Lakhovsky George: Mathematician's review (Step 2)

Reviewer: PhD mathematician. Date: 2026-10-08. Scope: derivations, notation, units, hidden assumptions, and every number that can be recomputed. Physics verdicts (what mainstream physics says about cells, radiation and cancer) belong to the Physicist; I touch them only where a calculation settles the matter.

Tags: [read] from a document (page given), [web] with URL, [memory] with confidence, [calc] my own arithmetic (numpy, shown with inputs). Source gap: none for the five core texts, but figures are missing from The Secret of Life copy, and every cited primary paper (Roffo, Metalnikoff, Nodon, Rivera, Rigaux, de Cigna) is known only through Lakhovsky. The books are written by the inventor, who sold the apparatus (stake); the 1929 booklet was published by his own company C.O.L.Y.S.A.

Symbols used throughout. f = frequency (Hz); lambda = wavelength (m); c = 2.998e8 m/s; L = inductance (henry); C = capacitance (farad); R = resistance (ohm); Q = quality factor (how many radians of oscillation before the energy dies by a factor e; Q = omega L / R for a series circuit); k_B T = thermal energy (4.3e-21 J at body temperature 310 K); skin depth delta = 1 / sqrt(pi f mu0 sigma), the depth at which a radio wave in a conductor of conductivity sigma (siemens/m) has dropped to 1/e.

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Part A. Corrections to Step 1 (and to earlier documents)

1. claims.md, claim 4: "patent US 1,962,565, 1931". The French priority date is 2 May 1931 and the US application was filed 13 Nov 1931, but the patent was granted 12 June 1934 [read: deep/US1962565, p.1-2]. Likewise US 2,351,055 was filed 21 Nov 1941 and granted 13 June 1944, after the inventor's death (administratrix Anne-Marie Louise Lakhovsky, assignor to Henry S. Blum) [read: deep/2351055, p.2]. "1931" should be read as a filing/priority year only. The Winey application's "Lakhovsky, 1934" citation matches the grant year.
2. deep/Radiation-and-Waves, section 3 (Rivera table), "26.8 octaves ... internal inconsistency". The inconsistency is real in the 1941 text but its origin can now be shown. Lakhovsky's own 1927 L'Universion prints the same scale: "Ondes radioelectriques (22 octaves connues): 30.000 metres ... a quelques millimetres", then Nichols and Tear rays "6 mm a 0,3 mm", then infra-red "(8 octaves) 0,3 mm a 0,8 micron" [read: research/files/1927-LUniversion_djvu.txt, "Echelle complete des radiations connues"]. [calc] 30,000 m divided by 2^22 = 7.2 mm, so 22 octaves ends at "a few millimetres", exactly as printed in 1927. The 1941 table's lower limit "0.00025 m" (0.25 mm) overlaps the infra-red row and gives 26.8 octaves; it is a misprint or mis-translation of the 1927 lower limit (about 7 mm). The other rows (X 11.3 vs 12 octaves, UV 4.7 vs 5, visible 1.0, IR 8.4-8.5 vs 8) are fine. Rivera's table is therefore not "internally inconsistent" in the original; the 1941 English copy is garbled.
3. deep/Radiation-and-Waves section 7 (lead thickness). The deep read reports "cosmic rays traverse at most 11 m of lead" and notes the Secret of Life objection of "7 m or more". The earlier book gives a third value: L'Universion (1927) says "un ecran de plomb de 2 m d'epaisseur, une couche de terre de 10 m suffisent a peine" and, from a lake experiment, "35 m d'eau, soit l'equivalent de 1,80 m de plomb" [read: 1927-LUniversion_djvu.txt, lines ~1044 and ~4531-4536]. So the same author's lead figure for the penetrating cosmic radiation is 2 m (1927), 7 m (The Secret of Life, objection as quoted) and 11 m (1941), while the 25 m lead sphere stays fixed. The number drifts with the literature, the conclusion does not. Add to claim 5.
4. claims.md, claim 3 and facts.json (Radiations and Waves). Agree with the Investigator: the 1924 note has a fourth author (J. Magrou, Pasteur Institute) and the design was 3 treated plants against 16 controls [web: https://www.colysa.ch/le-radio-cellulo-oscillateur-originel/; seller, interested party]. I add that the three plants received different doses (2x3 h = 6 h, 11x3 h = 33 h, 3x3 h = 9 h; a 5.5-fold range) yet necrosis began "about 16 days" after the first session in all three (see claim 3).
5. Investigator lead on the Kotzareff "40 W". Verified in the OCR of the thesis: "il nous apprit que son emetteur avait quarante watts de puissance absorbee", preceded by "il a travaille avec des ondes relativement petites" [read: research/files/1931-Kotzareff-these..._djvu.txt, lines 3738-3741]. It is 40 W absorbed (input) power, not radiated. Kotzareff's own sets were 20 W, 100 W and 150 W at 4-5 m (same file, lines ~6120-6310), so 40 W is a modest transmitter of the time.
6. deep/The-Secret-of-Life section 3.4 (b) (lumped LC estimate L = 8.6e-12 H, C = 1.2e-16 F, f = 4.9 THz, lambda = 61 micrometres): I reproduced these numbers [calc]. The note is right that the deep reader supplied the inputs, not Lakhovsky. Add one fact the deep read left out: with the electrolyte core Lakhovsky himself specifies ("conducting fluid containing all the mineral salts found in sea water" [read: Waves That Heal p.13-14]) the resistance of that same filament is of order 10^7-10^8 ohm, which kills the oscillation (see claim 1).
7. Investigator "Corrections" 2 and 3 (undated Societe de Biologie note; Gosset and Gutmann only) are accepted and checked against CH 135072, which cites "C. R. de la Societe de Biologie, 26 juillet 1924, page 626" [read: research/pages/gp-CH135072A.html]. The same patent states the plant spiral's own wavelength is "a peu pres 2 m" and cites C. R. Acad. Sci. 188, p.657 [read: same file].
8. Credit where due (a check that passed). The Secret of Life digest says penetration depth in soil is "inversely proportional to the square root of the product of vibration and soil conductivity" and gives 16,000 m waves reaching 80 m in sand or limestone, 2 m in seawater, a few dozen metres in clay [read: digest The-Secret-of-Life.04, p.53]. That is the skin-depth law. [calc] f = c/16,000 m = 18.7 kHz; delta (seawater, sigma = 4 S/m) = 1.84 m; delta (dry soil, sigma = 2 mS/m) = 82 m; delta (clay, sigma = 0.05 S/m) = 16 m. All match his figures. Lakhovsky's electrical arithmetic is usually right; the problems are in the model it is applied to.
9. No error found in facts.json's arithmetic ("750,000 to 3 billion Hz" for 400 m to 10 cm: 0.75 MHz and 3 GHz, correct; "trillion" = 10^12 is used consistently, e.g. visible light "375 to 750 trillion" in 1927 matches 3.75e14-7.5e14 Hz).

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Part B. Per-claim review

Claim 1. Every cell is an electrical oscillating circuit (chromosome filament with L and C, radiating in the infra-red at about twice its length, powered by cosmic radiation)

a. Is the mathematics internally consistent? The pieces are, one at a time; the whole is not. - Lakhovsky writes no formulas. The relation behind his words is f = 1 / (2 pi sqrt(LC)) and lambda = c / f [memory, high]. His two recipes are (i) estimate L from a micrograph (Corynactis viridis, magnified 1,000 times), guess C ("taking certain average values") and find "a radiation localized in the infra-red region"; (ii) lambda = 2 x (filament length), "the half-wave principle ... as the electric dipoles of Hertz" [read: Secret of Life p.23]. No filament length, L, C or wavelength is printed. There is nothing to reproduce; he calls both methods inaccurate. - Recipe (ii) is the half-wave rule for a thin perfect conductor surrounded by empty space. [calc] For filament lengths 1-10 micrometres it gives lambda = 2-20 micrometres, f = 15-150 THz, infra-red, consistent with his qualitative statement. - Hidden assumption 1: the core conducts like metal. He says the filament is "filled with a conducting fluid containing all the mineral salts found in sea water" [read: Waves That Heal p.13-14; Secret of Life p.23 "mineral conductors"]. Seawater has sigma of about 5 S/m, thousands of times worse than copper (6e7 S/m) [memory, high]. [calc] Take a straight filament of length 10 micrometres and radius 0.1 micrometre (my inputs). L = (mu0 l / 2 pi)(ln(2l/a) - 1) = 8.6e-12 H; C = 2 pi eps0 l / ln(l/a) = 1.2e-16 F; f0 = 4.9 THz. Its resistance is R = l / (sigma pi a^2) = 6.4e7 ohm. The critical resistance for oscillation is 2 sqrt(L/C) = 534 ohm. R is 1.2e5 times too large: Q = omega L / R = 4e-6, and the energy decays in 2L/R = 3e-19 s, about a million times faster than one period (2e-13 s). A shorter, thinner filament (1 micrometre, 0.05 micrometre) is no better (Q = 7e-6). Putting water around it (relative permittivity 80 [memory, high]) lowers f0 by about 9 and Q by another factor of 9. So, with Lakhovsky's own material (salt water inside an insulating sheath), the "circuit" is overdamped by five to six orders of magnitude: it cannot oscillate even once. - Hidden assumption 2: surrounding the filament by empty space. A filament inside a cell is in water and protein at 310 K; radiated power (radiation resistance of a 10-micrometre dipole is far below a milliohm) is irrelevant next to the loss in R. - Hidden assumption 3, scale mismatch: the cell's circuit lives at micrometre wavelengths (10^13-10^14 Hz), his apparatus at 2-10 m (30-150 MHz), a ratio of 10^5-10^7 in frequency [calc]. The text never says how a 2 m wave drives a 10 micrometre resonator at its own frequency. The only link offered is "harmonics" (see claim 4), which for a 10 cm to 400 m device is the 10^5-th harmonic of the highest fundamental [calc: 3e14 Hz / 3e9 Hz = 1e5], and one that nothing in the patents produces. - Hidden assumption 4, the infinitesimal-energy argument (Secret of Life pp.62-63: "Australia to Europe"). A tuned receiver can use a very weak signal: true. But the argument omits noise. [calc] Take the strong field of 1 V/m at about 100 MHz (a few hundred metres from a broadcast transmitter [memory, medium]) and a cell volume (10 micrometre)^3 with relative permittivity 80: the electrical energy stored is (1/2) eps0 eps_r E^2 V = 3.5e-25 J, which is 8e-5 of k_B T = 4.3e-21 J. For natural fields (atmospherics, microvolts to millivolts per metre) the ratio is 10^-11 to 10^-17 [calc, E = 1e-3 and 1e-6 V/m]. A resonance can raise the amplitude only by Q, and Q of a salt-water circuit is below one. This is the standard "k_B T problem" argued by R. K. Adair, Phys. Rev. A 43, 1039-1048 (1991) [web: https://authors.library.caltech.edu/records/dgkv5-5pg42 and bibliographic record in search results; I confirmed the citation and the thermal-noise thrust, not the full text]. - Hidden assumption 5, spectral coincidence. At 310 K the body's own thermal radiation peaks at 2.898e-3 / 310 = 9.3 micrometres [calc, Wien's law] and radiates sigma T^4 = 524 W/m^2 [calc]. His predicted cell band (2-20 micrometres) is the band of the body's and the surroundings' ordinary heat radiation, which dwarfs any external "cosmic" input at that wavelength (see claim 5). Any 10-micrometre resonance would be driven by heat, not by cosmic rays. - Energy source contradiction: the model says the cells' oscillation is "maintained by cosmic radiation" and that body temperature (98.2 F) is Joule heating of the cellular circuits by cosmic, atmospheric and telluric waves; at death we cool "because the cells have ceased to oscillate" [read: Radiations and Waves pp.49-50 (deep read)]. [calc] Cosmic-ray secondaries at sea level deliver about 170 muons per square metre per second at roughly 4 GeV each [memory, medium; order of magnitude only, the web check did not return the PDG page]: 170 x 1.8 m^2 x 6.4e-10 J = 2e-7 W on a whole body, against a resting metabolic output near 100 W [memory, high]. The ratio is 5e8. The body's heat has a known source (metabolism); it stops when metabolism stops and the body cools by Newton's law. The statement is refuted by the energy budget, and a factor of 10 error in my flux changes nothing. - Nature's contemporary review identifies Fig. 14's "oscillating circuit" as "no more than part of a stinging cell" (the thread of a cnidocyst) [web: via research/2026-10-08-internet-archive.md, Nature 146, 26 Oct 1940, J. Ritchie]. This bears on the premise that the picture shows a filament circuit, not on the arithmetic.

b. Consistent with established experiments? No measurement of any resonance, inductance or capacitance of a chromosome exists in the sources. Electrical properties of tissue are measured and are those of lossy electrolytes (conductivity about 0.7 S/m at 150 MHz in muscle [memory, medium]; skin depth [calc] about 5 cm at sigma = 0.7 S/m). Nodon's electrometer results (discharge in 25-500 s, 3-15 times uranium for insects) are reported second-hand with no controls [read: Secret of Life p.24]; I have not seen the original [Revue Scientifique, 22 Oct 1927, p.609].

c. Tested? Replicated? The circuit picture itself was never tested; no person is recorded as measuring a cell's filament resonance. Mitogenetic rays (Gurwitsch), cited as confirmation, are cited without reference. Related later ideas (Frohlich's coherent membrane oscillations near 100 GHz, cited by Winey) are a different mechanism and differ in frequency by three orders of magnitude from Lakhovsky's [read: deep/piezo, p.34].

d. If true, what in mainstream physics would change? One would need cells whose internal structure behaves as high-Q resonators in water at 310 K: ionic conduction would have to behave like metallic conduction at THz; thermal noise at 310 K would have to be absent from the signal path; and ambient fields far below k_B T would have to be capable of driving it. Condensed-matter and biophysics (dielectric relaxation of water near 20 GHz [memory, high], thermal equilibrium) would have to be revised, not just extended.

Verdict: split. (i) The specific model (salt-water-filled insulated filament as an LC circuit at infra-red wavelengths, powered by cosmic radiation, heating the body): Refuted by calculation (Q of order 10^-6 with his own assumptions; energy supply short by 10^8-10^9). (ii) The vague form (cells have characteristic oscillations that outside fields can affect): Not testable as stated (no frequency, amplitude or coupling is ever given).

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Claim 2. Disease is oscillatory disequilibrium; cancer cells oscillate at a different frequency; a "war of radiations" (microbe, irradiated cholesterol, altered cell constants)

a. Internal consistency. - No frequency is attached to any disease, microbe or cancer cell. The one worked example is the mineral ratio: a cell has 29 % mineral content (Aethalium septicum, an unrelated slime mould); a hypothetical microbe has 40; "its oscillating power and consequently its frequency are not the same" [read: Secret of Life p.65]. No function maps mineral ratio to frequency, so 29 versus 40 yields no number and no prediction. This is a verbal example, not a calculation. - The typhoid/coli experiment shows two bacteria migrating to opposite electrodes of a battery (electrophoresis: opposite surface charge sign) [read: Secret of Life p.27]. Opposite DC charge does not imply different oscillation frequency; this is the step "charge to frequency" that is missing. No voltage, field or direction is given. - Mechanism in 1941: radiation from a hydrocarbon of "very high frequency" reinforces the chondromes (mitochondria) by resonance but interferes with chromosomes "whose frequency is much lower"; the result is a membrane-bound chondrome, "a neoplasic cell" [read: Radiations and Waves p.56 (deep read)]. A resonance reinforces only a signal at the resonant frequency, so a single "very high frequency" source cannot be simultaneously resonant with the chondromes and non-resonant with the chromosomes unless their frequencies are given. They are not. He also needs the chromosomes (about 10-100 micrometres of filament) to resonate lower than the much smaller chondromes, which is consistent with the half-wave rule; but no numbers are put to it. - Rivera's table: Lakhovsky says all radiations above yellow are "malevolent and cancerogenic" and all below stimulate division [read: p.53 (deep read)]. The printed table gives effects "for doses existing in ambient centers" and calls gamma "definitely stimulative on vegetables"; it never names yellow. The dividing line at yellow (about 580 nm, photon energy 2.1 eV [calc, hc/lambda]) is his gloss. Also the age of onset is stated as 45 (heading), 40 (conclusion) and 50 ("the most common age"), p.57-61. - Roffo's formulae: "irradiated cholesterol (C21H36O8) and non-irradiated cholesterol (C27H46O)" [read: page image p.54, per deep read]. Cholesterol is C27H46O [memory, high]. Irradiation (photo-oxidation, UV) would be expected to leave the 27 carbons in place. Going from C27 to C21 loses 6 carbons while adding 7 oxygens and removing 10 hydrogens; no reaction is offered and no source for the formula is cited. It may be a mistranslation, but as printed it is unbalanced chemistry offered as evidence. Mass check [calc]: C27H46O = 386.7 g/mol, C21H36O8 = 416.5 g/mol. Roffo's own quoted letter (27 Sept 1938) says only "photo-active and luminescent agent"; "radio-active" is Lakhovsky's word. - Roffo's 55 %: "more than 55 % of thousands of mice" (1941) versus 55 % of "rats" with sarcoma (Secret of Life) [read: deep reads]. The species differs between the two retellings. No counts. [calc] If "thousands" meant 2,000 animals, 55 % is 1,100 tumours and the controls' zero is trivially significant; the issue is not statistics but whether such an experiment exists in the primary literature, which I could not check (Gallica/BHL blocked for the Investigator). - Thorotrast: 11 of 17 rats (65 %) and 4 of 5 (80 %) with sarcoma [read: p.55]. Arithmetic fine. With n = 5 the 80 % has a wide uncertainty (exact 95 % interval about 28 % to 99 % [calc, Clopper-Pearson]). This is a result about a radioactive drug, and says nothing about oscillation. - Metalnikoff's quoted conclusion (radon alters Sarcina morphology) contains no mention of oscillation. Roffo's 1931 sentence says the results "could be interpreted as" and are "compatible with" cellular oscillation. Neither is a quantitative confirmation.

b. Consistent with established experiments? UV and ionising radiation as carcinogens are established [memory, high]; the "interference of frequencies" mechanism for them is not. The yellow dividing line contradicts the fact that visible light and infra-red are not carcinogenic while photon energy is what matters. A single RF photon at 150 MHz has 6.2e-7 eV, 2.3e-5 k_B T [calc]; it cannot break a chemical bond.

c. Tested? The "different frequency" statement could be tested if a frequency were specified; none is. The animal experiments attributed to Roffo are reported but unchecked here (the Investigator found no copy; PubMed's coverage before 1945 is thin).

d. If true: A quantum-mechanical description of carcinogenesis through DNA damage and promotion would be replaced by a classical-resonance picture; that is far outside current physics and biology and, for the reasons in claim 1, has no workable mechanism.

Verdict: Not testable as stated (no frequency or amplitude for any state; mineral-ratio example is non-numerical). Sub-claims by others reported through Lakhovsky (irradiated cholesterol tumours, thorotrast sarcomas) are Never properly tested in this record (primary papers not obtained); the printed chemical formula for irradiated cholesterol is unbalanced and unsourced.

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Claim 3. Restoring oscillation cures; the 2 m Radio-cellulo-oscillator and an open copper spiral necrosed crown-gall tumours on geraniums; collars and belts relieved illnesses

a. Internal consistency. - Frequency conversion: 2 m gives 150 MHz [calc: 2.998e8 / 2 = 1.50e8 Hz]. 2-10 m is 30-150 MHz. The bracketed "[approximately 100-200 million cycles per second]" in the Hatonn-edited copy is wrong (corresponds to 1.5-3 m) and is not Lakhovsky's [read: deep read]. - Half-wave rule applied to the open spiral: lambda about 2 m means about 1 m of wire. [calc] A circle of circumference 1 m has diameter 0.32 m, a plausible plant collar, so the patent's "a peu pres 2 m" is arithmetically consistent with his own half-wave rule [read: CH 135072, via research/pages]. A human collar of one turn (circumference about 0.4 m) would resonate near 0.8 m (about 370 MHz), not 2 m, unless it has several turns; the patent calls the clasp the tuning element, and changing the wire length by a few centimetres changes lambda by only a few percent. - A spiral with "no source of energy of its own" ("aucune source d'energie propre") can only scatter ambient fields. [calc] Even an ideal resonant small loop at 2 m has a maximum effective aperture of 3 lambda^2 / (8 pi) = 0.48 m^2; in a strong 1 V/m field (S = E^2 / 2 Z0 = 1.3 mW/m^2) the most it can intercept is about 0.6 mW, and typical fields are 10^6 times weaker. It cannot deliver therapeutic energy; a mechanism by "tuning" would need to be informational, and none is described. For a conducting body (sigma about 0.7 S/m, skin depth 5 cm at 150 MHz [calc]) the collar's Q would be small as well. - The geranium note (1924): 3 treated plants (6 h, 33 h, 9 h of exposure), 16 untreated controls, necrosis starting about 16 days after the first session in each [web: https://www.colysa.ch/le-radio-cellulo-oscillateur-originel/; seller, an interested party, quoting the 1925 press article that quotes the note; same citation in Roussakow 2013]. Statistics [calc]: if 3 of 19 plants were chosen for treatment at random and all 3 necrosed while all 16 controls did not, the chance of this arrangement is 1 / C(19,3) = 1/969 = 0.001 (exact test). That number is only meaningful if allocation was random; the quoted text does not say how plants were chosen, whether the controls were of equal age, or whether the experimenter knew which plant was which (no blinding). The result that matters is the dose independence: onset at about 16 days after a total of 6 h, 9 h and 33 h. A thermal or cumulative-dose effect would be expected to vary with dose; a fixed lag suggests either a threshold triggered at the first exposure or a process unrelated to dose (for example, a natural time course of tumours at that stage). Three animals cannot distinguish these. I record it as a pertinent pattern, not as a verdict. - Thermal explanation: the only power figure is 40 W absorbed (input) power [read: Kotzareff, as above]. [calc] Heat balance for a 5 mm stem, 10 cm exposed length: surface 1.6e-3 m^2; with natural convection h about 10 W/m^2K [memory, medium] the loss is 0.016 W per kelvin. A rise of 10 K needs 0.16 W absorbed, that is 0.4 % of the input. At lambda = 2 m a few-centimetre-thick plant is electrically small, so the absorbed fraction could be well below that, or, near a resonant spiral, higher. The data in the sources do not allow heating to be excluded or confirmed; the missing quantity is the specific absorption rate (W/kg) in the tumour, which was never measured. Roussakow (a non-thermal-effect advocate, with a stake in that argument) calls the effect "mainly or entirely not temperature dependent" [web: https://onlinelibrary.wiley.com/doi/10.1155/2013/428027]. Schereschewsky, who reported the same band killing mouse tumours in the 1920s, abandoned his non-thermal reading in 1933 under criticism (same source). - Clinical statements: durations are checkable. Mme C.: 8 Sept to 19 Nov 1931 = 72 days [calc]. M.M.: caption says "three months" and "six months" in the same caption [read: Waves That Heal p.32]. Mme S.: 26 Apr to 12 May = 16 days, to 30 May = 34 days; 1941 caption says "thirty-five days later" (25 April to 30 May = 35 days); the booklet's "three weeks" and "cured within a few weeks" do not match either set of dates. Treatment counts: "four treatments" versus three listed (26 Apr 15 min, 28 Apr 10 min, 12 May 20 min). Goitre: 11 Jan to 4 Mar 1938 = 52 days = 7.4 weeks, consistent with "seven weeks". These are small inconsistencies in the record, not evidence either way. - Selection and denominators: the books present about 6 documented cancer cases (Mme C., M.M., Mme S., J.S., plus the goitre and prostate cases) out of "hundreds" (de Cigna); the number treated and the number of failures are not given. The 1941 book admits "some failures" and Dr. H.'s "few negative results", while the 1941 booklet says "without exception all the patients remained well" (discrepancy 14 in the deep read). The proportion cured cannot be computed: the denominator is missing, which is the key quantity. Confounds present in the text: prior X-ray and radium (Mme S., Mme C.), silver nitrate dressings applied under the apparatus by Rigaux, hospital care, biopsy possibly removing tissue. Rodent ulcer (basal cell carcinoma) is slow-growing and local [memory, high]; stated durations of 3 and 15 years fit that. - Collars and belts: reports are testimonials from practitioners 1928-29; symptoms "returned when the collar broke" is an observation with no blinding and no control; the same wire shape is claimed to cure rheumatism, phlebitis, haemorrhage and sarcoma of the ovary [read: Waves That Heal pp.25-29, deep read section 8].

b. Consistent with established experiments? Radio-frequency heating of tumours (hyperthermia) works and is established at levels that heat tissue; whether a non-thermal 150 MHz effect exists in plants is the open question the Investigator documented (Schereschewsky, Roussakow). The Physicist and Contrarian should weigh the dispute; the arithmetic cannot decide it without a dose.

c. Tested? By whom? Replicated? The 1924 experiment was done by four authors including a Pasteur Institute plant pathologist (Magrou) and a Paris surgeon (Gosset); Kotzareff (1931), a competing physician, calls it "seule et unique au point de vue experimental" [read: Investigator note]. I found no replication in the sources. Lakhovsky's 1941 retelling mentions pelargoniums "in 1924"; the Society note gives 26 July 1924. No controlled clinical test of the collars or the MWO is recorded; a New York hospital table and Italian and Belgian physician reports exist only as Lakhovsky reports them.

d. If true: A non-thermal, tuned-frequency effect at 150 MHz on tumour tissue would require a field-sensing mechanism whose energy per event is far below k_B T (see claim 1); that would force a re-examination of RF dosimetry limits and of the thermal-only picture of RF interaction.

Verdict: Never properly tested. Geranium experiment: a documented small controlled experiment (3 vs 16) with no recorded randomisation, no dose measurement, no replication and no thermal control; not dismissed by a failed replication, simply never repeated in the record I have. Collars and belts and the case reports: Never properly tested (uncontrolled; confounded; denominators missing). The mechanism by which a passive unpowered spiral "supplies" energy: Refuted by calculation as an energy supply (microwatts at most); it could only act as an informational or field-shaping device, for which no model is given.

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Claim 4. The Multiple Wave Oscillator emits 10 cm to 400 m plus harmonics to the infra-red from concentric split rings; has cured cancer and other conditions; no contra-indication

a. Internal consistency. - The patents: US 1,962,565 gives no number (no wavelength, ring size, voltage or spark gap) and no medical word [read: deep/US1962565, "Numbers: none"]. US 2,351,055 also gives no numbers [read: deep/2351055]. Both are structurally coherent: a spark gap exciting a coil; the coil coupling to open rings; passive rings re-radiating at their own resonance. The patent's rationale ("among these waves or their harmonics there is almost always one ... capable of producing the best effect") is a shotgun argument and mathematically fine as a statement about broadband emission. - The spectrum figures come from the books only. Check the geometry. A split ring behaves as a resonator when its wire length is about lambda/2 [memory, high]; for diameter D that gives lambda about 2 pi D. [calc] D = 0.1 m gives 0.63 m (477 MHz); D = 0.4 m gives 2.5 m (119 MHz); D = 1.0 m gives 6.3 m (48 MHz); D = 1.5 m gives 9.4 m (32 MHz). A 10 cm fundamental needs a ring of diameter 1.6 cm; a 400 m fundamental needs one of diameter 64 m. The patent's apparatus is a table or floor-stand device (Fig. 3); its biggest ring is of the order of 1 m. Sphere capacitances across the gaps (spheres of 1 cm radius give about 0.6 pF in series [calc]) do not rescue the long end: to bring a 1 m ring to 400 m (0.75 MHz) needs 13 nF, equal to the capacitance of an isolated sphere of 230 m radius [calc: C = 1.28e-8 F]. Therefore: the rings of the patent give fundamentals from decimetres up to a few metres at most; "10 cm to 400 m" (a span of 12 octaves [calc: log2(4000) = 12.0]) cannot come from the rings. The 400 m end might refer to the driver circuit (a coil with a large condenser), but then it is not radiated by the rings. The books do not distinguish. - Different statements of range in the same paragraph: "all frequencies from 3 metres to the infra-red" versus "all fundamental wave lengths from 10 cms. to 400 metres" [read: Waves That Heal p.30, 1941 book p.35]. 3 m and 10 cm differ by a factor of 30; and Dr. Rigaux's 1933 report gives "from 10 to 400 metres" with "750,000 to 3 billion", which would be 30 MHz to 0.75 MHz, not 3 GHz [calc]. The unit slipped somewhere between Lakhovsky and Rigaux; the arithmetic of frequency versus wavelength inside each phrase is right (400 m = 0.75 MHz, 10 cm = 3 GHz). - Harmonics to light: "numerous harmonics which, together with fundamental waves ... may extend as far as the infra-red and visible light (1-300 trillion vibrations per second)" [read: Waves That Heal p.30]. From the 3 GHz top fundamental to 300 THz is the 100,000th harmonic [calc]. A damped spark discharge has a continuous spectrum that falls off with frequency; no ring "harmonic" reaches infra-red. What the photographs show (white brush discharge on the rim, Fig. 3 [read: page image p.32]) is real light, but light from the glowing discharge (thermal and atomic emission), not a harmonic of the ring. - "Many wavelengths at once" as a quantitative claim: a spark-gap circuit emits damped waves. If each ring has Q of 10-30 [memory, medium; typical for loaded rings of that era], its band is f/Q, or 3-10 % wide [calc]. For rings with diameters running from 0.1 m to 0.5 m (a 5:1 range, 2.3 octaves) one needs about ln 5 / ln 1.1 = 17 rings spaced by 10 % to cover the band without gaps [calc]. The figures in Fig. 1 (six rings a-f) cover less than that. So the device spans roughly one decade of wavelength at best, from decimetres to a few metres, with gaps; a continuous "all wavelengths" statement is an overstatement. This is a calculation from the patent's geometry with my assumed Q; the patent gives no Q. - Safety statements: "cannot cause any injurious effects. As all the radiations ... are of an electrostatic nature they cannot over-heat or burn" [read: Waves That Heal p.33]. The same page calls the field "electro-magnetic" (p.31) and the photograph shows a visible brush discharge (corona), i.e. a heating discharge, though tiny. A time-varying electric field at 30-150 MHz is accompanied by a magnetic field and a displacement current; "electrostatic" is a misnomer. The absorbed power is set by the field strength (never stated), not by the adjective. Also tissue absorbs RF at these frequencies with skin depth about 5 cm at 150 MHz [calc], so "heating" depends on dose, and 15-minute sessions at 4-5 feet from a transmitter of unspecified power are not shown to be harmless or harmful. "No contra-indication ever reported" is an absence of reports from the manufacturer's own users, not a safety trial.

b. Consistent with experiments? The patents describe an operable spark-gap radio-frequency generator, and the physics of coupled resonant rings is textbook (Hertz, Marconi). The clinical reports are not assessable by arithmetic. Hyperthermia and short-wave diathermy, the neighbouring practices, are in medicine; the MWO's dose is unmeasured.

c. Tested? Hospital St-Louis (Paris), Calvaire clinic, Genoa (de Cigna), the Institut de Physique Biologique, Brussels (Dr. H./Henry) report cases [read: deep reads]. No controlled study. PubMed search by the Investigator found no clinical trial of the MWO (Farrell 1982 is a history note; Reid 1989 a hypothesis) [web: research/2026-10-08-pubmed.md]. The MWO patent family (FR 732276, filed 2 May 1931; certificate of addition FR 42146E; CH 164027) is confirmed by the Investigator; the Swiss text names therapy and adds gas tubes while the US text does not [web: https://patents.google.com/patent/CH164027A/fr].

d. If true: Cancer cures at unmeasured, probably sub-thermal doses of metre-wave fields would require an unknown field-to-biology transduction; and the energy/noise argument of claim 1 would have to fail.

Verdict: split. (i) The apparatus as a spark-gap multi-ring radiator: Plausible, untested as a producer of several simultaneous damped frequencies in the decimetre-to-metre range (what the patents describe; no measurement of its spectrum is on record). (ii) The stated spectrum "10 cm to 400 m plus harmonics to infra-red/visible": Internally inconsistent (ring geometry cannot reach 400 m; "3 m" and "10 cm" for the same range; 1e5-th harmonic). (iii) Cures of cancer, goitre, prostate, ulcers: Never properly tested (case series with no denominators; confounders; one-sided source). (iv) "No contra-indication": Not testable as stated (an absence of reports).

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Claim 5. Universion/cosmic waves drive radioactivity and cancer geography: lead shielding would remove radium's radioactivity; young skin screens radiation until 40-45; soil and water conductivity set cancer incidence

a. Internal consistency. - Lead sphere: "a solid 25 meters in diameter" and then "25 meters thick"; "cosmic rays can traverse a maximum thickness of 11 meters of lead"; "If that thickness were reduced to 1 meter, radio-activity would be increased considerably" [read: Radiations and Waves p.59 (deep read)]. Radius of a 25 m diameter sphere is 12.5 m, still above 11 m, so his reasoning holds with either wording. But the logical structure fails on three counts. (1) The lead figure drifts: 2 m (1927), 7 m (Secret of Life), 11 m (1941), see Correction 3. (2) Progressively thinner shields should give progressively less radioactivity loss, so "1 m" gives "considerably increased" (OK in his scheme), yet the sentence ends by asking only "a lead plate a fraction of a millimetre thick" to stop the hydrocarbons' radiation, a different role for lead (blocking an emitted radiation instead of an incoming trigger). The passage shifts between the two without notice. (3) The premise (cosmic rays set the decay rate) is a quantitative claim with a testable consequence: radium's activity would be proportional to the cosmic flux. - Energy budget [calc]: 1 g of Ra-226 has 3.7e10 decays/s with a 4.78 MeV alpha each: 0.028 W from alphas alone (about 0.11 W with the daughters, I took four decays per chain as an estimate). The cosmic flux through a 1 cm^2 area is about 1 muon per minute (0.017 per s) at about 4 GeV [memory, medium; the web check did not retrieve the PDG page]: 1.1e-11 W. The radium emits 2.6e9 times more power than the cosmic radiation falling on a 1 cm^2 target. The radiation cannot be the energy source. Lakhovsky's wording is "removed from the action of cosmic waves and the compression of its spacial void would be eliminated", i.e. a trigger, not a supply, but then a trigger of this size would have to act on 10^21 nuclei per gram through a flux of order 1/cm^2/min, which is 10^-12 events per nucleus per second against a decay rate of order 10^-11 per nucleus per second for Ra-226 [calc: 3.7e10 per s / 2.7e21 atoms per g = 1.4e-11 per s]. The numbers aren't absurd by themselves for a trigger; the point is that the activity would then vary by orders of magnitude between sea level, mountain top and mines. Sea-level muon intensity is reduced by a factor of order 10^6 by deep underground sites, and Ra sources are used in underground low-background laboratories [memory, medium]. The 11 m lead equivalent is about 125 m of water [calc: 11 m x 11.34 g/cm^3 = 12,470 g/cm^2], far less than the 1,000-3,000 m water equivalent of deep underground labs [memory, high]. If decay rates depended on cosmic flux, there would be a measured depth dependence; none is known to me [memory, medium; for the Physicist to confirm]. - Earth's rotation as the cells' energy source [read: Secret of Life p.64]: the equator moves at 465 m/s, which he states as 27 km per minute (27.9 km/min [calc]), correct. The motional field is v x B with the interplanetary field of about 5 nT [memory, medium]: 465 x 5e-9 = 2.3 microvolts per metre, a DC field, so about 2e-11 V over a 10-micrometre cell. No oscillation, no energy. "All the energy of radiation at the earth's surface comes from electromagnetic induction brought about by the rotation of the earth" is contradicted by the sun's thermal input (about 1,000 W/m^2 [memory, high]) alone. - Raphael Dubois "250 million years at one million per second" for the units in a silkworm egg [read: Secret of Life p.64]: [calc] 250e6 years x 3.156e7 s x 1e6 per s = 7.9e21 units. One milligram of water holds 3.3e19 molecules [calc, 1e-3 g / 18 g/mol x 6.022e23]. A silkworm egg weighs about a milligram or less [memory, low-medium]. Then the stated number of "units" exceeds the number of molecules in the whole egg by a factor of about 240. Either the quoted sum is a rhetorical figure or the units are not molecules and aggregates, in which case the count would be even smaller. The quotation is Dubois' as relayed; as a number it cannot be literal. - Skin argument: young skin has "all the cells tightly pressed", old skin "widely spaced" and filled with "inert" connective tissue, which "permits the infiltration of all kinds of detrimental radiations" [read: Radiations and Waves p.58]. [calc] Skin is 1-3 mm thick. At 150 MHz tissue skin depth is about 5 cm; at the infra-red wavelengths of his own cell model (micrometres) absorption by water is complete within micrometres to a fraction of a millimetre [memory, medium]. Radiation that does penetrate skin (RF, gamma) is not stopped by the gaps between cells, whose scale (micrometres) is much smaller than any penetrating wavelength except visible and UV. No wavelength-dependent attenuation is computed. Age at onset is given as 45, 40 and 50 in different places. - Soil and cancer: the skin-depth law is right (see Correction 8). The data: the Contributo (1927) prints "densite de mortalite par cancer pour mille habitants et par an" for the 80 Paris quartiers (annual deaths per 1,000 inhabitants, mean of 1921-22) from the Paris statistical yearbook of 1925 [read: research/files/1927-Contributo-eziologia-cancro_djvu.txt, lines ~185-200 and the table at ~1385-2020]. Values he cites: low (sand, Beauchamp/Fontainebleau): 0.3 (Gaillon), 0.5, 0.6, 0.8, 1.04-1.1; high (plastic clay and gypsum marl): Auteuil 1.76, Javel 1.61, Grenelle 2.08, Saint-Lambert 1.57, Saint-Vincent-de-Paul 1.97, Hopital Saint-Louis 1.44, Pere-Lachaise 1.58, Charonne 1.41. [calc] Means 0.72 (n = 6) and 1.68 (n = 8); the groups do not overlap, so the exact rank test gives p = 1/C(14,6) = 3.3e-4 (one-sided) for this selection. This is statistically striking but is a post hoc selection of the extreme quartiers described by Lakhovsky after seeing the data, not a pre-specified comparison. Confounders visible in the data itself: (i) the quantity is deaths per resident, not age-standardised; (ii) three of the "high" quartiers contain hospitals or a cemetery (Saint-Vincent-de-Paul, Hopital Saint-Louis, Pere-Lachaise), so deaths are recorded where patients die; Lakhovsky explains Saint-Louis by X-ray apparatus whose "waves interfere with cosmic waves" [read: Contributo line ~223], an ad hoc rescue that he does not apply to the other two; (iii) the highest value in the whole table is Salpetriere at 2.80 (a hospital quartier, and the hospital where Lakhovsky himself worked), and I could not find it discussed in the Contributo text; (iv) the clay districts (Grenelle, Javel, Auteuil) are the industrial riverside; income, occupation, age and diagnosis practice differ with geology in Paris. Wealth, age and hospital location are standard ecological confounders. Also the OCR table could not be aligned name by name with the geology column, so I could not test all 80 values, only the ones he quotes in the text. - The mechanism chain "soil conductivity -> screening of cosmic waves -> cancer" uses the skin-depth law for 16,000 m waves (18.7 kHz) while the cell resonance is at micrometres; two spectral regions are linked only by the phrase "cosmic waves cover the whole range".

b. Consistent with experiments? Lakhovsky's own statement of the objection (cosmic rays passing 7 m of lead cannot drive a nucleus-sized circuit) is a fair one, and his reply (cosmic waves include radio atmospherics of thousands of metres) changes the meaning of "cosmic": atmospherics come from lightning [memory, high]. His calculation of penetration depth is correct; his link to cancer is not shown.

c. Tested? The soil-cancer correlation has not been tested with adjusted data in the sources. The shielded-radium prediction has not been tested by him: it is a thought experiment ("If you enclosed...").

d. If true: Radioactive decay rates would depend on the ambient cosmic flux; nuclear half-lives, precisely measured in different laboratories, would not be constants. Mass-energy bookkeeping for radioactive decay (binding energy, Q-values) would need replacing.

Verdict: split. (i) Cosmic "universion" driving radioactivity (shielding would stop radium): Refuted by calculation (energy budget 10^9 short if the radiation is the supply; a trigger idea implies depth-dependent decay rates that are not observed [memory, medium; Physicist to confirm]); the lead thickness is stated inconsistently (2, 7, 11 m). (ii) Skin screens cosmic/carcinogenic radiation until 40-45 and the oscillator restores it: Internally inconsistent (45/40/50; the screen is invoked both for incoming triggers and outgoing emission; cell spacing irrelevant at the relevant wavelengths). (iii) Cancer geography follows soil/water conductivity: Never properly tested (a clear tabulated claim from 1921-22 Paris deaths; not age-adjusted; hospital quartiers; post hoc grouping; pre-specified analysis never done).

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Claim 6. Treating prospective mothers gives healthier children; heredity is fixed by the nucleus's vibration

a. Internal consistency. The heredity model: "the hereditary substance is ... in the nucleus; and the specific differentiation of the nucleus is due to its power of vibrating in accordance with a wavelength determined by the diameter of the circuits ... and the value of the nuclear electrical capacity"; the dominant sex cell is that whose wavelength "approximates most closely to the normal standard typical of its sex" [read: Secret of Life p.62]. No wavelength, tolerance or probability is given, so no prediction can be derived. The 1941 book uses 48 chromosomes (24 per gamete) [read: p.62], the count accepted in 1941; 46 was established in 1956 [memory, high]. That is a period fact, not an error. Treatment numbers: three cases of treated mothers ("Child of the Waves") [read: claims.md]; three cases give no statistics; a case in which a tubercular woman of 24 was treated during pregnancy [read: p.62]. A healthy outcome has a high base rate.

b/c. No test is on record. Pour rester jeune a 100 ans (1939) opens with cellular oscillation but is not read beyond the opening [Investigator].

d. If true: Heredity would be carried by a physical resonance property of nuclei, replacing DNA sequence inheritance; the whole of molecular genetics (not yet established when he wrote) would have to be reinterpreted. Mendelian ratios and mutation are not addressed.

Verdict: split. Heredity by nuclear wavelength: Historically superseded but instructive (a physical-resonance theory of heredity proposed before DNA was identified as the carrier, later replaced by molecular genetics) and Not testable as stated (no wavelength or tolerance). The "healthier children" treatment claim: Never properly tested (three anecdotes).

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Claim 7. Later citation (Winey, US 2007/0258329): Lakhovsky treated and cured cancer patients until 1942; a precursor of Rife and Priore; MWO listed among treatments supplying missing frequencies of a "polychromatic" healthy spectrum

a. Internal consistency. Winey writes that Lakhovsky "treated and cured a number of cancer patients" until his death in 1942, citing "Lakhovsky, 1939" [read: deep/piezo, [0109]]. A 1939 book cannot document 1939-1942 treatments; no other source is given. The range statement "from 3 meters to infrared" is given as a wavelength range, consistent with Lakhovsky's own 3 m phrase, and carries the same internal tension as in claim 4. "Polychromatic" (Mae-Wan Ho) is a verbal model: a healthy spectrum is one where all frequencies in range are equally represented; no frequency range, spectral density or metric is defined, so "missing frequencies" cannot be measured. Winey himself writes a conditional ("If Lakhovsky's theory is correct ...").

b/c. The statement is a citation and an interpretation. It adds no experiment [read]. The applicant has a commercial stake (golf shafts with phi tunings) and his Lakhovsky paragraph is background for his own application.

d. Same as claims 1 and 4.

Verdict: Not testable as stated for the "polychromatic spectrum" framework (no definition); the sentence "cured cancer patients until 1942" is an uncorroborated secondary characterisation: its only citation predates part of the period. It is recorded here as that applicant's characterisation, beside Lakhovsky's own claims (claim 4), and weighed there.

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Part C. Calculations in one place (inputs and results)

Part D. The Investigator's findings, as they bear on the mathematics (kept in the record whatever the verdict)

• Kotzareff 40 W absorbed power and the 2-10 m band for the Radio-cellulo-oscillator: first power number on record; needed for any heating estimate (claim 3). Verified in the OCR.
• 1924 note (n = 3 treated, 16 controls, dose-independent onset at about 16 days; co-author Magrou): the single most concrete experimental datum in the project (claim 3). Source is a seller's reproduction of a 1925 press quote (interested party) plus Roussakow and Kotzareff citing the same note; the original (C. R. Soc. Biol. 91, 626-628) was not obtained (BHL, Gallica blocked).
• Schereschewsky 1926-1928 (8-135 MHz, mouse tumours, 23 % complete remission at 67 MHz per Roussakow) and his 1933 change to a thermal reading: related work in the same band; relevant to the thermal question (claim 3). Described by Roussakow, who argues for non-thermal RF effects.
• CH 135072 / FR 657295 (open circuit, 2 m): confirms the half-wave arithmetic (Part B claim 3). CH 164027 (MWO family; names therapy, adds gas tubes) shows the medical purpose absent from the US text was stated in the Swiss one; this affects the interpretation of claim 4 (the US text's silence is a drafting choice, not a sign that the inventor did not intend therapy).
• Nature 1940 (Ritchie): "apparently has worked marvellous cures ... but that does not prove that its electric radiations acted upon hypothetical cell-oscillations"; Corynactis "oscillating circuit" is a cnidocyst thread. A contemporary biologist separated clinical report from theory in the same way I do (claims 1 and 4).
• L'Universion (1927) and Contributo (1927): used above for lead thickness drift, the 22-octave scale and the Paris mortality table (claim 5).
• Absence of evidence: the record after 1942 (Institute of Multiple Waves 1945; Blum, Chicago; no clinical trials; a mainstream hyperthermia history note by Farrell 1982 unread). A mathematician can say what would settle the numerical matters: the original 1924 note; measured field strengths and spectrum of a surviving MWO; SAR measurements; and age-standardised cancer mortality by quartier from the 1925 Paris yearbook.

Part E. What would settle the open numerical questions

1. Measure the output spectrum and field of an MWO built to the patent (spectrum analyser, near-field probe, 0.1 MHz to 3 GHz): settles claim 4 (ii) and gives a dose for claim 3.
2. Repeat the geranium experiment blinded, with randomised allocation, measured specific absorption rate and a heated-only control, with at least about 10 plants per arm (Part B claim 3, exact test).
3. Re-analyse the Paris data: take all 80 quartiers from the 1925 yearbook, age-standardise, exclude hospital and cemetery quartiers, and test geology (pre-specified) against mortality (claim 5 iii).
4. Compare measured decay rates of a Ra-226 source at the surface and in a deep underground laboratory (claim 5 i).
5. Measure the impedance of an isolated chromosome in electrolyte up to THz (claim 1).

Summary verdict table