Rankings Primer Arbiter Physicist Mathematician Contrarian QE Sites log Claims Sources Research notes

Tesla: Mathematician's review (Step 2)

Reviewer: PhD mathematician. Scope: derivations, notation, units, hidden assumptions, and numerical checks. I read facts.json, claims.md, all deep/ notes, the Investigator's sites.md "Leads for review", the research/ notes, and the Colorado Springs Notes and 1904 article in research/files/ (the 3 July 1899 entry and the 1904 text). I re-did the key arithmetic myself (Python, numpy; results quoted below as [calc]). Tags: [read] from a document; [calc] my own arithmetic; [web] with URL; [memory] with confidence.

Stance. Tesla is mostly a words-and-experiment author: the patents and lectures contain very few equations. Where numbers are given I checked them. Where they are not, I say "no derivation exists to check" and do not substitute a verdict on the physics (that is the physicist's job). Several secondary sources (Nichelson, Valone, Bass) do contain algebra, and that is where most of the mathematical defects are.

Source gap carried forward: no Tesla laboratory data for the earth-resonance claim other than the Colorado Notes diary (now obtained, see Corrections). Anything on the Allis-Chalmers (Dahlstrand) report, Trump's 1943 report and Eyring Corp.'s raw coil data rests on a secondary quotation or on the advocate's printed figure.

Corrections to Step 1 (and to the Investigator)

claims.md Source gap is out of date (Investigator agrees). The Colorado Springs Notes and the 1904 Electrical World article are now in research/files/. I read the 3 July 1899 entry myself: "It showed clearly the existence of stationary waves ... It would be difficult to believe that they were reflected from the opposite point of the Earth's surface, though it may be possible. But I rather think they are reflected from the point of the cloud where the conducting path began; in this case the point where the lightning struck the ground would be a nodal point." [read: Colorado Springs Notes, research/files/colorado-springs-notes-1899-1900_djvu.txt, entry of 3 July] The 1904 article instead says the storm had "retreated to a distance of about three hundred kilometers" and concludes "this planet, despite its vast extent, behaved like a conductor of limited dimensions ... to transmit power, in unlimited amounts, to any terrestrial distance and almost without loss" [read: research/files/tesla-1904-transmission-without-wires.pdf]. Claim 1 should be split as the Investigator suggests: (1a) stationary waves seen with a receding storm; (1b) the antipodal/zero-loss generalisation. Tesla's own diary does not make 1b.
claims.md claim 1 and the "Marinčić/Jackson 6.6 Hz" lead. The Investigator quotes Marinčić quoting Jackson that the lowest mode of a perfectly conducting sphere is "6.6 Hz". I cannot reproduce 6.6 Hz. Lowest-mode estimate for a wave circling the earth: f = c/(2 pi R) = 7.49 Hz [calc] (R = 6371 km). For the n = 1 TM mode of a thin concentric shell cavity the standard formula f_n = (c / 2 pi R) sqrt(n(n+1)) gives 10.6 Hz [calc]; real Schumann peaks are about 7.8, 14, 20 Hz (lower than the ideal formula because of finite conductivity) [memory, high]. The "6.6" and Tesla's "6, 18, 30 Hz" are reported second-hand; the Jackson wording was not checked. Treat 6-6.6 Hz as unverified. Valone's 7.5 Hz is correct as c/(2 pi R) (wavelength 39,970 km versus circumference 40,075 km, as the Valone deep read notes).
Fuelless vs voltage-gain coil size. Fuelless-ed9194 footnote 15 says the Eyring measurement used two 3-inch-diameter coils of 43 turns of No. 20 wire; voltage-gain.pdf and Thermodynamics2 say 4-inch coils of 43 turns, dated 10 Dec 1990. If these are one measurement, one is a misprint; if two, only one has its numbers on record. Not resolved. facts.json should not merge them.
claims.md claim 5, "about 10 hp per pound". The sources give three different mass bases: the 1906 model "30 horsepower ... a little less than ten pounds" (about 3 hp/lb), Tesla's slogan "twenty horsepower per pound" and "ten horsepower to one pound", and the Waterside test "200 horsepower ... weighed 400 pounds" (0.5 hp/lb for the whole machine) [read: Hayes deep read p.138, 146, 150]. The 10 hp/lb figure can only mean rotor (or a stripped-down) mass and is not demonstrated by any measurement on record. Claim 5 should say so.
Hayes/steam check, agreed. I redid the ideal-expansion estimate independently with an ideal-gas isentrope (gamma 1.3) from 140 psia saturated steam to 14.7 psia: enthalpy drop about 157 BTU/lb, ideal rate 2545/157 = 16.2 lb per hp-h, so 38 lb/hp-h is about 43 per cent. The deep read has 15.5 lb and 41 per cent from steam tables [memory, medium]; the two agree within about 4 per cent. Tesla's own "130 BTU" gives 2545/(38 x 130) = 51.5 per cent [calc]. The 27-52 per cent band stands. Caveat: "efficiency ratio against ideal expansion" (what the deep read computes) is not the same quantity as Dahlstrand's "mechanical efficiency" 38 per cent and not Tesla's "80-90 per cent"; the claims.md sentence puts them together and should name the definitions.
Nichelson "voltage gain" (claims.md claim 3 and corrections). Agreed with the deep read, and I add the main point it stops short of (Claim 3, a below). 0.18 and 0.97 are network-analyzer S21 transmission ratios, which for a passive part cannot exceed 1; they are not comparable to X_L/R.
Famous Illusions Fig. 7, Mars article, Fuelless 1.6 x 10^11 J, Valone Eq. (5), FBI items. I checked these arithmetic corrections and agree: (1/4)^30 x 1 MW = 8.7e-13 W [calc]; 0.5 x 0.25 F x (3.6e5 V)^2 = 1.6e10 J [calc]; 19 MeV is per second, not an energy; 250,000 sq ft mirror gives 1.64e13 (deep read) .

Claim 1: transmission through the earth/air by resonance, "true conduction", small loss to the antipodes

a. Mathematical consistency. - Patent arithmetic is internally consistent. A 925 Hz wave at 185,000 mi/s has wavelength 200 mi and a quarter wave of 50 mi [calc, exact]. The model plant: L about 10 uH (10,000 cm), C = 0.04 uF gives f = 1/(2 pi sqrt(LC)) = 252 kHz [calc], inside the stated 230-250 kHz. Stored energy per discharge is (1/2) C V^2 = 0.5 x 0.04e-6 x (50e3)^2 = 50 J, so 5,000 breaks per second is 250 kW of capacitor throughput [calc]. The patent gives no efficiency, no received power, no lamp count (deep read p.357); "fair economy" and "considerable amounts" are unquantified. - Hidden assumption 1: the quarter-wave rule uses the speed of light on the conductor (185,000 mi/s). For a long helical coil the effective phase speed along the wire is lower than c (the axial speed is much lower); Tesla's examples assume c without argument. I leave the physics of that to the physicist, but note that the 50-mile coil is a design statement, not a derivation. - Hidden assumption 2: the earth is a conductor of "ideal geometry". Skin depth in ground, delta = sqrt(2/(omega mu sigma)) = 503 sqrt(rho/f) m: for rho = 100 ohm-m it is 10 m at 250 kHz, 165 m at 925 Hz, 1.8 km at 8 Hz; for sea water (rho = 0.25 ohm-m) 0.5 m at 250 kHz [calc; typical resistivities memory, medium]. The patent gives no earth resistance, no return-path analysis, and the earth is mentioned only as a ground connection (deep read 1.5). The claim "true conduction, not radiation" is a classification without a calculation behind it. - Tesla's "a small fraction of one per cent" loss to 12,000 miles has no derivation anywhere in the folder (Tesla Said p.90). The Colorado diary itself calls antipodal reflection "difficult to believe". - Quantitative check of the loss claim with a number from outside: Schumann-mode quality factors measured by a Kyoto study were Q = 3.30, 4.52, 4.92 for the first three modes [web: https://repository.kulib.kyoto-u.ac.jp/dspace/handle/2433/178586, via search summary; I did not open the paper]. Taking Q = 3.3 at f = 7.8 Hz, amplitude decays as exp(-pi f t / Q); in the 0.067 s a wave needs to reach the antipode, amplitude falls by exp(-0.50) = 0.61, power to 0.37, i.e. about 63 per cent loss (4.3 dB) [calc, rough: it treats a free-mode decay time as a propagation loss]. That is two orders of magnitude from "a small fraction of one per cent". ELF attenuation about 1 dB/Mm at 76-82 Hz in the same literature gives about 20 dB over 20 Mm at those frequencies [web: https://springerplus.springeropen.com/articles/10.1186/s40064-016-1742-3/tables/2, via search summary; not opened; my multiplication]. - The 3 July observation: the Investigator's suggested test works out as follows [calc]. Stationary-wave maxima along a path are half a wavelength apart. If the storm moved at 20-50 km/h and the receiver peaked about every half hour, the maxima are 10-25 km apart, so wavelength 20-50 km, f = c/lambda about 6-15 kHz. The Corums say Tesla's receivers worked "primarily in the 8 kHz to 22 kHz range" [web: research/files/corum-2003-tesla-planetary-radio-signals.pdf, from NASA Radio JOVE Tesla.pdf]. These agree. So the 3 July pattern is internally consistent with a kHz-range standing/interference pattern over a few hundred km. It is not evidence for a 6-10 Hz whole-earth resonance: half-wavelength at 8 Hz is about 19,000 km, so a 300-km storm retreat could never run through several nodes. Marinčić makes the same point (frequency "much lower" than Tesla worked at). A competing explanation I can compute: ground-wave versus ionospheric-reflected-wave fading (d >> h, path difference about 2h^2/d, fringe spacing in d about lambda d^2 / 2h^2 = 80 km for lambda 20 km, d 200 km, h 70 km [calc; geometry assumed, h memory, medium]) gives a pattern with the same order of spacing as the storm travel. I do not claim it is the right explanation; the point is that the repeated maxima do not single out reflection from the lightning-strike point or the antipode. Lightning is a broadband impulsive source, not the CW source a stationary-wave picture needs. - Tesla's 1916 interview (Investigator): "18 million horsepower" is capacitor energy dumped in very short times; an undamped wave "gives you only ... 1 kilowatt". That is a statement about peak, not average, power and is mathematically unobjectionable as such. Check with the Corum figure in Valone (6.5 hp, 7.5 Hz): 4.85 kW / 7.5 Hz = 646 J per pulse. To reach 18e6 hp = 13.4 GW the pulse would last 646 J / 13.4 GW = 48 ns [calc], shorter than one period of any kHz oscillation. So either the pulse energy was far larger than 646 J or "activity" is defined differently; unresolved. - Atmosphere-conduction scheme (patent 645,576 p.357): "15 to 20 million volts" and "many thousands of horse-power" over "many hundreds and even thousands of miles" with terminals 30,000-35,000 ft up. Energy budget [calc, assumptions flagged]: 5,000 hp = 3.7 MW at 20 MV needs 0.19 A. A sphere of radius 25 m held at 20 MV stores (1/2)(4 pi eps0 a) V^2 = 0.56 MJ [calc]. A conductive column that actually carries 0.19 A along, say, 1,600 km needs a field of only about 12 V/m, many orders below the 10^5-10^6 V/m at which air ionises at that altitude (breakdown about 0.8 MV/m at 0.26 atm [memory, medium]), so the ionisation must be held by a different mechanism or continuously re-created; electron attachment in air at that pressure removes free electrons in tens of nanoseconds [memory, medium: three-body attachment coefficient not looked up]. I cannot make this a proof, but the patent supplies no number for the ionised volume or the power needed to maintain it, and every estimate I can make is enormous. This is left to the physicist; I report that the claim has no energy budget.

b. Consistency with experiment. What the patents describe in an evacuated tube (energy moving between two terminals 50 ft apart through air at 120-150 mm Hg) is ordinary glow/streamer conduction between electrodes and is not in dispute (no data given). Experiments that bear on long-distance ELF/VLF transmission through the earth-ionosphere guide are mainstream: Project Sanguine / ELF facilities. The Wisconsin Test Facility radiated about 1 W for about 2.6 MW input power [web: https://en.wikipedia.org/wiki/Project_Sanguine and https://apps.dtic.mil/sti/pdfs/ADA126212.pdf; the two numbers come from different documents; ratio about 4e-7, order of magnitude only, my division]. That is a working earth-ionosphere system, delivering signal, with efficiency about 7 orders of magnitude below "a small fraction of one per cent" loss. It is also a point in Tesla's favour: signals do propagate globally at ELF through the earth-ionosphere cavity (the Schumann resonances are the natural version). The 2003 Corum reconstruction of the receivers is an independent technical analysis (Corums have a stake via CPG/Viziv [memory, medium; not verified]); I could not check their negative-resistance coherer model.

c. Tested? By Tesla (1899-1900), with the only quantitative evidence a diary entry and the 1904 article; observer Fritz Lowenstein is named by Tesla as having made "similar observations" (no account found). No replication of the power-transmission claim is documented. Wardenclyffe (1901-1905) never ran a long-distance power test on record. Later: CPG/Viziv Zenneck-wave tower (Texas) has more than 100 patents and no published field measurement found; absence of data there is an anomaly to analyse (secrecy, negative results, unsearched venues), not a finding (Investigator, research/2026-10-08-viziv-cpg.md). MIT/WiTricity resonant coupling (patents citing US 787,412) is a replication only of the short-range, non-radiative near-field coupling, not of earth transmission.

d. If true: the earth-ionosphere cavity would need to deliver power with about 99 per cent efficiency over 20 Mm, i.e. a Q far above the measured 3-5, and the mainstream account of ELF attenuation would have to be wrong by two orders of magnitude; a conduction path through the stratosphere at about 10 km would need a standing ionisation source with no known power supply.

Verdict (split). - Claim 1a (signals propagate at a distance through the earth-ionosphere system by resonant, non-Hertzian, near-field-coupled apparatus; stationary-like patterns seen with a receding storm): Plausible, untested in the sense the diary describes; the pattern is consistent with VLF in the kHz range; it was never reproduced under control. - Claim 1b (power transmitted with "a small fraction of one per cent" loss to any distance by conduction): Never properly tested (no data, no replication); my loss estimate from Q gives about 60 per cent to the antipode at ELF, and the ELF facility record is about 1e-7 efficiency, so the numerical form is Refuted by calculation as stated for whole-earth power delivery, with the caveat that my estimate uses a free-mode Q as a proxy. I would rate that part Tested, not supported only if Wardenclyffe-class apparatus had run; it did not.

Claim 2: electrostatic effects, "electric sound waves", transverse ether wrong; later "longitudinal/scalar"

a. Consistency. - The letter of 21 Dec 1892 contains no equations. The one scaling law, dissipation proportional to (surface charge density)^2 times frequency, is introduced by "Assuming ..." and Tesla himself lists a contrary observation (maximum charge per atom). A dimensional reconstruction (charge per air molecule proportional to density sigma, energy per impact proportional to sigma^2, impacts per second proportional to frequency) reproduces his law, so it is self-consistent as an assumption, not derived. The same scaling (loss linear in frequency) is the form of the empirical corona-loss formulas for conductors, a mainstream result [memory, medium]. - Famous Illusions p.4-5, the lightning-rod "heat analogy": sphere of radius r and charge q has potential q/r (Gaussian units); medium at the pin tip at q/(r+h); difference qh/[r(r+h)]; for the larger sphere R = nr at equal potential (Q = nq) the difference is qh/[r(nr+h)], smaller by (r+h)/(nr+h). The algebra is correct [calc, checked symbolically by hand]. It is an electrostatic argument about equal potential spheres; no equation connects it to lightning behaviour. - 1891 lecture: "electrostatic effects diminish nearly with the cube of the distance ... electromagnetic ... simply with distance". That is the dipole static field (1/r^3) versus radiation field (1/r), correct as stated [memory, high]. Note that it argues that, at a distance, the electromagnetic term wins, which is the opposite of the use he makes of it. - "Longitudinal" is a later gloss. Tesla does not use the word (deep reads). Bass's chapter in Valone is rigorous as mathematics but the fields have E parallel to B, div E = div B = 0 and zero Poynting vector (Valone deep read); in Fourier terms every plane-wave component is transverse, and no energy is transported. So "longitudinal compression waves" in the editor's sentence do not follow from Bass's equations. In a plasma, longitudinal electrostatic (Langmuir) modes exist at the plasma frequency, a mainstream result [memory, high], but they need charge density, which Tesla's "electrified air" would supply only at very high field strength. - Valone's Nichelson chapter: the 7.5 Hz is the editor's insertion; Eq. (5) drops "per second". The chain E = hf per electron per second gives 3.1 pW (19 MeV/s), and per electron 3.1e-14 V, not 19-190 MV [calc, agreed with deep read]. Internally inconsistent (units).

b. Experiment. Corona/ionic-wind loss and Maxwell-stress acoustic emission from alternately charged plates are real and measured; Tesla's examples (a glass plate near a condenser emits sound; condenser ringing; a hole in oil under a point) are standard electrostriction/ion-wind effects [memory, medium]. The claim that these are the main light and heat mechanism is not what the experiments in the folder show; the Hertz experiments (transverse polarised waves, reflection, interference) are established.

c. Tested? The dissipation law was never measured quantitatively (Tesla: "accurate quantitative determinations ... are as yet wanting"). No documented test of "electric sound waves" as a distinct wave mode.

d. If true: Maxwell's transverse-wave picture of light, with all of optics and the Hertz results, would have to be replaced; mainstream already includes the near-field and plasma longitudinal parts.

Verdict (split). Tesla's weak form (air dissipates RF energy by charged-particle motion): Established in its mainstream form (corona loss, ion wind), Never properly tested as the specific sigma^2-f law. Strong form (transverse ether wrong, energy carried as electrostatic sound waves): Superseded, Historically superseded but instructive. Later "longitudinal/scalar wave" power-transmission readings (Valone/Bass/Nichelson): Internally inconsistent (Valone's unit error) and Refuted by calculation for energy transport in Bass's waves (zero Poynting flux).

Claim 3: resonant circuits, voltage rise, bifilar coil, "voltage gain"

a. Consistency. - Resonant voltage rise Q = omega L / R is textbook; Tesla's patent 685,012 (cooling to lower R) states "gain proportional to frequency times inductance divided by resistance" (deep read 3.3) and gives no numbers. Correct as a Q statement. - Patent 512,340. Tesla's arithmetic: 100 V over 1000 turns is 0.1 V between adjacent turns; with the two strands in series the difference is 50 V; the squared ratio (500)^2 = 250,000 [calc, correct]. He then slides between "energy stored" and "capacity" (deep read s.3d): the energy ratio 250,000 holds for the same capacitance per adjacent pair. As an effective lumped capacitance, C_eff = 2E/V^2 = N C_p (dV/V)^2: for the single coil C_p/N, for the bifilar N C_p/4, ratio N^2/4 = 250,000 for N = 1000 [calc]. For the 43-turn coil measured by Nichelson the same model predicts N^2/4 = 462, so a resonant-frequency ratio of about 21 (19 MHz down to about 0.9 MHz) [calc]. - The measurement does not show that. From the printed L and f the implied lumped capacitances are 0.34 pF (single) and 1.02 pF (bifilar), ratio 3.0 (frequency ratio 1.73) [calc]. The simple inter-turn model overstates the effect by about 150 times. That is a quantitative disagreement between Tesla's reasoning and the only measurement in the folder; the likely reason (that a real coil's self-capacitance is dominated by other contributions, and that the "resonances" read from a through-measurement may not be the first self-resonance) is a hypothesis I cannot check. Also, 0.34 pF seems low for a 4-inch diameter coil by Medhurst's rule of thumb (a few pF) [memory, low-medium; not web-checked], which suggests the 19 MHz peak is not the coil's first parallel resonance. - Nichelson's "voltage gain" V.G. = X_L / R: with R silently set to 1 ohm (the denominator is "1"), the number is 2 pi f L = 24,822 and 14,173 [calc], i.e. an inductive reactance in ohms, not a gain. The ratio "theory 0.57" is (f L)_b/(f L)_s, independent of R only if both coils have equal resistance, which is unmeasured. The "929.3%" should be 945.6%. - The measured quantity is S21 (labelled "Vin/Vout", scale 0 to 1) from an HP network analyzer. For a series element Z between two 50-ohm ports S21 = 100/(100+Z) [calc, standard two-port; ports 50 ohm memory, medium]. Then 0.97 requires |Z| of at least 3.1 ohm; 0.18 requires |Z| of at least 455 ohm [calc]. So the bifilar coil at 11 MHz is simply a low-impedance (near series-resonant, nearly transparent) element, and the single coil at 19 MHz presents hundreds of ohms. That is what a larger distributed capacitance and a different resonance would do; it is not a voltage gain. A passive S21 cannot exceed 1, so "0.97" is a transmission and the ratio 5.39 compares transmission with a reactance ratio (different quantities). Result: the paper's headline "greater than expected by theory" compares unlike quantities. Internally inconsistent as a gain claim. - Thermodynamics2 Step 13: the sizing example gives C = 5 uC / 5000 V = 0.001 uF and stored energy (1/2) C V^2 = 0.0125 J [calc, agrees with deep read]; this is a finite store, not a source. The text says "several orders of magnitude" while its own Fig. 6 shows a ratio of about 5.

b. Experiment. High-voltage resonant rise, wireless lighting of vacuum tubes in an alternating electrostatic field and single-wire power transfer are demonstrated and mainstream. Series-wound close-spaced coils with higher self-capacitance and lower self-resonant frequency are mainstream. Nothing in the folder measures a net voltage gain exceeding Q.

c. Tested? Tesla's lighting demonstrations (1891-93) are described in the lectures; the bifilar coil was measured once, in 1990, by Nichelson and M. King at Eyring Corp.; Nichelson is the advocate. No independent replication found.

d. If true (net gain): nothing could produce more output than input without violating energy conservation; a mere resonance rise would change nothing.

Verdict (split). Resonant voltage rise, high-frequency lighting, built-in capacitance: Established. Bifilar coil "neutralises self-induction by capacity" at its self-resonance: Plausible, untested (consistent with ordinary self-resonance; no neutral test of Tesla's stated purpose). "Voltage gain" as measured/calculated by Nichelson: Internally inconsistent (reactance with R = 1 compared with an S21 ratio) and Never properly tested.

Claim 4: energy from the ambient medium ("fuelless" generator, radiant energy)

a. Consistency. - Tesla's 1900 water-tank/sink argument: pumping out what flows in costs as much as is recovered "not a bit more" unless the inflow is converted into something else for free. Thermodynamic check [calc, in Thermodynamics2 deep read]: sink at 77 K in 300 K surroundings: engine efficiency 1 - 77/300 = 0.743; reversible refrigerator pumping the rejected 0.257 J out needs 0.257/0.345 = 0.743 J: exactly cancels. Tesla's escape route needs eta = 1, i.e. T_cold = 0 or conversion of single-reservoir heat entirely into work. The rods-to-space example is an ordinary engine between two naturally maintained reservoirs, not a violation. Nichelson's quote of Kelvin drops "of the coldest of" (deep read). - Elsasser's criteria for a self-sustaining disk dynamo quoted by Nichelson (conductor 1000x better, much faster, or "many miles" large) and his own admission that the unipolar dynamo runs "until heat losses ... equal the rotational energy of the flywheel" concedes finite stored energy. - Fuelless arithmetic: 0.5 x 0.25 F x (3.6e5 V)^2 = 1.6e10 J = 4.5 MWh (the paper prints 10^11). A spherical-shell estimate for the earth to a 60 km shell is C = 4 pi eps0 R^2 / h = 0.075 F [calc], so Nichelson's 0.25 F from q/V is a measured-ratio estimate, same order of magnitude. - Energy-availability estimate for collecting charge from the air (patent 685,957 "radiant energy" collector): fair-weather vertical field about 130 V/m and air-earth current density about 2e-12 A/m^2 [memory, medium]. A collector of 1 km^2 at 100 m height: I = 2e-6 A, V = 13 kV, P = 0.026 W [calc]. Cosmic-ray power at ground level is of the order 1e-5 W/m^2 at most (energy density about 1 eV/cm^3 times c/4; sea level much lower after atmospheric absorption) [calc, memory medium], i.e. at most about 10 W per km^2. So the ambient electrical and cosmic sources available to a ground collector are at the watt level. - The 1893 Fuelless "250,000 times" energy figure is the stored-inter-turn-energy ratio described above, not an extractable energy.

b. Experiment. No output measurement of any such device is in the folder. Tesla's patent says "as I have actually observed" a condenser charging "even to the point of rupturing the dielectric" (no values). Gray's and Moray's device claims are unmeasured in the folder; Edwin Gray tests are quoted second hand.

c. Tested? Never by an independent neutral test on record. Tesla's 1931 (Brooklyn Eagle) and 1933 (New York American) statements about harnessing cosmic energy are interview quotes without a device description. The Pierce-Arrow car story appears nowhere in the folder's Fuelless paper (deep read).

d. If true: the Second Law in its Kelvin form (single-reservoir conversion) or energy conservation would have to fail; mainstream thermodynamics already allows using natural gradients (solar, thermal, atmospheric electric) at their measured power levels.

Verdict (split). Using a natural gradient (earth-to-space thermal rods, sunlight, wind): Established. A self-acting machine that derives net work from a single ambient medium without a maintained gradient: Refuted by calculation (Second Law, Carnot case above). Tesla's own caveat (continuous pumping, "no known perfect converter") puts his 1900 text on the Established side. "Radiant energy" particle collector delivering useful power: Refuted by calculation in magnitude (watt-scale per km^2); Never properly tested as a device (no measurement). Nichelson's bifilar-coil-as-extractor reading: Not testable as stated (he calls the key step speculation).

Claim 5: Tesla disc (bladeless) turbine

a. Consistency. - Rim speeds: 18 in at 9,000 rpm gives 215 m/s = 707 ft/s [calc, agrees]. Power 200 hp = 149 kW at 942 rad/s gives torque 158 N m [calc]. The torque-spring scale "22 hp per degree" is Tesla's calibration, seen by reporters only as a beam position (deep read). - Efficiencies see Corrections 5: 27-52 per cent of ideal single-stage expansion against Tesla's "80-90". Tesla's own figures (38 lb/hp-h, 130 BTU) give 52 per cent by his own heat drop [calc]. - Mass basis of "10 hp/lb": see Corrections 4. - Dahlstrand's 38 per cent and the claim "projected 8-12 lb if compounded" are two versions of one projection that differ by 50 per cent (Stockbridge "not more than eight", O'Neill "less than 12").

b. Experiment. Later independent measurements: Armstrong 1952, 14 per cent at 1.11 hp; Rice 1965 and others, 20-35 per cent; Hoya and Guha 2009, 140 W, about 25 per cent, with nozzle losses dominant [web: research/files/hoya-guha-2009-tesla-turbine-test-rig.pdf, from the author's posted copy of Proc. IMechE Part A 223, 451-465]. They sit in or below Tesla's recomputed band and far below 80-90.

c. Tested? Yes, by Tesla's associates (Lowenstein, Richards, Waterside Station, no neutral tester named) and independently by later groups; the Allis-Chalmers report exists only as a quotation in O'Neill; no primary copy found (Investigator).

d. If true (80-90 per cent): nothing fundamental, since it would only mean the boundary-layer drag mechanism works better than all later measurements show; it is a matter of engineering performance, not new physics.

Verdict (split). "The disc turbine works as a viable machine": Established (but at 20-52 per cent of ideal, not 80-90). "80-90 per cent efficiency": Tested, not supported. "10 hp per pound": Never properly tested (mass basis undefined).

Claim 6: teleforce "death ray" and the fate of the papers

a. Consistency. No equations or apparatus exist. A column-density check [calc; memory, high for the constants]: a horizontal path of 400 km (250 miles) at 10 km altitude (air density about 0.41 kg/m^3) has column density about 1.6e8 kg/m^2 = 1.6e4 g/cm^2, against nuclear interaction length about 90 g/cm^2 in air, so about 170 interaction lengths; charged particles of any energy would not reach the target as a beam, and a neutral-particle beam would diverge. Energy to melt about 100 kg of aluminium engine is about 1e8 J [calc: 100 kg x (0.9 kJ/kgK x 640 K + 397 kJ/kg)], so a beam that melts it in a second needs about 100 MW delivered, no source or beam stage is described.

b. Experiment. None. The FBI file (1998 release, 265 pp.) contains the 1943 OAP seal, Trump's report quoted second-hand, the 1983 airtel. Trump's sentence ("did not include new, sound, workable principles or methods") is reported by PBS and others; Trump was an institutional evaluator with a stake (Investigator). "Project Nick" is documented only in a popular chain (PBS 2000, probably Cheney 1981); no primary record found.

c. Tested? Never: no device exists on record. Whether the papers contained a design is unknown; the deciding records are NARA RG 131, RG 342, and the Tesla Museum archive in Belgrade (163,911 call numbers, Investigator).

d. If true: a long-range high-power particle beam within the atmosphere would require physics beyond known beam-transport and scattering/absorption laws; mainstream uses exo-atmospheric particle beams only.

Verdict. The beam as described to the press: Not testable as stated; its strong form (conventional charged or neutral particles through 400 km of low-altitude air at lethal power) Refuted by calculation (column density above). The fate-of-papers part is a historical question with no mathematical content; the scale does not fit; open records listed above. Absence of the papers' contents is an anomaly (seal, evaluation, release to Kosanovic in 1952 per the 1981 FBI letter) with several explanations and no neutral record.

Claim 7: priority and discovery claims (cosmic rays, radioactivity, X-rays, radio, etc.)

a. Consistency. No derivations; the claims are attributions. Documented and mathematical content is the rotating-field motor and polyphase system, which is standard: two sinusoidal currents at 90 degrees through perpendicular coil pairs give a constant-magnitude rotating field [memory, high].

b-c. Radio: the US Supreme Court in Marconi Wireless Telegraph Co. v. United States (1943) held Marconi's four-circuit tuning patent invalid for anticipation by Tesla, Lodge and Stone, noting Tesla's 645,576 disclosed the four-circuit tuned system; it did not rule on who invented radio, as many popular pages claim [web: https://mercurians.org/antenna-newsletter/rereading-the-supreme-court-teslas-invention-of-radio/ via search results, which are partly advocacy; read the opinion 320 U.S. 1 for the final wording]. Other priority claims (cosmic rays, atom smasher, electron microscope, artificial radioactivity) rest on O'Neill's and Valone's statements about 1891-92 lecture observations; O'Neill concedes the records "are not available". The Corum analysis of 1899 signals as possibly Jovian is an interpretation by interested parties (CPG), not a test.

d. If true: mainstream attribution of these discoveries (Hess for cosmic rays, Becquerel/Curie for radioactivity, Roentgen for X-rays) would be revised.

Verdict (split). Polyphase/rotating-field motor and tuned-circuit prior art: Established. Radio priority: Established in the limited legal sense above; the broader "Tesla invented radio" is Not testable as stated (depends on definition). Cosmic rays, atom smasher, electron microscope, artificial radioactivity: Never properly tested (no experimental records; one document in the folder quotes lecture phrases only).

Investigator's findings I keep in the record (whatever the verdict)

Colorado Notes 3 July 1899, 30 July, 6 Aug (water-pipe stationary waves) [Correction 2026-10-08, arbiter F9.3: the diary does not record this experiment; the 6 August item is listed among entries omitted for want of time, and the only datum is a 550 ft distance (Notes lines 6320-6327, 6993).]: the diary is more cautious than the 1904 article; antipodal reflection "difficult to believe".
1904 Electrical World article: Lowenstein named as second observer; retreat of the storm to about 300 km; "any terrestrial distance and almost without loss".
1916 legal interview: peak "activity" versus average power (undamped wave gives only 1 kW); bears on "thousands of times greater" in US 787,412 and on claim 4.
Foreign patent family members (GB 1901/11293, GB 1901/13563, AT 16480, FR 354,791; 1921 turbine patents) not read by me; they bear on claims 1, 2, 5.
Marinčić/Wait/Jackson: Tesla's earth-as-resonator idea (Wait: "intriguing similarity with later developments in ELF communications") and Marinčić's list of Tesla's errors; frequency 6.6 Hz not reproduced (Corrections 2).
Corum & Corum receivers (negative-resistance coherer, 8-22 kHz) and Jovian-signal analysis; stake noted.
CPG/Viziv Zenneck-wave tower: >100 patents, no field data found; absence is an anomaly to analyse.
WiTricity resonant transfer: near-field relative of claim 3.
Turbine measurements by others (Armstrong 14%, Rice 20-35%, Hoya and Guha ~25%) and the missing Allis-Chalmers primary report.
Wardenclyffe 1917 demolition accounts (The Sun: Federal Government, wartime precaution; Anderson 1992: commercial development after Boldt's foreclosure); FBI Vault holds three parts (about 354 pp.) against 265 pp. in the folder; Trump report and "Project Nick"; Tesla Museum archive. None bears on the mathematics; they are the deciding records for claim 6.

What would settle the main open points (calculation or measurement)

Claim 1: a controlled field-strength-versus-distance measurement from a Tesla-type grounded resonator over ranges of 10-1000 km at VLF, and the corresponding ground-wave/sky-wave fading prediction; the receiver frequency of the 3 July 1899 entry (from the Notes) would let one compute the node spacing and compare it with the half-hour period and the storm's speed.
Claim 3: repeat the 1990 measurement with both ports calibrated, measure R, report Z (not just |S21|), and compare the 150-fold gap between the N^2/4 capacitance model and the observed 3-fold ratio.
Claim 4: any open-loop output measurement with energy accounting.
Claim 5: the Allis-Chalmers test log.