Mendeleev-D: Mathematician's review (Step 2)

Reviewer: PhD mathematician. Scope: derivations, notation, units, hidden assumptions. Physics verdicts belong to the physicist; where I use physics I say so. Source tags: [read] from the project documents (PDF page numbers of the 1904 Longmans copy; printed page = PDF page - 8), [calc] my own computation (Python/NumPy, reproduced below), [web] with URL, [memory] with confidence.

Source gap: none for the essay. The two PDFs are one book (Step 1 deep read, checked: text of pp.35-51 read by me from the extracted text; the periodic table p.34 read from a rotated page image). Claim 6 rests on a secondary article (Stewart 2019). The Russian 1905 text is the Investigator's download; I used its footnotes only as reported in research/2026-10-06-ru-wikisource.md [web] and did not re-read the Russian.

0. What the mathematics of this project actually is

The essay has four computational parts. Everything else (radioactivity, fluorescence, rejection of electrons) is qualitative and has no equations to check.

1. Extrapolating atomic-weight ratios down the periodic table to get y (at most 0.4) and x (at most 0.17), pp.35-36, 42.
2. Kinetic theory: speed of a gas particle of atomic weight x, v = 1843 sqrt(2(1+at)/x) m/s (eq. I), which at t = -80 degC becomes v = 2191/sqrt(x), x = 4,800,000/v^2 (eq. II), pp.43-44.
3. Escape velocities of Earth, Sun and a 50-solar-mass star, converted into upper bounds on x, pp.45-50.
4. The adopted result: x about 1e-6 (of hydrogen), v about 2,250 km/s, p.51.

1. Verification of the arithmetic (Step 1 checked)

[calc] unless stated.

Result for question (a), narrow sense: every number printed in the essay's calculations follows from its stated formulas, apart from the one misprint (a). The arithmetic is sound. The problems are in the premises and in what the numbers are used for, set out in section 2.

Notation and units (checks against the text): - "Mean velocity" of 1843 m/s for hydrogen is really the root-mean-square speed. [calc] sqrt(3kT/m) for H2 at 273 K is 1838 m/s; the true mean speed is 1694 m/s. He computes it from "a litre weighs 0.09 g", which gives 1838 (p.43). The label is loose, the number is right for the rms speed. - "Mass of the earth = 398.10^12" is GM (units m^3/s^2), not a mass in kilograms. He calls it "mass in absolute units" (p.45). Fine if read as G = 1 units. - Density x/2 for a monatomic gas relative to H2 = 1 is correct, and v proportional to 1/sqrt(density) at fixed temperature is correct (kinetic theory). - The Step 1 deep read's reading of pp.28, 31, 32, 34, 44, 47 from page images holds up: the table on p.34 (seen by me) shows series 0 with x in the zero group, series 1 with y in the zero group and H in group I, He in series 2, Ne in series 3, Ar in series 4. [read: page image p.34]

2. Hidden assumptions and weak points in the derivations

2.1 The estimate of y (at most 0.4), pp.35-36

How it works, in my reading (the text is terse). The ratios of weights of the series-3 to series-2 elements in the same group (Cl:F ... Na:Li, and Ne:He for the zero group) rise as one moves to lower groups: 1.86, 2.00, 2.21, 2.37, 2.45, 2.67, 3.28, 4.98. The matching ratio for series 2 to series 1 is known in group I only: Li:H = 6.97. For the zero group he needs He:y (series 2 to series 1). He says it is "considerably greater" than 6.97, "at least 10". [calc] One way to reach 10: Ne:He/Na:Li = 4.98/3.28 = 1.52 is the jump from group I to the zero group at the series 3:2 level. Apply the same jump to 6.97: 6.97 x 1.52 = 10.6, so y <= 4/10.6 = 0.38, which rounds to his 0.4. This is my reconstruction, not Mendeleev's stated procedure. The Investigator's lead, Lente (2019), reconstructs it differently (a near-geometric series with factor about 1.4, intercept about 9.5 rounded up to 10) [web: link.springer.com/article/10.1007/s40828-019-0092-5, via research/2026-10-06-springer-lente.md].

Hidden assumptions: 1. One data point (Li:H) carries the whole 2:1 level. Hydrogen is the odd element of the table (group I and VII both claimed), so it is the weakest anchor possible. 2. The jump from group I to group 0 at one level is assumed to equal or exceed the jump at another level. Nothing in the data tests that. 3. The words "at least" are not earned. The trend in the 3:2 ratios is increasing, but the steps themselves grow (x1.07, 1.10, 1.07, 1.03, 1.09, 1.23, 1.52), so one can extrapolate in many ways and get many "at least" values. 4. The inequality is an extrapolation, not an interpolation. Mendeleev says so himself: gallium, scandium and germanium were predicted "by interpolation", now he extrapolates and "risks" something (pp.24-26 [read]). The strength of the 1871 predictions does not transfer.

2.2 The estimate of x (at most 0.17), p.42

[read: p.42] He takes Xe:Kr = 1.56, Kr:Ar = 2.15, Ar:He = 9.5 and gets He:x = 23.6, so x <= 4.0/23.6 = 0.17, "the maximum possible value". [calc] This is a second-order (quadratic) extrapolation through three equally spaced points: first differences 0.59 and 7.35, second difference 6.76, next difference 14.11, next value 9.5 + 14.11 = 23.61. It reproduces 23.6 exactly. This agrees with the German text's phrase "aus einer Parabel zweiter Ordnung" and with Lente's reconstruction, both reported by the Investigator (the German phrase is Lente's quotation of the Rundschau text, not read by me [web]).

Sensitivity [calc]. The same three points, extrapolated four ways, give:

Three points cannot choose between these. The data move by a factor of 8 in x with the model. Modern atomic weights (Ar 39.95, Kr 83.8, Xe 131.3, He 4.003) change the quadratic result only a little (He:x = 25.2, x = 0.159), so the problem is the model, not the data. Also: the point Ar:He = 9.5 jumps by a factor 4.4 from the previous ratio. That is a sign that He sits in the anomalous first period, and a cubic or higher fit through it would not be trusted.

Most important logical point. The 0.17 bound plays no role in the final answer. The final x (about 1e-6) comes from the kinetic argument alone (section 2.3), six orders of magnitude below 0.17. So the "periodic extrapolation" argument for x is an upper bound that is never active. Mendeleev knows this: "most probably the atomic weight of x is far less" (p.42). The zero series is placed in the table by the periodic law, but its numerical entry comes from a different, non-periodic argument.

2.3 The kinetic argument for x about 1e-6, pp.42-51

The logic: x must be light enough that its particles escape the gravity of every body in the universe, "as otherwise it would accumulate about the largest mass and not fill all space" (p.42). Escape speed gives v, v gives x by eq. II.

Hidden assumptions, with my calculations:

1. Threshold, not tail. He sets the typical (rms) speed equal to the escape speed. In a Maxwell gas a fraction of particles always exceeds any speed; whether a gas is retained depends on v_esc compared with the typical speed over very long times. The condition "rms speed equals escape speed" is a convention, not a derivation. It changes x by a modest factor (the bound goes as 1/v^2), not by orders of magnitude, so this does not break the number. [memory, high: the Jeans escape criterion in standard texts].

2. One temperature for all of space, -80 degC. [calc] Since v is proportional to sqrt(T), his range -150 to -40 degC changes x by factors 0.64 to 1.2, which is negligible beside his six orders of magnitude. This part of the choice is harmless, and he is right to say so (p.44: "the value of t ... has hardly any significance"). But the assumption that the ether has a temperature of its own, not that of the matter it permeates, is a real gap. Mendeleev himself says the ether permeates all bodies and "sets up a rapid state of interchange with the surrounding medium" (p.58 [read]), and that the temperature "probably varies in different localities" (p.44 [read]). If ether atoms exchange energy with matter, their temperature in a hot body is the body's temperature. [calc] For x = 1e-6 (atom mass 1.7e-33 kg), v_rms = sqrt(3kT/m): at 193 K (-80 degC) 2,185 km/s (0.7% of c); at 300 K 2,725 km/s; at the Sun's surface, 5,800 K, 11,980 km/s (4% of c); at the Sun's core, about 1.6e7 K, 623,000 km/s, which is 2.1 times the speed of light. The core temperature of the Sun was not known in 1902 (so this is not a charge that he ignored a known number), but it shows that the single "x and v" derived from a fixed -80 degC is internally consistent only if the ether never thermalises with matter. His own rejection of the lower limit x = 5e-11 was exactly that its speed would reach c (p.51). With thermalisation his adopted value would reach it too in a hot star. Flag: this uses modern solar temperatures [memory, high]; the dependence v proportional to sqrt(T) is his own eq. I.

3. Stars of the same mean density as the Sun and masses up to 50 solar masses. The cube-root scaling v_esc proportional to n^(1/3) is correct under equal density [calc]. But the stop at 50 solar masses is arbitrary (he allows nothing larger than what double stars showed in 1902), and equal density is an assumption. [calc] A white dwarf shows how much hangs on it. Sirius B: mass 1.018 solar masses, radius 5,635 km [web: https://en.wikipedia.org/wiki/Sirius_B]. Escape speed = sqrt(2GM/R) = 6.9e6 m/s (6,900 km/s), three times his adopted ether speed. By his own criterion x would then have to be <= 4.8e6/(6.9e6)^2 = 1.0e-7, ten times smaller than adopted, and a neutron star would push it near the speed of light (that last step is [memory, high] for neutron-star escape speeds of order 0.5c). Sirius B's density was not known until after 1915 [memory, medium], so this is "historically superseded", not a mistake by 1902 standards. It still shows the logic has no stopping point: each newly discovered dense body forces x smaller, and the hypothesised answer has no principled upper limit on v other than c (the limit he rejects as "emission theory", p.51).

4. No confinement. A gas whose particles all exceed the escape speed of every star has no gravitational confinement at all. It would not have a density that is "all-permeating and uniform" except by being continuously supplied. Nothing in the essay says what keeps ether in the universe, or what its density is, beyond Lord Kelvin's lower bound of 1e-16 g/m^3 as printed (p.10), which I could not verify [memory, low: Kelvin's estimate]. At that density, x = 1e-6 gives 6e13 atoms per cubic metre [calc], with free path unknown because no collision cross-section is given.

5. Gas speed against wave speed. This is the one inconsistency inside the essay itself. Mendeleev adopts v about 2,250 km/s because the ether must carry light, and says "the velocity of the ether atoms is not very much less (130 times) than that of the propagation of their transverse vibrations" (p.57 [read]; [calc] 3e8/2.25e6 = 133). - In kinetic theory the speed of any wave in a gas is of the order of the particle speed. [calc] For a monatomic gas the sound speed is 0.745 times the rms speed, here about 1.6e6 m/s, not 3e8 m/s. A signal cannot be carried faster than the particles carry it unless there is a rigid medium. - Transverse waves (light is transverse) cannot propagate in a gas at all. Linearise the equations of fluid motion for a transverse velocity u: du/dt = nu d2u/dy2 (nu = kinematic viscosity). That is a diffusion equation. Its solutions decay, they do not travel. A medium carrying transverse waves needs a shear stiffness (an elastic solid). A free-streaming gas with no collisions (needed for permeating bodies without friction) carries no waves at all. So the ether as a gas with Mendeleev's own speed cannot do the job he created it for. He half-admits this in the 130-times remark and the "emission theory" remark, but it is left undeveloped. This is a mathematical, not an experimental, obstruction, and it is the clearest "internally inconsistent" finding in the project.

6. Drag. Not computed: it needs a cross-section that the essay does not provide (it says only that ether permeates bodies like water through a sieve, p.19). If the ether does not interact with matter, it cannot be "dissolved" by uranium (p.54); if it does, drag on planets and thermalisation follow. The essay needs both properties.

2.4 Solar energy from accumulated ether (Russian 1905 footnote 20, Investigator lead; not in the English)

Mendeleev asks whether ether piling up around the Sun could explain the Sun's excess energy [web: research/2026-10-06-ru-wikisource.md]. [calc] energy budget with his own numbers: - Gravitational energy per ether atom at the Sun's surface: GMm/R = m v_esc^2/2 = 3.1e-22 J, against thermal energy 1.5kT = 4.0e-21 J at 193 K. So ether atoms hardly feel the Sun's gravity: in an isothermal atmosphere the density enhancement at the surface would be only exp(0.12) = 1.12. - Energy delivered if ether falls onto the Sun: even the ether in the Sun's volume at Kelvin's 1e-19 kg/m^3 is 1.4e8 kg, giving 2.6e19 J, that is 7e-8 s of the Sun's output (3.8e26 W [memory, high]). - To supply the Sun for 1e9 years (about 1.2e43 J) with infall energy 1.85e11 J/kg, one needs 6e31 kg, about 30 solar masses of ether. Conclusion: the speculation is ruled out by energy accounting for any density near Kelvin's bound. It is a footnote and Mendeleev's phrasing is a question ("whether this could help"), so I judge it a stated hypothesis, not a claim. I keep it in the record as the Investigator's lead.

2.5 The periodic-law ideology (claim 5)

[read: p.39-41] "No elementary substance is conceivable which is not subject to the periodic law" yet "the unity of the substance of the ether" is retained while "unity of matter" is rejected. That is a logical tension: placing x in the periodic table (a table whose columns relate elements through a common structure) while denying that it relates to other matter. He is aware of it ("Of course, this is a hypothesis", p.39). It is a tension, not a contradiction, because the argument runs "x is the first member", but he rejects the very explanation (Prout-type unity) that gives the table a reason for existing. Lente suggests the rejection of Prout's hypothesis was a personal motive [web, as his characterisation].

3. Claims, with questions a-d and verdicts

Claim 1. The ether is a real chemical element: a monatomic gas, with weight, permeating all bodies, unable to combine.

a. Mathematics: no equation states it directly. As a physical model it fails internally on one point: a gas cannot carry transverse waves at the speed of light, and the ether speed he derives is 1/133 of c (section 2.3, item 5). The kinetic parts (speed, escape) are arithmetically correct. The claim that ether is "like hydrogen in palladium" is an analogy with no calculation. b. Experiment (physicist's domain, brief): Michelson-Morley (1887) and later ether-drift tests, and special relativity (1905), remove the need for a medium [memory, high]. A particle gas at 2,250 km/s would give detectable drag and Doppler effects, not computed by Mendeleev [calc, no data]. c. Tested? Mendeleev proposes no experiment. The Investigator found no replication or test of a gaseous element-ether ("no replication or experimental test ... found this run", sites.md). He did earlier low-pressure gas work (1874, Kirpichev; deviations from Boyle's law, p.11) that he says was meant to reach the ether, and calls it impossible to measure below tenths of a millimetre of mercury. d. If true: relativity's relative-motion postulate would need a preferred rest frame, with ether atoms at 0.7% of c carrying light at c by a mechanism the kinetic theory forbids; a particle lighter than hydrogen by 6 orders of magnitude, with no place in atomic structure. Verdict (split): - Strong form (a monatomic gas ether that carries light, with the derived properties): Internally inconsistent (gas waves cannot be transverse or exceed particle speed). - Weak form (the ether is a material substance worth a chemical search): Historically superseded but instructive. In 1902 it was a respectable position; it was replaced by field theory and relativity. Investigator leads kept: the Russian text says the ether may be a mixture of gases (footnote 9). That weakens "one x" but leaves each bound as written. Mendeleev's motive since the 1870s (footnote 5) and the 1905 preface ("serving the cause of unity in natural philosophy", Principles of Chemistry 3rd English ed. [web via research/2026-10-06-internet-archive.md]) are the inventor's statements. Yarkovsky 1901 (an independent ether density) was not located, so nothing can be said of its mathematics.

Claim 2. The periodic system needs a zero group of inert gases and a zero series with x, and a second element y (possibly coronium) between x and hydrogen.

a. Mathematics: the zero group of noble gases is a sound extrapolation of the table (Ne:He 4.98 continues the ratio trend; Errera 1900, Ramsay 1900 per the Investigator). The y and x estimates are extrapolations with unstated and untestable assumptions (sections 2.1, 2.2): one data point for y; three points and a quadratic for x, with results spread by a factor 8 under reasonable alternatives. b. Experiment: the noble gases are established (He, Ne, Ar, Kr, Xe, found 1894-1898 [memory, high]). Nothing lighter than hydrogen has been found; modern atomic number counts from Z = 1 (physicist's domain). The coronal green line: Mendeleev gives 531.7 nm, modern 530.3 nm (5303 Angstrom), identified as Fe XIV (Grotrian 1939, Edlen 1940) [web: https://en.wikipedia.org/wiki/Solar_corona]. So coronium is not an element and y has no observational basis. c. Tested? The coronium identification was tested by spectroscopy in 1939-40 and settled. No test of x. (The Investigator found no sub-hydrogen element tests; the later "element zero" proposals by Antropoff 1926 and the "100 Years of Element Zero" ZAAC paper of June 2026 were not read, only their titles; I can say nothing of their mathematics.) d. If true: atomic number and nuclear charge would have to start below 1; Moseley's (1913) charge ordering would have a place for Z = 0 [memory, high], which it does not need. Verdict (split): - Zero group of noble gases: Established. - y = coronium: Tested, not supported (the green line is Fe XIV; spectroscopic identification 1939-40). - Zero series with x, and sub-hydrogen elements in general: Never properly tested; as derived from the periodic ratios, the numbers are Not testable as stated (extrapolation choice decides the answer). Note for the table dispute: ru.wikipedia says x and y were absent from the tables of the essay (Investigator). The project's English page image p.34 [read: page image p.34] prints both, so that statement is contradicted by the primary document. Whether the 8th edition (1906) prints them is not established (not obtained).

Claim 3. The ether atom weighs about one millionth of hydrogen and moves at about 2,250 km/s, from kinetic theory at -80 degC and escape velocities.

a. Mathematics: arithmetic correct (section 1), except the printed a = 0.000367. The reasoning has gaps that matter: threshold-not-tail (minor), single temperature and no thermalisation (the speed would exceed c in the Sun's core if the ether thermalised), the stopping point at 50 solar masses (a white dwarf of 1 solar mass would require x about 1e-7), and no confinement. The result is a bound with a free choice, not a derivation: the range spans 9.6e-7 to 5.3e-11, five orders of magnitude, and the choice of 1e-6 is an adopted value ("I think it is impossible ... to admit the latter value", p.51) justified by avoiding the emission theory. b. Experiment: no measurement of any ether-atom property exists. Not applicable. c. Tested: not by anyone. Mendeleev gives no test. d. If true: a gas of particles of mass 1.7e-33 kg at 2,250 km/s has energy 4e-21 J per atom, the same as any gas at 193 K; no part of mainstream mechanics changes, but a medium like that would have to be nearly non-interacting yet able to dissolve in uranium (claim 4). Verdict (split): - The arithmetic from stated premises: Established as consistent. - The conclusion x about 1e-6 and v about 2,250 km/s as properties of a real ether: Internally inconsistent in its use (the temperature assumption conflicts with his own exchange mechanism; the speed assumed to carry waves cannot do so), and Never properly tested. No role of the number can be separated from the premises, so the strongest honest label is Internally inconsistent for the combined framework. Investigator leads kept: the Russian text doubts absolute zero (footnote 23) and plans a separate article, which explains the -80 degC choice. His own bound shows the choice does not matter by orders of magnitude.

Claim 4. Radioactivity (U, Th, Ra as ether-accumulating "suns"), the Curie ZnS flask demonstration and Dewar's low-temperature phosphorescence support the ether.

a. Mathematics: there is none. No quantity is predicted: no solubility, binding energy, flux or luminosity. [calc] The only number one can derive: an atom of 1.7e-33 kg at 2,250 km/s carries 0.025 eV of kinetic energy, comparable to the energy that binds light noble gases to surfaces. The text gives no value to compare. Pass over one worked check: he argues "solubility of the ether in paraffin increases in extreme cold" (p.57-58) and that is consistent with ordinary gas physics in form (gas solubility rises as T falls) but with nothing derived. b. Experiment: radioactivity and fluorescence of zinc sulphide under radiation are established; the interpretation is the point. Mainstream, in physicist's domain: the flask result is an effect of radiation and emanation from radium passing through the tube [memory, medium: Rutherford and Soddy's emanation, 1902-03]. c. Tested: Mendeleev reports what the Curies showed him (p.55-56 [read], an observation by witnesses, not his measurement). No controlled test of the ether interpretation was found in the record, and the Investigator found no replication. d. If true: radioactive emission would be an inflow or exchange of a sub-hydrogen gas rather than the decay of atoms, which would have to be reconciled with alpha, beta and gamma radiation and with later nuclear physics. Verdict: Not testable as stated (as offered, no measurable prediction distinguishes it from the standard picture). The weaker reading, that radioactivity in 1902 was puzzling enough to motivate it: Historically superseded but instructive. Interested parties: none of the sources here has a commercial stake. Mendeleev is the author and the only witness of how he was shown the demonstration in the Curies' and Becquerel's laboratories.

Claim 5. Matter has no single primary substance; atoms are not divisible into electrons or transmutable; only the ether is a unity.

a. Mathematics: a logical, not a computational claim. There is a tension (section 2.5): the table is used to place x, while the explanation for why elements form a table is rejected. No equations. b. Experiment: contradicted by later experiments. Atomic structure and electrons, transmutation (Rutherford 1919) and subatomic particles [memory, high]. Mendeleev's dismissals of Emmens (silver to gold) and Fittica (phosphorus to arsenic, 1900) were well-founded in their cases (those reports are not accepted today [memory, medium]); the general ban was not. c. Tested: the specific transmutation reports he rejected were tested by others and not confirmed [memory, medium; not checked on the web]. The general statements were superseded. d. If true: the electron and nuclear models would be wrong. Verdict (split): - Rejection of the two particular 1900-era transmutation claims: not assessed here (Never properly tested in this project, no sources). - General denial of atomic subdivision and transmutation: Historically superseded but instructive.

Claim 6 (secondary, from Stewart 2019). Mendeleev's earlier predictions (Ga, Sc, Ge) were largely successful; the ether and coronium additions were not predictions from element properties.

a. Mathematics: the reported figures are consistent: Re 190 vs 186.2, Po 212 vs 209, Fr 220 vs 223, Pa 235 vs 231 [read: digest of Stewart pp.4]. The difference from the 1871 predictions is structural: Ga, Sc, Ge were interpolations between known neighbours, with several properties each; the ether is extrapolation, one property (atomic weight), from a few ratios (sections 2.1, 2.2). I agree with Stewart's reading in method terms, and Mendeleev himself says as much (pp.24-26). b. Experiment: gallium (1875), scandium (1879) and germanium (1886) were found, with weights close to the predictions [memory, high; not rechecked]. Coronium: Fe XIV [web, above]. c. Tested: yes, by Lecoq de Boisbaudran, Nilson and Winkler (as named by Mendeleev, p.24 [read]). d. Nothing mainstream changes. Verdict (split): - Ga, Sc, Ge predictions: Established. - Ether and coronium additions as "predictions": not predictions in this sense; y = coronium Tested, not supported; x Never properly tested.

4. Corrections to Step 1

1. Stewart attributes the coronium identification to "Gruenwald" (facts.json, mainstream_conflict and summary; claims.md). Standard accounts credit Grotrian (1939) and Edlen (1940) with identifying the green line as Fe XIV at 5303 Angstrom [web: https://en.wikipedia.org/wiki/Solar_corona]. The Step 1 text records Stewart's statement faithfully (digest p.6); the discrepancy is in Stewart as digested, or in the digest. Keep as: Stewart gives the name "Gruenwald"; other sources give Grotrian and Edlen. I did not check Stewart's original page.
2. Stewart's "six more blanks between hydrogen and helium" in the 1904 table (digest p.6, facts.json summary): the project's English table (page image p.34, which I read) has only x (series 0) and y (series 1) in the zero group. Six blanks are not visible in it. The digest's sentence may describe Werner's 1905 table (Stewart's next sentence says Werner "included spaces for elements lighter than hydrogen"). Treat the claim as unverified for Mendeleev's table. The Russian footnote 17 (a possible light halogen between H and He, [web]) is the nearest thing in the record.
3. "Mean molecular velocity" in facts.json: the 1,843 m/s is the rms speed (1838 [calc]); the true mean is 1694 m/s.
4. claims.md claim 3 says x is "derived from kinetic theory ... and from escape velocities". That is right, but the same claim should say that the periodic extrapolation to x <= 0.17 (claim 2) plays no part in the 1e-6 figure. Add that the escape criterion has no upper stopping rule except c and the stellar masses known in 1902.
5. Deep read, section 4 "130 times" remark was left as a question mark. The printed text is clear: ether atoms move at 1/130 of the speed of propagation of their transverse vibrations (p.57), and [calc] 3e8/2.25e6 = 133. This is a self-consistent number and the basis of my inconsistency finding, not an unclear passage.
6. Deep read says "x <= 0.17 from noble-gas ratios" without giving the method. Method reproduced here: quadratic extrapolation through Xe:Kr, Kr:Ar, Ar:He, exact to 23.61.
7. Deep read, the Earth escape check gives 11,176 m/s; I get 11,180 [calc]. The difference is rounding, no correction needed. The Earth/Sun/50-sun numbers all agree with the essay to within 0.1% (section 1).
8. Step 1 confirmed: the misprint a = 0.000367 (versus 0.00367); two PDFs are one text (I did not redo the whole comparison); table placement of x and y (page image p.34).
9. Name corrections from the Investigator (Kirpichev, Rogovsky, Nasini-Andreoli-Salvadori 1898 instead of Anderlini 1893) come from the Russian text and are not checked by me. For the English 1904 text, the printed forms stand; the Russian forms should be used when citing people.

5. What would settle the open questions (mathematics)

• Whether the x extrapolation is meaningful: apply the same quadratic method to other groups where the answer is known (hold back one element, predict it); if it fails, the 23.6 is unreliable. My sensitivity table already shows the dependence on the model.
• Whether the ether temperature assumption is consistent: Mendeleev's own text on exchange with matter (p.58) against eq. I at stellar temperatures. A reading of the Russian text (footnote on temperature variation) is the place to look.
• Whether x and y survive in the 1906 8th edition table: obtain Osnovy khimii 8th ed. (RGB/NEB, not yet reached by the Investigator).

6. Reproducing the numbers

All [calc] figures in this file can be recomputed with: eq. I and II (section 1); GM = g R^2 for Earth; v_esc = sqrt(2GM/R); v scales as n^(1/3) at equal density; v_rms = sqrt(3kT/m) with m = 1e-6 x 1.008 x 1.6605e-27 kg; quadratic extrapolation of three equally spaced points (second difference). Constants: k = 1.380649e-23 J/K, c = 2.998e8 m/s.

7. Summary of verdicts

1. Ether as a gas element: strong form Internally inconsistent; weak form Historically superseded but instructive.
2. Zero group: Established. y = coronium: Tested, not supported. Zero series x: Never properly tested, numbers Not testable as stated.
3. x about 1e-6, v about 2,250 km/s: arithmetic Established; conclusion Internally inconsistent as a property of a light-carrying ether, and Never properly tested.
4. Radioactivity and phosphorescence as support: Not testable as stated.
5. No primary matter, no transmutation: Historically superseded but instructive.
6. Earlier predictions Ga, Sc, Ge: Established; ether and coronium additions are not predictions of that kind.