Arbiter (Step 3): Summerfield (Arnold Sommerfeld), Atomic Structure and Spectral Lines

Date: 2026-10-09. Inputs read in full: claims.md, physicist.md, mathematician.md, contrarian.md, sites.md. Not re-read: facts.json, digests, the deep reads (I used the reviewers' quotations of them). Primary sources I read myself: the 1923 English book as a single-page OCR scan saved by the Investigator (research/files/Sommerfeld-1923-AtomicStructure-IA-...djvu.txt; its OCR is far more legible than the folder PDF's), Ch. V sections 6-7.

Tags: [read] from a document or saved file; [web] with URL; [memory] with confidence; [computed] my own calculation (scripts in /tmp/arb, project .venv).

0. Source gap and what rests on what

Naming. The folder says "Summerfield"; the author is Arnold Sommerfeld. The book is the standard textbook of the old (Bohr-Sommerfeld) quantum theory, written by one of its creators, not an outsider's work. There is no suppression story of the usual kind; the interesting history is one dispute (the size of the hydrogen doublet), the later discovery that the formula is right for reasons the book does not give, and the Nobel silence. Claims are judged twice: against 1921-23 knowledge, and against today's physics.

Source gap (repeated from claims.md). The folder PDF has no text layer; all 324 pages were OCR'd and most equations are garbled. Page-image deep reads cover Ch. IV, Ch. V part 1 (Stark), Ch. VIII and Notes 1-17. Still digest-only, with no deep read: Ch. I-III (including Moseley's law itself), Ch. VI (series spectra, inner quantum numbers, anomalous Zeeman effect, doublet-atom model) and Ch. VII (bands). New in this step: I read Ch. V sections 6 and 7 (the normal Zeeman effect by Larmor precession, and the adiabatic hypothesis, book pp. 296-311) in the Investigator's cleaner OCR, so those two items no longer rest on a digest alone (see claim 4). Ch. VI statements in claim 7 and the "peripheral claims" still rest on digests plus [memory]. Where a verdict below depends on [memory] alone it says so. (Update: the QE-ordered deep reads of Ch. I-III and Ch. VI were done later; see section 12.) The same cleaner OCR file would make the remaining deep reads cheap (recommended in section 5).

1. Corrected claim list (final verdicts, Step 2 scale)

Each verdict is for the claim as the book states it, on the three reviewers' evidence. The weighing of how much the formula's later survival outweighs the wrong picture is left to the reader.

2. Where the reviewers agree

All three reviewers agree on the shape of the whole record, and I find no reason to doubt it: - Sommerfeld's 1916 energy formula is numerically the Dirac formula, while the picture behind it (spinless precessing ellipses) is wrong. Physicist and contrarian cite Kragh ("correct for the wrong reasons", [web: https://www.ilnuovosaggiatore.sif.it/article/402]), Heisenberg's "miracle" and Schroedinger's "fortuitous computational accident" [web: arXiv 2401.07485; https://pmc.ncbi.nlm.nih.gov/articles/PMC7418904/]. The mathematician and physicist each re-derived the expansion independently and got the same terms. - The book flags its own weak steps (degeneracy conceded to Bohr, n = 0 by "permanence of atoms", "not inevitable", "certainly rather risky", "hopeless", two retractions of earlier German editions). - Misprints (alpha^4/4 for alpha^2/4; 191/32 for 2807/1024; Note 13 and Note 17 numbers; sign slips) change no physical conclusion. - Stark tables are reproduced (mathematician enumerated modern parabolic states; same component sets). - Millikan's e was low by about 0.6 per cent, so the "truest" h is low by about 1.2 per cent. - The Weber point (Frauenfelder-Weber 2019) is real and limits the "experimentum crucis" claim.

3. Disagreements, and how I settle them

Claim 1 (orbits)

No real disagreement. Physicist and mathematician split exactly as above; contrarian adds the "Tested, not supported" part for many-electron extensions. I adopt all three parts. The mathematician's extra point is worth keeping: the Langevin average replacement (1 -> (n+1)(2n+1)/(6n^2) in the magneton section) vanishes in quantum mechanics (the mean of m^2 over m = -J..J is J(J+1)/3 [computed by the mathematician]), so the 8.7 and 13.7 Weiss-magneton agreement was partly coincidental (O2 has S = 1, 2.83 Bohr magnetons = 14.1 Weiss magnetons [memory, medium; consistent with the mathematician's and physicist's accounts]). The author hedged.

Claim 2 (fine structure): the one real disagreement

• Physicist: the 1920s direct hydrogen values (0.27-0.33) are "Tested, not supported at the time"; a simple two-line Doppler model keeps the peaks at the true spacing, so blending of two lines does not explain 25 per cent; the shortfall is "not explained by any source I have".
• Contrarian: a seven-component Dirac-plus-Lamb-shift Doppler simulation gives apparent splittings of 0.31-0.32 cm^-1 (equal populations) and 0.28-0.30 (d states down-weighted); so "largely a blend artefact".
• Mathematician: tabulates the book's selection-lag prediction (0.329 for Halpha) against each observer: Michelson and Fabry-Buisson close, Gehrcke-Lau 17-19 per cent low; says the direct data could not by themselves separate relativity (0.365) from Abraham (0.292).

My recomputation [computed, /tmp/arb/blend.py]. I built the seven allowed Halpha components (3s1/2->2p1/2, 3s1/2->2p3/2, 3p1/2->2s1/2, 3p3/2->2s1/2, 3d3/2->2p1/2, 3d3/2->2p3/2, 3d5/2->2p3/2) from the Dirac energies (reduced-mass Rydberg), Lamb shifts for s levels (+1057.8 MHz x 8/n^3) and p1/2 (-12.8 MHz x 8/n^3) [memory for the n-scaling and the p1/2 value; the 2S value is web-confirmed below], line strengths from 6j symbols and exact hydrogen radial integrals, and each component as a Gaussian with Doppler FWHM 7.16e-7 x nu x sqrt(T/M). Dirac 2p3/2-2p1/2 came out 0.36504 cm^-1 = 10943 MHz, equal to R alpha^2/16 = 0.36503. Separation of the two maxima of the blend:

Findings: (1) The contrarian's numbers reproduce (0.314 at 320 K, equal populations). (2) The physicist's two-line model left out the five other components; with all seven, the apparent doublet is 0.26-0.32 cm^-1 over plausible conditions, which covers Michelson (0.32), Fabry-Buisson (0.306), Meissner-Paschen (0.288) and, only if the d levels are strongly under-populated (x 0.1-0.3 at 320 K), Gehrcke-Lau (0.271-0.283). (3) The result depends on source temperature and on level populations that nobody measured in the 1920s tubes, and on broadening I did not model (Stark and pressure broadening in a discharge, instrument function, the observers' fitting method). It is a plausibility check, not a fit of their data. (4) The residual for Gehrcke-Lau is therefore "explainable but not shown", not "unexplained" and not "resolved". I did not verify the contrarian's further claim that the same model gives nearly equal Delta nu for Halpha, Hbeta and Hgamma (as Gehrcke and Lau found) and that the book's own selection-lag scheme does not; I computed only Halpha.

Verdict on the dispute. The direct measurements were real, repeated by several observers and correctly reported by Gehrcke and Lau ("etwa um 25 Proz. kleiner ... ein Irrtum bei der Messung nicht ersichtlich" [read by physicist, mathematician and contrarian: Ann. Phys. 65, p. 575]). Their stake: Gehrcke, Lau and Lenard were prominent opponents of relativity (the Investigator's source for Gehrcke is de.wikipedia; Sommerfeld's own private letter calls them "fanatisch", which is his characterisation, written to Paschen, not neutral). The data are weighed on merit: the measured peak separation of a blended multi-component line is not the 2P3/2-2P1/2 spacing, so the data do not refute 0.365. They also do not confirm it; at 1921 resolution the direct data could not decide between relativity (0.365) and Abraham (0.292), because the observed mean 0.297 [computed from the four H-alpha values 0.32, 0.306, 0.288, 0.272] sits near the midpoint of the two predictions (0.296) after allowing for blending. The deciding evidence in 1923 was the He+ pattern and scale (Paschen) and the X-ray L-doublets.

The Paschen He+ evidence, weighed fairly. In favour: a risky pattern (12 components, ratios fixed in advance) matched on the plate; Sommerfeld to Paschen 1924 notes that the 1:3 ratio holds "very exactly" for higher He+ analogues. Against or qualifying: Paschen's own text says the unreduced hydrogen data gave e/mu = 1.74 against 1.76 +/- 0.01 and that with "freundlicher Hilfe ihres Schoepfers" (the friendly help of its creator) the material was brought into satisfactory agreement [read by physicist, mathematician, contrarian: Paschen 1916]. Both parties had a stake. I read it as: the pattern is a genuine test, the absolute scale 0.3645 +/- 0.0045 is a one-parameter fit to a pattern the theory fixes. The scale then proved right (modern 0.36503). The book also records the intensity rule as "contradicted by experience" and Bohr's remark that failure of the selection principle in Paschen's photographs was "a piece of good fortune".

Sommerfeld's 1924 letter. The contrarian corrects the Investigator: the letter is dated 23 May 1924, after the 1922 text of the folder, so the book cannot contain that doubt; the Investigator's phrase "answered privately with doubt about 0.365 rather than in the book" should not be read as concealment (accepted; see section 4). The fair statement: in the book the dissenting values sit in a footnote table with "partly explained"; two years later, in a private letter, he separates the secure structure ("durch die Anzahl und Lage der Komponenten sichergestellt") from the less secure size ("die Groesse von 0,365 moechte ich nicht mit gleicher Sicherheit behaupten") and asks for new measurements. Whether the 4th edition said so is unknown.

Modern comparison of the n = 2 interval [computed + web]: measured 2P3/2-2P1/2 = 10969.127(95) MHz and 2S1/2-2P1/2 Lamb shift 1057.8514(19) MHz in a German thesis [web: search result, arXiv 0705.2508, not opened]; R alpha^2/16 with R_H = 10943 MHz, so the measured interval exceeds the formula by 0.24 per cent [computed]. The physicist attributes this excess to the electron's anomalous magnetic moment [memory, medium; not web-confirmed; I did not find a source and leave it as the reviewers' inference]. The physicist's correction that the "Lamb shift" is the 2S-2P1/2 splitting and not the 0.2 per cent excess is right (the first deep read lumped them).

Claim 3 (conservation laws)

Differences are only in labels. Physicist: the rule Established, derivation superseded. Mathematician: whole claim "Historically superseded but instructive", with the outcome Established and the ether wording Not testable. Contrarian: adds "Plausible, untested" for the Delta n = 0 derivation and "Tested, not supported" for no-recoil. I adopt the split in section 1. On "Plausible, untested" for the Delta n = 0 step: the result agrees with parity (Laporte), so the outcome is established; the author's argument for it is flagged by himself as "not inevitable" and I do not count it separately. On recoil: the book itself footnotes Einstein's directed-quantum view (which needs recoil) and sets it aside as an assumption. Evidence against no-recoil (Compton 1923; photon deflection of atoms, Frisch 1933) is [memory, high] and post-dates or is outside the book; I did not web-check it, and the verdict does not depend on a replication I cannot source. The Bohr-Kramers-Slater route (average energy conservation) the author rejected was refuted in 1925 [memory, high; contrarian]; not web-checked. "Ether": the book defines it as having "no properties other than such as light itself possesses" [read: p.139 per mathematician and contrarian]; it is the optical field, not a medium. A reader should not take it as an endorsement of the older ether.

Claim 4 (Stark, Zeeman, adiabatic)

Disagreement was about the digest-only parts. I read them (book pp. 296-311) [read: research/files IA OCR, lines ~27685-28700]: - Larmor precession: the book states the precession rate omega = eH/(2mc) and the restriction that the centrifugal term be small; shows the Coriolis force cancels the magnetic force when 2 omega_0 = eH/(mc); applies the Kepler quantisation in the precessing frame; gets Delta W = n_phi h omega/(2 pi) and Delta nu = (n_a - n_e) eH/(4 pi m c) with h cancelled ("the quantum theory has in a certain sense become latent"). With the selection rule Delta n_phi = +/-1, 0 this is the Lorentz triplet. The contrarian's numeric check (4.70e-5 cm^-1 per gauss from e/m = 1.77e7) [computed by contrarian]; I confirm 1.7588e7/(4 pi x 2.998e10) = 4.67e-5 [computed, mental arithmetic from modern e/m]. The book admits the extension of the selection and polarisation rule from the electric to the magnetic case rests on a "not quite inevitable reasoning". It also notes extra "double and triple resolution" components in Paschen-Back photographs, ascribed to unintended electric fields (Bohr). - Adiabatic hypothesis: the book states three necessary conditions (infinitely slow; the influence acts on a parameter, not directly on the coordinates; unsystematic with respect to phase, with the example that hauling the pendulum thread only at the turning points would violate it, Warburg's remark), works the pendulum example (E/nu constant), states that "transitions through degenerate systems are excluded, as Bohr has shown" and defers the general proof to Note 12. It then applies it to the circular orbit entering a magnetic field adiabatically: radius unchanged, rotation rate shifted by exactly the Larmor rate, and by continuity of Stark to Zeeman orbits it excludes the zero equatorial quantum number in the Zeeman effect. - So the contrarian's "Plausible, untested" for the proof, and the physicist's "Established for non-degenerate systems", both hold: the theorem is established (quantum adiabatic theorem [memory, high]) while the book's Note 12 proof is incomplete (small divisors untreated, per the deep read). I give both parts. - Absolute field scale: Stark's measured 104,000 V/cm against Epstein's 107,000 inferred from the theory is 2.9 per cent [computed by physicist; I confirm (107-104)/104 = 2.9 per cent], larger than the author's own "hardly 1 per cent" measurement uncertainty. It is a consistency check at about 3 per cent, not an independent test of the coefficient. Position of lines relative to each other is the real test. - Unobserved and extra components: the mathematician's modern enumeration gives Halpha p-components at {2,3,4,8} and s at {0,1,5,6}, so positions 5, 6, 8 are allowed in modern theory and the book's account (faint by Kramers' intensities) is the right kind of explanation; but I have no source for how the faint components were later observed. The four extra Hbeta components (p at 4 and 12; s at 0 and 8) are not in the first-order pattern; modern explanation unknown to me (second-order or field-inhomogeneity effects, [memory, medium], from physicist). Left Never properly tested in the record I have.

Claim 5 (X-rays)

Mostly agreed. Physicist and mathematician: Established/superseded; contrarian: orbit models "Tested, not supported". The three differ only in wording; the author's own calculation of the K-excitation mismatch is a test against observation (6.4 / 8 / 45 units against at most 0.2), so I use Tested, not supported. The mathematician's independent result is worth keeping and I verified its reasoning (not the fit itself): with seven rows of Table 53 and constant s, the slope of s against Z vanishes near alpha^2 = 5.33e-5, so the L-doublets alone fix alpha^2 to about 1 per cent, independently of the optical doublet [computed by the mathematician; seven rows only, illustration not a re-analysis]. "Doublet riddle": in 1921 the X-ray doublets were read as relativistic and the optical ones as magnetic; Millikan's 1924 ultraviolet data led to a single relativistic law, Sommerfeld writing to Lande that the formula "extends its validity into the optical realm" [web: Eckert chapter, MPRL Studies 2, via the Investigator; not re-read]. Moseley's law is digest-only.

Claim 6 (Abraham, constants)

Agreed on the arithmetic. I recomputed the table [computed, modern e = 4.80320e-10 esu, h = 6.62607e-27 erg s, alpha = 7.29735e-3]:

Propagation checked: Paschen's 0.0045/0.3645 = 1/81 = 1.23 per cent in the doublet; the doublet goes as alpha^2, e as alpha^3, h as alpha^5, so e has 1/54 and h 5/162; h goes as e^(5/3) at fixed R and e/m0, so Millikan's -0.61 per cent gives -1.01 per cent in h [computed], close to the actual -1.2 per cent. The failure of the narrow bars is a systematic (Millikan's air viscosity; [web: https://en.wikipedia.org/wiki/Oil_drop_experiment], reported by physicist and contrarian), not arithmetic; the wide bars were honest and contain the truth. alpha comes out nearly right because errors in e and e/m0 nearly cancel in (e x e/m0)^(1/3) [computed by physicist, mathematician, contrarian].

The 4/5 test. Abraham's rigid sphere gives gamma = 2/5 where relativity gives 1/2 in p^2 = 2 m0 T (1 + gamma T/m0c^2); the fine-structure terms scale as 2 gamma alpha^2, so every interval is 4/5 as large (0.292 against 0.365) and the same alpha would need alpha^2 higher by 1.25 for Abraham [computed]. Glitscher's 1917 dissertation (Munich, after a suggestion by Lenz) did exactly this: alpha^2 = 5.32 +/- 0.22e-5 (relativity) against 6.65e-5 (Abraham, a factor 1.25, as expected) compared with alpha^2 from independent e and h [read by contrarian, poor OCR; exact figures for the independent value need checking]. The Abraham alternative is excluded by (0.3645 - 0.292)/0.0045 = 16 sigma [computed] if Paschen's error is not underestimated, and on modern alpha^2 = 5.325e-5. The deep reader's note that the test "needs an independent alpha" is answered by Millikan's e with Planck's h, the X-ray fit and Glitscher; the contrarian's correction that the book's cited source (Glitscher) does raise this is accepted.

Weber. Physicist, mathematician and contrarian all read Frauenfelder-Weber 2019 [read: arXiv 1902.09612]: Sommerfeld's quantisation of Weber's 1846 force law gives -1/(2n^2) - alpha^2/(2 n^3 l) + alpha^2/(2n^4) hartree, against Sommerfeld's -alpha^2/(2n^3 l) + 3alpha^2/(8n^4); the l-dependent term, which sets the splitting inside a level, is identical. I checked the first-order step independently: the Weber Hamiltonian's perturbation is -alpha^2 p_r^2/(2r); the Kepler averages <1/r> = 1/n^2, <1/r^2> = 1/(n^3 l), <1/r^3> = 1/(n^3 l^3) give = 1/(n^3 l) - 1/n^4 [computed by hand], so the shift is -alpha^2/(2 n^3 l) + alpha^2/(2n^4), matching. The two differ by alpha^2/(8n^4) hartree, R alpha^2/(4 n^4) in wave numbers, 0.0913 cm^-1 at n = 2 [computed]. For n = 2 the interval l = 1 vs l = 2 is alpha^2/32 hartree = R alpha^2/16 = 0.365 in both. So the 1916-24 fine-structure intervals cannot separate relativity from Weber's law; only an absolute shift of each n-level could. [QE revision, section 11: I wrote "that was not measurable then"; wrong. Paschen 1916 reported one such measurement at low significance.] Weber's law is superseded by Maxwell's theory and has no spin or radiation; modern evidence excludes it on other grounds [memory, high; physicist]. The Frauenfelder-Weber authors state the difference "seems difficult to actually measure". The claim "experimentum crucis for relativity" is therefore an experimentum crucis between Abraham and relativity, not between relativity and all velocity-dependent dynamics. Sommerfeld's 1924 suspicion that the mass formula for an isolated electron might need an electron-nucleus interaction term [read by physicist, mathematician, contrarian: letter of 23 May 1924] is a further, unchecked, alternative.

Claim 7 (limits)

Agreed. The contrarian adds from the 1924 letters that carbon's p ground term was "ein harter Schlag fuer Bohrs sonst so ueberzeugende Modellfolge" and that arithmetic regularities "reach much further than the model pictures lead one to expect" (Sommerfeld to Lande, 20 April 1924) [web: https://sommerfeld.userweb.mwn.de/TransDat/02183.html, reported by contrarian and Investigator]. Kept in the record as evidence of how the author saw the theory two years after the book.

4. Corrections to earlier documents

State plainly what was wrong and what is corrected: 1. sites.md, Leads item 4 (Investigator): "answered privately with doubt about 0.365 rather than in the book" is misleading. The letter (23 May 1924) post-dates the 1922 text in the folder and says the new edition is being printed ("an der fleissig gedruckt wird", per the contrarian). The book printed the dissenting values and said "partly explained". Correct reading: no concealment is shown; whether the 4th edition recorded the doubt is unknown. (Contrarian correction, accepted.) 2. claims.md claim 2 and facts.json: "all below 0.365 and only partly explained." Correction: replace with "largely explainable as a blend of seven fine-structure components (apparent splitting 0.26-0.32 cm^-1 depending on temperature and unmeasured level populations); residual for Gehrcke-Lau open." My recomputation supports the contrarian over the physicist's "unexplained" and "Tested, not supported at the time" labels. The physicist's own two-line model was incomplete (it omitted five components). 3. claims.md "Source gap" and recommended deep reads (a): Ch. V sections 6-7 are no longer digest-only; claim 4's Zeeman and adiabatic parts are now read (above). Update the gap note; Ch. VI-VII and Ch. I-III remain digest-only. The cleaner IA OCR should be used for the remaining deep reads. 4. Deep read, Ch. VIII section 4.1 (physicist's refinement): the 0.2 per cent excess of the measured 2P3/2-2P1/2 interval is not the Lamb shift (which is the 2S1/2-2P1/2 splitting, 1057.85 MHz = 0.035 cm^-1). Measured 10969.13 MHz against 10943 MHz [computed] is +0.24 per cent; the cause is the reviewers' inference ([memory, medium]) and is left unconfirmed. 5. Deep read Ch. V section 6.5 (field term "m e E"): page images and Note 11 give "mE" (already corrected in claims.md; mathematician confirms the earlier text self-corrected). 6. Mathematician's correction to sites.md: in Sommerfeld's 29 Dec 1915 letter, 27/36 divided by 7/16 = 1.714, not 1.73, and "0.159 x 16/25 = 0.102" uses 0.159 where the line above computes 0.169 (0.169 x 16/25 = 0.108). The slip is Sommerfeld's (a working draft), not the Investigator's. Neither affects the point (absolute values off by more than a factor of two for that draft calculation, a different quantity from the fine-structure splitting). 7. sites.md, lead on Frauenfelder-Weber ("Not read"): now read by all three reviewers and by me for the key step (above). 8. Physicist's note, deep read Ch. V section 5.2 vs Ch. VIII section 4.4: the symbol n maps to l + 1 in the selection rule and to j + 1/2 in the energy formula. This is why the book's 5 allowed He+ 4686 components differ from the 8 (j, j') pairs allowed by Dirac rules (Delta l = +/-1, Delta j = 0, +/-1) that the contrarian counted from 12. Count confirmed by the contrarian [computed]; mapping of Sommerfeld's labels I-III, a-d to j values not verified. Before the primer uses "5 vs 8", it should check a modern He II treatment ([memory, medium], unchanged). 9. Retained: no correction to the digest-based corrections in claims.md; the reviewers confirm them where checked.

5. What would settle it

6. Open questions

1. How much of the 1916 agreement was fit and how much was prediction? Paschen's data were re-reduced with Sommerfeld's help; the pattern (ratios 1:3, 1:2:6) is a prediction, the scale is a one-parameter fit. A neutral reduction does not exist in the record.
2. Why does a spinless relativistic ellipse give the Dirac energy? The "Sommerfeld puzzle": Keppeler's semiclassical explanation (Maslov index and spin cancel) is cited second-hand; Biedenharn (1983) traced it to a symmetry; Suslov (2024) and Barley, Ruffing and Suslov (2025) revisit it [web: arXiv 2401.07485, 2506.00408; only abstracts or snippets read by the reviewers, not verified by me].
3. Was Kragh's "minority" dissent (1985 HSPS 15:67-125, paywalled, not obtained) as small as Kragh 2025 says ("did not affect the consensus view" [web: ilnuovosaggiatore.sif.it/article/402])? Robotti 1986 not obtained.
4. Does the claim that the He+ 4686 pattern has 12 (book) / 8 allowed (Dirac) components change which weak components are "forbidden"? Unverified.
5. How the 1920s Hbeta extra Stark components were settled.
6. Anything in Ch. VI-VII that would change verdicts: the doublet-atom model, Lande's formula, band spectra (digest only). (Update, section 12: Ch. VI, including the doublet-atom model and Lande's formula, was deep-read and changed no verdict; Ch. VII band spectra remain digest only.)

7. Absence-of-evidence analysis

(a) The Nobel Prize that never came. Sommerfeld is widely reported as the most-nominated physicist never to win: 84 nominations 1917-1951 by one count [web: https://www.bbvaopenmind.com/en/sommerfeld-the-eternal-nobel-candidate/ (search summary; page redirected, not read)]; other sources say 81 (Einstein Papers, citing Eckert 2013) or 74 (Physics World) [web: same search result]. The count is therefore uncertain by about ten; a 1926 nomination jointly with Paschen is from Google snippets via the Investigator only; the nomination texts were not read (the nobelprize.org archive refused automated access, 403). This is a record that goes dark where it would be expected to continue (a famous, much-cited theorist whose formula was vindicated in 1928). Competing explanations, none yet tested against records: - (i) The committee saw the old quantum theory as provisional and replaced by quantum mechanics before it could be honoured; for: the theory was superseded in 1925-26, and Dirac's and Heisenberg's later work reproduced his formula. Against: fine structure and the X-ray relations survived unchanged, and nominators kept nominating until 1951. - (ii) The prize went to experimenters (Stark 1919, Siegbahn 1924, Franck-Hertz 1925, Compton 1927) and to later theorists; for: Stark, Siegbahn and Bohr's atom were rewarded for the topics he developed; against: no source here says the committee judged his work as theory-only. - (iii) Politics or personalities (the relativity dispute with Stark, Lenard, Gehrcke; Sommerfeld as DPG chairman defending Einstein in 1920); for: Stark and Lenard were Nobel laureates and anti-relativists who also became Nazi supporters ([memory, medium], not in the review files); against: no source ties a committee decision to this. - Deciding record: the Nobel committee's evaluation reports for 1919-1951 (the Nobel archive; sealed 50 years, now open), and the nomination texts. Not obtained.

(b) Lau's 1924 report and the plates. The claim that higher He+ analogues show smaller Delta nu in ratio 1:3 rests on Sommerfeld's letter alone (one documented source; no second source found). It is evidence and is kept: it came from the author himself, who found it made him "sehr nachdenklich", which is not what an interested defender of 0.365 would be expected to write. Explanations: unresolved components, Stark or pressure effects, or a real departure. Deciding record: Lau's report and the plates (location unknown).

(c) 1925-33: the hydrogen doublet goes quiet. Houston and Hsieh (1933) found Balmer fine-structure separations "far too small" by about 3 per cent, Gibbs and Williams the same, Pasternack 1938 suggested a 2S1/2 shift; the work "received little attention for more than a decade" until Lamb and Retherford (1947) [web: https://en.wikipedia.org/wiki/Lamb_shift, read by physicist and contrarian]. So the record shows a gap in hydrogen fine-structure work between the 1920s dispute and 1933, and then a small genuine departure (Lamb shift) that the old theory could not have contained. The 25 per cent shortfall of 1921 and the 3 per cent shortfall of 1933 are different sizes; I do not claim the second explains the first. For the 1920s blend explanation see section 3, claim 2.

(d) Misprints and the German original. Whether the 24 misprints in Ch. IV and the sign errors in the Notes are in the German original is unchecked; both German editions are available.

8. Interested parties

• Sommerfeld: author; stake in his own theory's success.
• Paschen: measurements reduced "with the friendly help of" the theory's author.
• Gehrcke, Lau, Lenard: opponents of relativity; Gehrcke's data stand on their own; Sommerfeld's "fanatisch" is his private characterisation.
• Millikan: his e carried a systematic error (viscosity of air); his stake was his own measurement.
• Frauenfelder and Weber: authors writing in favour of Weber's electrodynamics; their math was checked by the mathematician and by me for the first-order step, their physical interpretation was not.

9. Web verification log (budget used: 3 searches, 2 fetches of 8)

• 2P3/2-2P1/2 interval of hydrogen 10969.127(95) MHz and Lamb shift 1057.8514(19) MHz in the n = 2 literature [web: search result for arXiv 0705.2508 (not opened); tag retained as web, single search]. Retagged from [memory, high]/[memory, medium] (physicist, mathematician).
• Nobel nominations: counts differ by source (84, 81, 74) [web search, URLs in section 7].
• Millikan and Lamb shift items carried from the reviewers [web: Wikipedia pages].
• Not web-checked (memory): anomalous magnetic moment as source of the 0.24 per cent excess; Frisch 1933; Keppeler's explanation; BKS refutation; O2 spin; the n-scaling of Lamb shifts for n = 3 in my simulation; Stark 1919 and Lenard Nobel dates.

10. Interest-weighted reading guide for the primer author (no new verdicts)

Teach: (1) the formula is right and the picture wrong, with the worked n = 2 doublet 0.365 cm^-1 = 10.94 GHz; (2) the 7-component blend example for why a measured peak separation is not a level spacing; (3) the error-bar lesson (wide bars right, narrow bars wrong: Millikan); (4) the Weber caveat on "crucial experiments"; (5) the Larmor derivation in which h cancels. Keep the dissent, the 1924 letter, the Gehrcke-Lau quote, the Weber result and the Nobel silence in the record, as they are pertinent whatever the verdicts.

11. QE revision (after qe.md, verdict MINOR; 2026-10-09)

Inputs: qe.md in full; the new deep read deep/Paschen-1916-BohrsHeliumlinien-AnnPhys50-iss16.md (Opus, page images checked) in full; the Investigator's Paschen text for the passages the deep read quotes. The other QE deep reads (Ch. III sections 4-7, Ch. VI, Ch. I-II, Ch. VIII footnotes, Ch. II section 6) were done afterwards; section 12 (QE revision 2) gives their results, and the former "pending" marks below now point there. I did no new web search for this section. Tags as in section 0.

11.1 What QE found wrong in this file, stated plainly

1. I wrote that the absolute (l-independent) level shift "was not measurable then" (section 3, claim 6; section 5 row 2). That was wrong as a statement of the record, and I repeated the physicist's unsourced sentence instead of checking it. The book (p.488) says deviations "of the order of magnitude of our relativity correction R alpha^2/64 ... and bear the correct sign" had been observed (Paschen), and Paschen 1916 prints a number [read: qe.md section 3.1, IA OCR lines 42967-42984; deep read section 3]. I had said at the top of this file that I did not re-read the deep reads; that is how it was missed. The two places are now annotated in the text above.
2. I labelled the 1920s direct-hydrogen challenge "Tested", which is not on the scale. Replaced (section 1, claim 2) by Tested, not supported, for the claim "0.365 is 25 per cent too large".
3. I left Paschen's own 1916 remark on blending out of claim 2 (11.3 below).
4. Smaller items from qe.md section 7, accepted: Glitscher figures confirmed (11.5); letters merged (11.6); Bucherer not cited (11.7); five withdrawals, not two (11.8).

11.2 The circular-orbit term: what Paschen measured, and what it can and cannot separate

[read: deep read Paschen 1916, sections 2-5, page images pp.907-908, 931-936; arithmetic re-computed there and spot-checked by me.]

Why this term matters. In atomic units Sommerfeld's level shift is -alpha^2/(2N^3 n) + 3alpha^2/(8N^4) and Weber's is -alpha^2/(2N^3 l) + alpha^2/(2N^4) [section 3, claim 6]. For a circular orbit (n = l = N) Sommerfeld gives -alpha^2/(8N^4) and Weber gives zero [computed by QE; I confirm by substituting n = l = N]. For N = 2 the relativistic circle term is R alpha^2/64 = 0.091 cm^-1. It shifts a whole line and does not split it.

What was measured. Paschen compared the effective Rydberg constant from the circular component of He+ 4686 (transition 4 -> 3) with that from the circular component of He+ 4339 (10 -> 4), each divided by its Bohr factor. Observed N(4686) - N(4339) = 0.569 +/- 0.25 cm^-1; calculated with "Bohr's Relativitaetsglied" 0.590 (1.0125 - 0.423) [read: p.933]. His own summary: "findet die richtige Groessenordnung" (finds the right order of magnitude) [read: p.932]. Without the term, "4339 wuerde um 0,025 A.-E. niedriger berechnet" and group 1,3 "would have no place for the observed weak component"; he says the same for 5411, 4859 and 4541 without numbers [read: pp.931-932]. R_He (his N_He) = 109722.144 +/- 0.04 cm^-1, fitted by inspection ("Diskussion") to all helium groups with formula (Ib), which already contains the relativity term; the 4686 group alone preferred a scheme 0.003 A higher [read: pp.911, 932].

What the figure supports, by the face-value arithmetic of the deep reader [computed]:

• Against Weber (zero term): about 2.3 possible-errors. The +/-0.25 is a linear sum of setting errors (0.07 + 0.18), a "possible error" in Paschen's sense, not a standard deviation.
• Relativity versus Abraham: this test cannot separate them, in principle. Paschen took alpha^2 from the measured He+ splitting (Delta nu_H = 0.3645 +/- 0.0045 gives alpha^2 = 5.3146e-5). Every fine-structure quantity, the circular shift included, scales as 2 gamma alpha^2, so under Abraham the same splitting fixes the same product and the prediction is again 0.590. The test sees the ratio of circle shift to splitting, which is the same for all gamma [read/computed, deep read section 5]. Abraham is excluded only where alpha is fixed independently of the splitting, through e and h (p.936: e = 4.776 +/- 0.07e-10, h = 6.553 +/- 0.16e-27; under Abraham e would be about 1.4 times and h about 1.75 times larger [computed by the deep reader]). This sharpens my "4/5 test" in section 3, claim 6 and does not change its conclusion.
• Hydrogen does not test it. Paschen subtracts the term (0.526 to 0.405 cm^-1) line by line to get N_H = 109677.691 +/- 0.06. The deep reader's refit of his own table gives a coefficient 0.51 +/- 0.48 times the relativity term: consistent with the full term, 4/5, or none [computed]. Paschen's Halpha comparison is built from Meissner's red component, corrected by -0.0039 A.
• Caveats that belong beside the figure.
• It is one measurement, with no replication on record.
• The author (Paschen) was an interested party in the sense of 11.3 (the theory's creator helped classify his plates, "mit freundlicher Hilfe ihres Schoepfers"; replate study "foerderte ... Feinheiten zutage, die vorher nicht erkannt waren" [read: pp.906-907]).
• The 0.569 rests on one weak component at 4339 (4338.536 +/- 0.007, assigned to group 1,3) and on the position of the circular component in the fitted 12-component 4686 scheme. The error bar does not include the assignment or the dependence on the fitted scheme (which itself uses alpha^2 and N_He). With Paschen's "less certain" 4338.732 the difference becomes about 0.516.
• Weber's circular term is zero only if Weber's formula is the whole story for the other terms; both theories share the l-dependent part, so the fitted scheme is not neutral between them.
• Counter-evidence kept in the record. Sommerfeld's 29 Dec 1915 draft found absolute Balmer differences off by more than a factor of two (0.102 against 0.24). The primer ties this draft to the circular-orbit shift [qe.md section 3.1, from the primer]; I have not re-read the draft to confirm which quantity it concerns. It was a working draft, before the final scheme.

Revised verdict, claim 6, "fine structure singles out special relativity among all velocity-dependent dynamics": - Splittings alone: cannot separate relativity from Weber (unchanged). - Circular-orbit term: tested once, by Paschen 1916, an interested party, with a result at about 2.3 possible-errors from the Weber value of zero and consistent with relativity (and with Abraham, because of the construction above); never repeated. As an independent, neutral discrimination between relativity and Weber: Never properly tested. I keep the scale name and put the one result beside it, not behind it. - "Experimentum crucis" between relativity and all velocity-dependent dynamics: still not supported by the splittings; weakly supported by one low-significance measurement. - Abraham versus relativity: Tested, not supported for Abraham (unchanged), by the absolute scale, not by the circular term.

11.3 Paschen's own statement on blending (claim 2)

[read: p.933, quoted in the deep read.] "Die Messung war aber schwierig, und es ist wegen der uebergreifenden Raender der unscharfen Komponenten wahrscheinlich, dass der Abstand der eingestellten schwaerzesten Stellen zu klein ist." (The measurement was difficult, and because the edges of the blurred components overlap, it is probable that the distance between the darkest points set on is too small.) His resolved Halpha values: 6562.853 and 6562.732 A (0.121 A = 0.281 cm^-1); Meissner, interferometer: 6562.849 (intensity 7) and 6562.725 (5) (0.124 A = 0.288 cm^-1), 7:5 centroid 6562.797, equal to Paschen's unresolved maximum. Theory: 0.1571 A (0.3645 cm^-1). Both measured spacings are about 21-23 per cent low [computed by the deep reader].

Weighing: in 1916 the experimenter who made the measurement gave the blending explanation before the dispute began. That supports the contrarian's and my simulation (section 3, claim 2) and shows the explanation was not invented after Gehrcke and Lau's 1921 paper. It does not prove the Gehrcke-Lau residual is blending (my table covered 0.26-0.32 cm^-1 under unmeasured temperature and populations). Stake: Paschen was the theory's collaborator; Meissner's interferometric value is a second observer, in Paschen's own lab, whose spacing agrees with the first. Both are low by the same amount, which is what a shared blending effect would produce and also what a shared real shortfall would produce; the data cannot separate these. The book also names Doppler and Stark broadening as the cause of blurring [read: qe.md section 3.2, IA OCR lines 42402-42424].

11.4 Other QE items taken into the record (deep reads now done: see section 12.1)

All [read] by QE from the digests and IA OCR; I have not re-read them myself except where noted. 1. Electron-impact checks of the He+ energy formula (Ch. VI section 3): predicted second ionisation potential of He 25.3 + 54 = 79.3 V (not 79.8, digest misprint); observed 79.5 +/- 0.3 V (Franck and Knipping) and 80 V (Horton). 4686 from neutral He predicted 75.4 V; Rau saw it absent at 75 V and at full intensity at 80 V [qe.md 3.3, IA OCR lines 31190-31236]. These are non-spectroscopic support for claim 1 (hydrogen-like Z^2 formula): independent of the doublet and of the Paschen plates. Weight: confirmed by deep read 2 (section 12.1 item 1; the book's 4686 figure should be 75.9 V); supports Established for the energy formula, which is already the verdict. 2. Field-blamed selection breaks: n -> n-2 series (Stark, Liebert, Hardtke), d -> s (Foote, Meggers, Mohler), b-p combinations in Li, Na, K, Al, Tl, Ag, He 1S-2S at 585 A, doubled and tripled Zeeman components, Paschen's forbidden lines in a magnetic field; each attributed to intense discharge fields, none with a measured field [qe.md 3.4]. This is the context for claim 2e (Never properly tested, stray field explanation). Today many are allowed or field-induced in the modern scheme [memory, medium, QE]. Confirmed by deep reads 2-3, with corrections (section 12.1 item 2). 3. Barkla's J-radiation (1917): reported by a leading experimenter; the book says "repeated tests by other observers have not been able to confirm" it and the theory has no place for it [read: IA OCR lines 10915-10919]. A reported anomaly set aside by non-replication and by theory; both reasons stated. Verdict, if scored: Tested, not supported on the book's account; confirmed by deep read 4 (section 12.1 item 3). 4. Nebulium, coronium, protofluor: the nuclear model was used to exclude claimed elements lighter than helium; the lines were later shown to be forbidden lines of ionised O, N and Fe (now web-checked for O and Fe; the book itself does not identify them: section 12.1 item 4). Instructive case of theory excluding an observation correctly. 5. Sommerfeld-Debye accumulation theory (1913): the author's own classical alternative to light quanta; predicted accumulation times of years for X-rays; experiment shows immediate emission. A tested-and-failed alternative bearing on claim 3. Tested, not supported. Confirmed by deep read 6 (section 12.1 item 5). 6. Doublet-atom model numbers (with Heisenberg): Li Delta nu = 0.32 cm^-1 (p-term), 0.035 (d-term), "in excellent agreement"; a forerunner of spin [memory, high]. Fues's half-integer quantum numbers rejected by the author and later vindicated [memory, high; corrected in 12.1 item 6]: a second case, beside the ellipse formula, of the right numbers and the wrong reasoning, in the opposite direction (the author held the integer picture; the half-integers were right). Corrected in section 12.1 item 6: this overstated the record; the vindicated half-integers are Landé's j and m, which the book's Addendum partly uses. 7. Records named but not examined: Birge, Phys. Rev. 17, 589 (1921), cited by the book for the centre-of-gravity point (not a doublet retest; section 12.1 item 7); Birge's alpha^2 = 5.308e-5 (digest .23 p.274); Grossmann 1921 on Newcomb's Mercury figures; Franck and Knipping 1919; Horton 1920. Added to "what would settle it" (section 5) and to the leads for the next Investigator run.

11.5 Correction to my earlier use of Glitscher (resolved)

My section 3, claim 6 text, the contrarian and the primer all flagged Glitscher's independent alpha^2 as "needs checking" [contrarian cited about 5.8e-5]. QE read the dissertation: alpha^2 = (5.32 +/- 0.22)e-5 (relativity) and (6.65 +/- 0.28)e-5 (Abraham) from the spectra, compared with the value from Sommerfeld's e, h (alpha = 7.29e-3, alpha^2 = 5.30e-5). Even with the extreme values e = 4.8e-10 (taken as surely too large) and h = 6.3e-27 (surely too small) he gets alpha = 7.65e-3, alpha^2 = 5.86e-5, below the Abraham minimum 6.37e-5 [read: qe.md section 7, correction 2; Glitscher lines 4355-4436; QE recomputed 2 pi e^2/(h c) = 7.66e-3; I confirm 5.86e-5 as 7.66e-3 squared]. The contrarian's 5.8e-5 is the extreme bound, not the best value (5.30e-5). The conclusion stands and is firmer: Abraham fails by the author's own extreme assumptions.

11.6 Letter attributions (corrected)

"Ein harter Schlag fuer Bohrs sonst so ueberzeugende Modellfolge" is in Sommerfeld to Paschen, 23 May 1924 (TransDat/02183), not to Lande. "Die arithmetischen Regelmaessigkeiten [reichen] viel weiter ... als nach den Modellvorstellungen zu erwarten waere" is in Sommerfeld to Lande, 20 April 1924 (SBPK Nachlass Lande, via the Investigator's note). My section 3, claim 7 merged them; the text is read as two letters. The 1924 "doublet riddle" passage also comes from the Paschen letter (the formula's "ungeheure Erweiterung des Gueltigkeitsbereiches" and "jedenfalls ebenso unverstaendlich") [read: qe.md section 5].

11.7 Bucherer

The primer's claim that Abraham is also excluded by "Bucherer; Guye and Lavanchy 1916 [read: book cites]" is wrong in part: the book cites Schaefer and Neumann (Ann. d. Phys. 45, 529, 1914) and Guye and Lavanchy, not Bucherer [qe.md spot check 9]. My section 3 text did not repeat it. Evidence for Abraham's exclusion in this file is the spectroscopic constants (section 3, claim 6 and 11.2), not mass-velocity experiments.

11.8 Withdrawals recorded by the book

(Corrected in section 12.3: the A-doublet and its "false interpretation" are one episode, and "refuted by Hoyt" refers to the extra L-limits; the triplet Runge denominators complete the five. The paragraph below is kept as written.)

"Two retractions" in section 2 should read "several (at least five)": the L1/L2 levels in different atoms; Kossel's defects; the A-doublets "refuted by Hoyt's measurements and falls to the ground"; the identification theta = x'; one doublet's "false interpretation" (digests .08 p.89, p.95; .09 p.101; .23 p.262, p.272) [qe.md section 7, correction 5]. They support the point that the book flags its own errors; they were not hidden.

11.9 Small points

• Section 3, claim 2 table, 77 K / "d states x 0.1" cell: "third peak appears (0.42 from outer peaks)" and the primer and example.py figure "0.300*" (gap between the two highest peaks) describe the same profile. Use the example.py wording in the primer; no numbers change.
• Sommerfeld himself wrote "If, then, Paschen has not under-estimated the accuracy of his measurement ..." in the e/h passage [read: qe.md 3.11, IA OCR line 47880]: the author flagged the error bar of the key datum. Add it to the primer's error-bar lesson.
• Translation slip in the book: "the higher the temperature the lower the velocities due to heat motion" (should be "higher"), IA OCR lines 42415-42416. For the misprint list.
• Digest .16 p.182: the second ionisation potential is printed 79.8 V in the digest, 79.3 V in the text.

11.10 Revised and unchanged verdicts

No verdict moves to a different category. The changes are the scale-name fix for 2d and the replacement of an unsourced negative in 6b by the source's figure with caveats. The overall QE verdict (MINOR) is accepted. The weighing of how much one weak, interested-party measurement counts is left to the reader.

11.11 What would settle it (additions to section 5)

11.12 Corrections to earlier documents (additions to section 4)

10. arbiter.md section 3 claim 6 and section 5 row 2, physicist.md line 85: "not measurable then / not measured at the needed accuracy then" is wrong; Paschen 1916 and the book (p.488) report one measurement (11.2).
11. contrarian.md / primer.md Glitscher "needs checking" and "about 5.8e-5": resolved, 11.5.
12. arbiter.md section 3 claim 7 and contrarian.md: two letters merged; 11.6.
13. primer.md section 6 row 6a: Bucherer not cited by the book; replace with Schaefer and Neumann 1914 and Guye and Lavanchy 1916; 11.7.
14. arbiter.md section 2 and primer.md "two retractions": at least five; 11.8.
15. claims.md claim 2 and facts.json: apply section 4 correction 2 (blend wording) and add Paschen 1916's own remark; claims.md claim 6: add the circular-orbit discriminator and Paschen's figure.
16. primer.md section 4 and Check-yourself answer 3 ("Weber ... only an absolute shift could, not measured then"): replace with 11.2; add the 1924 "doublet riddle" bullet and the J-radiation and nebulium bullets (qe.md section 8).
17. Paschen text OCR (research/files): page images show Table I e/mu for Halpha 1.760 (OCR 1,750) and Hbeta/Hgamma 1.753 (OCR 1,758); 4686 scheme shift 0.003 A (OCR 0,008); Halpha resolved 6562.853/6562.732 (OCR truncated). Anyone quoting the OCR should use these.

12. QE revision 2 (orchestrator fix, 2026-10-09)

Inputs: QE's recheck (qe.md, "Remaining" item 1); the six new deep reads, read in full by me: deep/Sommerfeld-AtomicStructureAndSpectralLines-0c4636.ch6-part1.md (PDF pp.167-187), .ch6-part2.md (PDF pp.188-216), .ch3.md (PDF pp.76-106), .ch8-notes.md (PDF pp.250-257), .ch1-2.md (PDF pp.29-32, 41-45) and .ch2-s6.md (PDF pp.54-58). The deep readers (Sonnet) worked from the IA OCR and the folder OCR, with page images for every table and formula they cite. I checked three load-bearing numbers myself against the IA OCR or a page image: 79.3 V and the "50.1 volts" step (IA OCR lines 31197, 31221-31225), and the printed "6.3 . 10^9 major cals" (page image PDF p.57, book p.95). Web checks: 5 searches and 7 fetches (3 failed), listed in 12.6. Tags as in section 0; "[read: deep ch6-1, book p.N]" means read by the named deep read at that book page.

This section replaces the "pending" marks in section 11. Where it corrects section 11 or earlier files, it says so.

12.1 What the deep reads settle (section 11.4, item by item)

1. He+ electron-impact checks (11.4 item 1): confirmed, with one slip in the book. The book prints "I + 4 . 13.5 = 79.3 volts" for the second-order ionisation of He, against 79.5 +/- 0.3 V (Franck and Knipping, Phys. Zeitschr. 20, 481, 1919) and 80 V (Horton, Proc. Roy. Soc. 95, 408, 1920) [read: deep ch6-1, book p.344, page image]. For He+ 4686 it prints 25.3 + 50.1 = 75.4 V; Rau saw "no trace" at 75 V and full intensity at 80 V [read: deep ch6-1, book pp.344-345]. New: the book's 50.1 is an arithmetic slip. Lifting He+ from its ground state to the 4-quantum orbit takes (1 - 1/16) x 4 x 13.5 = 50.6 V, so the prediction is 75.9 V [computed; printed the same in the folder OCR, the IA OCR and the page image]. Both values lie in Rau's 75-80 V window, so the check stands. Caveat from the deep read: the He table (Table 38) is partly built on the observed 20.45 V, so only the double-ionisation and 4686 figures are independent predictions. Modern check: the total binding energy of He is 79.0 eV [memory, high; not web-checked] (24.59 + 54.42). Verdict for claim 1a unchanged: Established.
2. Field-blamed selection breaks (11.4 item 2): confirmed, and sharpened. No field strength is quoted for any of them [read: deep ch6-1 Q2; deep ch6-2 Q1]. The author's evidence is of three grades: - an applied field in the laboratory: Stark, Liebert and Hardtke called up the anomalous He series (2p-mp, 2s-md, 2s-ms) "only when strong electric fields are applied" [read: deep ch6-1, book pp.332-333]; - an assumed field: "it is quite possible that ... intensities of field due to condensed discharges may have played a part" (b-p combinations in Li, Na, K, Al, Tl, Zn, Cu, Ag) [read: deep ch6-1, book p.333]; He 1S-2S at 585 A is "presumably to be expected only in intense electric fields" [read: deep ch6-1, book p.350]; - a different excuse: the Hg 1S-2p1 and 1S-2p3 steps seen in electron-collision curves (not 2P0/2P2, as QE wrote) are excused because the selection principle "restricts only the phenomena of monochromatic emission, not the effects of electronic collisions" [read: deep ch6-1, book p.348]. The doubled and tripled Zeeman components are in Ch. V section 6, not Ch. VI: Bohr ascribed a faint line at twice the normal splitting on a Paschen-Back plate "to the unintended simultaneous action of an electric field", and the book adds that lines at three times are visible on the original [read: IA OCR lines 28030-28100, book p.~301]. Paschen's magnetically "called forth" forbidden lines (p_i d_j, i > j) are cited from Paschen and Back 1921 with no field values [read: deep ch6-2, book p.363]. Verdict for 2e unchanged: Never properly tested. The deep reads strengthen its basis: the field explanation was tested only where a field was deliberately applied (Stark's He series), and assumed everywhere else.
3. Barkla's J-radiation (11.4 item 3): confirmed. "Barkla believed in 1917 that he had detected signs of a radiation still harder than K-radiation ... Repeated tests by other observers have, however, not been able to confirm the existence of this radiation." The footnote cites Richtmyer's summary, Phys. Rev. 17, 433 (1921); no other observer is named [read: deep ch3, book p.145]. Verdict on the book's account: Tested, not supported (unchanged). Single-source note: the replication failures are known here only from the book's sentence and Richtmyer's citation; the papers themselves were not obtained.
4. Nebulium, coronium, protofluor (11.4 item 4): read, and the [memory] item web-checked. The book says only that the lines "seem to hint at elements having an atomic weight of 2 or 3", and excludes such elements because He would then need nuclear charge 4 or 5 and the alpha-particle would carry outer electrons, "irreconcilable with our general experiences of a-rays" [read: deep ch1-2, book pp.70-71, page image]. It does not say what the lines are; a footnote leaves "isotopes of H or He of atomic weight 2 or 3" open [read: deep ch1-2, book p.71]. Later identifications: Bowen (Nature 120, 473, 1927) showed the nebulium lines are forbidden lines of doubly ionised oxygen in very thin gas [web: https://en.wikipedia.org/wiki/Nebulium; https://en.wikipedia.org/wiki/Doubly_ionized_oxygen]; Grotrian and Edlén showed the coronium lines come from iron ions that have lost about half their electrons [web: https://en.wikipedia.org/wiki/Walter_Grotrian], the green line being Fe XIV (Edlén, about 1941-42) [web: search summary only, https://sciencenotes.org/element-coronium-sun/ and arXiv 2103.02369, not fetched]. "Protofluor" was not checked (no source found); the earlier "ionised O, N and Fe" stays [web] for O and Fe and [memory, high] for the nitrogen lines. Both hedges in the book turned out right in their way: no element lies between H and He, and isotopes of mass 2 and 3 (deuterium, helium-3) exist [memory, high].
5. Sommerfeld-Debye accumulation theory (11.4 item 5): confirmed. The atom is supposed to "pile up energy of radiation to a limiting amount determined by the constant h"; Einstein's law follows "under certain conditions (chosen in rather an artificial manner)"; for X-rays "impossible 'accumulation times' come out, times of the order of magnitude of years!", while emission "commences immediately" [read: deep ch1-2, book pp.44-45]. The footnote is Ann. d. Phys. 41, 873 (1913) (folder OCR misprints 1918). Verdict Tested, not supported (unchanged); the author himself states the failure.
6. Doublet-atom model and the half-integers (11.4 item 6): numbers confirmed; my "later vindicated" sentence was too simple and is corrected here. - Numbers: Li Delta nu_p = 0.32 cm^-1 and Delta nu_d = 0.035 cm^-1, said to "agree excellently" [read: deep ch6-2, book p.409, page image]. The p-value is compared with Kent's 0.306 and 0.326 and the 0.34 of the II N.S.; the d-value has no measurement in the book (Table 39 has a dash), so it is a prediction, not an agreement [read: deep ch6-2, book pp.361, 370]. The trunk's moment is "half that of a Bohr magneton", and the 1/2 : 1/2 split of the ground-state moment is a "fundamental assumption, which is justified by its success" [read: deep ch6-2, book pp.406-407]. The author's own reservation: "we should scarcely need to regard our model of the doublet atoms as inevitable, if it were not linked up with the whole realm of the Zeeman effect" [read: book p.409]. - Fues: the author rejects Fues's half-integer definition of the s-term constants because "this regular course would be destroyed" in his Table 43 [read: deep ch6-2, book p.377]. But the same book's Addendum (with Heisenberg) itself uses half-integer phase integrals n = n - 1/2 and half-integer magnetic numbers m, keeping only the total integral [read: deep ch6-2, book pp.406-407, page image]. - What was vindicated [web: H. Schmidt-Böcking and B. Friedrich, arXiv 2203.07833, https://arxiv.org/abs/2203.07833]: Landé's 1921 half-integer angular-momentum quantum numbers (j, m), which "Sommerfeld and Bohr" did not accept at first ("neither agreed with his ideas, in particular those concerning half-integer quantum numbers"), and which spin (1925) explained; the paper also records that Sommerfeld had considered m + 1/2 for the s-term and "then found reasons to exclude half-integer quantum numbers based on experimental data". - Correction to 11.4 item 6 (and to qe.md 3.8, facts.json, claims.md and primer section 7): "Fues's half-integers rejected by the author and later vindicated" overstates the record. The half-integers that won are Landé's angular-momentum numbers, which the book's own Addendum partly adopts as m*. Fues's half-integers concern the numbering of the s-term constants; in modern notation that is quantum-defect bookkeeping, and whether it maps onto anything physical (the l + 1/2 of later semiclassical theory is one candidate) is not settled by our sources [memory, medium]. Revised statement: the author resisted half-integers in his empirical tables while his model in the Addendum already used them; the half-integer angular momenta were vindicated by spin in 1925.
7. Records named but not examined (11.4 item 7): one correction. The book cites Birge (Phys. Rev. 17, 589, 1921) only for testing "again very carefully in detail" that the weak satellite of the long-wave H-alpha group does not shift its centre of gravity; it gives no Birge number and does not call it a retest of the doublet [read: deep ch8, book p.487, page image]. Sections 11.4, 11.11 and the primer described it as "his retest of the Halpha components"; that wording goes beyond the book. What Birge found is unknown here; the paper is still not obtained.

12.2 Claim 5: Moseley's law (no longer [memory])

• As printed (eq. 7, book p.157): nu/R = (3/4)(Z - 1)^2 = (Z - 1)^2 (1/1^2 - 1/2^2), i.e. sqrt(nu/R) = sqrt(3/4) (Z - s) with slope 0.866 and s "almost equal to 1", for K-alpha [read: deep ch3, page image PDF p.88].
• Test data: Table 11 (Na to Nd, and W). The author's verdict: a straight line "except for a small curvature for larger Z's", and "we may no longer regard Moseley's equation (7) as exact"; the curvature is assigned to relativity (Ch. VIII) [read: deep ch3, book pp.156-158]. The deep reader's check of the printed table: 0.866 (Z - 1) fits within about 0.1 up to Cu and is 3 units (4.5 per cent) low at W [computed].
• Gaps predicted: Z = 43, 61, 75, 85, 87, "the number of gaps must be five" [read: deep ch3, book p.150, page image; the IA OCR drops 61].
• Modern check [web: D. C. Wheeler et al., "Observation of relativistic corrections to Moseley's law at high atomic number", arXiv 1809.10480, https://arxiv.org/abs/1809.10480]: K-alpha for 13 elements from Z = 29 to 92 follows the Bohr square law "fairly well at low Z", the deviations "become larger with increasing Z", and a relativistic Bohr-Sommerfeld model fits the whole range (probability value 0.20) where the plain Bohr model does not.
• Verdict: Established as an approximate empirical law, exactly as the book states it (not exact; curvature at high Z from relativity). Unchanged in category; the tag moves from [memory, high] to [read] plus [web]. The modern finding that the high-Z curvature is relativistic and is fitted by a relativistic Bohr-Sommerfeld model agrees with what the book says on p.158.

12.3 Withdrawals: the list of five is corrected (section 11.8)

The deep reads show two of the five items in 11.8 are one episode, and one was misattributed: - The "A-doublets" (phi' phi)/(chi psi) were shown wrong by G. Hertz's L-absorption limits, confirmed by Smekal and Coster; the book says this was "owing to a false interpretation by the author" [read: deep ch3, book pp.171-172, page image]. So the "A-doublets" and "one doublet's false interpretation" in 11.8 are the same withdrawal. - "Refuted by Hoyt's measurements" refers to the extra L-limits Lambda1 and Lambda2, which "fall to the ground with the 'A-doublet'" (F. C. Hoyt, Proc. Nat. Acad. 6, 639, 1920) [read: deep ch3, book p.181, page image]; not to the A-doublets themselves. - theta = chi' (following Friman in earlier editions) is withdrawn [read: deep ch3, book p.159]. - New from Ch. VI: the author's first Runge-denominator scheme for triplet terms is called "erroneous" and replaced [read: deep ch6-2, book p.392, page image]. - With the two Ch. VIII items (the L1/L2 levels in different atoms; Kossel's defects), the author's own withdrawals are still at least five: L1/L2; Kossel's defects; the A-doublet with its Lambda limits; theta = chi'; the triplet denominators. The book also reports others' withdrawals: Franck's 11 V H2+ interpretation "no longer tenable" [read: deep ch6-1, book p.344]; Rutherford's "He of mass 3" inference "not inevitable" [read: deep ch2-s6, book p.93]. The count "at least five" in sections 1, 2 and 11.8 stands; the list is corrected.

12.4 Other new items from the reads (no verdict depends on them)

• Mass defect of He, printed ten times too large. The book gives Delta m = 4 x 1.0077 - 4.00 = 0.03 per gramme-atom and Q = 0.03 c^2/(4.19 x 10^10) = "6.3 . 10^9 major cals" [read: deep ch2-s6, page image PDF p.57, book p.95; I checked the image]. The arithmetic gives 0.03 x 9 x 10^20 / 4.19 x 10^10 = 6.4 x 10^8 kilocalories [computed]. The modern binding energy of He-4, 28.3 MeV per atom [memory, high], gives 6.5 x 10^8 kcal per mole. The book's conclusion (far more stable than any chemical compound, about 100 major cals) is unaffected. Whether the slip is in the German original or the translation is unknown.
• Bohr's He model, three printings. Ch. II gives 28.8 V against 25.4 V observed (book p.69); Ch. VI gives 28.7 against 25.3 (p.349); Note 14 gives 28.75 against 25.3 (p.602) [read: deep ch1-2]. The book is not uniform in rounding; the review files use 28.75 and 25.3, which match Note 14's own calculation ((17/8) x 13.53 = 28.75 V [computed]).
• Lenz's He-nucleus model ("an inverted oxygen molecule": four H nuclei circling in a plane, two electrons on the axis) is reported "without committing ourselves", with data "admittedly rather unconvincing" and an admitted stability problem [read: deep ch2-s6, book pp.96-97].
• Rutherford's disintegrations (1919-21) as printed: N, B, F, Na, Al, P shattered; Li, Be, Mg, Si, Cl, K uncertain; none from C, O, S, Ca, Ti, Mn, Cu, Sn; the Al H-rays carry about 28 per cent more energy than the incident alpha, which the book reads as energy released by the nucleus [read: deep ch2-s6, book pp.92-93].
• The direct H-alpha table (book p.482) prints seven rows from four groups (Michelson 0.32 and 0.33 for H-beta; Fabry and Buisson 0.306; Meissner and Paschen 0.288; Gehrcke and Lau 0.272, 0.283, 0.271) with no years [read: deep ch8, page image]. The book itself names Doppler broadening and the Stark effect "as a further reason for the blurring of the lines", and says of the shortfall that it "is partly explained by the influence of the variable term" [read: deep ch8, book p.482]. This supports 11.3 and adds that the author, not only Paschen, named blending causes.
• The p.488 passage on the circular-orbit term gives no numbers and does not name the paper ("which we shall presently quote"); the numbers are in Paschen 1916 (11.2) [read: deep ch8]. No change to 11.2.
• Selection rule for the inner quantum number as printed: n_i -> n_i +/- 1 or n_i; Delta n_i = 0 allowed; 0 -> 0 excluded "by a special convention" from Landé, not derived [read: deep ch6-2, book pp.365-367]. Landé's splitting factors are printed as level spacings 2, 4/3, 2/3, 6/5, 4/5, 8/7, 6/7 for s, p1, p2, d1, d2, b1, b2 [read: deep ch6-2, Fig. 100, page image], the modern g-values of those terms [memory, high].

12.5 Verdicts after revision 2

No verdict changes category. The evidence did not require any change: every deep read confirmed the digest-based statements it was asked to check, apart from the corrections listed in 12.1 (items 6, 7), 12.3 and 12.4. Nothing in sections 1-11 now rests on "pending" deep reads. Still digest-only: Ch. I sections 1-5, Ch. II sections 1-2, 4-5, Ch. III sections 1-3 and Ch. VII (bands); no verdict depends on them.

12.6 Web verification log for this revision (5 searches, 7 fetches; slightly over the budget of 8)

• Searches: Bowen 1927 nebulium O III; coronium Fe XIV Edlén Grotrian; Moseley's law modern deviations; Landé half-integer history; the 2021 Landé g-factor paper by title.
• Fetched: arXiv 1809.10480 (abstract, then PDF text via curl to /tmp); en.wikipedia.org/wiki/Nebulium; en.wikipedia.org/wiki/Walter_Grotrian; arXiv 2203.07833 (abstract, then PDF text via curl to /tmp). Failed: doaj.org (403), the Harvard CfA listing (redirect) and the Wiley page for Natural Sciences 1, e20210068 (no text).
• Not checked: protofluor; the 79.0 eV He total binding energy and the 28.3 MeV He-4 binding energy (both [memory, high]).

12.7 Corrections to earlier documents (additions to sections 4 and 11.12)

18. arbiter 11.4 item 6, qe.md 3.8, facts.json, claims.md, primer section 7: "Fues's half-integers ... later proved right" overstated; see 12.1 item 6.
19. arbiter 11.4 item 7 and 11.11, facts.json, claims.md, primer sections 7 and 9: Birge 1921 is cited by the book for the centre-of-gravity point, not as a retest of the doublet; see 12.1 item 7.
20. arbiter 11.8, facts.json, claims.md, primer section 7: the five withdrawals list conflated the A-doublet with its "false interpretation" and attached "refuted by Hoyt" to the wrong item; see 12.3.
21. qe.md 3.4 and arbiter 11.4 item 2: the Hg collision lines are 1S-2p1 and 1S-2p3, not 2P0/2P2; the Zeeman 2x and 3x components are in Ch. V section 6, not Ch. VI.
22. The book (for the misprint list): 4686 excitation printed 50.1 / 75.4 V, should be 50.6 / 75.9 V; He heat of formation printed 6.3 x 10^9 major cals, should be about 6.4 x 10^8; p.71 "doubly ionised helium" in the Li++ paragraph should be lithium; Bohr's He model printed 28.8/25.4, 28.7/25.3 and 28.75/25.3 in three places.