Summerfield (Arnold Sommerfeld): Mathematician's review (Step 2)

Reviewer: PhD mathematician. Date: 2026-10-09. The folder name "Summerfield" is a misspelling of Sommerfeld; this file uses "Sommerfeld".

Source tags: [read] from the project documents (deep reads, page images as they report them, files in research/files/); [computed] my own calculation in this review (scripts in /tmp/sf, run in the project .venv); [web] with URL; [memory] with confidence.

Source gap (repeated from claims.md). The only primary document is the 1923 English translation of Atomic Structure and Spectral Lines, scanned without a text layer. Equations come from page images read by the deep readers, who recomputed numbers. Digest only, no deep read: Ch. I-III, Ch. V sections 6-7 (Zeeman effect, adiabatic hypothesis), Ch. VI (series spectra, Landé, inner quantum numbers), Ch. VII (bands). Wherever my verdict touches those, I say so. I did not re-render page images myself; I verified the deep readers' equations by deriving or recomputing them independently where listed below. I did read the Investigator's saved sources that bear on the mathematics: Paschen 1916 and Gehrcke-Lau 1921 (OCR text of the originals), the Sommerfeld-to-Paschen letter transcription of 29 Dec 1915, and Frauenfelder-Weber 2019 (arXiv:1902.09612).

Reader's glossary (symbols used below)

• n: azimuthal quantum number (book notation; modern k = j + 1/2 in the fine-structure formula). n': radial quantum number. n + n' = "quantum sum" (the modern principal number N).
• alpha = 2 pi e^2/(h c) = 1/137.04 ("fine-structure constant"); alpha^2 = 5.3e-5.
• R: Rydberg constant in wave numbers (109,678 cm^-1 for hydrogen). Delta nu_H = R alpha^2/16 = 0.365 cm^-1: the hydrogen n = 2 doublet, the unit of every fine-structure interval in the book.
• "Phase integral": the integral of momentum p over the coordinate q around one full cycle, set equal to an integer times h.
• "Degenerate": different motions with the same energy. The Kepler (1/r) problem is degenerate, which is why the quantum conditions are ambiguous (see claim 1).

Summary of verdicts

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Claim 1: Quantisation of orbits

Statement (claims.md 1): phase integrals around each coordinate equal n h; hydrogen energy depends only on n + n'; n = 0 excluded; spatial orientation quantised, cos alpha = n1/(n1+n2).

a. Is the mathematics internally consistent? Yes for the energy, with gaps the book itself names. - Chain of equations [read: Ch. IV deep read, section E]: the Kepler conic is assumed, not derived (Sommerfeld: "obtained in this somewhat indirect and artificial way"). The radial integral gives 1/sqrt(1-eps^2) - 1 = n'/n, so 1 - eps^2 = n^2/(n+n')^2; the energy is W = -2 pi^2 m e^2 E^2/(h^2 (n+n')^2). The deep reader checked the integral numerically (agreement 1e-12) and the algebra with SymPy. The equation list for the energy is consistent. - Worked number (from the book's result): for n + n' = 3 the allowed orbits are n = 3, 2, 1 with b/a = 1, 2/3, 1/3 and eccentricity 0, sqrt(5)/3, sqrt(8)/3. All reproduce; the misprint "b = a sqrt(2)" (for a/2, n + n' = 2, n = 1) is flagged by the deep reader and is a simple error (b cannot exceed a). - Stated gaps (all conceded by Sommerfeld): (i) no reason is given for the choice of polar coordinates; (ii) the Kepler problem is degenerate, so parabolic coordinates select different quantised ellipses; "Bohr denies that both conditions are necessary" and Sommerfeld replies "we must admit that this view is incontestable" but appeals to the relativistic limit; (iii) n = 0 is excluded by "permanence of atoms", n1 = 0 "taken for granted" until Ch. V; (iv) all spatial orientations are assumed equally probable. - Hidden assumptions I add to the deep reader's list. 1. The positive integer ranges. The old theory counts n1 = 1..n for orientations (cos alpha = n1/n). Modern quantum mechanics counts m = -l..+l, that is 2l + 1 = 2n - 1 states for l = n - 1, in which the orientation ratios are m/sqrt(l(l+1)), and none points exactly along the field. In the old counting an atom with n = 1 has cos alpha = 1 and a nonzero angular momentum h/2 pi along the field; the modern ground state has none. 2. The number of states per quantum sum. The old theory gives s(s+1)/2 (s = n + n'); the modern count is s^2 orbital states (2 s^2 with spin) [memory, high; textbook counting]. This matters wherever statistical weights enter. 3. Test case (the magneton correction, Ch. IV). Sommerfeld replaces the classical Langevin average = 1/3 by (n+1)(2n+1)/(6 n^2): 1 for n = 1, 5/8 for n = 2, 14/27 for n = 3 (I recomputed these). This changes the inferred magneton number by sqrt(3) for n = 1 (8.7 against 5 Weiss magnetons) and sqrt(15/8) for n = 2 (13.7 against 10). In quantum mechanics, for any angular momentum J with m = -J..J, = J(J+1)/3, so the average of cos^2 is exactly 1/3 [computed: mean of m^2 over m = -J..J is J(J+1)/3] and the "correction" disappears; the effective moment is g sqrt(J(J+1)) mu_B. For O2, S = 1, g = 2: 2.83 mu_B = 14.1 Weiss magnetons, close to the observed 14 and to Sommerfeld's 13.7, but for a different reason (electron spin, discovered 1925) [memory, high]. So the 1921 agreement was partly coincidental, and the book's own hedge ("our numbers 8.7 and 13.7 should [not] be accurately true") was prudent. 4. Unit slip, not conceptual: the book writes the magnetic moment in "gauss x cm"; the cgs unit is erg/gauss = gauss cm^3; and mu = (e/2) omega a^2 needs e in emu while the rest of the chapter uses esu (the deep reader notes both). The printed value 5584 per mole matches the modern 9.274e-21 x 6.022e23 = 5585, not the book's own constants (5600) [read + memory, high].

b. Consistent with established experiments? The energy formula reproduces the Balmer, Lyman, Paschen and Pickering series to the accuracy of 1921 (R_H = 109677.69, R_He = 109722.14 from the deep read). Spatial quantisation was tested by Stern and Gerlach in 1922 [memory, high] (silver atoms split in two beams): consistent with quantised orientation, but the old-theory prediction for silver (n = 1 gives cos alpha = +/-1... in the book's counting, positive only) differs in detail from the modern spin-1/2 explanation. I did not check this by [web]; no verdict depends on it.

c. Tested? Yes for the energy levels (spectroscopy, since Balmer 1885). The phase-integral rule was proposed independently by Sommerfeld (Dec 1915), W. Wilson (Phil. Mag. 31, 1916) and Ishiwara (Proc. Tokyo Math.-Phys. Soc. 8, 1915); the book credits Wilson (footnote) [read]. The Investigator confirms Ishiwara's priority claim [web, arXiv:1708.04676 per research/2026-10-09-arxiv.md]. The rule itself was superseded by the Schrödinger equation, but its semiclassical form (Bohr-Sommerfeld/EBK with Maslov index) survives; Barley, Ruffing and Suslov (arXiv:2506.00408) re-derive the old-theory levels from modern methods [web, per the Investigator, not re-read in full].

d. If true, what in mainstream physics changes? Nothing: mainstream replaced this with wave mechanics. The old orbit picture is false (no definite orbits; the electron has l = 0 states reaching the nucleus; spin). The energy formula is retained.

Verdict: split as in the table. Hydrogen energy formula Established; orbit model, n = 0 exclusion, spatial quantisation counts: Historically superseded but instructive.

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Claim 2: Relativistic fine structure

Statement (claims.md 2): quantised precessing ellipses give the doublet R alpha^2/16 = 0.365 cm^-1 for n = 2, a 1:3 triplet for n = 3, 1:2:6 for n = 4, 3:5:10:30 for n = 5; evidence: Paschen's He+ 4686 photographs, 0.3645 +/- 0.0045 cm^-1 indirectly, X-ray doublets; counter-evidence: direct hydrogen values 0.27-0.33, intensity rule contradicted.

a. Internal consistency. - I re-derived the formula myself. With the exact energy W/(m0 c^2) = [1 + alpha^2/(n' + sqrt(n^2 - alpha^2))^2]^(-1/2) - 1 (nuclear charge 1; the book's eqn 23), expansion in powers of alpha with n + n' = N gives [computed, SymPy]: W/(m0 c^2) = -alpha^2/(2 N^2) - alpha^4/(2 N^3 n) + 3 alpha^4/(8 N^4) + ... In atomic units this is -1/(2N^2) - alpha^2/(2 N^3 n) + 3 alpha^2/(8 N^4), identical to the expression Frauenfelder and Weber quote as Sommerfeld's [read: arXiv:1902.09612, eq. 1.2] and equal to the second-order Dirac result [read: same, p.2; memory, high]. So the book's "term" (n, n') = R(E/e)^2 {1/N^2 + alpha^2 (E/e)^2/N^4 (1/4 + n'/n)} agrees (the deep reader's check of the n'/n structure agrees with mine). - The component counting is pure arithmetic and is right: with n'/n steps, N = 3 has intervals 1/2, 3/2 (1:3); N = 4 has 1/3, 2/3, 2 (1:2:6); N = 5 has 1/4, 5/12, 5/6, 5/2 = 3:5:10:30 [read: deep read, recomputed exactly]. - Numbers: Delta nu_H = R alpha^2/16 with the book's alpha^2 = 5.32e-5 and R = 1.097e5 gives 0.36475 cm^-1; with modern alpha^2 = 5.3251e-5 and R_H = 109677.58, 0.36504 cm^-1 [read: deep read]. The printed alpha^2 = 5.32e-5 does not follow from the printed e = 4.77e-10 and h = 6.55e-27 (they give 5.29e-5, 0.6% lower); this is a rounding gap, size well under the quoted error. - Misprint: the ratio printed as "alpha^4/4 = 1.3e-5" should be alpha^2/4 = 1.33e-5 (alpha^4/4 = 7e-10) [read: deep read, p.248]. The sign "-3/8 beta^4" for "+3/8" in the mass expansion is also a misprint. Neither changes a physical result. - Sommerfeld's claim that the relativistic radial quantisation is "free from objection" because the problem is not degenerate: this is right in the sense that the (n, n') pair now has different energies, so the two quantum conditions both matter. But note a subtlety I verified: the formula depends on (n, n') only through the combination in the expansion above; the shift -alpha^2/(2 N^3 n) depends on n alone and the 3 alpha^2/(8 N^4) only on N. Levels with the same N and n are the same regardless of which "orbit" they came from. The Dirac theory has the same degeneracy (2S1/2 and 2P1/2 share an energy), which is why the Lamb shift (below) is a real departure. - A hidden assumption for the doublet scale: the nucleus is held fixed ("we disregard the counter-motion of the nucleus"). Darwin's correction is of relative order alpha^2 m/M (footnote, p.248). For hydrogen that is about 0.365/1836 = 2e-4 cm^-1, 20 times smaller than the +/-0.0045 error on Paschen's value [computed from the deep read].

b. Consistent with established experiments? - The He+ 4686 pattern: Sommerfeld's formula places 12 components over 3.40 cm^-1 = 0.747 Angstrom (the book says "does not even amount to 0.8 A") [read + computed, deep read]. The book calls the spark photograph "a striking confirmation". - Evidence against, printed in the book itself [read: deep read section 4]: (1) every direct hydrogen value (Michelson 0.32 and 0.33, Fabry-Buisson 0.306, Meissner-Paschen 0.288, Gehrcke-Lau 0.272, 0.283, 0.271 cm^-1) is below 0.365, "partly explained"; (2) the intensity rule 2:1, 3:2:1, 4:3:2:1 is "contradicted by experience" (component II b is the strongest, not I a); (3) weak "forbidden" components are blamed on stray fields (Kramers: about 100 V/cm DC, 300 V/cm spark), an after-the-fact explanation with no independent field measurement; (4) Paschen's 0.3645 was obtained by fitting the He+ pattern with the theory's own rational interval scheme. - My own test of how far the book's explanation of the shortfall goes. The deep reader gives the lag from the selection principle: for H-alpha the measured interval is IIb - Ia = Delta nu_H (1 - 8/81) = 0.329 cm^-1; for H-beta IIc - Ib = 0.350. Dividing each observer's value by this expected value [computed]:

So the book's explanation brings Michelson and Fabry-Buisson close, and leaves the Gehrcke-Lau values 17-19% low. For comparison, the Abraham electron (claim 6) predicts a uniform factor 0.8 on every interval, which is exactly where the Gehrcke-Lau ratios lie. I report this as arithmetic, not as a conclusion: the spread of the direct data (0.81 to 0.97) is wider than the difference between the two hypotheses, and Doppler broadening alone is of the order of the doublet itself (hydrogen at 300 K, H-alpha: full width about 0.08 Angstrom = 0.19 cm^-1 [computed with the standard Doppler formula, coefficient 7.16e-7 [memory, high]]). The unresolved blending that Sommerfeld invokes is therefore plausible, but the direct hydrogen data could not by themselves distinguish. The discriminating evidence is He+ (4 times heavier, Z = 2, a 16-fold larger interval), where Paschen's scale is 0.3645 +/- 0.0045. - Gehrcke and Lau 1921 in their own words [read: Gehrcke-Lau, Ann. Phys. 65, p.575, OCR]: "Bemerkenswert ist, dass der Betrag fuer Delta nu etwa um 25 Proz. kleiner gemessen wird, als er aus der theoretischen Feinstrukturkonstanten sich nach Sommerfeld berechnet ... in Anbetracht der Groesse der gefundenen Abweichung ein Irrtum bei der Messung nicht ersichtlich" (the interval is measured about 25% smaller than Sommerfeld's theory gives; given the size of the deviation, a measurement error is not apparent to them). They also found the intensity ratio of the two components depends on the source. They draw no conclusion about the cause. 25% below 0.365 is 0.274; the book did not discuss this paper's number beyond the footnote table. - Sommerfeld's private letter, 23 May 1924 [web, https://sommerfeld.userweb.mwn.de/TransDat/02183.html, as transcribed by the Investigator]: he regards the number and position of the components as securing the relativistic theory, but "die Groesse von Delta nu = 0,365 moechte ich nicht mit gleicher Sicherheit behaupten", proposes a mass effect that could make it smaller, and is made "sehr nachdenklich" by Lau's result for higher He+ analogues. This sits beside the book's statements (kept in the record whatever the verdict). - Letter of 29 Dec 1915 [web, https://sommerfeld.userweb.mwn.de/TransDat/00585.html]: Sommerfeld compares differences of Balmer lines of Paschen 0.16 : 0.24 : 0.26 (1 : 1.5 : 1.62) with the theory's 7/16 : 16/25 : 27/36. Two arithmetic points [computed]: 27/36 divided by 7/16 is 1.714, not 1.73 as written; and the absolute value is wrong by more than a factor two (0.102 against 0.24). The letter says "Das stimmt vortrefflich" about the ratios and "Dagegen stimmen die Absolutwerte nicht" about the sizes, and he tries a factor 5 in the correction term. This is a working draft about the circular-orbit shift (Bohr's relativistic term), not the fine-structure splitting itself; it shows a calibration problem at the start of the Paschen collaboration, resolved when Paschen's data were reduced with Sommerfeld's help [read: Paschen 1916, p.~904 OCR: "mit freundlicher Hilfe ihres Schoepfers war es moeglich ... das gesamte Beobachtungsmaterial ... in genuegende Uebereinstimmung zu bringen"; before that, his hydrogen data gave e/mu = 1.74e7, against the accepted 1.76 +/- 0.01]. I read this as the data being re-reduced, not as fraud; the point for a student is that the agreement was partly constructed by fitting, which the book half-admits ("served Paschen as a means of calculating the ideal hydrogen doublet"). - Modern outcome [memory, medium; the web search for the exact figure returned no number, so this stays [memory]]: the measured 2P3/2 - 2P1/2 interval of hydrogen is about 10.97 GHz = 0.366 cm^-1. [computed] Dirac's value Delta nu = R alpha^2/16 with modern constants is 0.36504 cm^-1 = 10.944 GHz; the excess is of the order of the electron's anomalous magnetic moment (about 0.1%, [memory, high]). The 2S1/2 - 2P1/2 Lamb shift (about 1057 MHz = 0.035 cm^-1, Lamb and Retherford 1947 [web: the search confirmed the definition, not the number; the figure is [memory, high]]) is about 10% of the doublet. It lies in the sense that would shorten an apparent 2S-2P3/2 spacing, so part of the 1920-21 shortfall may be this effect. I cannot apportion it between Doppler blending, unresolved components and the Lamb shift: that needs a line-shape model, which I did not build. - Lyman alpha is a doublet in fact (2S1/2 and 2P1/2 -> 1S1/2 and 2P3/2 -> 1S1/2 split by 0.365 cm^-1) whereas the book says the Lyman series consists of "strictly simple lines" [memory, high; deep read]; that statement of the book is wrong by modern theory, in Sommerfeld's scheme the selection principle left one transition. - The 12-component pattern: using the modern rules (Delta l = +/-1, Delta j = 0, +/-1) the deep reader's inference is that 8 positions, not 5, are allowed for He+ 4686 [memory, medium; the deep reader flagged this as unchecked and so do I]. If right, one of the weak components attributed to a stray field would be an allowed line. I did not verify.

c. Tested? By whom? Replicated? Paschen (1916) for He+ and H; Michelson, Fabry-Buisson, Meissner-Paschen, Gehrcke-Lau for H (results above); Hansen and Jacobsen (1921) for the Zeeman effect of He+ 4686; later, Millikan and Bowen 1924 for extension to optical spectra (cited via Kragh per the Investigator, not read); Houston, Williams and Pasternack in the 1930s for hydrogen H-alpha, not obtained; Lamb and Retherford 1947 (microwave). Dirac 1928 reproduced the formula. The Investigator also lists Kragh 1985 as the standard history; it is paywalled and I did not read it. The Kragh 2025 article [web, research/2026-10-09-nuovo-saggiatore.md] says that the agreement "was far from perfect and questioned by a minority", but did not alter the consensus that fine structure was solved.

d. If true, what changes? Nothing; it is true as a formula. What is false is the mechanism. The formula is identical to the Dirac energy with Sommerfeld's n read as j + 1/2. This is called the "Sommerfeld puzzle" [web, Vickers 2018, https://pmc.ncbi.nlm.nih.gov/articles/PMC7418904/; Suslov arXiv:2401.07485; Kragh 2025: "correct for the wrong reasons"]. Keppeler's explanation, which Frauenfelder and Weber cite [read: arXiv:1902.09612, p.3], is that the Maslov index and the effect of spin cancel in the semiclassical calculation [secondhand; I did not read Keppeler].

Verdict (split). Formula and interval ratios: Established. Mechanism: Historically superseded but instructive. Intensity rule: Tested, not supported. The sentence "direct measurements confirm 0.365": Tested, not supported as stated.

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Claim 3: Wave and quantum pictures joined by conservation laws; selection principle

a. Internal consistency. Note 9 (angular momentum of a spherical wave) checks out [read: deep read; and I cross-checked the physical result]: the angular momentum radiated is N = (W/omega) 2ab sin(gamma)/(a^2+b^2), maximal for circular polarisation, so one emission with W = h nu carries at most h/2 pi. This agrees with the standard result that circularly polarised dipole radiation carries angular momentum W/omega [memory, high]. The inequality |Delta n| <= 1 follows from this and the postulate that the atom's angular momentum n h/2 pi is passed entirely to the radiation. - Postulates: energy, momentum and angular momentum conserved in a single act; no recoil of the atom; wave propagation "repress all doubts". Sommerfeld's own score: five of seven determining elements fixed, two (including coherence length) not [read]. - Weak steps he flags himself: Delta n = 0 "not inevitable"; the invariable-plane assumption "certainly rather risky"; the atom and ether cannot be wholly decoupled in stationary orbits (footnote). A further point on which I add: the atom is treated as a closed system with a fixed invariable plane, which holds only in the absence of external fields; the book handles fields separately and finds the rule "rescinded" at high current density. - Errors: Note 10(f) says "Delta n = 0" where "+/-1" is meant; misprints "i.e. pi = 0" and "a = 0 and b = 0" [read: deep reads].

b. Consistent with experiment? The selection rule Delta l = +/-1 and the circular/linear polarisation pattern are experimentally established today [memory, high], though the quantum number in modern theory is l, not Sommerfeld's n, and the fully correct derivation is quantum electrodynamic. The statement that radiation is "emitted in a single act" with no recoil conflicts with Einstein's momentum-carrying light quanta (the book's own footnote concedes recoil in Einstein's view); the recoil is real but small [memory, high]. Observed Zeeman and Stark polarisations agree with the rule [read: deep read].

c. Tested? The polarisation rule by Stark-effect observations (Stark, Lo Surdo); the weakened rule in discharges by Paschen's photographs; Rubinowicz derived the rule in 1918, Bohr by the correspondence principle. Replicated broadly.

d. If true, what changes? The "ether" is defined by Sommerfeld as having "no properties other than such as light itself possesses and such as arise out of the laws of the optical field" and "every system of reference its own ether" [read: p.139]. That is the electromagnetic field, not a medium. Not testable as stated (an interpretation with no measurable difference from the field). A reader should not take this passage as an endorsement of an aether in the older sense.

Verdict. Historically superseded but instructive. Selection rule outcome: Established; the derivation's postulates: superseded; the ether wording: Not testable as stated.

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Claim 4: Electric and magnetic field effects

a. Internal consistency. Stark effect [read: deep read section 6]: first-order energy -W = Rh (E/e)^2/(n1+n2+n3)^2 + (3 h^2 F/(8 pi^2 m E)) (n2 - n1)(n1+n2+n3), with the correction that the page image has "m E", not "m e E" (so the earlier deep read in this same file was wrong at that point and corrected itself in a later section; the printed "2 pi" in the first term is a misprint for 2 pi^2). I confirm the algebra from the formula as printed. The coefficient equals (3/2) e a0 F n (n1 - n2), the wave-mechanical first-order result (Schrödinger and Epstein 1926) [memory, high; confirmed by the deep reader with SymPy]. - My independent test of the component tables. I enumerated all transitions n = 3, 4, 5, 6 -> 2 using modern parabolic quantum numbers (n1, n2, m) with the dipole rule Delta m = 0 (p-polarised) or +/-1 (s-polarised) and the shift n (n1 - n2) [computed, /tmp/sf/stark.py]. Result for non-negative positions: - H-alpha: p {2, 3, 4, 8}, s {0, 1, 5, 6} - H-beta: p {0, 2, 6, 8, 10, 14}, s {2, 4, 6, 10, 12} - H-gamma: p {2, 5, 8, 12, 15, 18, 22}, s {0, 3, 7, 10, 13, 17, 20} - H-delta: p {0, 4, ..., 32}, s {2, 6, ..., 30} These agree with the deep reader's recalculation of Sommerfeld's Tables 27-34 and with the claim of the book. The two clerical slips in the tables (rows 111 -> 101, and the N values in rows 211 of Table 29) are confirmed by the formula and change no set of components. Note the ranges: in Sommerfeld's labelling n3 >= 1 corresponds to |m| + 1, and "Delta n3 = 0, +/-1" to Delta |m| in {0, +/-1}, which gives the same sets as the modern rule. This is why the old theory reproduced the Stark patterns exactly. - Admitted mismatches [read: p.155-156]: H-alpha components at 5, 6, 8 predicted but not seen (Sommerfeld cites Kramers's intensities for faintness); for H-beta four observed components (p at 4 and 12; s at 0 and 8) are not predicted, some "questionable". The field scale was corrected from the theory itself: 107,000 against Stark's measured 104,000 V/cm, a 3% consistency check, not an independent absolute test. - Parabolic vs polar quantisation: Sommerfeld states plainly that for a weak field (Delta nu_F < Delta nu_H) the parabolic method is not valid and that "the present method of quantising the Kepler motion for the case when no external forces are acting is wrong"; for strong fields (all Stark observations) parabolic is right. This is an honest statement of the degeneracy ambiguity (claim 1). - Worked number [read: deep read]: at F = 104,000 V/cm the smallest interval C = 3 e a0 F/(2 h c) = 6.66 cm^-1, about 2.9 Angstrom per unit at H-alpha. - Zeeman effect and adiabatic hypothesis: digest only for the book chapters (Ch. V sections 6-7). The deep read of Note 12 [read] finds the Burgers adiabatic-invariance argument correctly structured but silent on near-resonances ("small divisors", the technical obstruction in perturbation theory), and the degeneracy gap is stated. The normal Zeeman triplet from Larmor precession: standard classical result with h cancelling [memory, high]. I have not checked Ch. V eqns (14)-(16). I leave those two items as resting on the digest and [memory].

b. Consistent with experiment? Yes for Stark first order (Stark 1913, Lo Surdo 1913; the book's comparisons for H-gamma and H-delta are described as "identical in every detail"). The anomalous Zeeman effect is not explained in the book's pages (Ch. VI, digest only) and indeed was not explained until spin (1925) [memory, high].

c. Tested? Yes, by Stark and Lo Surdo in 1913-14; Epstein and Schwarzschild's 1916 papers gave the theory; Kramers computed intensities. Replicated in Stark-effect work thereafter, and in wave mechanics.

d. If true, what changes? Nothing.

Verdict. Stark: Established. Zeeman and the adiabatic hypothesis: Established, flagged as digest-based. Old-theory ambiguity: Historically superseded but instructive.

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Claim 5: X-ray spectra mirror the hydrogen fine structure

a. Internal consistency. The deep reader reproduced the Z^4 scaling (92^4 = 7.2e7), every coefficient in eqns (1) and (2) (1/4, 1/8, 5/64; 5/4, 21/8, 429/64; 53/8), the series inversion (10) with coefficients 19/32, and the stated sums [read: deep read, Ch. VIII part 2]. The sample standard deviation of the 38 printed screening numbers is 0.043 about the mean 3.49; one row (76 Os) is inconsistent with its own input (3.57 against 3.49). - My independent fit. I inverted Sommerfeld's exact doublet formula (the same expression as in claim 2, with nuclear charge Z - s) for seven rows taken from the deep read (Nb 41, Mo 42, Ce 58, W 74, Bi 83, Th 90, U 92) and recomputed Z - s [computed, /tmp/sf/xr.py]: 37.50, 38.53, 54.44, 70.50, 79.60, 86.61, 88.56, in agreement with Table 53 (37.50, 38.53, 54.44, 70.49, 79.58, 86.57, 88.51; small differences at high Z come from the book's approximate inversion). I then varied alpha^2 and fitted the slope of s against Z:

A constant screening number requires zero slope, which occurs near alpha^2 = 5.33e-5 (modern 5.325e-5). So the X-ray L-doublets, with the single assumption that s is constant, themselves determine alpha^2 to better than about 1%, independently of the optical hydrogen doublet. This is a stronger statement than the book makes and is not circular (only the shape of the Z dependence is used). Seven rows is a small sample; it is an illustration, not a re-analysis of Table 53. - The doublet formula's hidden assumption: the same effective charge Z - s is used for both levels of the doublet. The book itself says (p.509) that two different screening constants appear in Delta nu and in nu. The modern reading: "regular" doublets are spin-orbit doublets (same l, different j), and "irregular" doublets are screening doublets (same j, different l) [memory, high; deep read]. Sommerfeld's n = j + 1/2 explains why his relativistic formula worked without spin. - The book's statement that atomic models for the screening numbers are "hopeless" (105 atomic species required for different-atom assignments; grouped ellipses cut the K shell [read]) is internally consistent and includes an explicit retraction of earlier German editions. The 105 = 1.3.5.7 counting is simple combinatorics.

b. Consistent with experiment? Yes: Moseley's law, the Z^4 growth of the L-doublet (Table 52: Delta nu_H rises from 0.38 to 0.50 as Z goes from 41 to 92, attributed to the dropped higher relativistic terms), the level scheme (Table 55) agrees with modern absorption edges to about 0.2% for uranium (L3 and K, [memory, medium; deep reader's spot check]).

c. Tested? By the X-ray spectroscopists (Siegbahn, Hjalmar, Coster, de Broglie, Dauvillier, Duane, Stenstroem, Hertz, Wentzel, Smekal) over 1913-1921. Replicated by many laboratories. Moseley's law is itself digest-only (Ch. III).

d. If true, what changes? Nothing; the Z^4 law and screening are in textbooks. The orbital explanation of screening is superseded by shell theory with spin and the Pauli principle.

Verdict. Established as an empirical law; orbital models Historically superseded but instructive.

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Claim 6: Relativity versus the "absolute" electron; universal constants

a. Internal consistency. - The 4/5 factor. I re-derived it. The book gives the Abraham kinetic energy T = (m0 c^2 beta^2/2)(1 + (3/5) beta^2 + ...) against relativity's coefficient 3/4. With dT = v dp (so p = integral of (dT/dv)/v dv), the momentum for the rigid sphere is p = m0 v (1 + (2/5) beta^2), the "transverse mass" expansion 1 + (2/5) beta^2 of the book's eqn (16a). Expressing p^2 through T gives p^2 = 2 m0 T [1 + gamma T/(m0 c^2)] with gamma = 1/2 for relativity and gamma = 2/5 for the rigid sphere [computed, SymPy: gamma = 1/2 and 2/5 exactly]. Because the fine-structure terms scale as 2 gamma alpha^2, every interval is reduced by 2 gamma = 4/5 for the rigid electron. The ratios (1:3, 1:2:6) are unchanged. All of this is consistent with the book's eqns (17a), (20) and (21) [read]. Sommerfeld's remark that agreement beyond first order in beta^2 for gamma = 1/2 "must be regarded as an accident" is correct: the calculation keeps only first powers of beta^2. - The test logic. The deep reader notes that a single measured doublet cannot fix both alpha and gamma. Correct. The book escapes this because alpha can be fixed otherwise: Millikan's e = 4.774e-10 and Planck's h = 6.55e-27 give alpha^2 = 5.32e-5 (so gamma = 1/2 predicts 0.3650, gamma = 2/5 predicts 0.292); and, as I show under claim 5, the X-ray L-doublets themselves give alpha^2 near 5.33e-5. The He+ scale of 0.3645 +/- 0.0045 then excludes 0.292 by (0.3645 - 0.292)/0.0045 = 16 standard errors [computed], provided that Paschen's error is not underestimated. That is, the Abraham alternative is excluded under the book's own error model; the contrary evidence is the direct hydrogen data (see claim 2), which scatter in a range from 0.27 to 0.33. - What the test does not exclude. Frauenfelder and Weber [read: arXiv:1902.09612, Theorem A and eq. 1.2] apply Sommerfeld's quantisation to Weber's Hamiltonian H = (1/2) [r/(r + alpha^2)] p_r^2 + p_phi^2/(2 r^2) - 1/r (atomic units; from a Lagrangian L = T + (1/r)(1 + v_r^2/(2c^2))). They obtain, to order alpha^2, -1/(2 n^2) - alpha^2/(2 n^3 l) + alpha^2/(2 n^4), against Sommerfeld's -1/(2 n^2) - alpha^2/(2 n^3 l) + 3 alpha^2/(8 n^4). I checked this myself two ways [computed]: (i) first-order perturbation theory: the perturbation is -alpha^2 p_r^2/(2r), whose classical average is -(alpha^2/2)(1/(n^3 l) - 1/n^4); (ii) numerical quantisation of the radial phase integral at alpha = 0.01 and 0.02 for (n, l) = (2, 1) and (3, 1): shifts -0.031250 and -0.012346 in units of alpha^2, equal to the Weber formula to 4-5 digits, and different from Sommerfeld's -0.039063 and -0.013889. The two formulas differ only by a term that depends on n alone, so every interval inside one n-manifold is the same in the Weber and in the relativistic theory (for n = 2: both give alpha^2/32 in atomic units, the 0.365 cm^-1). The fine-structure splittings therefore do not distinguish the Weber force law from special relativity at this order; the only difference is an overall shift of each n-level, which no pair of lines within a manifold measures. What the Sommerfeld-Lenz-Glitscher test excludes is the specific Abraham rigid-sphere kinetic energy, not all velocity-dependent dynamics. (The Weber model has no spin and no relativity; Frauenfelder and Weber themselves say the absolute n^-4 difference is hard to measure. I did not check whether the Weber result survives at the next order or in X-ray doublets where alpha^2 (Z - s)^2 is not small.) This is a lead the Investigator marked "not read"; I have read it and it bears directly on the claim that fine structure is the "experimentum crucis" for relativity. I record it whatever one thinks of Weber electrodynamics. - Constants e and h. Error propagation [read: deep read; I verified exponents]: e proportional to alpha^3, so 3/2 x 1/81 = 1/54; h proportional to alpha^5, so 5/2 x 1/81 = 5/162 (6.526 +/- 0.200 e-27); the semi-spectroscopic h proportional to e^(5/3), so 5/3 x 0.004/4.774 gives +/- 0.009. All propagation arithmetic is right. Weaknesses: one error source per route (R_inf and e/m0 declared "absolutely exact"); the +/-0.005 on alpha in eqn (15) is not derivable from Millikan's +/-0.004 alone (that gives +/-0.002) and is unexplained [read: deep read]. - Modern comparison [memory, high; deep read]: Millikan's e is 0.6% low, so the semi-spectroscopic h = 6.545e-27 is 1.2% low (modern 6.62607e-27) and lies 9 sigma away on the book's own bars; the wide purely spectroscopic bars contain the true values. The cause is the viscosity of air in the oil-drop analysis [web, https://scipython.com/blog/measurements-of-the-electron-charge-over-time/, which quotes Feynman on the viscosity error; also the Illinois lecture slides it lists]. alpha itself comes out within 0.03% because the e and e/m0 errors nearly cancel in e (e/m0) to the 1/3 power [computed: (0.9939 x 1.0058)^(1/3) = 0.9999]. - Mercury (Note 16 and p.244). The special-relativistic perihelion advance, Delta phi = 4 pi^3 a^2/(tau^2 (1 - e^2) c^2) (per revolution), gives 7.16 seconds of arc per century against 43 observed; general relativity gives six times more, 42.98 [read + computed by the deep reader; the book's "24" instead of "4" is the general-relativity factor]. Consistent. The book's use of the Coulomb-form law for gravity is illustrative only (the deep reader's point).

b. Consistent with established experiments? The relativistic mass increase: yes (Kaufmann's early data were ambiguous; Bucherer, Guye-Lavanchy and later work support relativity, [memory, high]; the book cites Schaefer-Neumann and Guye-Lavanchy). Abraham's rigid electron was abandoned for this reason and because it is not Lorentz covariant [memory, high].

c. Tested? Lenz suggested the test; Glitscher (Munich, 1917) did the calculation (Ann. Phys. 52, 608), saved by the Investigator; the experiments are Paschen's He+ photographs and the X-ray L-doublets. Opposition came from Gehrcke, Lau, Lenard and others, who were also anti-relativity campaigners (Sommerfeld's letters record this [web, research/2026-10-09-sommerfeld-projekt-lmu.md]); the Investigator records the dissent next to their data. Gehrcke and Lau's measurement of a doublet 25% too small is real data; their interpretation was not offered in the paper. Sommerfeld's 1924 letter calls "the possibility of putting Abraham in the place of Einstein" "indiscutabel" and attributes it to the "fanatical spirit of Lenard, Gehrke, Lau" while taking Lau's helium result seriously. The 25%-small hydrogen number lies near the 20%-small Abraham prediction (0.274 against 0.292); I report that arithmetic coincidence and also that the He+ and X-ray evidence, where the same arithmetic does not hold, outweighs it.

d. If true, what changes? If the rigid-electron factor 4/5 had held, special relativity's mass-velocity law would be wrong at order beta^2 (which is equivalent to saying Abraham's mass function is right). Nothing in mainstream physics changes now because relativity is in use.

Verdict (split). Abraham alternative: Tested, not supported. "Fine structure uniquely selects relativity among all velocity-dependent dynamics": not supported (Weber gives the same intervals at order alpha^2); a limit of the test, not a refutation of the book. Constants: Historically superseded but instructive.

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Claim 7: Honest limits and unexplained facts

a. Consistency. The list is accurate to the deep reads: the Bohr He model gives 28.75 V where 25.3 V is observed (Note 14: computed again by the deep reader, every number except a small calorie figure reproduces); the Weinacht result on non-convergent expansions (Note 13; the book's n-power errors in eqns (10)-(12) mean the third-order formulas as printed are wrong) [read]; the gamma-ray scattering 0.04 against 0.2.

b-d. These are statements about 1921-22 knowledge, all still correct as history. The "unexplained" items were later explained by spin (anomalous Zeeman, the magneton numbers), by the quantum treatment (He ionisation potential 24.6 V in wave mechanics [memory, high]) and by Compton scattering (the gamma-ray anomaly) [memory, medium]. Not independently re-tested by me.

Verdict. Historically superseded but instructive.

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Errors and misprints in the book that affect arithmetic (verified or restated)

• Wrong as printed: Ch. VIII "alpha^4/4 = 1.3e-5" (alpha^2/4); mass expansion "-3/8 beta^4" (+3/8); eqn (14) bk.507 coefficient 191/32 (derived 2807/1024); Table 53 row Os; Note 4 eqn 22c sign; Note 6 eqns 27, 29 sign; Note 11 eqn 28 sign; Note 13 eqns 10-12 powers of n; Note 17 +4c xi^3 and b^2 sign; cerium c:a 1.62 (1.633); antimony "563 degrees". The Note 13 and Note 17 numbers are wrong as printed. None changes a physical conclusion in the chapters read. Not checked: whether any are in the German original; the Investigator has now located the German 1921 text (archive.org) and probably the 1922 text (HathiTrust), so the check is possible (see Corrections).

Corrections to Step 1

1. claims.md claim 2, "Evidence offered ... X-ray L-doublets": no error, but the statement that Paschen's value is an independent confirmation needs the qualifier already in the claim ("obtained by fitting the theory's own interval scheme"). The ratios (1:3, 1:2:6) test the pattern; the scale 0.3645 tests the scale only given the pattern. I add: the X-ray L-doublets (claim 5) give an independent scale check (my fit, alpha^2 near 5.33e-5, seven rows).
2. claims.md claim 6 says "the deep reader notes the 4/5 test needs an independent alpha". Correct. I add the resolution (Millikan + Planck, and the X-ray fit) and the limit: the same intervals are produced by Weber's law (Frauenfelder-Weber 2019). The claim as worded in the book ("experimentum crucis for relativity") should not be repeated without that limit.
3. facts.json / claims.md claim 4 says the Zeeman effect rests on a digest. Agreed; I did not check it either. This is unchanged and remains the main unread item among my claims.
4. sites.md "Leads for review", item 1, Sommerfeld to Paschen 29 Dec 1915: the summary states the ratios 1 : 1.46 : 1.73; Sommerfeld's own letter has this slip (27/36 divided by 7/16 = 1.714). The Investigator transcribed it faithfully; the error is Sommerfeld's. Also in the letter "0.159 x 16/25 = 0.102" uses 0.159 where the line above computes 0.169 (0.169 x 16/25 = 0.108). Neither affects the point (the absolute values disagree by more than a factor of two).
5. sites.md lead on Frauenfelder and Weber ("Not read"): now read. Result in claim 6 above. Their abstract's statement that the formula "coincides with Wesley's Schrödinger result" is not a statement about agreement with hydrogen data; it differs from Sommerfeld's formula in the n^-4 term (1/2 against 3/8).
6. deep read, Ch. VIII, section 3.3 / claims.md claim 2 wording "Counter-evidence ... all below 0.365 and only partly explained": correct; I add the quantitative breakdown (table in claim 2): the book's selection lag accounts for the Michelson values but leaves the Gehrcke-Lau values 17-19% low.
7. claims.md "Recommended further deep reads": still needed: (a) Ch. V sections 6-7 pp.157-166 (Zeeman, adiabatic), (b) Ch. VI pp.167-216. I would add (e): the German 1921 or 1922 edition, to check whether the misprints listed are translation or original, and whether the 0.365 discussion differs.
8. No error found in the Step 1 treatment of the Ch. IV, Ch. V part 1 and Ch. VIII calculations; the deep reads' recalculations that I redid (series coefficients, Z - s inversion, component sets, 4/5 factor, error exponents, Langevin averages) agree.

What a reader should take from the mathematics

• A formula can be right and its derivation wrong: Sommerfeld's 1916 formula equals Dirac's exact result, yet it came from orbits with no spin.
• Intervals inside a manifold constrain less than they seem: Weber's force law and special relativity differ only in a term that does not depend on l.
• The quality of the arithmetic in the book is high; the quality of the error bars is mixed (single source of error per route, systematic errors not carried).
• Sommerfeld's open admissions (degeneracy, n = 0, "hopeless" models, five of seven elements) are in the book itself.