LeBon-G: Mathematician review (Step 2)

Reviewer: PhD mathematician. Date: 2026-10-08. Source: source folder: LeBon-G (read only). Tags: [read] from the documents (PDF page numbers; I read the extracted text and page images p.73, 74, 76 of the Matter PDF myself), [calc] my own arithmetic (Python, .venv), [web] with URL, [memory] with confidence. No source gap: all three documents have text. The Matter PDF is OCR, so every number I rely on was checked against a page image or against an independent recomputation.

0. What this review is and is not

Le Bon (1841-1931) wrote almost no equations. His "mathematics" is (i) a handful of unit-conversion and kinetic-energy calculations in Matter Book II (PDF pp.66-80, the part Step 1 left as a deep-read gap, which I read in full), (ii) the electroscope tables in Matter Second Part (deep-read), (iii) decay-time and ratio statements in Forces Book IV, and (iv) a verbal critique of conservation of energy (Forces PDF pp.63-68, 80-84) and of the dynamo (Forces PDF pp.157-159). I checked every number I could. The physics verdicts belong to the Physicist; I judge consistency, units and hidden assumptions, and I say where a conclusion does not follow from a correct calculation.

Terms used: kgm = kilogram-force metre (old work unit, 1 kgm = 9.807 J); hp = metric horsepower = 75 kgm/s = 735.5 W; "great calorie" = kilocalorie (kcal); c = speed of light (3.0e8 m/s); E = mc^2 is the mass-energy relation (E = energy, m = mass).

1. Worked check of the "intra-atomic energy" arithmetic (claim 2)

Le Bon's premise [read: Matter PDF p.71]: emitted particles move at 100,000 km/s (v = 1e8 m/s, a third of c); take 1 g (m = 1e-3 kg) and suppose it entire is "dissociated" and moving at that speed.

Conclusion on (a): the arithmetic is internally consistent and carefully done. The calculation is correct; the question is what it is a calculation of.

Errors and slips I found (not in Step 1)

1. Mechanical equivalent of heat, Matter p.73 (page image checked). "transforming these calories into kilogrammetres at the rate of 1125 kilogrammetres per great calorie". The correct value is 427 kgm per kcal (4186 J / 9.807) [calc]. Le Bon's own other book gives 425 [read: Forces PDF p.65, "a weight of 425 kilogrammes" falling 1 m to heat 1 kg of water 1 degree]. So the two books contradict each other, and Matter's 1125 is wrong by a factor 2.6. The figure is not carried into a result, so no conclusion changes. There is a second ambiguity on the same page: "100 calorie-grammes an hour" is 100 small calories, so the printed 876,000 "calories per annum" are small calories (0.876e3 kcal), and "great calorie" in the next clause would then need dividing by 1000. As printed the sentence mixes the two units.
2. Earth as a charged sphere, Matter p.74 (page image checked). "barely a twentieth part" of the 96,000 coulombs (= 4,800 C) "would suffice to charge a globe the size of the earth to a potential of 6000 volts". The earth's capacitance is C = 4 pi eps0 R = 7.1e-4 F [calc], so 4,800 C gives V = Q/C = 6.8 million volts, not 6,000. The printed "6000 volts" corresponds to Q = 4.3 C, i.e. one twenty-thousandth, not one twentieth. The error is a factor of 1000 (a misprint in the French or the English; I could not check the French). It understates Le Bon's point and changes nothing downstream. The "thirty years" (96,000 C at 1e-4 C/s = 30.4 years) is correct [calc].
3. Units in the English. "100,000 kilogrammes per second" for particle speed (Matter p.78, 79) should be kilometres; "6 kilogrammes per ton per second" (p.72 footnote) should be per ton (a force, not a rate). Translation or print slips.
4. Locomotive count "exceeding all the locomotives on all the French lines" (p.79): not checkable from the text.

Hidden assumptions in claim 2 (this is where the mathematics stops following)

• A. Total conversion at c/3. KE = 1/2 m v^2 for the whole gram assumes the whole gram turns into particles at 1e8 m/s. Nothing Le Bon measured gives that. What was measured is the speed of the few particles emitted per second; the energy per gram that results depends on how many are emitted, which is the whole question. Le Bon himself admits the fraction: "Rutherford calculates the number of molecules ... having undergone a commencement of dissociation, at 1 in 100,000,000 ... Were it otherwise, it would be easy to extract from atoms a colossal amount of energy" [read: Matter PDF p.427].
• B. Newtonian formula at relativistic speed. At v = c/3, gamma = 1.061 and the relativistic KE = (gamma - 1) m c^2 = 5.46e12 J, 9% above Le Bon's 5.0e12 J [calc]. Small at c/3, but his rotating-sphere case "at the speed of light" (p.79) is meaningless (infinite energy for a massive body; also a rigid rim at c/3 needs tensile strength of order rho v^2 = 8.8e19 Pa [calc], about 1e10 times steel).
• C. A different, correct bound exists and Le Bon's figure sits below it. The mass-energy relation gives the maximum energy of 1 g as m c^2 = 9.0e13 J [calc]. Le Bon's 5.0e12 J is 5.6% of that. His 1905 text predates Einstein's June 1905 paper in print only by months and does not use E = mc^2 (the book never states it as far as the digests and my reads show; I searched PDF pp.66-82 only, not the whole book). So "a gram of matter holds an enormous energy" is correct in order of magnitude; his figure is not derived from the right relation.
• D. What is observed, per gram. Curie's measured heat of radium, 100 cal/h per gram [read: p.73], is 0.116 W [calc], about 3.7e9 J over Le Bon's assumed 1,000-year life. Radium-226 has a half-life of about 1,600 years and defines the curie, 3.7e10 decays/s per gram [web: https://en.wikipedia.org/wiki/Radium]. Counting all alphas in its decay chain to lead (about 28 MeV per atom, [memory, medium]; I did not confirm the chain energy on the web) gives roughly 1.2e10 J per gram, which is about 1e-4 of m c^2 [calc]. So the best natural source, radium, delivers about 1/400 of Le Bon's headline 5e12 J, and ordinary metals deliver nothing detectable beyond background. For scale [memory, high]: fission of U-235 is about 8e10 J/g, hydrogen-to-helium fusion about 6e11 J/g. Le Bon's Curie figure (his own "different method") gives a number three orders below his headline, and he calls it "inferior ... but still colossal" [read: p.73] without noting the gap.
• E. "Energy out exceeds the trigger" (spark on gunpowder, p.54, p.173). As a statement about a stored reserve released by a small trigger this is mathematically unobjectionable (any explosive, any chain reaction). It is not over-unity unless the reserve is renewed for free. Le Bon himself, for the sulphides, says the reserve is "a limited residual charge, which does not renew itself spontaneously" [read: Forces PDF p.328].

2. The conservation-of-energy and dynamo arguments (claims 2, 3)

Poincaré quotation (Forces PDF pp.66-67, 80-84). Le Bon says conservation of energy is a "dogma" and quotes Poincaré to the effect that energy "can no longer be defined" outside simple cases, and that "the abyss of potential energy provides it with an inviolable shelter" [read]. What Poincaré actually wrote (Science and Hypothesis, ch. 8) [web: https://en.wikisource.org/wiki/Science_and_Hypothesis/Chapter_8]: applied to the whole universe the principle shrinks to "there is something which remains constant" (nearly tautological); once one allows any function of a conserved quantity the choice of "energy" has no guide; but he explicitly rejects that the law "is reduced to mere tautology", says that for a limited system "it has a perfectly clear meaning", and treats the universal postulates as conventions that are "not absolutely arbitrary" because experiment shows them convenient. So Poincaré's remark is a point about the logical status of the law in its most general form, not a finding that energy is created. Le Bon's use of it ("we are compelled to acknowledge that the second postulate [energy can be lost] would have in its favour at least as many facts") is a reading Poincaré's text does not support. Mathematically, the charge that potential energy "can always be given the value necessary" is false for a conservative force field: U(x) is fixed (up to a constant) by independent force measurements, U(x) = - integral of F . dx, path-independent exactly when curl F = 0. A hidden "abyss" is therefore testable: it must match measured forces.

Dynamo (Forces PDF pp.157-159). Le Bon: a coil cutting a magnet's lines of force produces "an unlimited quantity of electricity" from "a finite quantity of magnetism"; the usual account (work turned into electricity) "would be as marvellous as the transmutation of lead into gold by simply shaking it in a bottle"; instead the lines are "constantly replaced at the expense of the intra-atomic energy" [read: p.157-158]. This is a testable energy balance, and standard electromagnetism gives a quantitative prediction that his hypothesis does not:

Worked example [calc]. A straight conductor of length l = 0.5 m moves at v = 10 m/s across a field B = 1 T, closing through a load R = 5 ohm. - EMF (electromotive force) = B l v = 5 V; current I = EMF / R = 1 A; electrical power = EMF x I = 5 W. - The current-carrying wire feels a force F = B I l = 0.5 N opposing the motion (Lenz's law). Mechanical power needed to keep v constant: F v = 0.5 x 10 = 5 W. - Mechanical input = electrical output (ideal, no losses); in a real dynamo the difference is Joule heat and friction, all accounted for. - With a larger load (R = 2.5 ohm), I = 2 A, F = 1 N, input power 10 W. The braking torque on the shaft rises with the electrical load.

If instead the output came from a hidden atomic reservoir, the shaft would feel no extra load as current is drawn. Every engine driving a generator shows the opposite: fuel consumption tracks electrical load. Le Bon concedes "a determinate quantity of movement and consequently of work is ... required" [read: p.158] but calls it a mere condition. He gives no measurement. The magnet's own energy is not consumed either: the magnetic force on a charge is perpendicular to its velocity and does no work. So "a finite quantity of magnetism gives unlimited electricity" is correct, and also unremarkable: the energy source is the mover. This is a refutation by calculation of the "intra-atomic source" explanation for the dynamo (the claim as Le Bon states it), and it needs no new experiment; the experiment (torque against load) has been done on every generator.

3. Consistency checks on the experimental chapters (claims 3, 4)

Done by the deep readers; I re-derived the ones that carry weight.

• Electroscope units. The reading is "degrees per minute" of gold-leaf fall. Without the electroscope's capacitance and the starting voltage this cannot be converted to ionization current, so every figure is relative to its own table. No blank (no-source) leak rate is reported anywhere in the read pages [read: deep/Matter §8.1]. Le Bon says himself: "no measurements ... by the electroscope have any very precise quantitative value" [read: Matter p.421]. Cross-substance comparisons ("ordinary metal 1-2 deg/min" after 8 days, p.430, versus 2 g uranium oxide 37 deg per 10 min = 3.7 deg/min, p.419) cannot be turned into activities.
• Internal slips (confirmed by the deep reader's page images): electrode table "single to triple" (data span x6 for the positive column); lamp-black positive 7 versus negative 61 against "markedly superior positive"; thorium chloride "70 per minute ... 7 times" a 1 degree baseline (7 fits); receptacle "same dimensions (12 x 17 cm)" versus earlier 10 cm squares; quinine, which of phosphorescence and discharge outlasts the other, stated both ways (p.184-185 versus p.413). None invalidates the qualitative claims; together they show the tables were not proof-read and the numbers should be read as orders of magnitude.
• Ratios in the hydration table (my recomputation): dried/hydrated = 3.7, 7.4, 2.6 and 3.0 for uranium nitrate, red uranium oxide, thorium oxide, radium bromide [calc, from the table in deep/ §5.1]. Le Bon's own caveat that water also screens the particles means the table does not isolate chemistry.
• Afterglow decay series (Forces PDF p.322; confirmed against the primary 1899 note, CR 128 pp.? Jan 16 1899, [web: research/files, IA scan]) Exposure needed for an image: 2 h at 3 days, 12 h at 15 d, 30 h at 25 d, 40 d at 6 months, traces after 60 d at 18 months. The book reproduces the 1899 numbers exactly [read, both]. Taking intensity proportional to 1/exposure, the apparent time constant (time for I to fall by e if the decay were exponential) grows: 3 to 15 d gives about 6.7 d, 15 to 25 d about 11 d, 25 to 180 d about 45 d [calc]. A single exponential does not fit; a log-log fit of the first four points gives I ~ t^-1.5 (first three: t^-1.2) [calc]. A power law in time is what one expects from a spread of trap depths in an impure phosphor [memory, high], and it is not exotic. Caveats: plate reciprocity failure means late intensities are underestimated (the true decay is slower still) [memory, high]; the five data points are eyeballed exposure times with no error bars; no blank plate (see physicist for the experimental view).
• Transparency method, Forces PDF p.363 (deep reader's finding, I confirm the logic). Equal darkening needs 20 s through the test body and 5 s through the 1 mm ebonite standard. By his own "time ratio" rule the body transmits 5/20 = 1/4 of the ebonite, i.e. four times less transparent, not "twice as transparent". Direction and factor are both wrong. The table's ratios ("half as transparent", "one-fourth") then have unknown direction; only the verbal ordering is reliable.
• Wien check [calc; law: peak wavelength = 2898 micrometre-K / T]: 373 K (100 C): 7.8 um; 310 K (37 C): 9.3 um. So Forces p.339 "not less than 5 to 6 um" for the 100 C chimney is correct, and p.362 "bodies under 100 C emit radiations hardly exceeding 5 to 6 um" is wrong as printed (deep reader's flag confirmed). His glass-strip control still works as designed because glass cuts off beyond a few micrometres [memory, high].
• Fig. 35 (candle brighter than sunlit card). Deep reader's estimate: candle about 5,000-10,000 cd/m^2; white card in full sun L = rho E / pi = 0.8 x 1e5 / 3.14 = 25,000 cd/m^2 [calc, E and rho from memory, medium]. On visible luminance the card is brighter, and on a blue-sensitive plate more so. Le Bon gives no measurement. The qualitative point he needs (a self-luminous source is bright per unit area) is right.
• Other numbers I verified: Moon at 1e5 km/s in 3.8 s [calc]; Rutherford's 1 in 1e8 with 3.6e16 molecules per mm^3 gives 3.6e8, as printed (modern Loschmidt number 2.7e16 per mm^3 at 0 C [memory, high], so his density is a third high; harmless).

4. Corrections to Step 1

1. claims.md "Deep-read gaps" (first bullet). I read Matter PDF pp.66-82 myself; the intra-atomic energy gap is closed by section 1 above. The request "reproduce each calculation with inputs; check units" is answered there. Pages 99-108 (the objections chapter) I did not read in full; they remain digest-only.
2. claims.md claim 2 states "6,800 million horse-power if one gram of dissociated particles were stopped in a second": correct, but add that it assumes total dissociation at 100,000 km/s, a premise, not a measurement (p.427, Le Bon's own 1 in 1e8).
3. facts.json / deep Forces §3.1. Add that the decay-series numbers equal those of the primary 1899 note (CR 128, 16 Jan 1899, presented by Poincaré; research/2026-10-08-archive-org.md), so the book is not a later alteration. The 1899 note says the screens were kept "two years" in the dark; the book says 18 months for the last step: not a conflict (last point is 18 months).
4. sites.md says CR 128 (30 Jan 1899) was "downloaded, not read". I read its opening (research/pages/...1899-01-30_128_5.txt lines 1809 ff.): Le Bon states that ZnS "illumination spectrum" runs from near line G past violet with a maximum near H, the "extinction spectrum" from F into the infra-red, and that the screen's sensitivity to the extinction spectrum is "at least equal to that of photographic plates for ordinary light"; CaS is "more than 300 times" less sensitive than ZnS; fogged fine-grain film is "at least 5000 times" less sensitive than ZnS [read, OCR]. This agrees with Forces pp.337, 357, 361 (CaS gives no image in an hour). The OCR of the extinction maximum ("lambda = 14,5"?) is garbled; Le Bon's letter of 9 Jan 1899 puts the maximum near 1 um [read: Investigator note], the book gives the active band as 0.8-3 um: three expressions of the same region, not contradictory.
5. deep/Matter §7.1 cites "Revue Scientifique, 22nd November 1902, p. 650"; the Poincaré Papers editors cite the memoir at Revue scientifique pp.614-625, 15 Nov 1902 (letter of 29 Nov 1902 refers to its p.624). These may be two issues of one series of articles; unresolved, for the Investigator.
6. Investigator's Moray note (John E. Moray: "'1,340,000 barrels of gun powder' ... a nuclear fission reaction") has already been noted by the Investigator as Le Bon's figure for the speed of the particles. My check confirms it: the 1,340,000-barrel figure is the powder needed to give a 15 g bullet 100,000 km/s (section 1), not a fission estimate.

5. The Investigator's findings that bear on my claims (kept whatever my verdict)

• Poincaré as questioner of protocol (letters 1896-1903): asked whether the 1896 effect was "diffusion par l'air", about equal exposure times, cleaning, de-fogging only behind ebonite; his 1903 question on Le Bon's reservoir, "si cette combinaison est endothermique, comment peut-elle être si extraordinairement stable ?", is the right mathematical question: if atoms were compressed sub-atoms, forming them stores energy, and a stored configuration stable for 1e9 years needs a barrier (this question was answered later by nuclear binding and tunnelling [memory, high]). [web: research/2026-10-08-henripoincarepapers.md]. Poincaré's 1903 framing treats the reservoir as potential energy, which sits comfortably with the conservation law Le Bon attacks.
• Berget's reproduction (1897, Lippmann's laboratory): concerns the light-on-metals discharge experiment (photoelectric class), not the energy claims. Documented in Le Bon's own CR note and Poincaré's letter as summarised by the editors.
• Izarn letter (Forces p.350 footnote): one positive repetition report of the infra-red/ZnS experiment, documented in one source (Le Bon's book), and Le Bon's account of why others failed (red darkroom lamps quench ZnS) is physically sound [read; memory, high]. Not found elsewhere on the web by the Investigator.
• Moray's own writings cite Le Bon (1926 Radiant Energy pp.41-42; Sea of Energy bibliography; Air Force proposal "as early as 1897"): confirms the core of claim 6 from Moray's side. Moray's 1926 text paraphrases the Matter introduction, which contains no equation; nothing I checked mathematically passes from Le Bon to Moray except the verbal "energy exceeds the trigger".
• Kragh and Nye group black light with N-rays (a label of those sources). In my checks, the black-light chapters' quantities hold together better than the N-ray literature's (those relied on eye judgement of faint sparks; Le Bon's use plates and exposure times), but the missing blanks apply to both.
• "Girard Le Beau" third book: no trace; nothing mathematical to check.

6. Questions a-d per claim, with verdicts

Claim 1: Matter is perishable (all atoms slowly dissociate)

a. Mathematically consistent? No equations, no rate law, no decay constant for "ordinary" matter and no definition of the end state "ether"; so neither consistent nor inconsistent. The one number he gives (1 in 1e8 molecules ionised, p.427) is a statement about ions in air, which he then identifies with dissociation (p.423), an identification, not a derivation. b. Experiments? Radioactive elements do decay (Established). For ordinary stable matter, bounds from absence of decay products are far tighter than anything in 1905 (stable isotopes; proton lifetime bounds above 1e30 years [memory, medium; not web-checked]). His electroscope leaks from metals are accounted for by photoelectric, thermionic, contact and background-ionization effects and by the penetrating radiation Elster and Geitel and Thomson discussed (Legge's notes, [read: Matter pp.144, 177-178, 428-429]). c. Tested? Le Bon and, per Legge's notes, some others (Kalähne, Gates, Ramsay) on electroscope effects; J. J. Thomson found emanation in ordinary substances and attributed it to radioactive impurity [read: Legge's note, Matter p.428]. No test of "all atoms dissociate" in a controlled blank-compared way is on record in the sources I have. I have not read Kalähne, Gates or Ramsay (cited through Legge only). d. If true: stable nuclei would not be stable; every gram of lead or copper would be a radiation source on the scale of radioactive ores; also the observed isotopic ratios and the age of terrestrial stable matter would be unexplained. Verdict (split): weak form, some elements disintegrate, Established (not a Le Bon discovery in the form claimed: see claim 5). Strong form, universal spontaneous dissociation of ordinary matter toward ether, Tested, not supported. "Ether" as the end state Not testable as stated (no measurable prediction).

Claim 2: Atoms are reservoirs of intra-atomic energy; conservation of energy is a dogma

a. Arithmetic: consistent (section 1). Two misprints/unit slips (1125 kgm/kcal; 6000 V). The step from the arithmetic to "energy of one gram of any metal" rests on assumption A (total conversion at c/3). The criticism of potential energy misreads Poincaré (section 2). The dynamo claim contradicts energy accounting. b. Experiment: radium's measured heat (0.116 W per gram) is consistent with a stored reserve of order 1e10 J per gram; ordinary matter yields none. Every dynamo shows input torque rising with load. Nuclear energy (fission, fusion) later confirmed enormous stored energy in nuclei, at 1e-3 to 1e-4 of m c^2 per gram of fuel [memory, high]. c. Tested? The dynamo energy balance is tested daily; the "intra-atomic source" explanation of it was not tested by Le Bon (no torque or power measurement). Le Bon's reservoir for radioactivity was never tested as "release on demand from ordinary bodies". d. If true (as stated for dynamos and ordinary bodies): the first law of thermodynamics and Noether's theorem (time-translation symmetry gives conservation of energy [memory, high]) would fail; generators would not load their prime movers. Verdict (split): "matter holds a very large reservoir of energy" Established (order of magnitude right; the mass-energy relation is the right tool, which he did not have); the figure 6,800 million hp as what a gram of ordinary matter can yield, Refuted by calculation against the radium budget and the measured background (0.116 W/g and about 1e10 J/g for radium; nothing for ordinary matter), with the arithmetic itself correct; "conservation of energy is a dogma that survives by inventing potential energy" Refuted by calculation (potential energy is fixed by force measurements; Poincaré quoted out of context); "dynamos draw on intra-atomic energy" Refuted by calculation (section 2 worked example).

Claim 3: Light, heat and chemistry release the energy in ordinary bodies

a. Tables are not proof-read (section 3) but nothing is self-contradictory in the main claim; the inference from "charge leaks" to "atoms dissociate" is the unexamined step, and one controlled alternative (heat acting by expelling a store) is argued, not measured. b. The negative leak under ultra-violet and its dependence on metal and cleaning is the photoelectric effect (Hertz, Hallwachs, Lenard; Einstein 1905) [memory, high]; positive leak at short range is photo-ionization of air by vacuum ultra-violet (Lenard, as Le Bon himself reports, pp.400-405); flame, reaction-gas and phosphorus ionization are known ion-producing processes [memory, medium]. The "emanation" from metals matches closed-vessel background ionization (C. T. R. Wilson; McLennan and Burton, cited by Legge [read: Matter p.411]) and radioactive impurity (J. J. Thomson, per Legge). c. Tested by Le Bon; the translator names a check by others (Kalähne on quinine sulphate, Gates, Ramsay on the ultra-violet effect) [read: Legge's notes, secondhand]; the Faraday-cage experiment "would be satisfactory to have ... checked by an independent observer" (Legge, p.411); Campbell (J. J. Thomson's laboratory) attributed the chemical-reaction leak to heat (reported through Le Bon, p.417-418). d. If the strong reading were true: ordinary bodies would show nuclear-scale energies and decay products; Elster and Geitel's penetrating radiation would be only a part. Verdict (split): the observed leaks Established as phenomena, accepted under other explanations (photoelectric effect, flame and chemical ionization, background radiation); "this is atomic dissociation releasing intra-atomic energy" Tested, not supported on the record available (no blank leak rate, no energy accounting, rival explanations favoured by Thomson and others). Quinine-hydration and phosphorus effects: mechanism Never properly tested in the sources I have (effect reported as confirmed by Kalähne; the later literature is not in the folder).

Claim 4: Black light

a. The decay table, the glass-strip, ice-trough and ZnS/CaS opposite-effect arguments are mutually consistent. Found: transparency-ratio worked example reversed; p.362 Wien wording; Fig. 35 brightness claim unsupported; statuette caption versus text on timing. The "darkness added to darkness" and "small amplitude" statements are rhetoric, not mathematics: an afterglow too weak for the dark-adapted eye but integrated by a plate over days involves no new radiation. b. Consistent with: long afterglow with power-law decay; thermoluminescence; infra-red stimulation of CaS/SrS and quenching of ZnS:Cu; near-infrared transparency of ebonite and black lacquer [memory, high]. Deviations: ammoniacal copper sulphate "transparent" is surprising (it absorbs in the near-infrared [memory, medium]); the metal-plate "radiations" through mica match Russell-effect chemical fogging [memory, high]. c. Tested by Le Bon from 1896 to 1900; presented by Poincaré to the Académie in 1899 (a procedure that doubled as protocol review). Repeated: Izarn (one letter, in Le Bon's book), Berget (a different, 1897 experiment); critics who "tried them with a red lamp" would have quenched the ZnS (Le Bon's technical answer). No neutral, controlled replication in the folder. d. If the interpretation were true as a "new radiation of small amplitude": nothing in mainstream physics needs to change; the phenomena are reproduced by standard solid-state luminescence physics. Verdict (split): the phenomena (afterglow lasting months, IR stimulation vs heat, IR transparency of ebonite and paper, photography through them) Established in kind (with Le Bon's specific numbers Never properly tested by an independent party on record); the claim that this is a distinct "invisible light" or involves intra-atomic energy Not testable as stated; the naval searchlight idea, never tested (it was later realised with image converters [memory, high]).

Claim 5: Priority and resistance

a. Not a mathematical claim. Dates are consistent across documents: 1896 note (CR 122), 1897, 1899, 1902 (Revue scientifique, 15 and 22 Nov? see Correction 5), Matter 1905; Becquerel's uranium rays March 1896. Le Bon's January 1896 note is about light passing through iron and copper ("n'est qu'une question de temps"), not about uranium-type radioactivity [read: research/archive-org note]; the "priority on radioactivity of ordinary bodies" rests on the 1897-1902 notes. b. Documents show a mixed record, not silence: Poincaré presented four notes; Berget reproduced one experiment; Le Bon was right against H. Becquerel on polarisation of uranium rays (Martins, via the Investigator); and Poincaré judged the 1902 priority note against Nodon "plutôt nuisible". c. Replication: see claim 4 and above. d. Nothing in mainstream physics changes. Verdict: Historically superseded but instructive (a physicist-historian of the period, not a mathematician, should weigh priority). I flag only the dates, which are consistent.

Claim 6: Moray connection (secondary)

a. No mathematics. The only quantitative thing that travels from Le Bon to Moray's world is the 1,340,000-barrel figure, which John Moray attaches to fission (Investigator) but which is Le Bon's kinetic-energy estimate for particles at 100,000 km/s. The reviewer's items ("Tesla-coil engines ... release more energy than that used in the trigger"; Le Bon "altering the half-life of radium"; "resonant nuclear disintegration") give no figures. Le Bon's own text for the half-life: heating radium temporarily lowers its activity [read: via Vassilatos; not checked in Le Bon]; Rutherford and Soddy's later work says nuclear decay rates are not changed by heat [memory, high]. b. Documentary: Moray cites Le Bon by name (Investigator, [read: Morray-TH]); FBI/library story is a first-person family recollection (John E. Moray, 1978 and 1980) and I have no record of it. c. Not tested; nothing to test mathematically. d. If the energy claims were true as the reviewer states them (net energy from small high-frequency bombardments): conservation of energy, as in section 2. Verdict (split): "Moray read and used Le Bon" is documented in Moray's own writings (and Vassilatos's inference of "very deep secrets" has no numbers behind it); the energy-gain claims attributed to Le Bon through the reviewer Not testable as stated (no data) and, if read as net energy beyond the stored reserve, would fall to the calculation in section 2. The "Girard Le Beau" book: no evidence found.

7. Take-aways for the learner (mathematical lessons)

• A correct calculation can sit on a premise nobody measured. Le Bon's kinetic-energy sums all check; the question is whether the whole gram can be converted (it cannot, in any process then or now observed for ordinary matter).
• The right tool, E = mc^2, bounds the answer at 9e13 J per gram; real processes reach 1e-4 to 1e-2 of it. Run Section 1's table with your own mass and speed.
• Always carry units through a sentence. Two of Le Bon's errors (1125 versus 425 kgm per kcal; 6000 volts for the earth) are catchable in one line each.
• Ask what a hypothesis predicts that its rival does not. "Dynamo output from hidden atoms" predicts no change in shaft torque with load; the standard theory predicts torque proportional to current (F = B I l). The measurement already exists.
• Ratios need a stated direction (his 20 s versus 5 s example) and the data tables need blanks.

8. What would settle remaining questions

• A blank-run table of the electroscope with no source, in the same apparatus, over the same days (not on record).
• Independent repeat of the ZnS/CaS and ice-trough tests with a spectrally resolved source and calibrated photodiode or plate.
• The French originals (IA 1905-1907, gustavelebon.org) to settle the 1125 and 6000 V misprints.
• The three Phil. Mag./Physical Review papers (Kalähne 1905, Gates 1904, Ramsay and Spencer 1906) to see what they actually confirmed.