Peek-Frank: Arbiter (Step 3)

Source reviewed: F. W. Peek Jr., Dielectric Phenomena in High Voltage Engineering, 2nd ed., McGraw-Hill, 1920 (325-page scan). Peek worked for General Electric (GE), a maker of transmission equipment; GE's Consulting Engineering Department ran the experiments. That stake applies to every GE-sourced comparison below [read: preface; facts.json].

Tags: [read] from the extracted text or deep reads; [web] with URL; [memory] with confidence; [calc] my own arithmetic (numpy venv).

Source gap note (carried forward)

The text is OCR on 324 of 325 pages. Chapter III (PDF pp.60-92), Chapters VIII-IX (pp.227-251) and the appendix pp.295-311 were read from page images by the deep readers. Chapter V (Lines A and B, the experiments on which the loss law is fitted, PDF pp.154-188), the sphere-gap tables, the oil and solid chapters and the loss-data appendix (pp.277-294) rest on digests; the Physicist and Mathematician each read parts of the Chapter V text layer themselves, and I re-read the single-phase statements (below). Nobody has read the original papers (Peek 1911-1915, 1927 Part IV), Fitch's thesis page images, the 1929 third edition or Hartmann's full paper; the Investigator saved no files in research/files/. Every verdict below rests on the book plus cited web sources, not on [memory] alone, except where a [memory] tag is shown.

Where the three reviewers differ, and what I recomputed

1. Does ionization physics reproduce Peek's curve? Physicist vs Contrarian. The Physicist reports the r = 1 to 0.01 cm rise as a factor 2.9-3.3 against Peek's 3.08, with absolute values "30-50% too high"; the Contrarian reports a constant ratio of 1.46-1.57 (K = 18) or 1.26-1.37 (K = 10). I re-ran a Townsend integral (alpha = A p exp(-B p / E), A = 15 /cm/torr, B = 365 V/cm/torr, no attachment, 1/x field, p = 743 torr) [calc]. Ratio to Peek's law: K = 18 gives 1.57, 1.55, 1.50, 1.45 at r = 0.01, 0.1, 0.5, 1.0 cm; K = 10 gives 1.26, 1.35, 1.35, 1.32. Rise from r = 1 to 0.01 cm: 3.34 against Peek 3.08 [calc]. The reviewers do not actually conflict: they are two statements of the same result (right shape, absolute level 25-55% too high with crude coefficients). The A, B form is a standard air fit [web: search result, not confirmed for these exact constants: https://homepages.cwi.nl/~ebert/AvStrJPD06.pdf and others]; the K = 18-20 criterion is the Raether-Meek value [web: same search, "avalanche gain around 10^8", e^18.4]. Conclusion: shape supported, absolute value not derived by anyone here.
2. Fit constants. Mathematician/Physicist/Contrarian agree and I confirm from Table III [calc]: OLS on all 19 points gives g_o = 30.4, slope 8.71, ratio 0.286, rms misfit of Peek's law 3.49%, max 6.17%. A fit weighted by relative error gives 28.96 and ratio 0.312 [calc]. So the ratio ranges 0.286-0.312 with the fitting recipe, g_o 29-30.4. At r = 0.01 cm: Peek 119.5, OLS 117.5 kV/cm [calc], 1.7% apart. The discard of 9 of 19 points is a real method flaw with a small practical effect; the constants are good to about 5%, so quoting "0.301" to three figures is unjustified.
3. Line A single or three-phase. Physicist says single-phase; Contrarian says unknown; claims.md says unstated. I read the extracted text: "Tests were made on single-phase lines unless otherwise noted" (Table of Symbols page) and, in the loss-data section, "As these measurements were made on a single-phase line, the voltages between wires were twice the value given" [read: _text lines 472 and 13487]. Line A (3/0 7-strand cable, 1.18 cm diameter, spacing 310 cm, 109,500 cm total length) is not marked as an exception [read: Fig. 121 caption]. Physicist is right. Residual caveat: the "unless otherwise noted" sentence is general; I found no Line A-specific statement.
4. Claim 4, air films. Physicist: "break down at the gradient of a free gap"; Contrarian: claims.md overstates. Both quote the same table [read: p.300]: agreement from about 0.01 cm up (35.6 vs 36.8 kV/cm at 0.455 cm), film lower at the thinnest (161 vs 196 at 0.0025 cm; 101 vs 143 at delta 0.74, 0.0051 cm). Contrarian's correction stands; claims.md's 196 kV/cm is the free sphere-gap figure.
5. Paschen cross-check. Mathematician says Table IX (497 V peak at 0.00254 cm) is 6% above the appendix Paschen table (about 470 V); Physicist says Peek's sphere gap sparked at 363 V rms = 513 V peak, "35% above" a 380 V uniform-gap estimate. Recomputed [calc]: pd = 76 cm Hg x 0.00254 cm = 0.193 cm Hg cm = 1.93 torr cm; Townsend-form Paschen with gamma = 0.01 gives 384 V; 363 x sqrt 2 = 513 V; Peek's own table interpolates to about 467 V. So Peek's measurement is about 30% above an idealised Paschen estimate and 6% above his own table. Both reviewers' numbers are right; they compare against different references. The Paschen table in the book is also about 7% high in voltage at its minimum (350 V vs 327 V at 0.567 torr cm) [web: https://en.wikipedia.org/wiki/Paschen%27s_law, via Mathematician's search]. A 25-micrometre gap set by counting screw turns has large uncertainty the book does not give. Verdict unaffected: Paschen behaviour, right order of magnitude.
6. A = 241 vs 244. Contrarian: a genuine misprint; Mathematician: not an error of substance, both within the fit's own 7% scatter (refit exponent -0.498 +/- 0.021, A = 243 at fixed exponent) [Mathematician, calc from Table LV, p.171]. I accept the Mathematician: a 1.2% rounding difference inside 7% scatter; call it an inconsistency without effect.
7. Claim 3 verdict: Physicist and Mathematician "Established", Contrarian "Never properly tested" by a neutral party. Not a disagreement about the data but about the word. The formula was fitted to Peek's own two lines (rms 7.4% in c^2, 16% worst [Mathematician calc]); comparisons with Leadville, Hendricks and Au Sable are plots or one rough pair of numbers (15 kW/mile vs 15-20) chosen by Peek; no neutral controlled test is on record in what we hold. Against that, it has been in engineering use for a century [web: https://en.wikipedia.org/wiki/Peek%27s_law, per Contrarian; https://kpi.ua/nv-abstract-14-6 citation only]. The scale has no "adopted but not neutrally tested"; I split below.
8. Humidity independence. All three agree it is asserted without data in pp.60-92. I checked the web: a streamer-criterion derivation of Peek's law (Hartmann, IEEE Trans. IA-20, 1984) uses an apparent ionization coefficient that varies with humidity, pressure and temperature [web: search summary of the abstract, https://ieeexplore.ieee.org/document/4504655, abstract not fetched by me]; corona onset is reported to rise with humidity (electron attachment to water molecules) and loss near inception is reported to depend on relative humidity, approximately cubically [web: search summaries, https://typeset.io/topics/peek-s-law-2vkw38v0 and https://digital-library.theiet.org/content/journals/10.1049/piee.1971.0131; snippets only, full texts not read]. The direction of the effect is therefore indicated by later literature but its size at inception is unknown to me. Peek states he found no effect; his data are not in the pages read.
9. Fitch 1913 (new, my check). The fetch of the archive.org full text returned: Fitch's modified equation is g = 33.65 (1 + 0.00295 p / sqrt r) kV/cm (p in cm Hg, r in cm), with the conclusion that critical intensity "is independent of the absolute density of the gas, but depends on the number and spacing of the molecules, in accord with the theory of secondary ionization", from CO2 tests (CO2 about 1.5 times denser than air, negligible change) [web: https://archive.org/stream/influenceofdensi00fitcrich/influenceofdensi00fitcrich_djvu.txt; tool-summarised text, not checked on a page image, so the formula's printing is unverified]. Taking the formula as returned and p = 76: 33.65(1 + 0.2242/sqrt r) gives 109 vs Peek 119.5 at r = 0.01 cm (-9%), 57.5 vs 58.2 at r = 0.1 cm, and 41.2 vs 38.8 at r = 1 cm (+6%) [calc]. So the one independent non-GE pressure-range study supports the general form but with different constants and a second term linear in pressure, where Peek's goes as sqrt(delta) [calc, from the two formulas]. This refines the Investigator's "supports Peek": it supports the type of law (1 + c/sqrt r, rising as r falls) and disagrees with Peek's numbers by up to about 9%. The CO2 result is expected from Townsend scaling, where ionization depends on E/p [memory, high]. Both points stay in the record.
10. EHV corona. Contrarian cites web sources that corona loss and radio interference are design limits at EHV, managed by bundled conductors [web: https://eepower.com/technical-articles/understanding-corona-discharge-in-high-voltage-transmission-lines/]; the Physicist had this as [memory, medium]. Retagged [web] via the Contrarian's source, which I did not re-fetch.

Corrected claim list

(Differences from claims.md.)

1. Peek's law of visual corona. Correct: constants are about +/-5% (g_o 29-30.4, coefficient 0.286-0.312 depending on the fit); "9 of 19 points discarded" is an inference from matching values, the book names no discarded points and no rule; the "independent of humidity" and "down to delta 0.02" statements are assertions (printed tables stop at delta 0.069, and the two lowest points are 7.7-7.8% below the formula); "independent of material" rests on one reading per diameter of five metals. Three different constants (29.8-30, 31, 27.2) are used for three geometries.
2. Energy-storage distance. Correct: it is an identity with a free function, a = r(g_v/g_o - 1), and "bears out the theory" is not a test. Peek wrote the density form as "found experimentally" (p.67) and as the "expected form" (p.76).
3. Loss law. Correct: Line A was single-phase (above; direct statements p.20 and p.286, see QE revision); the small-conductor law (33a) carries a third term, c^2 = 20,500 sqrt((r + 6/s + 0.04)/s) x 10^-5 per km, not (r + 0.04); A = 241 is 20,500/85; q and h of the probability curve are defined (Ch. V p.182); the 100 kHz Alexanderson check exists (Fig. 140); A = 241/244 is rounding inside 7% scatter; the Au Sable number is quoted from an N.E.L.A. article, not measured by Peek; Leadville and Hendricks are plotted only.
4. Small gaps and films. Correct: 196 kV/cm is the free sphere-gap value; the film value is 161 kV/cm at 0.0025 cm; films equal free gaps only from about 0.01 cm up.
5. Impulse time lag. Correct: "55-900 kc" labels are single half-sine pulses, labels within about 20% of the undamped LC frequencies (45, 101, 285, 781 kHz) [calc, Physicist and Mathematician]; the rain figure rests on two insulators of one type plus one cask series.
6. D.c. versus a.c. No change; the "1.41" is d.c. over a.c. rms, i.e. about 1.0 against a.c. peak. QE revision: the heating explanation does have measurements behind it for solids (Chapter VII, Rayner 1912), though none that isolate heating as the sole cause.
7. Energy (claims 2 and 5). Correct: the book does give measured energies, but they are circuit energies (0.45 J air, 1.04 J oil, Hayden and Steinmetz 1910), not the energy in Peek's ionization zone; the oil-to-air ratio from those two numbers is 2.3, while the text elsewhere says about 30 (unreconciled).

Verdicts per claim

Claim 1. Peek's law of visual corona

• Agree: all three accept the fit, spacing independence (Table I, 96-100 kV/cm over 42-fold spacing change), and a.c. agreement with Whitehead's independent compilation within about 1.5% [read: p.296]. All three note the weak spots above.
• Disagree: the Contrarian grants d.c. as "Established with contested exceptions"; the Physicist says d.c. polarity matters (2-11% below a.c. peak, Watson; 4-6%, Whitehead).
• Stronger argument: the Physicist and Mathematician, because the discard test [calc] shows that the selection did not manufacture the law; the Contrarian's point that agreement is in-sample for Tables VI-VII is also correct and I keep it.
• Settle it: refit the original 1911 Part I data (free on archive.org [web: via Investigator]) with all points and a stated rule; read Fitch's thesis page images; a humidity series on polished wire at fixed delta.
• Final: Established (a.c., smooth wires, about 1 atm, +/-5%). Splits: humidity independence Never properly tested in the book, with later literature indicating a dependence; material independence Plausible, untested; extension to delta 0.02 Plausible, untested; d.c. Established only to within 4-11% depending on polarity and source.

Claim 2. Energy-storage distance

• Agree: all three say it is a restatement of the fit. The Mathematician's small-excess derivation (alpha_eff linear in E - E_o gives E_s = E_o(1 + c/sqrt r) and, with Townsend scaling, Peek's density form) and my Townsend integral [calc] both show the 1/sqrt r shape follows from a finite ionization zone in a 1/x field.
• Disagree on the scale label for the physical content: Physicist "Historically superseded but instructive"; Mathematician "Plausible, untested" by Peek's route; Contrarian both.
• Stronger: the Physicist, since the algebraic content is trivially true and the physics came from Townsend-Meek-Raether theory. The sphere cross-check (X = 0.54 sqrt R from onset spacing versus 0.54 from the fit) is the one place where the idea does more than relabel the fit [read: p.87-90].
• Settle it: read Peek 1927 Part IV in full for any statement of the rupturing energy; compare a streamer-criterion calculation with documented air coefficients and attachment against g_v(r) over 0.01-1 cm.
• Final: Not testable as stated (the literal "energy" claim and the "bears out" argument); Historically superseded but instructive (as mechanism); the geometric content (finite ionization zone growing as sqrt r) is Plausible, untested by Peek and consistent with later theory.

Claim 3. Corona-loss law

• Agree: arithmetic recomputes (e_o = 51.9-52.0 kV); the sqrt(r/s) exponent is supported (-0.498 +/- 0.021 over 12 tests); the formula is a fit good to about 10%; the 1927 oscillograph paper is Peek confirming Peek.
• Disagree: label (item 7).
• Stronger: the Contrarian on the evidential point (no neutral controlled test in our record), the Physicist and Mathematician on use and range. Both hold.
• Settle it: deep read pp.154-188 and 277-294; compare the formula with an independently measured loss set (for example the Watson and later line data) not used in the fit.
• Final: Plausible, untested by a neutral party for the loss law as a general predictor, with the qualification that it fits its own data to about 10% and has been used for a century (use is not a test). Storm rule (x 0.8) and altitude rules: Never properly tested in the record (rough comparisons only; not dismissed). "Corona will rarely limit line voltage": Historically superseded but instructive (a reasonable 1920 forecast; at EHV loss and radio interference limit design [web: eepower link above]).

Claim 4. Small gaps, thin films, Paschen

• Agree: Paschen behaviour reproduced, small-gap strength far above 30 kV/cm. Peek's table is consistent with textbook Paschen to about 7% in voltage at the minimum.
• Disagree: how far films follow free gaps (item 4).
• Final: Established (strength rises at small gaps; Paschen branch). Thin films equal free gaps: Plausible, untested below 0.01 cm (book's own data show 18-29% lower). "Energy distance cut short" as the explanation: Not testable as stated.

Claim 5. Impulse time lag

• Agree: qualitative time lag established; sphere gaps work as impulse voltmeters within a few percent (modern standard IEC 60052 accuracy about +/-3% at S <= 0.5D [web: https://iteh.es/catalog/standards/sist/c2e5bb77-cee9-434c-8385-19e10adef442/sist-en-60052-2004, per Physicist]); the numbers are single-lab (Peek 1915, GE).
• Contrarian anomaly kept: for thin wires (r = 0.0318 and 0.0573 cm) 100-kc impulse spark-over is below the 60-cycle value (for example 100 vs 135 kV), uncommented by Peek [read: p.304 page image]. Mathematician's circularity warning kept: needles are measured against a sphere gap whose own ratio is up to 1.06, and below 5 kV the book itself says accuracy is poor.
• Settle it: obtain the 1915 Trans. AIEE paper (p.1857) and compare volt-time curves with modern needle-gap data; replicate the thin-wire 100 kc anomaly.
• Final: Established (time lag, impulse ratio above 1 rising for steeper pulses, sphere-gap voltmeter). Quantitative ratios (needle 1.00-2.34, oil 2-3.3, solids 2.2-2.5, up to 5.8 at low density) and the rain figure: Plausible, untested by independent replication in our record (single source, Peek 1915, GE; not contradicted). "Minimum energy" explanation: Historically superseded but instructive.

Claim 6. D.c. versus a.c.

• Agree: air and oil d.c. equals a.c. peak within about 5% in tabulated cases, with exceptions in the book's own tables (thin wires in cylinders 1.2-1.5 times a.c. peak; long needle gaps 0.92-0.93).
• Solid insulation: d.c. 1.27-2.59 times a.c. peak (varnished cambric, Peek 1916, one material) rising with thickness and time; heating explanation has no measurement.
• Final: Established (air and oil, uniform conditions, with exceptions). Solid-insulation ratios: single-source data (Peek 1916); the heating-only cause is Plausible, untested.

Absence-of-evidence analysis

The record does not go dark for Peek himself: papers 1911-1927, lightning generators 1928-1932, Chief Engineer 1931, death 27 July 1933 [web: ethw.org, per Investigator; its death year is inconsistent within one source, 1931 vs 1933]. The anomalies are narrower.

A. The "energy distance" is never specified and apparently not developed. The 1920 book says the form of the rupturing energy "has not been considered as it was unnecessary" [read: p.234]. Candidate explanations: (i) it was superseded by Townsend/Meek ionization theory before Peek returned to it (Peek's 1927 title says "mechanism"; abstract only read); (ii) Peek treated it as a heuristic and moved to engineering work; (iii) the elaboration is in a paper we have not read (1927 Part IV, 3rd edition 1929); (iv) GE-internal reports. Evidence for (i) and (ii): the later literature cites the law, not the energy picture. Against (iii): unknown, because the full 1927 text and 1929 edition were not read. Deciding record: Peek 1927 Part IV and the 1929 edition (on HathiTrust and archive.org [web: per Investigator]). No sign of suppression: nothing in the record shows the corona work being rejected or blocked; the only unresolved matter is acceptance without scrutiny of the data selection.

B. Discarded data points. Nine of 19 points were dropped by an unstated rule; the 1911 Part I paper likely holds them. No competing hypothesis beyond "no record kept in the book"; the decisive record is freely available and unread.

C. Humidity data. The book claims no humidity effect twice but prints no humidity series in the pages read. Explanations: tests were run but unprinted; only an informal check was made; the claim is carried from the original papers. Decisive record: Peek 1911-1913 papers. Later literature reports a dependence near inception, so the claim should not be relied on [web, snippets].

D. Missing numbers in field comparisons. Leadville and Hendricks agreement is plotted only; Au Sable is a quoted "preliminary rough" figure. Peek says needed data "have not been recorded" [read: p.247]. Ordinary incompleteness; decisive records: Faccioli's Leadville report, the N.E.L.A. 1912 article, GE test files (not obtained; GE is an interested party).

E. Unobtained Investigator leads. Union College (84 items) blocked, Chronicling America blocked, NYT not reached, Espacenet, Stanford (the 1931 chair offer), Berkshire Eagle and miSci not reached; the "Wrestling with Nature" Steinmetz passage (1916) is a truncated snippet and unread. They bear on biography and how GE presented the work, not on the physics verdicts.

What would settle it (one item per disagreement)

1. Constants and discards: refit Peek 1911 Part I with all points, or a fresh a.c. onset series on polished wires, 0.01-1 cm radius.
2. Mechanism: a streamer-criterion calculation with documented air coefficients and attachment against Peek's g_v(r), plus Peek 1927 Part IV text.
3. Humidity: an onset series at fixed delta over 20-90% relative humidity.
4. Loss law: compare against an independent loss data set not used in the fit; deep read pp.154-188.
5. Thin-film and thin-wire impulse anomalies: repeat with calibrated gaps (micrometre screw error stated).
6. Fitch's formula: page image of the thesis to confirm 33.65(1 + 0.00295 p / sqrt r) and its range.

Open questions

• Does Fitch's different density scaling (linear in p in the second term) or Peek's sqrt(delta) fit the pressure data better over delta 0.07-1.3? The book's Table VIII is within 2% for delta 0.146-1.0 and 8% low below.
• What was the discard rule in the 1911 paper?
• Did Peek ever specify the rupturing energy (1927, 1929)?
• Is the thin-wire 100 kc impulse anomaly a space-charge effect (Contrarian's untested reading) or measurement error?
• How large is the humidity effect on visual onset on polished wires at 60 Hz?
• Why do Whitehead's d.c. positive-wire values sit 4-6% below the a.c. peak while Peek's own d.c. tables show equal values?

Web checks

Budget 8; used 4 (Fitch full text; Hartmann search; humidity search; Townsend constants and Meek criterion search). Reviewers' web-sourced items retained with their tags. Checks that stayed thin: Hartmann and the humidity literature rest on search summaries; the Townsend constants A = 15, B = 365 were not confirmed to those values (the form is confirmed); the Fitch formula comes from a tool summary of the OCR text.

Verdict summary

QE revision

Source: qe.md (verdict MINOR), and the new deep reads now appended to deep/ (Chapter V pp.154-189; Hayden-Steinmetz pp.196-202; Chapter IV pp.115-153; Chapter III continued pp.93-113; Chapter VII pp.202-227; loss-data appendix pp.277-294). Tags as above; [read: p.N] are PDF pages as given in the deep reads. Web checks this round: 1 (total used 5 of 8). I did not re-read page images myself; where I rely on a deep reader's page-image reading I say so, and the arithmetic I quote I recomputed from the numbers they print.

Verdicts that changed or were re-worded: no headline verdict changes. Five sub-verdicts change (claim 1 humidity wording and density scaling; claim 3 evidence; claim 5 "minimum energy" split; claim 6 heating wording). The QE's MINOR rating stands.

a. Claim 5, "minimum energy" and Hayden-Steinmetz (QE item a)

What the deep read of Figs. 147-151 and Table LXIX shows [read: page images p.199-202, deep reader]: - The passage is a 1910 General Electric investigation (Hayden and Steinmetz, A.I.E.E., June 1910; GE is an interested party, as for everything in this book). I have not seen the paper. A web search returned only a citation-level hit: Steinmetz, "Disruptive Strength with Transient Voltages", Proc. A.I.E.E. vol. XXIX no. 5, pp.1125-1158 [web: https://schenectadyhistorical.org/wp-content/uploads/2019/03/Charles-Steinmetz-Letter-Book-5-December-1910-January-1911.pdf, search summary only, title and pages not read on the page; authorship by Hayden and Steinmetz jointly is Peek's footnote]. Record to obtain. - The plotted parameter is the current admitted to the impulse, not energy. Figs. 148-151 are drawn curves with no data points. At low currents the spark distance is nearly independent of voltage: air spheres at 0.25 A hold 0.10-0.11 cm from 40 to 105 kV; oil spheres at 0.50 A hold 0.038-0.040 cm from 50 to 104 kV. At 2-4 A the curves bend but do not flatten by 100-105 kV (air needles at 4 A: 3.0 cm at 40 kV, 5.3 cm at 100 kV). So Hayden-Steinmetz's wording ("a voltage is reached where ...") is supported only for small currents. - 4 A (oil) is a plotted curve; 1.5 A (air) is an interpolation (about 1.37 A by the deep reader's reading). The conversion to 1.04 J and 0.45 J is not given and does not fit (1/2) L i^2 with one inductance (implied 0.13 H against 0.40 H) [calc, deep reader; I re-did the ratio: (4/1.5)^2 = 7.1 against the printed 1.04/0.45 = 2.31]. Peek himself says the circuit "does not give a circuit for which the constants can be definitely calculated". - The energy is "applied to the circuit", not absorbed in the gap. An assumed 5 pF gap stores about 0.002-0.03 J, 1/20 to 1/40 of the quoted joules [memory, medium: the 5 pF is the deep reader's assumption; not checked]. No timing measurement appears in the figures or Table LXIX. The link to time lag is Peek's inference. - The same curves are what a source-limited discharge would give (the source cannot sustain the spark); mainstream gas-discharge physics describes time lag as statistical plus formative lag [memory, high; not web-checked], with the external circuit deciding whether a discharge grows to an arc. Peek does not discuss this reading. - Internal inconsistency kept in the record: p.148 says the disruptive energy of oil "seems to be about 30 times" air; the two joule figures on p.202 give 2.3; the energy distances give about 4 (1.2/0.301 = 3.99 [calc]). Peek defers the 30 to Chapter VI and no derivation is printed. Unreconciled; I do not know which quantity the 30 refers to. - Table LXIX (Peek's own circuit, a different experiment): at equal 60-cycle strength oil needs 1.5-1.6 times more impulse voltage than air (103.3 against 67.8 kV at 50.2 kV; 321 against 198 kV at 108 kV) [calc from the table], impulse ratio 2.06-3.58 in oil, 1.35-1.83 in air. The impulse voltages are themselves read on sphere gaps, which have their own impulse ratio. The oil 60-cycle values disagree between Table LXIX (37.2 kV for 2.54-cm spheres at 0.25 cm), Fig. 150 and the appendix (70.0 kV), which the deep reader links to Peek's remark that oil values "vary to a considerable extent".

Re-weighed sub-verdicts for claim 5: - Spark-over under a limited supply depends on the admitted current/circuit energy and saturates with voltage at low currents: Plausible, untested (one 1910 GE investigation, drawn curves only, paper not obtained, not contradicted; consistent with circuit physics). - "The dielectric itself needs a definite minimum energy, which produces the time lag": Not testable as stated (gap energy not measured, no time measurement, circuit-limited reading gives the same curves). The Mathematician's and Contrarian's label therefore stays for the literal claim, but their statement that "no energy is ever computed" is wrong as a description of the book: energies in joules are quoted, and the criticism applies to the corona "energy distance" (claim 2), not to Chapter VI. - "Minimum energy" as the mechanism of time lag: Historically superseded but instructive (unchanged). - Qualitative time lag and sphere-gap voltmeter: Established, unchanged. Chapter IV adds a limit to the voltmeter statement: Peek's sphere impulse ratios (6.25-cm sphere against a 25-cm sphere, Table XXXVI) are 1.04-1.10; the pages give no independent absolute calibration (Chubb and Fortescue 1913 only cited); the 0.5% figure in the sphere tables is repeatability and the "under 0.5% for grounded spheres" figure is not stated in the book, where the grounded-sphere law is the case in which the calculated f fails and grounded values run 3-15% below non-grounded [read: deep read pp.115-138]. My "within a few percent" rests on modern IEC 60052 [web, per Physicist], not on Peek's pages. - Sphere-gap density law 27.23 delta (1 + 0.54/sqrt(delta R)): fits the cask data within 1.5% in 18 of 24 points (worst -3.8%); two sample runs sit 4-6% below the law against a stated 2%; the spacing limit appears as 2R, 3R and "diameter" in different places [read: deep read pp.115-138]. It is an empirical fit; my claim-2 remark that the sphere cross-check "does more than relabel the fit" is weakened accordingly (the 0.54 appears in both the sphere law and the wire law, and both are fits). - Rain figures, kept apart: sphere dry/rain spark-over ratio "about 2.5" at 0.2 in. per min (figure captions say 0.25) for 6.25-50 cm spheres, no table; wet 60-cycle spark-over of line insulators 50-90% of dry, impulse "generally not lowered by rain" [read: p.142, p.153]. These are separate from the Table VIII insulator figures discussed under claim 5 above (36-40% at 100 kc); I now state that the rain-on-spheres 2.5 is a bare average with no data shown.

b. Claim 3, loss law (QE item b)

New facts from the Chapter V and appendix deep reads [read: page images pp.154-189, 277-294, deep readers; arithmetic by them unless marked mine]: - How the constants were obtained. p = c^2 (e - e_o)^2 is "after trial", empirical. Sigma-Delta (two-group) reductions throw away points by eye: test 36 uses 6 of 14 points. c^2 = 20,500 sqrt(r/s) x 10^-5 per km comes from a 6 + 6 two-group reduction on 12 of the 21 tests of Table LIII (nine dropped without a stated reason; exponent -0.511 rounded to -0.50 before the constant was computed). Least squares on all 21 gives -0.496 and RMS scatter 10.1% (8.8% on the 12 used); the sqrt(r/s) exponent is therefore robust, in agreement with the Mathematician's -0.498 +/- 0.021. The nine dropped tests act as a holdout and do not change it. - A = 241 is 20,500 / 85 [calc: 241.2], which splices two separate fits: the indoor frequency offset and the outdoor s/r law. The 244 in the other printing remains an unexplained inconsistency, as the Mathematician said. - (f + 25). Indoor Table LI: least-squares frequency-axis intercepts -26.4, -22.7, -28.2 for three conductors at one spacing (61 cm), 30-120 cycles; so -25 fits those data. The outdoor Tables XLVIII-XLIX, fitted the same way, give intercepts from -22 to +13 cycles drifting with voltage, so the outdoor data do not determine the offset. I take this as a qualification on the form of the frequency law, not as a refutation: the outdoor frequency range is 47-115 cycles on only two sets. - Out-of-sample comparisons the book does not make. The formula gives 895 x 10^-5 per km at Line A (s/r = 525); Table LIV's delta-corrected mean is 1026, i.e. measured is 14.6% above the formula (formula 12.8% below) [mine, from the deep reader's numbers]. Only test 18 of those ten is in the fit. This is slightly outside the "about 10%" I gave earlier, for the line Peek studied most. - The density correction (c^2 proportional to 1/delta) is described by Peek as "not satisfactory" and "best until more complete data"; the correlation of c^2 with 1/delta over ten tests is +0.39 and scatter does not shrink after correction [deep reader, calc]. Printed delta values for tests 138, 125, 126, 80 and 79 differ by 0.02-0.04 from delta = 3.92 b/(273 + t) [read: p.153 for the formula] using the printed barometer, possibly a barometer misprint (72.5 for 75.2); a 3-4% effect on those corrected c^2. - Small-conductor law (33a) has the extra 6/s term; as printed it over-predicts Table LI's thinnest wire (r = 0.032 cm) at 60 cycles by 38%; without the term it under-predicts by 10%. Either the fit used other indoor tests or something is misprinted; unresolved. (The disruptive gradient law (36), g_d = g_o delta (1 + 0.30/sqrt(delta r) / (1 + 230 r^2)), reproduces its own indoor data within about 12% at worst, in-sample.) - Hidden geometry: Line A has four conductors in one plane at alternating polarity (1 and 3, 2 and 4) with one spacing s = 310 cm used in two-wire formulas; inner and outer conductors do not see the same field. Total length 109,500 cm = 4 x 27,375 cm (the deep reader's reading; the appendix reader found the convention unstated). - Probability curve (eq. 37) q and h are defined (spot number and size), fitted per test, in-sample, "probably of no practical importance" (Peek). Test 36's own data give h about 0.0075, a third of the h printed for tests 30, 103 and 105. This corrects the Mathematician's and the deep read's "undefined". - Temperature series. The 1/e_o line "cuts the axis at -273 C" is drawn, not fitted; fitted intercepts are -241 to -283 C depending on correction and points, over a 35-degree range. Reducing e_o to 25 C by absolute temperature leaves a spread of 1.4% (SD 2.0 kV on 139.5) against 140-161 kV unreduced [deep reader calc]. This supports e_o proportional to density over relative density about 0.87-1.0 on one line (Line A, GE, single-phase), not beyond. I add it to claim 1 below. - Alexanderson 100 kHz check (Fig. 140), the only out-of-sample test of the whole formula in Chapter V. Two wires, r = 0.127 cm, s = 67 cm, 200 cm long, f = 100,000, "Tests by Alexanderson" (a GE engineer [memory, high]), about five points read from a figure, no accuracy stated. The curve's slope equals the (34a) prediction with f + 25 = 100,025: 29.8 W per kV^2 against about 29 read off the figure [calc, deep reader]. But the drawn curve starts at about 8.3 kV to neutral, while e_d from (35a)/(36) is about 19.9 kV to neutral; the e_d used is not stated. So the check supports the linear frequency scaling over a factor of about 1,000 above the fitting range for one wire, not the onset voltage, and covers about three of five points. Peek: "complete conclusions cannot be drawn from this small amount of data". It is GE data; it is not a neutral test. Weighed in full, it is real out-of-sample support that moves nothing on the scale. - Weather (smoke, wind, frost, fog, snow, sleet): each is one occasion on Line A, figures only; "fair-weather" comparison curves are the model's own calculation, so they compare data to the model, not test it. The wind statement (no effect in practice) has no data on the pages and the book itself says laboratory tests show loss rising with air velocity. Snow gave the largest excess (about 10 kW near 205 kV between lines, from the figure). Sleet loss was as large next day with sleet still on the wires. - Watson d.c. data (Table LII): a.c. corona loss at equal maximum voltage is 1.52-1.70 times the d.c. loss (printed ratios 1.52-1.67; I get 1.54-1.70). Peripheral, kept in the record. - Test 36 is dated "10-6-10" on the data sheet and the weather matches the Table LXII entry; I read it as 6 October 1910 (the deep reader's inference, US date order).

Final for claim 3: Plausible, untested by a neutral party, unchanged. The sqrt(r/s) dependence and the quadratic form are well supported by Peek's own tests (scatter 9-10%); the frequency offset and the density correction are the weakest parts; the Line A comparison is 13-15% off and the 100 kHz check is GE-internal and slope-only. Storm and altitude rules Never properly tested, unchanged.

c. Citation correction (QE item c)

In item 3 above the second source (line 13487) is the note to the indoor-line tables (p.277). The direct statements for the outdoor lines are "Tests were made on single-phase lines unless otherwise noted" (p.20), "These tests were made on comparatively long single-phase lines out of doors" (p.286, line 13872), and, from the Chapter V deep read, every table gives kV between lines as exactly twice kV to neutral and the transformer winding was grounded at the neutral [read: p.156, p.159]. The residual caveat in item 3 is withdrawn: Line A was single-phase.

d. Paschen branch (QE item d)

pd = 760 torr x 0.00254 cm = 1.93 torr cm against an air minimum near 0.57 torr cm [web: https://en.wikipedia.org/wiki/Paschen%27s_law, as in item 5 and the QE]. This is the right-hand branch near the minimum. Where the Physicist's claim-4 wording ("qualitatively the left-hand branch") or the Mathematician's claim-4b ("left branch") is relied on, it is wrong. High V/d at small gaps at atmospheric pressure arises because V falls much more slowly than d on the right branch. Verdict for claim 4 unchanged.

e. Other items from the QE and deep reads, kept in the record

• Claim 1, humidity. My earlier statement that the book prints no humidity series is wrong for Chapter V: Table LXIII gives ten Line A tests (relative humidity 0.67-1.00; "vapor product" 0.06-0.55, units unstated) with the disruptive onset gradient reduced to 25 C and 76 cm of 18.73 kV/cm effective, SD 1.3% (SD 0.24); correlation with vapor product -0.08 [calc, deep reader]. Limits: every test at relative humidity 1.00 is a sub-zero test, so humidity and temperature are confounded; it is the disruptive onset on a stranded cable, not the visual onset on polished wire that the humidity claim concerns; GE data; the effect on loss is asserted, not tabulated. The visual indoor humidity test and the ozone test (wires run an hour in ozone, onset unchanged) print no numbers. Later literature indicating a humidity dependence at inception is unchanged [web snippets, item 8]. I keep Never properly tested (no controlled, neutral humidity series on polished wire), now with this in-book data stated and weighed: it shows no effect on stranded-cable onset at about 1-2% over a narrow, confounded range.
• Claim 1, density scaling: see the temperature series above; Established over about 0.87-1.0 relative density for onset, and Fitch's 1913 CO2 result that onset depends on molecule number and spacing is compatible [item 9]. Wider range stays Plausible, untested.
• Split conductors and the p.246 misprint. Two wires at one potential: onset 31.5 kV touching rising to 35.8 kV at best spacing (about 5% above the equivalent single wire, Fig. 56); three wires in a triangle 20-30% above a single wire of the same material (Peek); 16% for three and 20% for four (Whitehead, concentric-cylinder arrangement, reported by Peek) [read: p.97-98]. "May also be decreased by grouping" on PDF p.246 contradicts these and the sentence's own list; it is a misprint (word broken "de-/creased"). This is early evidence for the bundled conductors cited under EHV corona; I add it to the record for claim 1.
• Corona-driven vibration. Twenty-mil steel wires, 500 ft, about 10 ft apart, vibrating "in a few minutes" to amplitudes of several feet, one wire in the fundamental (period about 1 s), the other in the third harmonic; repeated in the laboratory (Figs. 83-84); Peek reads brightness varying over half a rotation as rotation at 60 Hz, giving a stroboscope of positive and negative corona [read: p.113]. No date, place, voltage or mechanism stated; the 1 s period and the 60 Hz reading are not reconciled in the text. Kept as an observation never followed up in these pages.
• Claim 6 and heating. Chapter VII gives measurements: Rayner (J.I.E.E., 8 Feb 1912) brass against wood terminals, time to break 570 s against 50 s at 9,000 V, converging to 1.16 at 14,000 V; recovery after rest only partial (60 s rest: 9.8 s against 20.5 s fresh); an air gap of 0.75 mm cut life by 42%, so corona is a second agent; sample temperatures reached 145-215 C at puncture in a 100 C oven (method of measuring not stated); loss rises steeply with temperature (Figs. 157-158). All but the temperature logs are Rayner's and cited only (J.I.E.E. not obtained); Peek's own experiments are not individually attributed. So the earlier "heating explanation has no measurement" is withdrawn. Heating is a documented contributor; heating as the sole cause is Plausible, untested since recovery is incomplete and corona acts too. The d.c. ratios themselves remain single-source (Peek 1916).
• Cumulative impulse damage. The "tenth impulse" insulator is a hypothetical ("It may be that ..."), not a reported experiment. The oiled-pressboard table (3.2 mm, 100 kV rapid 60-cycle strength; 2.5 microsecond impulses) reads: 100 kV, infinite; 140 kV, 100 impulses; 150 kV, 16; 155 kV, 2; 165 kV, 1 [read: p.215]. No experimenter, date or number of specimens is given. It is single-source and kept as such; I do not call it uncorroborated, I state that no second source is named in the book or found.
• Other Chapter VII material (thickness law g = g_0 (1 + a/t) with a garbled printed form, weakest-link area illustration, time-law constants, one 12% mismatch with the author's own table) is peripheral and stays in the deep read.
• Misprints found, none changing a verdict: Table XLVI (104.2 for 104.5; c^2 0.0115 against 0.0117), Fig. 117 caption, Table LV ln column, Table LVII "129" for 1297, Table LX average 61.5 for 51.5, Table LXI 11.7 for about 114.5, the p.153 worked example (delta2 0.61 against 0.69; 65 kV against about 74 kV), Table XXXVIII (0.87 for 0.77); Fig. 149 has two curves labelled "1.0".
• Retention (QE item e). The Investigator's continued-use items (Electronics 1950, Westinghouse Engineer 1951, worldradiohistory), the German VDI 1932 report of Peek's 10 MV generator and the 19 GE patents are added to the record as supporting a century of use and Peek's later work [sites.md; not re-fetched by me]. Hayden and Steinmetz 1910, Rayner 1912 (J.I.E.E.), Alexanderson (Proc. I.R.E. 1914), Watson 1910, Mershon 1908 and the Chubb-Fortescue 1913 sphere-gap paper are added to the records to obtain.

Settle-it items added

7. Hayden and Steinmetz 1910: how amperes were converted to joules, whether a time was measured, and what the 30-times figure refers to.
8. A humidity series on polished wire at fixed density over 20-90% relative humidity (item 3 above), now with the confounding in Table LXIII as the reason.
9. The 100 kHz Alexanderson data with stated e_d and accuracy, and a repeat on a second wire.
10. Independent refit of the 21-test outdoor set using the transcribed appendix (tests 86 and 77 of Table LV are not on pp.277-294) and the delta values recomputed from the printed barometers.

Web checks (revision)

Used 1 (Hayden-Steinmetz search, citation level only). Running total 5 of 8. Paschen minimum, IEC 60052 and Hartmann/humidity items keep their earlier tags.