Arbiter: Langmuir-Irving (Tonks and Langmuir, "Oscillations in Ionized Gases", Phys. Rev. 33, 195-210, Feb. 1929)

Role: Step 3. I read claims.md, facts.json, physicist.md, mathematician.md, contrarian.md, sites.md and the research notes, and recomputed the numbers where reviewers touched them. Tags: [read] document, [read: page image p.N] as reported by a reviewer who rendered the page (I did not re-render pages myself), [web] with URL, [memory] with confidence. Verdict scale is the Step 2 scale.

Source gap (repeated)

The project holds one document: a 17-page scan whose text layer is 100% OCR, so every equation was garbled. The theory (PDF pp.1-9) was transcribed from page images in Step 1 and re-derived independently by all three reviewers. The experimental part (pp.10-17) was a digest only in Step 1; the physicist read pp.13-17 from page images, the mathematician pp.13-17, and the contrarian pp.11-17. PDF pp.10-12 and Figs. 4-10 rest on the digest plus partial page reads. Nobody read the 1928 PNAS paper, the authors' June 1929 Note (Phys. Rev. 33, 990), Tonks's 1931 papers, or any later paper beyond abstracts or titles (Baalrud 2009: first page only). Statements about later work therefore rest on abstracts. Where a conclusion rests on [memory] alone, it is marked.

Recomputed numbers (numpy, Gaussian constants)

The one numerical difference between reviewers (physicist 2.5e-6 versus mathematician and me about 5e-6) is immaterial: both are negligible at the paper's wavelengths.

One observation neither reviewer made: the bulb comparison. The paper's 18 cm bulb as a half-wave gives 24.5 cm, while observed oscillations in the 27-81 cm band correspond to 370-1110 MHz [calculation]. The prediction is outside the observed band at its short end by a modest margin and the authors themselves reserved judgement ([read] per reviewers). So it is not a test.

Web checks done (budget 8; 3 used)

1. Merrill and Webb, Phys. Rev. 55, 1191 (1939) [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.55.1191 : abstract re-fetched. Confirms: mercury arc, movable probe, fast electrons scatter in planar regions tenths of a mm wide; crystal detector and Lecher wires; "stable periodic oscillations" in specific regions plus "turbulent" disturbances throughout; oscillation regions "generally coincided" with scattering regions but shifted slightly toward the anode; frequencies "agreed well with the formula derived by Tonks and Langmuir"; conclusion that plasma oscillations energised by fast electrons cause the scattering. Full text still unread. Authorship overlap with the 1928 oscillation papers (Webb and Pardue) is not shown by the abstract; Harold W. Webb is a listed author here, which makes the possible shared-school point the physicist raised worth keeping as unresolved (the 1928 Webb and Pardue authorship is [memory, medium]).
2. Alexeff and Neidigh [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.129.516 : correction to both reviewers. It is Phys. Rev. 129, 516 (15 Jan 1963), not 1962. Its summary says the "ionic sound waves predicted by Tonks and Langmuir" were observed in plasma columns and spherical discharge tubes, with fundamental frequencies and overtones matching the formula in terms of T_e, ion mass and dimensions. This upgrades the physicist's [memory, medium] to [web] and supports Claim 3 theory directly (abstract-level).
3. Bohm and Gross 1949 [web] https://doi.org/10.1103/PhysRev.75.1851 : confirmed as Phys. Rev. 75, 1851, "Theory of Plasma Oscillations. A. Origin of Medium-Like Behavior", with thermal motions included. A search-result summary also says the oscillations are characterised by zero group velocity; I treat that summary as unreliable for the full thermal result (the Bohm-Gross dispersion has non-zero group velocity at finite k, which I recomputed above). The reviewers' [memory, high] statement about the Bohm-Gross term stands; the specific "zero group velocity" line in the search snippet was not verified in the paper and I do not rely on it.

Not web-checked (do not change any verdict): priority before Thomson/Rayleigh [memory, medium]; Looney-Brown 1954 [memory, medium]; Wong-Motley-D'Angelo 1964 [memory, medium]; bistable discharge modes as a home for the two-state arc [memory, medium]; Mie/Tonks-Dattner sphere factor 1/sqrt 3 [memory, high].

Corrected claim list

Corrections agreed by all three reviewers, to be carried into the primer:

• Claim 4: "ratio near 1:2" must read "about 1.6:1 (the authors: 'almost twice'); 1080 MHz at 26 V against 660 MHz at 28 V", i.e. confounded with a voltage change [read: page image p.14, per three reviewers].
• Claim 4: the n^(-1/2) scaling rests on a relative ionization scale only; the authors write "no way of finding the absolute value of the ion density", "too much reliance is not to be placed on them", 13 of 15 points within error [read: page image p.15]. It is a consistency check over under one decade of abscissa, not a discriminating test.
• Claim 5: the authors call their correlation runs "inadequate" for three stated reasons; their only correlation is oscillations vanishing over the same 60-75 V anode range as the scattering [read: page image p.13/14].
• Claim 3: the 1.38 MHz resonance was not repeatable ("so elusive"); the amplifier showed a noise continuum [read: page image p.16]. The 1.5 MHz figure is an upper limit assuming n = 1e10; no density was measured.
• Claim 2: "zero group velocity" holds in the cold limit only; fine for the paper's wavelengths (shift below 1e-5).
• Digest/Step 1 errors: transit time word reversed (mathematician); wavelength threshold ~1e-1 cm, not 1e-5; ion spacing 4.6e-4 cm; two printed misprints (exp(r/lambda_D) sign; first "2(2 pi)^(1/2) lambda_D") that do not propagate; Alexeff-Neidigh date is 1963.
• facts.json how_read for experimental pages should say digest plus page images from the reviews.
• QE revision corrections (see the "QE revision" section below): the 300+ MHz control range is -80 V (to cathode) to 5 V above anode potential, not -80 to +5 V [read: p.13]; the crystal caveat is a "frequency conversion factor", not a "frequency variation factor" [read: p.10]; the "Webb" overlap between the 1928 and 1939 papers is withdrawn (J. S. Webb against Harold W. Webb); the Fig. 10 line labelled "SLOPE = -0.5" measures -0.67 and the data fit -0.60 +/- 0.16 (deep read); Fig. 10 spans 0.54 decade of relative ionization (the mathematician's "less than one decade" is right) and 26-81 cm of wavelength (the physicist's "roughly 15-90 cm" is too wide).

Claim 1. Plasma-electron frequency nu = (n e^2/(pi m))^(1/2) = 8980 n^(1/2) Hz

• Agreement: all three: mathematics correct (derived by Poisson plus Newton and by Thomson's divergence method); mainstream; the 1929 data are only a weak plausibility check (no absolute density; the bulb comparison is explicitly left open).
• Disagreement: none on the verdict. Emphasis only: the contrarian stresses that the 1929 experiments do not confirm the coefficient; later work does. I agree. The physicist and mathematician add that for a bounded plasma the observed resonance shifts (sphere surface mode nu/sqrt 3) [memory, high]; the paper's shape-independence statement is for the bulk oscillation and carries its own small-displacement caveat.
• Stronger argument: the decisive evidence is later and external: ionospheric critical frequencies, Merrill and Webb 1939 [web abstract], and the 1963 observation of the related ion branch.
• Settle it: already settled. For the 1929 data alone: absolute density by a microwave interferometer or probe in the same tube.
• Final verdict: Established. The paper's own experiments contributed weakly; the credit is for the derivation and the plasma-physics framework that later work verified.

Claim 2. Zero group velocity; EM dispersion nu^2 = n e^2/(pi m) + c^2/lambda^2

• Agreement: the dispersion relation is correct and confirmed; the zero group velocity is exact only for cold electrons.
• Disagreement: the physicist gives one "Established (cold limit)" verdict with a caveat; the mathematician and contrarian split the claim, putting the unqualified zero-group-velocity statement under "Historically superseded but instructive".
• Stronger argument: the split is more informative and matches the scale; the authors did not claim the strong form, but the claim as written in claims.md is unqualified. Pressure corrections appeared in 1936 (Linder, title only [web] Crossref) and Bohm-Gross 1949 [web]. Quantitatively the correction is below 1e-5 at 33 cm and order unity at 0.1 cm (recomputed).
• Settle it: a measurement of group velocity of Langmuir waves in a warm plasma (done in later work [memory, medium]).
• Final verdict: split. (i) EM dispersion relation: Established. (ii) Zero group velocity: Established as a cold-plasma limit; as a general statement Historically superseded but instructive.

Claim 3. Plasma-ion oscillations and electric sound waves, v = (kT_e/m_p)^(1/2), about 1.5 MHz upper limit for Hg

• Agreement: all three: derivation correct, equals the cold-ion ion-acoustic relation omega^2 = k^2 c_s^2/(1 + k^2 lambda_De^2); the paper's own detection (1.38 MHz) was a single unrepeatable observation; hidden assumption T_i << T_e (holds in an Hg arc [memory, high]).
• Disagreement: none on verdicts.
• Stronger argument: all consistent. New web support: Alexeff and Neidigh 1963 observed the "ionic sound waves predicted by Tonks and Langmuir" [web] https://journals.aps.org/pr/abstract/10.1103/PhysRev.129.516 (abstract-level). Baalrud et al. 2009 propose ion-acoustic instability as the cause of Langmuir's paradox [web, page 1 only], linking this branch to Claim 5.
• Settle it: a tuned receiver on a discharge with independently measured density and T_e, checking the dispersion near k lambda_De = 1. Alexeff and Neidigh did this kind of test in 1963.
• Final verdict: split. Theory: Established. The paper's own experimental identification of the 1.38 MHz line: Never properly tested (one irreproduced observation; not dismissed, simply not followed up by the authors in this paper).

Claim 4. Hg-arc RF noise 1e7-1e9 Hz; wavelength ~ (relative ionization)^(-1/2); two scattering states "1:2"

• Agreement: all three reject "1:2" as a number (1.64); all three say the scaling support is weak and self-qualified; all three say the two-state arc was reported with data and has no replication found.
• Disagreement: on the existence of RF oscillations. The physicist says Established (Penning, Dittmer, Webb and Pardue, Merrill and Webb). The mathematician and contrarian say Plausible, untested in the 1929 setting, supported by Merrill and Webb.
• Stronger argument: the physicist is right about the phenomenon and the others are right about this paper's data. The phenomenon (oscillations in the 1e8-1e9 Hz range in low-pressure arcs, localised to scattering regions) has an independent report with a movable probe, which also found "turbulent" noise throughout the tube, matching the paper's noise description [web abstract]. The caveat the contrarian adds is real: Merrill and Webb used the same detector type (crystal plus Lecher wires), so a shared detector artefact is not excluded; I only have the abstract. The 1929 amplitude data are relative, with no high-frequency detector check, and several features were irreproducible within the paper (the 2 C window at 19-21 C with non-monotonic 263/231/250 MHz; the 115 MHz beam identification fails the authors' own 26 % versus 10 % test; the 55 MHz Doppler identification is withdrawn).
• Settle it: (a) absolute density in an Hg arc by microwave interferometry, then check nu = 8980 n^(1/2) at several densities over more than one decade; (b) a modern tube, recording the two states at identical anode voltage (27 V, 20 mA) with an independently calibrated broadband receiver.
• Final verdict: split. (i) Existence of RF oscillations at 1e7-1e9 Hz in Hg arcs: Established qualitatively (independent group, abstract-level, with the shared-method caveat). (ii) Scaling lambda ~ n^(-1/2) in this paper: Plausible, untested (weak, self-qualified support; the underlying formula is Established from Claim 1). (iii) Two scattering states, switchable by a magnet, with frequencies near 1.6:1: Never properly tested (single laboratory, not dismissed, simply not followed up in anything found).

Claim 5. The oscillations may explain the anomalous scattering and energy gain ("Langmuir's paradox")

• Agreement: all three: Never properly tested; not dismissed without testing; the paper's correlation was "inadequate" by the authors' own account; no amplitude estimate reaches Dittmer's "20 volts or more" requirement; the record of work continues 1928-2015.
• Disagreement: only emphasis. The contrarian rejects both "Tested, not supported" and "Plausible, untested" because Merrill and Webb 1939 partly tested it favourably; the physicist and mathematician say the same in different words.
• Stronger argument: the contrarian's reasoning is the clearest. Evidence for: Merrill and Webb 1939 (oscillation regions generally coincident with scattering regions, shifted slightly toward the anode) [web abstract]; the 60-75 V coincidence in the paper; the two-state arc where scattering and frequency change together. Evidence against or complicating: competing mechanisms (sheath oscillations, Gabor-Ash-Dracott 1955; ion-acoustic instability, Baalrud 2009; non-local kinetics and striations); Godyak and Alexandrovich 2015 find non-Maxwellian distributions in 1-3 mTorr argon (abstract only), questioning how universal the anomaly is, though not specifically for Hg. Interested-party note: Merrill and Webb share a research line with the original oscillation work and Baalrud et al. propose their own mechanism; each is that party's characterisation, none a neutral resolution.
• Settle it: an Hg arc experiment with a movable probe measuring local oscillation amplitude and the electron energy distribution at the same point, and checking whether amplitude reaches the voltage required (Dittmer's 20 V); 1939 is the template.
• Final verdict: Never properly tested. Partly tested (1939, favourable at abstract level); unsettled; not refuted; not dismissed without testing.

What would settle it (summary)

1. Absolute electron density in the same kind of Hg arc (microwave interferometer or double probe), compared with frequencies: settles Claim 4(ii) and Claim 1's laboratory confirmation for this setup.
2. Movable-probe local amplitude plus electron energy distribution function: settles Claim 5, and quantitatively tests Dittmer's amplitude objection.
3. Two-state arc repeated at fixed anode voltage with a calibrated receiver: settles Claim 4(iii).
4. Tuned ion-range receiver with measured n and T_e: Claim 3 (done in other devices in 1963).
5. Reading of the 1929 Note, the 1928 PNAS paper and Collected Works vol. 5: whether misprints were corrected and whether the experimental caveats were stated differently.

Open questions

• Is the Phys. Rev. 33, 990 Note an erratum, a clarification, or new data? Unknown; paywalled.
• What do the Langmuir notebooks (LOC mm77029413) show about the 1.38 MHz line, the two-state arc and the 19-21 C window? Unexamined.
• Does Merrill and Webb's full text give an absolute density, and did they rule out detector artefacts? Only the abstract was read.
• Is there any overlap between the Merrill-Webb team and the 1928 Webb and Pardue work cited by Tonks and Langmuir (independence of the replication)?
• Does the Langmuir-paradox anomaly exist in Hg arcs of the 1925-29 type under modern probe diagnostics?
• Tonks's 1931 papers on shape dependence: does the claim of shape-independence survive for bounded plasmas as Tonks himself later treated them?
• Baalrud et al. describe a roughly 3 cm tube; this paper uses an 18 cm bulb (the 1925 apparatus is presumably the reference). A reader should not equate them.

Absence-of-evidence analysis

The project's main record does not go silent: the Investigator found continuous open follow-up (Tonks 1931-34; Linder 1936; Merrill-Webb 1939; Armstrong 1947; Bohm-Gross 1949; Hernqvist 1955; Gabor-Ash-Dracott 1955; Martin-Emeleus 1956; Alexeff-Neidigh 1963; Baalrud 2009; Godyak-Alexandrovich 2015). I found no sign of suppression, blocked funding or omitted data; the three reviewers agree. The theory entered the standard framework and credit is attributed.

There are, however, three narrow places where the record does go quiet where it would be expected to continue, and I treat them as anomalies, not neutral gaps:

1. The two-state (bistable) arc, switchable by a bar magnet, with hysteresis and abrupt oscillation amplitude jumps (Figs. 8-9).
2. The 1.38 MHz ion-range line that "had become so elusive" it could not be repeated.
3. The 2 C (19-21 C) temperature window for the critical 230-263 MHz oscillation.

Competing explanations for the silence:

• Not a robust phenomenon (a tube-specific, geometry-dependent marginal oscillator or circuit resonance). For: the 2 C window, vanishing with 2 mA of emission from filament g and with heating current reversal, non-monotonic frequency with temperature, and the authors' own words that the other evidence is "not clean cut"; 115 MHz and 55 MHz identifications failed by the authors' tests. Against: the stated controls (position dependence on the wall, two-electrode interference test, independence from collector potential) argue for plasma origin.
• Absorbed into later work under different vocabulary. For: the Investigator found striations and double-layer, sheath-oscillation and ion-acoustic-instability literature (Gabor 1955, Martin-Emeleus 1956, Baalrud 2009) covering the same ground; hysteretic discharge modes are known [memory, medium]. Against: none of the abstracts or titles seen mention this specific magnet-switched state or cite the 1929 figures for it. This was not searched for directly, which the contrarian notes.
• Superseded by the authors' own follow-up and companion papers, whose contents were not read (Phys. Rev. 33, 990; Phys. Rev. 34, 876, "A General Theory of the Plasma of an Arc"; and PNAS 14, 627). For: both were published within months. Against/unknown: contents unread.
• Raw data left in laboratory notebooks (the Langmuir papers at the Library of Congress, notebooks 1894-1957) and never published. Plausible given a company laboratory (General Electric) and the authors' wish for "experiments under essentially simpler conditions" ([read] per the contrarian). Evidence: finding aid not yet searched.
• Suppression or secrecy: no evidence of either was found; the work was published openly and corporate secrecy for a basic discharge study is not indicated. I record it as considered and unsupported, not as excluded.

What would decide between them: the LOC notebooks and finding aid; the Collected Works vol. 5 and the June 1929 Note; a citation search ("forward chain") on this paper for papers referring to the bistability or the 19-21 C window; and a modern replication.

Investigator findings: how I weighed and kept them

All kept: Merrill and Webb (support; abstract read twice); Baalrud et al. 2009 (alternative or related mechanism; first page only); Godyak and Alexandrovich 2015 (evidence against universality of the premise; abstract only; the "challenge the universality" phrase comes from a secondary summary); Gabor et al. 1955 and Martin-Emeleus 1956 (alternative oscillation types); Linder 1936 (thermal correction); Hernqvist 1955 and Peterson-Puthoff 1959 (ion oscillations; titles only); Tonks 1931-33 papers; LOC notebooks; Tonks, "The Birth of 'Plasma'" (1967), the first-hand history of how the work was received, unread. The Investigator's downloads awaiting approval (PNAS 1928, Baalrud copy, Collected Works vol. 5 text) remain the cheapest next reading.

Verdict table

Headline: the paper is a correct, historically important statement of plasma physics (Claims 1-3); its experiments are honest but weak, and the specific explanation of Langmuir's paradox remains open.

QE revision

Role: Step 4b remediation. Inputs: qe.md (verdict MINOR), and two new deep reads appended to deep/Osc-In-Ionized-Gas-a7691d.md: DR2 (PDF pp.5-6, thermal electrons and beam resonance, Opus) and DR1 (PDF pp.13-15, Figs. 8-10, Opus). DR3 (PDF pp.9-12, Figs. 2-7) and DR4 (abstract and p.16 conditions by page image) are not in the deep file. I did not read them and nothing below relies on them. I did not re-render pages myself; figure values are [read: page image p.N] as reported by the deep reader. Web checks this round: 2 more (arbiter total 5 of 8).

Summary of what changed

No label on the verdict scale changes. Three bases are restated (Claim 4(ii), Claim 4(iii), Claim 5), one claim gains a part (Claim 2(iii)), one earlier caveat is withdrawn (the "Webb" overlap), and several quotations are corrected.

A. Corrections to my own text and to the reviews (from qe.md)

1. The "Webb" overlap is withdrawn. My Web check 1 and Claim 5 said Merrill and Webb 1939 "share a research line" with the 1928 oscillation work, resting on [memory, medium] that the 1928 author was H. W. Webb. The paper prints "J. S. Webb and L. A. Pardue, Phys. Rev. 31, 1122 (1928) and 32, 946 (1928)" [read: page image p.8, per QE]. Crossref lists the 1939 authors as Harrison J. Merrill and Harold W. Webb [web] https://api.crossref.org/works/10.1103/PhysRev.55.1191 (I re-fetched this). The 1928 paper is Pardue and J. S. Webb, "Ionic Oscillations in the Glow Discharge", Phys. Rev. 32, 946 [web, per QE via Crossref] https://api.crossref.org/works?query.bibliographic=Pardue+Webb+1928. Different initials do not prove two different people, and nobody has checked affiliations, so the question of overlap stays open but has no support. I drop the "shared research line" caveat. I keep the separate caveat that both Merrill-Webb and the 1929 work used a crystal detector with Lecher wires, so a common detector artefact is not excluded (abstract only for 1939). Effect: the 1939 work is, on present evidence, an independent report, which slightly strengthens Claim 4(i) and Claim 5 evidence "for". Verdicts unchanged. In the Claim 5 interested-party note, only the Baalrud et al. clause stands.
2. Control range corrected. Frequency and amplitude of the 300+ MHz waves were "independent of the potential of the receiving collector over the whole range explored, namely from -80 v. with respect to the cathode to 5 v. above anode potential" [read: p.13; confirmed by DR1]. At the 33 V anode setting that is up to about 38 V, a wider range than "-80 to +5 V". It strengthens the plasma-origin control. Other controls DR1 confirms from p.13: same frequency with the beam on the collector, deflected past it, or the collector out of the beam; reversing the filament heating current changed nothing.
3. Crystal caveat quoted correctly. The text is "as if a frequency conversion factor for the crystal varied from time to time or with the crystal setting" [read: p.10, per QE]. It bears on how far amplitudes from different runs can be compared, not on the frequencies.
4. The 810 and 880 MHz entries are the cathode-g and cathode-d measurements at 33 V and 30 mA [read: p.13], not Fig. 10 points. My recomputed-numbers table, which lists them with the density calculation, does not say otherwise, but the example script labels them "Fig. 10 example" and should be relabelled (QE item).
5. Page labels. The 60-75 V correlation is on PDF p.13 (journal p.207); my "p.13/14" is replaced by p.13 [read: p.13, DR1].
6. Retag: the Baalrud et al. title is [web] (Investigator read it on the APS page); the affiliation "Research Laboratory, General Electric Company, Schenectady, N.Y., November 20, 1928" is [read: p.16], not [memory]. Two web checks for citations used in DR2, retagged [web]: Looney and Brown, "The Excitation of Plasma Oscillations", Phys. Rev. 93, 965-969 (1954), DOI 10.1103/physrev.93.965; Malmberg and Wharton, "Collisionless Damping of Electrostatic Plasma Waves", Phys. Rev. Lett. 13, 184-186 (1964), DOI 10.1103/physrevlett.13.184 [web] Crossref https://api.crossref.org/works?query.bibliographic=Looney+Brown+excitation+of+plasma+oscillations+1954 (titles and bibliographic data only; contents unread). The physics attributed to them is still [memory, medium].

B. Claim 2: new part, the paper's own thermal argument (DR2, pp.5-6)

What the authors say [read: page image pp.5-6, DR2]: electrons fast enough to cross one wavelength within one cycle (speed above v* = wavelength x frequency) do not contribute to the effective density of the oscillation. So the effective density, and with it the frequency, falls as the wavelength shortens (d(nu)/d(lambda) > 0). A "kind of cumulative action" is said to make oscillations below a certain minimum size impossible; the supporting calculations are unpublished and no number is given.

Mainstream comparison (DR2 [memory, high] for Bohm-Gross and Landau damping, computed here and by DR2 for the numbers):

My verdict on this part: the instinct (thermal electrons set a lower limit on the oscillation size) is right and prefigures Landau damping; the mechanism and the predicted frequency trend are wrong. Neither sign nor cutoff could be tested with 27-81 cm waves, because the Bohm-Gross shift is below 1e-5 at the paper's wavelengths (my recomputation above; DR2 gives under 1 percent for plasma wavelengths above a few mm). Part (iii), "the paper's qualitative thermal argument": Historically superseded but instructive (a reasonable position in 1928, replaced by Bohm-Gross and Landau).

The earlier reviews said only that the authors "gesture at thermal effects" (qe.md section 4, item 3); the argument is more specific than that.

Cold-limit statement of Claim 2(ii) is unchanged. Group velocity for Bohm-Gross is 3 k v_th^2/omega [memory, high]; my recomputed values stand.

C. Claim 4(ii): Fig. 10, quantitative reading (DR1, p.15)

Facts [read: page image p.15, digitised by the deep reader; fit computed by the deep reader; I checked the arithmetic of the spans]: - Axes: wavelength in air 10-100 cm (one decade), relative ionization 1-6 (0.78 decade). The 15 points span relative ionization 1.19-4.09 (factor 3.4, 0.54 decade) and wavelength 26.4-81.4 cm (370-1140 MHz). - Under lambda proportional to n^(-1/2), a factor 3.4 in ionization predicts a factor 1.85 in wavelength. The observed wavelength range is a factor 3.1. - Fit of log lambda on log (relative ionization): slope -0.60 +/- 0.16 (all 15; 95 percent interval -0.95 to -0.25); -0.61 +/- 0.09 without the two outliers. Slope 0: rejected (p about 0.003). Slope -1/2: consistent (p = 0.56; 0.23). Slope -1: rejected at p about 0.03 (all) and 0.001 (13 points). - The line drawn on the figure is labelled "SLOPE = -0.5" but measures about -0.67 against its own gridlines. Not explained in the paper. The authors' sentence that all but two points lie within the accuracy of the ionization values "of the line drawn" refers to this line [read: p.15]. - Error in the horizontal variable: the authors say the ionization scale is relative, "too much reliance is not to be placed on them", and they give no uncertainty. If the error is mainly in x, the ordinary fit is biased toward zero; the reverse regression gives -1.16. So the slope lies between about -0.6 and -1.2 depending on which variable carries the error, and the data alone cannot separate -1/2 from -1. They do rule out no dependence. - Outliers: #4 (relative ionization 1.47, 81.4 cm, 369 MHz) and #5 (1.47, 27.8 cm, 1079 MHz). They share a relative ionization and differ in frequency by a factor 2.9. #5 matches the low-scattering frequency 1080 MHz of p.14 and two other points match the high-scattering 660 MHz. That match is an inference by the deep reader (the paper does not say which points come from which run). If it holds, the two-state discharge itself is the source of the outliers, since one n cannot give two frequencies under Eq. (6) (this is DR1's point and I find it sound, but it is not confirmed).

Weighing. The verdict label for 4(ii) is unchanged, Plausible, untested, because the relative ionization values are not absolute and are admitted unreliable by the authors, and no test with a known density exists in the paper. The restated basis: the data show a real, negative dependence of wavelength on ionization, consistent with -1/2 but also with steeper slopes, over about half a decade of ionization. This is a modestly stronger statement than "consistency check over under one decade" and a weaker one than "confirms the coefficient". The 8980 coefficient itself is not tested by Fig. 10 at all; only the exponent. The mathematician's estimate of the abscissa span ("less than one decade") was right in direction (0.54 decade); the physicist's estimate of the wavelength range ("roughly 15-90 cm") was too wide on the short side (26-81 cm) and "a factor of about 3" is correct for wavelength.

What would settle it: unchanged from before (absolute density by microwave interferometry), now with a sharper requirement: an ionization scale with a stated uncertainty, so that an errors-in-variables fit can be done.

D. Claim 4(iii): two-state arc, Figs. 8-9 (DR1, pp.13-14)

Facts [read: page image p.14; values about +/-1 V and +/-5 uA for Fig. 8, +/-0.2 mV for Fig. 9]: - Bistable at 20 mA emission, 27 V anode. States distinguished "only by the volt-ampere characteristic of a collector and the frequency of the oscillations" [read: p.13]. - Fig. 8: at +10 V on the collector the high-scattering state carries about 333 uA against about 161 uA for the low-scattering state (ratio about 2.1); at +20 V about 503 against 380 uA. In the ion-saturation region both sit at -55 to -75 uA. The authors state the ion currents differ by about 3 percent (about 2 uA), below what the printed figure resolves, so the figure neither confirms nor contradicts it. - Fig. 9: mV on the crystal at most about 7 mV; the two states coexist at 20 and 25 mA emission, only low scattering at 10 and 15 mA, only high at 35 mA [read: p.14]. Hysteresis: anode brought to 27 V from high gives the high state, from low the low state. A bar magnet can switch states. Several back-and-forth transitions at a 30 V anode. - Frequencies 1080 MHz (low scattering, 26 V) and 660 MHz (high, 28 V), ratio 1.64, taken at different anode voltages [read: p.14]. Confounding with anode voltage is the corrected point already made. - One unresolved item: the text says raising the collector voltage causes a transition from high to low at about 22 V, but the arrow on Fig. 8 points from the low curve up toward the high curve. The deep reader flags this as unsettled (possible drafting slip). I record it as an open item and do not rely on direction.

Weighing. Verdict unchanged, Never properly tested. The new detail makes the claim better documented, not better tested: the figures show reproducible-looking, hysteretic states with the emission ranges stated, from one laboratory in one tube, and nothing found since. The absence-of-evidence analysis (item 1 under that heading) stands and gains the emission-range and magnet details as the targets for a replication.

E. Claim 5: authors' amplitude admission, the abstract/body inconsistency, and the beam-plasma route

1. The authors admit the amplitudes are small. "although their voltage-amplitude appears to be very small we believe that they play a fundamental role in high current gaseous discharges" [read: p.2, per QE]. Crystal voltages in the experiments: about 9 mV (p.10), 30-50 mV (p.11) and about 90 mV (p.16) [read, per QE]; Fig. 9 reaches about 7 mV [read: page image p.14, DR1]. Dittmer's requirement is 20 V or more [read per reviewers]. A crystal voltage is not the oscillation voltage in the plasma (the coupling is unknown, and the authors say the detector had no high-frequency calibration), so the gap of two to three orders of magnitude is an upper bound on the discrepancy only if the coupling is of order one. It is stated here as the reader's evidence both for and against: the authors saw the issue and held their view; the paper contains no number that reaches the required amplitude. Verdict unchanged.
2. Abstract against body. The abstract (p.1) says "Another observed frequency of 50 to 60 megacycles may correspond to oscillations of the beam electrons", and calls the correlation "tentative until simpler experimental conditions can be attained" [read: p.1, per QE]. The body (pp.15-16) concludes: "it seems to be impossible to identify the 55 with the theoretical 18 megacycle oscillation" [read, per DR2 and QE]. Both are the authors' statements and both stay in the record. The abstract was not updated; readers who summarise from the abstract will repeat a reading the authors withdrew. The 115 MHz beam identification is likewise weakly supported: 10 percent frequency rise found against 26 percent predicted for a 60 percent rise in emission, and the authors write "the other evidence is not clean cut" [read: p.12].
3. The beam-plasma idea and a possible mainstream route. DR2: the p.6 remark that beam waves, Doppler-shifted, may strike the ultimates' natural frequency and "give rise to resonance" matches the resonance condition of the beam-plasma (two-stream) instability (omega_p = k v_b +/- omega_b), but the paper has streaming electrons "withdraw energy" from the other species' oscillation and does not state that the oscillation grows at the beam's expense. DR2's number for the paper's beam (n_b/n_p = 3.8e-4): cold-beam growth rate about 0.05 omega_p, about e-folding in 3 plasma periods; resonant wavelength about 0.6 cm. The p.5 factor 3 (beam acceleration and retardation exciting neighbouring plasma) is velocity modulation, the picture used later in beam-plasma amplifiers. Beam-plasma instability as the mainstream explanation of Langmuir's paradox is [memory, medium] (DR2 itself says a web check is advised). Experimental support in the literature: Looney and Brown 1954 observed beam-excited plasma oscillations [web, title and bibliographic data only]. I do not lean on this: Baalrud et al. 2009 (first page only) propose ion-acoustic instability, a different mechanism, and I keep both as unsettled alternatives. A resonant wavelength near 0.6 cm is far from the 27-81 cm waves detected, so the paper's own detected waves are not the beam-resonance waves; the mechanism would act at shorter wavelength than anything measured. This is my reading of DR2's numbers, not stated by the authors.
4. Energy-transfer factors on p.5 (beam motion, fields outside the tube from unsymmetrical oscillations, fast electrons crossing local oscillations): the three "factors" are qualitative, none is quantified in the paper, and DR2 judges them qualitatively consistent with later theory (Trivelpiece-Gould and surface modes for the fringe field are [memory, medium]). Retained as the authors' proposed energy-transfer routes for Claim 5.

Claim 5 stays Never properly tested.

F. Items from qe.md not applied here

• DR3 (pp.9-12, Figs. 2-7 and detector calibration) and DR4 (abstract and p.16 conditions by image) are not yet in deep/. The 1.38 MHz line conditions (about 90 mV peak, 2 mH coil, Radiola 17 amplifier 550-1450 kc), the 19-21 C window details, the 115 MHz jump to about 900 MHz below 19.5 C, the 115 to 730 MHz change under a magnet, and the heating-current reversal control for the 300+ MHz waves are carried from qe.md's reading of the extracted text and are [read per QE], not independently verified by me. None changes a verdict.
• H. M. Mott-Smith Jr. (theory of infinite plane trains of oscillations; stationary observer sees nu_p as the minimum frequency) and S. Sweetser (experimental help) are credited [read: pp.15-16, per QE and DR2]. Kept.
• Pardue and J. S. Webb 1928 (earlier independent experimental work on ionic oscillations) is added to the related-work record for Claim 3. Not read.
• Tonks 1933 (Nature 132, 101, plasma frequency applied to the ionosphere) and Hernqvist 1955 are supportive leads for Claims 1 and 3 that the primer should carry (qe.md section 6).

G. Settle-it list after QE revision

1. Absolute density plus an ionization scale with stated uncertainty in an Hg arc, then an errors-in-variables fit of wavelength against density: settles 4(ii) between -1/2 and steeper slopes.
2. Repeat the two-state arc at identical anode voltage, with the emission range and the magnet switch documented, and record the frequency of each state: settles 4(iii) and tests whether Fig. 10's outliers are the two states.
3. Compare calibrated amplitude at the plasma with the voltage required to explain scattering (Dittmer's 20 V): settles Claim 5 on the authors' own admission.
4. Read PNAS 1928, the Phys. Rev. 33, 990 Note, and the Fig. 8 arrow direction question in the original journal figure if a better scan exists.

Revised headline

Unchanged: Claims 1-3 are correct and historically important; the experiments are honest but weak; Claim 5 remains open. The QE round adds that the paper's own thermal argument had the opposite sign to Bohm-Gross while correctly locating the size limit, that the 1929 Fig. 10 supports an inverse dependence on ionization but not the 1/2 exponent exactly, and that the authors themselves recorded the small amplitudes that cut against their explanation.