Soddy: Mathematician's review (Step 2)

Reviewer role: PhD mathematician. I check derivations, notation, units and hidden assumptions, and I re-did the arithmetic myself ([calc] = my own computation, Python in /tmp/soddy_m). Source tags: [read] from the project documents (PDF page = printed page + 44 for pp.153-165, + 48 for pp.268-300), [web] with URL, [memory] with confidence.

Scope note (from claims.md, repeated): the folder holds only two books, "Matter and Energy" (1911) and "The Interpretation of Radium" (4th ed., 1920, 1922 reprint). Text is readable. Soddy's economics books and his papers with Rutherford and Ramsay are not in the folder; the Investigator found some of them online but I did not read them. Nothing below about his economics is read from the folder. Both books are popular lectures: they state results in words, with almost no derivations ("skipping the mathematics", Interpretation p.157). So most of my work is to supply the missing mathematics and test the printed numbers against it.

Web budget used: 4 fetches (Wikipedia isotope pages for Ra, Th, U, Pb) for modern values.

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0. The mathematics, in the reader's terms (needed to follow the checks)

Symbols: N(t) = number of undecayed atoms; lambda = decay constant (fraction decaying per unit time, Soddy's "radioactive constant"); T = 1/lambda = mean life (Soddy's "period of average life"); tau = ln2/lambda = half-life (Soddy's "half-change period"). Subscripts 1, 2 = parent, product.

1. Decay: dN/dt = -lambda N, so N = N0 e^(-lambda t). Soddy's "geometrical progression" is this.
2. Mean life: the average of t over the distribution lambda e^(-lambda t) is 1/lambda. Mean/half ratio = 1/ln2 = 1.4427 (Soddy prints 1.45).
3. Secular equilibrium: with a very long-lived parent, dX/dt = lambda1 N - lambda2 X = 0 gives X/N = lambda1/lambda2 = T2/T1. Printed exactly this way, Interpretation p.162 [read: page image].
4. Growth from a purified parent through one long-lived intermediate (used in Soddy's ionium argument): if dIo/dt = lambda_U N_U (constant) and dRa/dt = lambda_Io Io, then Ra(t) = (1/2) lambda_Io lambda_U N_U t^2 for t much shorter than T_Io. Divided by the equilibrium amount lambda_U N_U / lambda_Ra this gives Ra/Ra_eq = t^2 / (2 T_Io T_Ra). I derived this; Soddy states only the result ("proportional to the square of the time", p.133 of the book).

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1. Claim-by-claim review

Claim 1. Radioactive decay is random, with a fixed chance per second, so amount falls exponentially (mean life 1/lambda, half-change 0.69 of it).

a. Mathematically consistent? Yes. Everything printed is correct: lambda definition (p.157), mean life = 1/lambda, 1/e = 0.368 after one mean life, halving progression (p.159), and the 1.45 ratio, which is 1/ln2 = 1.4427 rounded up [calc]. Small points: (i) the text says the half-change period is 0.69 of the mean life in claims.md, and "mean life is 1.45 times half-life" in the book: the same fact. (ii) Soddy's numbers for the emanation are not exactly linked: 481,250 s = 5.570 d [calc], times ln2 gives a 3.861 d half-life, but he prints 3.84 d, and 3.84 x 1.4427 = 5.54 d. A 0.5% mismatch, probably two measurements. Modern radon-222: half-life 3.82 d [memory, high], so his 3.84 is the better number and 5.57 d is about 1% high. (iii) Typographic slip p.159: capital N and lowercase n for the same exponent. (iv) The memorylessness claim ("independent of age") is the whole content of "exponential"; Soddy says it can be "directly proved" by comparing new and old emanation (p.158) but gives no data. Hidden assumption: lambda is independent of surroundings. True to very high accuracy for alpha and beta decay; small chemical-environment effects exist for electron capture (e.g. Be-7) [memory, medium]. Soddy's own text says "independent ... of every other consideration whatever" (p.157), which is slightly too strong in the strict form. b. Consistent with experiment? Yes: decay curves of emanation, radium D etc. measured by Soddy's contemporaries; modern half-lives agree to 1 to 10% (see table in 2b below). c. Tested? Constantly, since 1900, by Rutherford and Soddy and everybody after; the age-independence test (fresh vs old emanation) is described p.158 [read] without data. Replicated in every laboratory. d. If false? Nothing to change; it is mainstream.

Verdict: Established.

Claim 2. Radium's energy is internal atomic energy: about 133 cal/h per gram (about 100 in 1911), about 2.9 billion cal per gram over the complete change, more than a million times chemical fuel, consistent with conservation of energy.

a. Mathematics. Arithmetic checks [calc]: 133 x 8,760 = 1.165e6 cal per year; x 2,500 = 2.91e9 cal (printed 2.9e9). Multiplying by the mean life is exactly right (integral of P0 e^(-t/T) = P0 T); a half-life would be the wrong multiplier. Coal 8,000 cal/g; 2 2/3 g oxygen per g carbon (32/12 = 2.667); 8,000/3.667 = 2,182 cal/g of fuel plus oxygen (printed 2,200). Ratio 2.91e9/2,200 = 1.3e6 ("well over a million"): consistent. Against coal alone the ratio is 3.6e5, so the sentence "nearly a million times greater than ... any change known" (p.164) is loose; Soddy prints both bases so the reader can see it. Independent check of the 2.9e9 figure that does not use the mean life: one radium atom, with emanation, RaA and RaC' (the four alpha decays) plus the RaB and RaC beta decays, releases about 29 MeV in total (alpha energies 4.87, 5.59, 6.12, 7.83 MeV with recoil, plus beta Q-values 1.02 and 3.27 MeV) [calc; energies from memory, high for the alphas, medium for the beta Q]. One gram holds 6.022e23/226 = 2.66e21 atoms. 2.66e21 x 29.2 MeV = 1.2e10 J = 2.97e9 cal. So the "complete change" number (printed 2.9e9) agrees with a per-atom count to 3%. This also confirms the deep read's remark that 2.9e9 counts radium plus its quick products only (later RaD, E, F add roughly 6-7 MeV more, [memory, medium]). Power check: 1 g of Ra-226 has 3.66e10 decays per second [calc, with half-life 1,600 y]. Times 25 to 29 MeV gives 0.146 to 0.171 W = 125 to 147 cal/h [calc]. Soddy's 133 cal/h sits inside this range (some gamma and neutrino energy is not captured in a calorimeter). So the 1920 figure is right within the uncertainty of the measurement, and the 1911 figure of 100 is low by about 25%. The 1911 figure is stated inside the 1911 book, not only inferred: Matter and Energy p.210 says the heat in one day decomposes its own weight of CO2 and "in thirty-eight hours" its own weight of water [read, deep read]. Decomposing water takes 68.3 kcal/mol / 18.015 g = 3.79 kcal/g [calc, standard thermochemistry, memory high], and 3,790 cal / 38 h = 99.7 cal/h [calc]. One day at 100 cal/h = 2,400 cal decomposes 1.12 g of CO2 (2.14 kcal/g), at 133 cal/h it would be 1.49 g, so the 1911 sentences fit 100 and not 133. Ten years: 100 x 8,760 x 10 / 7,830 cal/g carbon = 1,119 times its weight in carbon ("over a thousand"). The quarter ton of coal: 100 cal/h x 8,760 x 2,500 / 8,000 = 0.27 tonne; 133 would give 0.36 tonne. All four 1911 sentences are consistent with 100 cal/h. Internal inconsistency in 1911 only: emanation "four days" half-activity and "5.3 days" mean life (p.220, 222) do not match (4 d x 1.4427 = 5.8 d; 5.3 d x ln2 = 3.7 d). b. Experiment. The 133 cal/h value is consistent with calorimetry [read: p.52; the 1920 text says "exact experiments"] and with the nuclear arithmetic above. Chemistry: 2.2e3 cal per gram of (fuel + oxygen) is about 9 kJ/g; radium gives about 1.2e10 J/g, ratio 1.3e6. c. Tested? Yes. Calorimetry (Curie and Laborde 1903, Rutherford, Soddy) [memory, high]. The challenge to "internal" energy came from Kelvin's 1906 Times campaign (helium pre-existing; external radiation or ether supply), recorded in Soddy's Nature 74, 516 (1906) [web via Investigator: files/nature-1906-...pdf, research/2026-10-08-internet-archive.md] and in Lucien Poincare 1907, Badash 1966 [web via Investigator, snippets only]. The external-source hypothesis was the live alternative to conservation of energy plus transmutation; it fell away as helium production and the exact energy balance per atom were established. Kelvin's own letters were not read here. d. If true, what changes? Nothing; it is the accepted picture, with one addition Soddy lacked: the source is the mass difference E = mc^2. Per atom 29 MeV out of 226 x 931.5 MeV is a fractional mass loss of 1.4e-4 [calc], versus about 1e-10 for burning carbon, which is where "a million times" comes from.

Verdict: Established (the 133 cal/h and 2.9e9 cal numbers are right to within the measurement; the 1911 value of about 100 cal/h is low by about 25% and was revised by Soddy himself). Split detail: "energy is internal" Established; "2.9e9 is the complete change" holds only for radium plus products through RaC' (Historically superseded as a statement of the whole series).

Claim 3. Radium survives because it is made: equilibrium ratio = ratio of lives; radium mean life about 2,500 y, uranium about 7,500 to 8,000 million y; chain U, UX, ionium, radium, emanation, lead (eight alphas).

a. Mathematics. The equilibrium law lambda1 N = lambda2 X is correct and its condition (parent much longer-lived than product) is the right one. Hidden points Soddy leaves out: the transient case X/N = lambda1/(lambda2 - lambda1) when the parent is not much longer-lived; the time to reach equilibrium (about 5 mean lives of the product, so about 28 days for the emanation) [calc from standard solution]. Atoms versus grams: the law gives a ratio of numbers of atoms. Soddy applies "3,200,000 parts of uranium to one of radium" (p.169) as if it were an atom ratio (3.2e6 x 2,500 = 8.0e9 y). Boltwood and Rutherford measured it by weight; if so, the atom ratio is 3.2e6 x 226/238 = 3.04e6 and uranium's mean life is 7.6e9 y [calc]. A 5% hidden assumption, inside the "few per cent" he claims for radium's life, but never mentioned. His own table (p.199) then uses the same number the other way (312.5 mg Ra per ton U = 1e9 mg/3.2e6, ionium 12.5 g = 40 x 312.5 mg), again equating weight and atom ratios. The radium mean life: Ramsay and Soddy 1904 got 1,150 y from 1.3 mm3 (implied), too short because gas volume was overestimated, as Soddy explains; with 0.6 mm3 per gram, N/X = (1/226)/(0.6e-3/22,400) = 1.65e5 and T = 5.57 d x 1.65e5 = 2,519 y [calc]. This reproduces the adopted 2,500 y, but the 0.6 mm3 may itself have been fixed using 2,500 (the book gives no raw data), so it is a consistency check, not an independent measurement. The ionium argument is checkable and works beautifully. Soddy states radium grows as t^2 and, in the 3 kg uranium preparation, after ten years the radium was "one five millionth of a milligram ... one part of radium from fifteen billion of uranium" (p.134). Fifteen billion here is the British 10^12 billion, so 1.5e13: the grams are 3,000 g / 1.5e13 = 2e-10 g, matching 2e-7 mg [calc]. Equilibrium would be 1 part in 3.2e6, so the fraction reached is 3.2e6/1.5e13 = 2.1e-7. My formula Ra/Ra_eq = t^2/(2 T_Io T_Ra) with t = 10, T_Io = 100,000, T_Ra = 2,500 gives 2.0e-7. Solving backwards for T_Io from his numbers gives 93,750 y, i.e. "almost exactly 100,000" within his rounding [calc]. Note what this measures: the product T_Io x T_Ra, not either alone (the uranium life cancels). Modern: T_Ra = 2,308 y, T_Io = 75,400/ln2 = 108,800 y [web: Wikipedia Ra and Th isotope pages, 1,600 y and 75,400 y], product 2.51e8 versus Soddy's 2.50e8: the product agrees to 0.4% even though his radium life is 8% high and his ionium life 8% low. The deep read did not catch this. Chain bookkeeping: U-238 to Pb-206 is 8 alpha decays (238 - 32 = 206) and 6 beta decays (92 - 16 + 6 = 82) [calc]; the 1911 text counts alphas only ("beta particles too small to matter", 238.5 - 32 = 206.5 against lead 207) and the 1920 text resolves this as an isotope of lead. Thorium to Pb-208: 6 alphas and 4 betas [calc]. Fine. Printed errors and inconsistencies: (i) p.243 says ionium "something like 12.5 grams per ton of uranium, or 58 grams per gram of radium" [read: page image PDF p.291]. Soddy's own table (p.153 of the book) has 12.5 g ionium and 312.5 mg radium per ton of uranium, which gives 40 g per gram, not 58 [calc]; the equilibrium ratio of lives is also 40 (100,000/2,500), or 40.7 with the mass correction 230/226. The number 58 is wrong by a factor 1.45, which happens to be the mean-to-half-life ratio; I only note the coincidence, I cannot tell whether that is the cause. This is an internal inconsistency in one sentence, not affecting the argument. Nobody in Step 1 flagged it. (ii) Uranium II "3,000,000 years (?)" with "400 grams (?)": 3e6/8e9 x 1 ton = 375 g, consistent with itself, but modern U-234 mean life is 245,500/ln2 = 354,000 y [web: Wikipedia, 2.455e5 y], about 8 times shorter. Soddy marks it with a question mark. (iii) Table rows: radium F 202 d gives 312.5 mg x 202/913,000 = 0.069 mg ("one fourteenth"), emanation 1/500 mg, UX 1/80 mg, RaD 3 mg: I recomputed all and they agree with his lives [calc]. The table is internally consistent. Fig. 44 periods are mean lives (the deep read says so), which is why they look larger than modern half-lives; my spot check against modern half-lives times 1/ln2: RaA 3.05 min gives 4.4 (printed 4.3), RaB 26.8 gives 38.7 (38.5), RaC 19.9 gives 28.7 (28.1), RaE 5.01 d gives 7.2 (7.2/7.5), RaF 138.4 d gives 199.7 (196/202), UX1 24.1 d gives 34.8 (35.5) [calc; half-lives from memory, high]. Soddy's short-lived lives are good to 1 to 3%. b. Experiment. Modern U-238 half-life 4.463e9 y [web: Wikipedia Isotopes of uranium] gives a mean life 6.44e9 y; Soddy's 8.0e9 (1920) and 7.5e9 (1911) are 24% and 16% high. Of the 24%: 8% comes from the radium life and the rest from the uranium:radium ratio (modern atom ratio 2.79e6, by weight 2.94e6 [calc], versus 3.2e6 printed, 9% high by weight). Soddy says he got the same value "by direct experiment from the rate of production of helium from uranium" (p.169): no data given. c. Tested? Yes: Boltwood, McCoy, Strutt (proportionality U:Ra), Rutherford and Boltwood (ratio), Soddy and Hitchins 1915-1919 (growth of radium from uranium), Boltwood's isolation of ionium. Replicated and refined by others [memory, high]. The 1911 book itself says the growth of radium from uranium "has not yet produced a definite result" (p.224). d. If true? Nothing; accepted.

Verdict: Established for the equilibrium law, the chain and the logic of the t^2 growth test. Historically superseded but instructive for the numerical values (radium 2,500 y is 8% high, uranium 8,000 million y is 24% high, uranium II 3e6 y is about 8 times too long). The sentence giving "58 grams per gram of radium" is Internally inconsistent with Soddy's own table (40 expected).

Claim 4. The alpha particle is a helium atom (charge 2, mass 4); radium becomes emanation by expelling one; helium accumulates in uranium and thorium minerals.

a. Mathematics. Mass bookkeeping is correct: 226 - 4 = 222. Charge 2 deduced from counting: with deflection one gets only e/m; mass 4 and charge 2 versus mass 2 and charge 1 is exactly the ambiguity he explains (Matter and Energy p.232 [read, deep read]). Velocity check: c/15 = 12,400 miles/s, c/20 = 9,300 miles/s; he writes "12,000" and "between 1/15 and 1/20 of light" [calc]; meteors 40 miles/s x 300 = 12,000 [calc]. Fine. Emanation volume arithmetic: 1e-12 g of radium gives 0.6e-15 cm3 of gas, 2.7e-20 mol, 1.6e4 atoms; Soddy says "less than a million", true but loose by about 60x [calc]. b. Consistent with experiment: yes (Rutherford and Royds thin-glass experiment, Ramsay and Soddy, Rutherford and Geiger counting) [memory, high; the book describes them without dates]. c. Tested and replicated, yes; Soddy was a party to the helium-from-radium experiment (he says so in the text of the book, p.230) [read]. d. Nothing changes.

Verdict: Established.

Claim 5. Isotopes: atoms of one element differ in mass; displacement rules; lead 206.05 and 207.9; Aston's whole-number masses; credit for 1913 generalisation.

a. Mathematics. Displacement rules: alpha loses 2 in atomic number and 4 in mass, beta gains 1 in atomic number and none in mass: one alpha plus two betas returns to the same place (Z same, A - 4) [calc]. Correct. The lead prediction: 206 and 208 mixture to give 207.2 needs 60% of 208 [calc], but real lead is 1.4% Pb-204, 24.1% Pb-206, 22.1% Pb-207, 52.4% Pb-208 [web: Wikipedia Isotopes of lead], which gives 207.24 [calc]. So "may well be a mixture of these two isotopes" (p.242) is a simplification; Soddy knew of the doubtful actinium-series end product. Aston's table: neon 90% of 20 and 10% of 22 gives 20.2 [calc]; chlorine 35 and 37 needs about 77:23 to give 35.46 [calc] (he says "similar proportion" and checks line intensities); argon 36 and 40 needs only 1.3% of the light one for 39.95, hence "barely detectable" [calc], though argon 38 exists [memory, high]. Ionium-thorium: (232.12 - 231.51)/(232.12 - 230) gives 29% ionium by atoms [calc; assumes ionium 230 exactly]. The hidden assumption in "all atomic weights are integers" is mass additivity: nuclear binding energy makes masses non-integral at the 1/1000 level; Aston found this himself in 1927 (packing fraction) [memory, high]. His own text is hedged: "to an accuracy of one part in a thousand on oxygen = 16" (p.248), but then "not too much to suppose that all atomic weights, except hydrogen, are exact integers". Errors in Aston's 1920 table as reproduced (sulphur 32 only, xenon 128-135, mercury 197-204) are measurement-stage errors [memory, high]; the deep read gives the page-image corrections (32.0, 75.0). b. Experiment: yes, lead atomic weights measured differ by origin (Richards, Honigschmid; Soddy and Hyman) [memory, medium; Soddy names no experimenters on p.242]; isotope identification by mass spectrograph [memory, high]. Densities proportional to atomic weight (p.243, atoms of equal volume) is consistent with nuclear structure. c. Tested and replicated, yes. d. Nothing changes; it is accepted. Credit and priority (an Investigator lead I checked in the Nobel lecture [read: files/nobel-1922-soddy-lecture.pdf]). The 1920 book says the complete generalisation was put forward "independently during the same month by A. S. Russell, K. Fajans, and the writer". The 1922 Nobel lecture says: Hevesy and Russell published within a fortnight "a form, only in part correct"; "a fortnight after Russell, K. Fajans ... published this generalization complete and accurate in every detail. A fortnight after Fajans, in a paper amplifying and amending Russell's scheme, I published exactly the same scheme"; Fajans's paper "only reached England after mine was written". So the 1920 "same month" is compatible with the 1922 order, but the 1922 version adds Hevesy and shows Fajans first. Soddy is an interested party in priority statements; both accounts are his own. I would report both without ranking them.

Verdict: Established for isotopes, displacement rules and the lead result. Historically superseded but instructive for "all atomic weights are whole numbers" and for the two-isotope description of common lead and for several entries of Aston's table.

Claim 6. Nuclear atom; nitrogen bombardment knocks out hydrogen (and a mass-3 particle); controlled release of nuclear energy is real but unsolved, with great danger.

a. Mathematics. Printed numbers I checked: 1/100,000 alphas making an H-particle x 10,000 hydrogen atoms in the path = 1 per 1e9 collisions [calc, consistent]; H-particle range about four times the alpha's: the H atom recoils with velocity 2 m_alpha/(m_alpha + m_H) = 8/5 = 1.6 times the alpha velocity and range scales as v^3 (m/z^2 equal for both) so 1.6^3 = 4.1 [calc, assumes the Geiger range law, memory medium]: consistent with "some four times". Alpha energy 4-8 MeV with charge 2e equals 2-4 million volts [calc]. Nuclear size: 1e5 times smaller than an atom of 1e-8 cm is 1e-13 cm, right. Departure: the speculation of a proton radius of about 1e-16 cm (p.226): from electromagnetic mass m = e^2/(r c^2) (up to a numerical factor), a particle 1,830 times heavier than the electron has radius 2.82e-13 cm / 1,836 = 1.5e-16 cm [calc]. The hidden assumption is that all of the proton's mass is electromagnetic. It is wrong: the modern proton radius is about 0.84e-13 cm, 500 times larger [memory, high; checked here not on the web], so the proton's mass comes mostly from strong-interaction energy, not from electrostatic energy. Soddy labels it "merely a suggestion". The 2-4 million volt requirement (p.250) assumed the classical barrier must be climbed; Gamow's tunnelling (1928) and Cockcroft and Walton (1932, lithium split by protons of about 0.7 MeV) show far lower energies suffice [memory, high]. His surmise N-14 to C-12 + 2H: charge balance 7 = 6 + 2 fails (7 protons versus 8) [calc], so it is wrong on charge counting alone before any experiment; the true reaction N-14 + alpha to O-17 + p balances: 14+4 = 17+1, 7+2 = 8+1 [calc]. b. Experiment: nuclear atom with alpha scattering, atomic number (Moseley), H from N: all established [memory, high]; Soddy's own text gives the limits ("though they do not yet rigorously prove", p.272; "the real problem ... remains completely unsolved"). The mass-3 doubly-charged particle (Rutherford, Bakerian Lecture June 1920) was not confirmed and is not what is observed [memory, high]; He-3 was identified in 1939 [memory, high]. These are not web-checked here. c. Tested? Nitrogen result: tested by Blackett's 1925 cloud-chamber photographs, which showed the capture mechanism [memory, high]. Mass-3: tested, not supported. Controlled energy release: achieved by fission (1938-1942), after Soddy's book [memory, high]. In 1920 it was "plausible, untested". d. If true? Nothing to change; this is the standard picture. The arithmetic of cost (2-4 MV) is the part that had to change. Record note: Soddy's warning about misuse (p.252, quoted in deep read) and the earlier Wells link and Guston 2012 / Sclove 1989 citations are Investigator leads [web via research notes]; they are not mathematical claims and I leave them to the other reviewers, but they belong in the record.

Verdict (split): nuclear atom, atomic number, hydrogen from nitrogen: Established. Mass-3 "helium isotope" from nitrogen and oxygen: Tested, not supported. Proton radius 1e-16 cm: Refuted by calculation and measurement (mass is not purely electromagnetic). Release of nuclear energy as real but unsolved: Plausible, untested in 1920 (later realised, which is outside the books). The 2-4 MV barrier estimate: Historically superseded but instructive.

Claim 7. Radioactivity changes the picture of earth and sun; civilisation depends on a diminishing capital of coal; "energy and wealth are synonymous"; speculations (gold-silver series, forgotten transmuting race).

a. Mathematics. Little mathematics is offered. Checkable parts: radium running down: e^(-25,000/2,500) = e^-10 = 4.5e-5, "less than one-thousandth" is true [calc]; "a hundred thousand years ago ... a thousand billion times as much" is understated: e^40 = 2.4e17 [calc], well beyond 1e12 or 1e15 (US or British billion). This is a conservative error that strengthens his point. The "radium heat of the earth" idea: not quantified in the folder. "Energy and wealth are synonymous" is a definition, not a derivation; I cannot test it mathematically. The gold-silver series: gold is Z = 79 and silver Z = 47. Alpha and beta steps change Z by -2 and +1 only, and mass by -4 or 0, so any chain from silver (mass about 107-109) to gold (197) would need a gain of about 90 mass units, which the decay rules never give; Soddy himself calls the idea "frank speculation". Alchemist recipes (pp.232-233, as quoted in the deep read): thallium (Z 81) minus an alpha is 79, correct; bismuth (83) minus two alphas is 79, correct; "lead losing an alpha and a beta" is 82 - 2 + 1 = 81, which is thallium, not gold [calc]. Either the book's sentence or the deep read's paraphrase has a slip; I did not re-read the page image (PDF pp.280-281), so this must be checked there before anyone relies on it. b. Experiment: radiogenic heat is a major part of the earth's interior heat budget [memory, medium], so "radium may supply the earth's heat" is on the right track; the claim that old Kelvin cooling-age calculations are not binding is established. Alternating geological and incandescent ages of the cosmos has no support [memory, high]. Gold from silver: no support. Coal as a non-renewable capital: consistent with fuel-reserve calculations [memory, medium]. c. Tested? The transmutation experiments of the time (Ramsay and Cameron 1907, copper to lithium, not confirmed by Curie and Gleditsch who traced lithium to glass; Soddy's own negative copper and gold tests, 1909) are Investigator leads [web via snippets, research/2026-10-08-google-search-leads.md]: a claimed transmutation by others that was tested and not confirmed. The gold-silver and forgotten-civilisation ideas were never tested because they are not testable as stated. d. If true? A silver-gold series would need a nuclear route that mainstream nuclear physics does not allow; a forgotten transmuting civilisation would need physics that has not been seen.

Verdict (split): radium heat as part of the earth's heat budget and the old cooling-age arguments: Established in outline (heat budget in detail: Plausible, untested within the folder). Cosmic alternation of ages: Historically superseded but instructive. "Energy and wealth are synonymous": Not testable as stated (a definition; the economics sources are outside the folder). Gold-silver series: Historically superseded but instructive (no test in the folder; the decay rules give no route). Forgotten transmuting race: Not testable as stated.

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2. Summary tables

2a. Soddy's numbers against modern values ([calc] and [memory, high] unless a URL is given)

2b. Printed errors and internal inconsistencies I found or confirmed

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3. Investigator's findings I weighed (kept in the record)

• 1909 first edition of The Interpretation of Radium states 100 calories per hour per gram (25 from radium, 75 from emanation) [web via Investigator: https://archive.org/details/interpretationof00soddrich]. Weight: confirms the 1911 value. I add that the 1911 book itself implies 100 cal/h through the "38 hours to decompose its own weight of water" sentence (see Claim 2), so the inference no longer rests on the "freezing to boiling" sentence alone.
• Nobel Lecture 1922 on the order of the displacement-law papers: read by me (see Claim 5); it adds Hevesy and the fortnight order.
• Nature 74 (1906) reply to Kelvin: bears on Claim 2; read by the Investigator, not by me; I rely on the Investigator's quotation. Kelvin's side is stated by his opponent only; the record should include Kelvin's own letters when they are obtained.
• Economics books, Wells 1914, Guston/Sclove, Morrisson/Brill, Daly/Merricks, Lindau lectures, Bodleian papers, the missing wartime files and 1953-54 Howorth letters: not mathematical; weighed only where they bear on claim 7 and 6. The Bodleian catalogue's remark that wartime envelopes were "probably disturbed, and some removed" is a documented gap in the record of Soddy's war work: absence of evidence to be analysed (commercial or personal loss, removal of classified material, selective use by the biographer) by the physicist and arbiter; I do not weigh it mathematically.
• Ramsay and Cameron (1907) copper-to-lithium claim, not confirmed by Curie and Gleditsch (lithium traced to glass), and Soddy's negative copper and gold tests (1909): single-source snippets via the Investigator; a transmutation claim tested and not confirmed, which gives context to the 1919-20 nitrogen result.

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Corrections to Step 1 (for the orchestrator)

1. claims.md, last bullet, and facts.json (Matter and Energy): the 1911 value of about 100 cal/h is not only inferred. Matter and Energy p.210 says the heat decomposes its own weight of water in 38 hours; 3.79 kcal/g / 38 h = 99.7 cal/h [calc]. Together with the 1909 edition (Investigator) this settles it. Also the 1911 CO2-in-a-day, ten-years-of-carbon and quarter-ton-of-coal statements all fit 100, not 133.
2. Deep read (Interpretation, section 3, "Numerical consistency checks"), missed items: (a) the printed "58 grams per gram of radium" (p.243) contradicts the book's own table (40); (b) British billion: "fifteen billion" (p.134) and "a thousand billion" (p.165) should be read with the long-scale billion; with it the 15-billion figure checks exactly; (c) the t^2 growth experiment reproduces T_Io = 100,000 y (my calc gives 93,750 y from his rounded numbers) and measures the product T_Io x T_Ra, which agrees with modern values to 0.4%.
3. Deep read states the 2.9e9 cal total counts only radium plus short-lived products: confirmed by an independent per-atom count (29 MeV x 2.66e21 atoms = 2.97e9 cal). The deep read's "27-28 MeV, 135-140 cal/h" was medium confidence: I get 25 MeV (alphas only, with recoil) to 29 MeV (including beta Q-values), 125-147 cal/h.
4. facts.json, Interpretation, "mainstream_conflict": the uranium life should also note that Soddy's 8,000 million years is 24% above the modern mean life (6.44e9 y) and that the ratio 3.2e6 is used as an atom ratio although it was a weight ratio (5%). The existing statement that radium's 2,500 y is 8% high stands.
5. claims.md claim 5 (credit): agree with the Investigator's correction; I verified in the Nobel lecture text that Hevesy and Russell came first (partly correct), Fajans second (complete), Soddy a fortnight after Fajans. Add Hevesy and the order, and say that Soddy's own two accounts (1920 "same month", 1922 fortnights) are both his characterisation.
6. claims.md claim 3: "eight alpha particles lost from uranium" is right; add "and six beta particles" so that the Z balance 92 - 16 + 6 = 82 is visible.
7. Possible slip to be checked on the page image (PDF p.280-281): the deep read quotes "lead losing an alpha and a beta" giving gold; the arithmetic gives thallium (Z 81). Either the book or the paraphrase is wrong; I did not re-render the page.
8. The deep read's Step 1 table line on Soddy's radius prediction: the ratio 1,830 applies to the electron radius (2.82e-13 cm), giving 1.5e-16 cm [calc]; Soddy's "about 1e-16" is a rounding, not an error.

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Verdict summary