission-lines/ (tags upgraded from [memory: medium] to [web]).
• Intracloud potential and leader propagation: https://arxiv.org/pdf/0906.0429 and https://pure.psu.edu/en/publications/production-of-very-high-potential-differences-by-intracloud-light/ (search results only, not opened in full).
• Duddell/Poulsen/Simon: https://en.wikipedia.org/wiki/William_Duddell (search summary only).
• Not checked: Debye/Maxwell-Wagner, Petersen coil date, short-antenna formula (all [memory: high] except as marked); the winding-resonance literature ([memory: medium]); the dependence of two-wire radiation on spacing ([memory: medium]).

Summary table