Maxwell: Arbiter (Step 3)

Inputs reconciled: physicist.md, mathematician.md, contrarian.md (plus facts.json, claims.md). Source tags: [read] = document itself; [read: page image p.N] = rendered PDF page (PDF page numbers); [web] = URL; [memory] = recalled, confidence given.

Source gap note (repeated from Step 1, updated)

All four PDFs have a text layer, but the two Maxwell scans are OCR with garbled equations. Step 1 did not check equations on page images. In this step I rendered and read PDF pp.29, 33, 34 and 52 of dynamicaltheoryo00maxw.pdf (the 1865 equations A, E, F, G, H, the twenty-variable list, the energy integral, and the absorption derivation, eq. 95-99). The 1855-56 paper (MaxwellOnFaradaysLines.pdf) was checked only through its prose summary and six laws; its fluid-analogy derivations and electrostatic/magnetic sections were not checked on page images by anyone. Belrose (2014) largely paraphrases Hunt (2012) and adds one antenna modelling case, so the Heaviside history rests essentially on one secondary source (Hunt, Physics Today). Experimental history (Hertz, Michelson-Morley, Aharonov-Bohm) is not in the folder; it comes from [web] and [memory] as tagged.

Disagreements found and how I settled them

1. Sign defects in the 1865 equations (mathematician only; physicist and contrarian did not check page images for this). I checked it myself on page images [read: page image p.33, p.34, p.52, p.29]. - Printed: (F) P = -rho p; (E) P = k f; (G) e + div f = 0; (H) de/dt + div p = 0; (29) P = -dF/dt (so P is the electric field in the modern sense). - (E) makes the displacement f parallel to P (correct). (F) makes the conduction current p antiparallel to P. That is wrong: current and displacement must both follow the field. Evidence inside the paper: eq. (95) printed as d2G/dx2 = +4 pi mu (G_tt/k - G_t/rho). I rederived it from (A), (E), (F): g = -G_t/k, q = +G_t/rho, so -4 pi mu (dg/dt + q) = 4 pi mu (G_tt/k - G_t/rho), matching the print. Physical conduction gives A_xx = mu eps A_tt + mu sigma A_t (damping term with the opposite sign). Substituting G = e^{-px} cos(qx+nt) gives p = 2 pi mu n/(rho q), his (97); with n/q > 0 the wave moves toward -x and e^{-px} grows along the motion, i.e. it amplifies. His text says the undulation "is thereby weakened" [read: p.52]. The magnitude |p| and so his gold-leaf estimate (99) are unaffected. Verdict: the printed (F) sign is wrong. Confidence high that the sign is as printed and inconsistent with his own prose; the mathematician's own caution that (A)(F)(G)(H) cannot all hold is right. - (G) is a second, separate sign problem. Taking e as positive free charge, (H) is correct (conservation) and (G) should read e = +div f. [web: https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field] says (G) as printed needs e to mean minus the charge density while (H) needs plus, and cites Arthur's conclusion that the (G) sign was wrong and corrected in the Treatise. Check with both corrected: div p' = div p + d(div f)/dt = -de/dt + de/dt = 0, as (C) requires. With the print, div p' = -2 de/dt. - Whether (F) is a print slip or Maxwell's convention I cannot tell; the Wikipedia fetch gave only the modern Ohm's law, not a transcription of his (F), so the claim that "later transcriptions silently repaired (F)" stays unverified. The mathematician's phrase "(A),(F),(G),(H) as printed" is right; the minimal repair is two sign flips (F and G), not one. - Importance: a real internal inconsistency, non-fatal, and not found by two of three reviewers. It does not touch the displacement-current idea, the wave speed (which uses no conduction term), or the energy integral. It does show the printed 1865 set is not literally "twenty consistent equations".

2. "Twenty equations = Heaviside's four in disguise?" Physicist: equivalent bookkeeping. Contrarian: not a disguise, since the twenty include constitutive and motional terms. Mathematician: redundancy and gauge freedom, and no well-posedness argument. Settled: all are partly right. The field content reduces to Heaviside's four vector equations plus constitutive relations; (H) follows from the others (once signs are repaired); potentials carry gauge freedom; and "20 equations in 20 unknowns" is bookkeeping, not an existence/uniqueness theorem. I counted again from the page image: 6 variable groups of 3 plus e and Psi = 20; equations B to F and A, 3 each = 18, plus G, H = 20 [read: page image p.34].

3. Claim 4 verdict wording. All three say Established. Reviewers agree the 1865 evidence alone was only suggestive. Recomputed against c = 299,792,458 m/s (defined value [web: https://en.wikipedia.org/wiki/Speed_of_light], [web: https://physics.nist.gov/cgi-bin/cuu/Value?c], cited by two reviewers): - Weber-Kohlrausch v = 310,740,000: 310.740/299.792 = 1.0365, +3.65 %. - Fizeau 314,858,000: +5.03 %. - Aberration 308,000,000: +2.74 %. - Foucault 298,000,000: -0.60 %. - Fizeau vs Foucault differ by 5.7 % (314.858/298), as large as the gap between the electrical number and any light number. Check: 1/sqrt(mu0 eps0) = 1/sqrt(1.2566e-6 x 8.854e-12) = 2.998e8 m/s (my arithmetic). All three reviewers' numbers agree with mine. Also agreed: Maxwell's own wording is hedged ("agrees sufficiently well") [read], and his strongest argument is that the two kinds of measurement are independent. The physicist and mathematician add that V = v holds only for mu = 1 (air); the plane wave he derives is stated entirely in magnetic terms (the modern symmetric E and B wave picture is later) [read: page image p.47 per mathematician, text layer per physicist and contrarian]. The n^2 = (dielectric)(magnetic) relation is a hedged low-frequency result (water: optical n^2 about 1.77 vs static relative permittivity about 80) [memory: high; the mathematician's example].

4. Claim 5 (ether). Physicist and mathematician split strong vs superseded; contrarian gave only "Historically superseded". The mathematician's extra point is useful: eq. (D) has a v x B term for conductor velocity with no stated reference body, and the equations are not Galilean invariant. Maxwell's para 73-75 (the conclusions are "independent of this hypothesis") means the ether is an interpretive layer, not a premise of the equations [read: text layer]. I adopt the split below. Michelson-Morley null result 1887 now web-checked [web: https://www.aps.org/apsnews/2007/11/november-1887-michelson-uminiferous-ether].

5. Claim 7 (Heaviside). Reviewers differ in verdict (history; "Plausible, untested"; "Established as history for Preece"). The scale is for physics claims, so I split into three separate episodes, per the contrarian's correction, and give each the closest label with its evidence status. See claim 7.

6. Belrose NEC-4 as a "test". All three agree it is a computer model compared with one antenna's measured data (per Belrose, citing Smith), not a test of the interpretation that the action lies in the field [read: Belrose]. Agreed; I keep that.

7. Hunt typo. Mathematician: Hunt's sidebar prints "grad(-dA/dt)" where curl is meant; result curl E = -mu dH/dt is unaffected [read: Hunt]. Minor.

Claim by claim

Claim 1: Faraday's lines of force given exact mathematical form via an incompressible-fluid analogy

• Agree: all three. Maths consistent (divergence-free flow plus a potential, same structure as steady currents and heat flow); Maxwell says it is "only a kind of artificial notation" [read: MaxwellOnFaradaysLines.pdf]; nothing in it conflicts with experiment (Coulomb, Ampere, Faraday) [memory: high]; nothing in mainstream needs changing.
• Not checked by anyone on page images (see Source gap).
• Contrarian's tested/never-tested/dismissed: not applicable. It is a representation, not a prediction.
• Final verdict: Established (as a mathematical representation). The fluid itself: Not testable as stated, and Maxwell disclaims it.

Claim 2: induction is governed by a point-wise vector (electro-tonic state = vector potential A)

• Agree: Faraday's law and E = -dA/dt - grad(phi), B = curl A are standard. The 1865 paper gives P = -dF/dt (29) [read: page image p.29], and calls electromagnetic momentum "the same thing to which Professor Faraday has applied the name of the Electrotonic State" [read: page image p.29].
• Mathematician's added caveat: A is fixed only up to a gradient; Maxwell sees this for closed circuits but does not flag it for F, G, H. Hence A as a directly measurable local quantity is not testable classically.
• Aharonov-Bohm: predicted 1959 (earlier Ehrenberg and Siday 1949); Chambers 1960 was the first claimed observation but later authors questioned the shielding; Tonomura 1986 was the unambiguous one [web: https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect]. This upgrades the reviewers' [memory] to [web]. Note the effect shows the loop integral of A is physical, not that A is an energy source.
• Contrarian distinction: Heaviside and Hertz eliminated potentials as unphysical; that dismissal was partly overturned. This is a case of an idea set aside and later vindicated in a precise, limited sense; it was tested (A-B experiments), not dismissed untested.
• Final verdict: Established. Strong form (A as local physical entity): Not testable as stated classically; the loop-integral form is Established by Aharonov-Bohm experiments. Maxwell's reading of A as a state of a medium: Historically superseded but instructive.

Claim 3: field energy lives in space; displacement current closes the equations (20 in 20)

• Agree: field energy and displacement current are Established (Hertz 1887-88 and all radio). The 1865 paper had no direct test of displacement current; its only evidence was the wave-speed argument (contrarian), so the claim was suggestive in 1865 and decisive only later.
• Energy integral re-checked on the page image: E = (1/2) sum (F p' + G q' + H r') dV, then, substituting (C) and integrating by parts with alpha, beta, gamma vanishing at infinity, E = (1/8 pi) sum (alpha mu alpha + ...) dV [read: page image p.34]. Correct, with Gaussian-type units (the 4 pi sits in (C)). Needs the surface term to vanish; Maxwell states the decay assumption [read]. Worked check by me: this is (1/8 pi) A.curl H integrated by parts into (1/8 pi) H.curl A = (1/8 pi) H.B per unit volume, as the mathematician found.
• Disagreement: where the energy is. Physicist, mathematician and contrarian all note the integral fixes only the total; a divergence-free vector can be added to S without changing any measurement. Maxwell himself drew the contrast: old theories put energy in electrified bodies, his in the field [read: text layer, para 74].
• The 1865 printed signs: Internally inconsistent (see disagreement 1).
• Final verdict, split: (a) field energy and displacement current: Established. (b) The printed 1865 equation set: Internally inconsistent as printed (sign errors in (F) and (G)); later repaired; the idea survives. (c) "Twenty equations in twenty unknowns" as a well-posed system: Plausible, untested in the mathematical sense (no existence/uniqueness argument; redundancy and gauge freedom). (d) Energy "located" in the field beyond the total: Not testable as stated.

Claim 4: waves at v match light; light is electromagnetic

• Agree: Established. Hertz 1887-88 produced and detected waves with reflection, refraction, polarisation, and light-like speed [web: https://ethw.org/Milestones:First_Generation_and_Experimental_Proof_of_Electromagnetic_Waves,_1886-1888]; FitzGerald called it the confirmation [read: Hunt]. Replication of Hertz by others: Hunt does NOT say Lodge replicated Hertz (see QE revision); Lodge's wire-wave work was parallel and overshadowed by Hertz [read: Hunt]; Lecher [memory: medium]. Not re-checked by me; no verdict hinges on the named replicators.
• Evidence ranking: in 1865, plausible-and-suggestive (about 5 % scatter, independent measurements, hedged wording); after Hertz, established. Contrarian's tested/never-tested/dismissed: tested, replicated, not dismissed.
• Weber-Kohlrausch numerical origin: contrarian computes 439,450,000 / sqrt(2) = 310,740,000 (equal to 310.74 x 10^6 to 5 figures; 439.45/1.41421 = 310.74). The 439,450 figure for Weber's constant is [memory: medium], the sqrt(2) relation was said to be supported by the speed-of-light Wikipedia page; I did not independently verify it and nothing hinges on it.
• Final verdict: Established. 1865 evidence alone: Plausible, suggestive; decisive confirmation by Hertz. Caveats: V = v holds for mu = 1; the n^2 = (dielectric)(magnetic) relation is low-frequency and non-dispersive only.

Claim 5: an all-pervading ethereal medium

• Agree: the equations do not need the medium (para 73-75). The conclusions are independent of the hypothesis [read]. Maxwell's introduction still asserts an ether [read].
• Tested vs never tested vs dismissed:
• Tested: a medium with a definite rest frame that moves relative to Earth: Michelson-Morley 1887 null [web: https://www.aps.org/apsnews/2007/11/november-1887-michelson-uminiferous-ether]; later Kennedy-Thorndike, Ives-Stilwell and resonator tests [memory: medium/high; not web-checked here; nothing depends on them].
• Not excluded: Lorentz-type ethers empirically equivalent to special relativity (physicist, contrarian) are not refuted by experiment, only unnecessary: Not testable as stated (no measurable difference).
• Contrarian's remark on positive ether-drift reports (Miller): no source in folder; I make no claim and do not rate it.
• Vacuum properties (permittivity, permeability, Casimir effect, QFT vacuum) are not a mechanical ether [memory: high, contrarian].
• Final verdict, split: strong form (mechanical medium with a detectable rest frame): Tested, not supported. Undetectable Lorentz-type ether: Not testable as stated. Maxwell's 1865 position: Historically superseded but instructive. The equations themselves: Established (claims 3, 4).

Claim 6: Heaviside's four vector equations, S = E x H, "real action is in the field"

• Agree: the equations are a correct reformulation of the field part; Poynting's theorem follows by div(E x H) = H.curl E - E.curl H; Poynting's abstract (Jan 1884) preceded Heaviside [read: Hunt]. My worked check: plane wave E = 100 V/m, H = 100/376.7 = 0.265 A/m, S = 26.5 W/m^2 (physicist; I reproduce it).
• Mathematician's useful link: Heaviside's set has matching signs in curl H = kE + eps dE/dt, i.e. it is effectively the repaired version of the sign problem in claim 3.
• Disagreement: none on verdict; contrarian flags Belrose as over-reading a NEC-4 modelling exercise. I agree.
• Contrarian's distinction: priority question is documented history, with Gibbs also reaching the vector form independently [read: Hunt]; Hertz's contribution [memory: medium]; not needed.
• Final verdict: Established (four-equation vector form and S = E x H as an integral conservation law). "Real action is in the field, not the wire" as a statement of local energy location: Not testable as stated. Priority (Poynting first, Heaviside independently): documented history [read: Hunt], not a physics claim.

Claim 7: Heaviside blocked by officials and journals, later vindicated

Three separate episodes, all [read: Hunt] with Belrose as secondary echo: - Preece and the 1887 loading paper. Preece (Post Office chief engineer) declared the paper worthless and blocked it via Arthur Heaviside, a Post Office employee; Oliver was not an employee. Inductive loading was later used successfully by AT&T after Pupin's 1899 patent. Physics verdict: the inductive-loading result is Established in engineering. History: well-documented but from one secondary source. Which test applies: the physics was tested in practice and worked; it was not dismissed untested at the end, but the obstruction preceded testing. - The Electrician. Editor Biggs was "abruptly removed" and the successor cancelled the series; Biggs "later hinted" the reason was support for Heaviside. Cause: Never established (hint only). - Royal Society 1893. Third operator installment rejected because pure mathematicians objected to divergent-series handling; the first two were published. This is a rigour dispute with a fair basis; the method was placed on rigorous footing later (Bromwich 1916, Carson, Wagner; Laplace transforms) [read: Belrose; date [memory: high]]. Not suppression of results, and the method was not "dismissed without testing" (it gave correct answers). - Heaviside was elected FRS in 1891 and received a pension in 1896 [read: Hunt, per contrarian]. - Final verdict: no scale label fits history exactly. By episode: Preece/loading obstruction: documented history, and the loading physics Established. Electrician cancellation: cause never established. Operator-calculus rejection: a partly fair rigour objection, later resolved in his favour on results. Overall "blocked then vindicated" must not be flattened to a general claim that "journals blocked him". Closest scale label for the physics content of the claim: Established (loading); Historically superseded but instructive (operator calculus rigour question).

Corrected claim list

1. Maxwell (1855-56) gives Faraday's lines of force an exact mathematical form through an incompressible-fluid analogy, explicitly as notation, not as a physical hypothesis.
2. In 1856 Maxwell identifies Faraday's "electro-tonic state" with a vector function A with B = curl A and EMF = -(1/4 pi) dA/dt (modern E = -dA/dt in his units), with explicit gauge freedom; he credits the mathematics to Thomson (1847, 1851) and Stokes (1849), so "origin of the vector potential" overstates it. A is meaningful only up to gauge, and its physical content is the loop integral (Aharonov-Bohm). [QE revised]
3. Maxwell (1865) states a field theory with displacement current, with energy in the field, as twenty equations in twenty variables. As printed the signs of (F) and (G) are inconsistent with his own prose, and the twenty are redundant and gauge-ambiguous. Its coil inductance calculation (430,165 m) sits between a fitted value (+6.2 %) and a rough measurement (-4.7 %), so laboratory scatter of about 5 % was normal in 1865.
4. The plane-wave solution of that theory (stated in terms of magnetic disturbance transverse to the direction of travel) has speed V = v, the ratio of electromagnetic to electrostatic units, for non-magnetic media; v and the measured speeds of light agree within about 5 % in 1865 (+3.65 %, +5.03 %, +2.74 %, -0.60 % from the modern value); Hertz 1887-88 made it a confirmed fact; Maxwell credits Faraday's 1846 "Thoughts on Ray Vibrations" with the same theory "in substance", lacking only the speed. Maxwell's own sunlight estimate (peak E about 600 V/m from P = sqrt(8 pi mu V W)) is 1.6 times below what his formula and stated input give (975 V/m; modern peak 958 V/m), unexplained, and his Sun-surface values are about ten times low; he also concedes that gold-leaf transmission of green light does not fit the theory without an extra assumption. [QE revised, Round 2]
5. Maxwell believed in an ether but explicitly made his conclusions independent of any mechanical model of it; a detectable-motion ether was later excluded by experiment. His stated evidence for a medium was indirect: optics, Thomson's energy/density argument, Faraday rotation with Verdet's reversal in paramagnetics. [QE revised]
6. Heaviside (1884-85) cast the field equations into four vector equations and obtained S = E x H, shortly after Poynting; the local location of energy flow is an interpretation.
7. Heaviside faced three distinct kinds of resistance (Preece's obstruction, an editorial cancellation of unproven cause, and a rigour objection at the Royal Society); the loading result was vindicated by practice, the operator calculus by later rigour.

What would settle it

• Claim 3 sign defects: already settled by calculation (done above). A further check: compare the page images of the Treatise (1873) equations with the 1865 set to date the correction; not done.
• Claim 3 energy location: no experiment distinguishes local energy density definitions that differ by a divergence-free term; only a theory with an observable that couples locally (for example gravity, in general relativity, which the reader is not assumed to know) could. Left open.
• Claim 4 (1865 strength): the independent verdict was settled by Hertz; a modern recalculation of the experimental errors in Weber-Kohlrausch and Fizeau would show whether the 1865 agreement was luck or earned. Not done here.
• Claim 5 residual: further Lorentz-violation null tests with modern resonators and clocks [memory: medium]; no source fetched.
• Claim 6 interpretation: a test showing wire heating through the sides vs through the wire ends. Both descriptions give the same integrated power, so nothing measurable separates them in circuits.
• Claim 7 causes: archival record of Electrician editorial changes in 1887 and Post Office correspondence. Hunt's account is single-source.

Open questions

• Was the (F) sign a typesetting error or a convention in Maxwell's mind (his statement that P is "difference of potential per unit of length" [read: page image p.29] could carry a sign convention)? Not settled.
• How much was 1865's "agreement sufficiently well" a stated confidence versus a conclusion after the fact (given Weber's constants were already known to be near light speed)?
• Physical reality of A beyond loop integrals; the local energy flow question.
• Heaviside's motivations and the Electrician episode: no independent source.

Web checks done (8 budgeted; 3 used by me, plus reviewers' 6)

1. https://en.wikipedia.org/wiki/A_Dynamical_Theory_of_the_Electromagnetic_Field: (G) printed with the wrong sign for e; (H) consistent with positive charge; sign fixed in the Treatise (Arthur). Confirms the mathematician's concern for (G); (F) was my own finding from his eq. (95) and page images.
2. https://www.aps.org/apsnews/2007/11/november-1887-michelson-uminiferous-ether: Michelson-Morley null result, November 1887.
3. https://en.wikipedia.org/wiki/Aharonov%E2%80%93Bohm_effect: 1959 prediction; Chambers 1960; Tonomura 1986 as unambiguous. Retained from reviewers: ETHW Hertz page (physicist); NIST/Wikipedia value of c. Still [memory]: Kennedy-Thorndike and Ives-Stilwell (high), Bromwich 1916 (high), Lecher (medium), Hertz's contribution to the vector form (medium), the 439,450,000 figure (medium). No verdict depends on them.

QE revision (after qe.md, verdict MINOR)

Sources: deep reads in deep/ (Faraday's Lines Part I pp.4-34 and Part II pp.35-55; Dynamical Theory pp.7-24; Hunt pp.5-7; Belrose pp.5-8) plus my own check of the text layer of Dynamical Theory PDF pp.53 and 60 (deep read 1 of the QE list was not delivered, so I did that part myself; equations there not re-imaged). No new web checks were made; items marked [memory] stay [memory].

Result: no verdict changes

Every final verdict above stands. Three attributions and tags change, and several pieces of evidence are added.

Corrections to tags and wording

1. Lodge did not replicate Hertz (claim 4 and the evidence tables). The deep read of Hunt (section 6) finds that Lodge appears only on pp.6-7. Hunt says Lodge's wire-wave experiments (Leyden-jar discharges along long wires, March 1888) were "overshadowed" by Hertz's waves in air at the 1888 Bath meeting. The word "replicate" is not in Hunt. So the earlier "[read: Hunt]" tag for replication was wrong. Correct form: Lodge ran parallel, independent wire-wave experiments [read: Hunt p.6]; whether he or anyone else repeated Hertz's air experiments is [memory: medium] and was not web-checked. This does not affect the claim 4 verdict, which rests on Hertz himself (web-checked ETHW) and on FitzGerald's reaction [read: Hunt].
2. Claim 2 attribution (the one wording change to the claim list). Part II of the 1856 paper (read on page images pp.36-54) does contain B = curl A (Law I), interaction energy proportional to the integral of A.J (Law V), and EMF = -(1/4 pi) dA/dt (Law VI), with the minus sign [read: page images p.47-52]. The 1/4 pi is Maxwell's unit convention (current measured by circulation of H with no 4 pi); my unit reconstruction gives modern E = -dA/dt, and Maxwell says he has "not discussed the values of the units" [read: p.52]. So Faraday's electro-tonic state is identified with the vector potential in 1856, not "later". But Maxwell credits the mathematics to Thomson (1847, 1851) and Stokes (1849) [read: p.55], cites Neumann [read: p.54], and calls the scheme "only a kind of artificial notation" [read: p.51]. The words "vector potential" do not occur in pp.35-55. The verdict (Established; strong form not testable classically; Maxwell's medium reading historically superseded) is unchanged. - Moving conductors: Law VI uses a convective derivative. It differs from the modern -dA/dt + v x B by a gradient term, which integrates to zero round a closed circuit, the only case Maxwell claims. Correct for circuits [deep read, SymPy-checked]. - Small unresolved sign wrinkle on pp.48-49 (force on a south pole, +dp/dx vs -m dp/dx); not load-bearing.
3. Claim 1 is now checked on page images. The Part I deep read confirms the refraction law tan(theta1)/tan(theta2) = k'/k, the source pressure p = k/(4 pi r) and uniqueness/reciprocity results [read: page images p.13, 14, 17, 22, 23, 28]. One correction to the dictionary I and the reviewers used: in the electric dictionary the force is X = -dp/dx = k u [read: page image p.23], so the fluid velocity u is the flux density (later D or B), and k plays the role of 1/permittivity (1/permeability). "Velocity = force" holds only for k = 1. Maxwell states electric/magnetic identity piecewise ("mathematically identical" for magnetism, p.24; "physically identical" for tension and potential, p.27), not as one three-way statement. This closes the mathematician's open note 5. Verdict unchanged. The paper has no new experiments and no numerical data; all experimental references are Faraday's, plus Ohm, Ampere, Weber and others.
4. Reviewer filename slip. The physicist's "[read: far.pdf ...]" means MaxwellOnFaradaysLines.pdf. Reviewer files are not revised.

New evidence, by claim

Claim 3 (and 4): size of experimental error in 1865. The Dynamical Theory compares calculated and measured self-inductance of the British Association coil: calculated L = 430,165 m, least-squares fit to the spinning-coil experiments 456,748 m (+6.2 %), rough electric-balance value 410,000 m (-4.7 %). Maxwell judges the calculation "probably much more accurate than either of the other determinations" [read: text layer p.60, my percentages]. This is the 1865 yardstick for scatter: about 5 % between theory and measurement in a well-run laboratory. That makes the 5 % agreement of v with the light-speed measurements (Weber-Kohlrausch 310,740,000 m/s, Fizeau 314,858,000, Foucault 298,000,000) suggestive rather than decisive, as the arbiter already judged. Not verdict-changing.

Claim 4: Maxwell's own sunlight estimate (para 108, PDF p.53). [read: text layer p.53; digits partly OCR-garbled, not checked on the page image]. With energy flux W through unit area and V the speed, Maxwell derives P = sqrt(8 pi V W) for the peak electromotive force per unit length, starting from Pouillet's value 83.4 ft.lb per second per ft^2 (as calculated by Thomson). He gets P = 60,000,000 in his units, "about 600 Daniell's cells per metre" at the Earth, about 13,000 per metre at the Sun's surface, a peak magnetic force of 0.193 (Kew horizontal field is 1.76 in the same units), and reversals "more than a thousand million million times in a second". - My check [my calculation; unit conversion from metre-gram-second electromagnetic units is my reading; Daniell cell about 1.1 V is [memory: high]]: 83.4 ft.lb/s/ft^2 = 1217 W/m^2. 600 cells x 1.1 V = about 660 V/m, which matches 60,000,000 units at 1 unit = 1e-5 V. Plugging 1217 W/m^2 into Maxwell's own formula gives about 975 V/m, and the modern E0 = sqrt(2S/(c eps0)) = 958 V/m. So Maxwell's printed 600 V/m is about 0.62 of what his own formula gives. The QE guess of a factor 1/sqrt(2) (RMS versus peak) does not fit (0.62, not 0.71). I cannot explain the gap from the text; it could be a unit-conversion slip for the foot-pound or an OCR misreading of the exponent. Not verdict-bearing (a worked exercise for the primer, with the discrepancy stated honestly as unexplained).

Claim 4: priority. Maxwell writes that "the propagation of transverse magnetic disturbances to the exclusion of normal ones is distinctly set forth by Professor Faraday in his 'Thoughts on Ray Vibrations'" (1846), and that Faraday's electromagnetic theory of light is "the same in substance" as his own, "except that in 1846 there were no data to calculate the velocity" [read: p.14]. Note: "magnetic", not "electric", disturbances. The quantitative theory and the speed link are Maxwell's.

Claim 5: what Maxwell actually offered for the ether [read: pp.8-12]. A chain of four indirect arguments: (a) light and heat cross vacua and transparent matter, so there is a medium that matter only modifies; (b) Thomson's 1854 energy argument that the medium has a density comparable to matter (no number given); (c) Faraday rotation, Verdet's opposite rotation in paramagnetic solutions, and Thomson's inference of a magnetization-linked motion in the medium, from which Maxwell has "some reason to suppose" rotation about the field direction; (d) dielectric polarization and absorption read as elastic yielding. He calls it a "datum" from optics, not a detected object. This supports "Historically superseded but instructive" and the observation that the equations do not need the medium. Verdict unchanged.

Claim 3 and the primer: the flywheel analogy for L, M, N (pp.14-20). Maxwell's mechanical picture: a flywheel geared to two cranks gives "reduced inertia" L and N and coupling M, momentum_A = L u + M v. The dictionary maps velocity to current, force to EMF, momentum to electromagnetic momentum (his electro-tonic state), and kinetic energy (1/2)L x^2 + M x y + (1/2)N y^2 to magnetic energy [read: pp.15-20, page images]. Single-circuit result: current x = b + (a-b)exp(-Rt/L), time constant L/R, equivalent to half the final current for time 2L/R [read: eqs (13)-(15), p.20]. Maxwell says it is an illustration, not a proof, and that the "mass" is in the field around the circuit, not in the wire. Equations on pp.21-23 were not image-checked in the deep read.

Weber and action at a distance (claims 3 and 5 context). Maxwell rejects Weber/Neumann on one stated ground: forces between particles that depend on velocity need "mechanical difficulties" the field view avoids; he calls the theory "exceedingly ingenious, and wonderfully comprehensive" [read: p.7-8]. In 1856 he calls Weber's a "really physical theory" and writes "it is a good thing to have two ways of looking at a subject"; his objection there is that conservation of energy needs forces along the line between particles and dependent on distance only [read: page image p.54]. For the alternative-science theme: Weber electrodynamics was a serious contemporary rival, judged on mechanical grounds, and Weber's constant is the very number that gave Maxwell the speed link.

Claim 6 and the longitudinal-wave theme. Belrose's only sentence is "Heaviside critized Helmholtz, because Helmholtz'e electromagnetic theory allowed for the existence of longitudinal rather than solely transverse wave propagation" [read: Belrose p.7 addendum]. It carries no date, citation or source; treat as a pointer, not evidence (the Helmholtz free-parameter detail is [memory: medium], unchecked). Maxwell himself restricts the 1865 theory to transverse disturbances (para 20, p.14).

Claim 6: Belrose's NEC-4 case. The comparison with measurements exists as one sentence: NEC-4D "did in indeed model this antenna system quite effectively" against data reported by C.E. Smith for station KVOK (IEEE Trans. Broadcasting 1989). No numbers or error margins are given; the detail is in Belrose's 1999 ACES and 2001 IEE papers, not seen [read: Belrose p.5-6]. The inference "as Heaviside concluded 130 years ago, the real action ... is due to fields that surround it" is rhetorical: a method-of-moments code solves for currents on wires and supports Maxwell's equations in general, not Heaviside's remark in particular. This agrees with the contrarian's flag. Belrose also quotes Arthur and Forfar that Hertz's 1890 paper was a "clarification" mathematically equivalent to Heaviside's version but written in components [read: p.7; second-hand, [memory: medium] on the Hertz titles]. Belrose's "fully compatible with special relativity" remark is unsourced and untouched here.

Claim 7: Hunt's text, verified passage by passage. - The 1887 loading paper was a joint paper by Arthur and Oliver Heaviside (April 1887). The block was Preece, as Arthur's superior at the Post Office, refusing clearance and calling it worthless; Oliver was not an employee [read: Hunt p.6]. This is not a journal rejection. - Electrician: Biggs was removed in October 1887, "a move he later hinted" was due to his Heaviside support; the new editor cancelled the series saying he found no one read them. Heaviside recovered with Thomson's help: Philosophical Magazine articles on waves, 1888 [read: Hunt p.6]. The Electrician series resumed in January 1891 [read: p.7]. Cause of the cancellation stays "never established". - Royal Society 1893: the first two installments of "On operators in physical mathematics" were published, the third was refereed and rejected because of the handling of divergent series. Heaviside was already an FRS (1891) [read: Hunt p.7]. Hunt says the methods "probably influenced" Dirac, who learned them as an engineering student [read: p.7]. - Pension: Royal Society Relief Fund grant refused in 1894; a government pension of 120 pounds a year (later 220) accepted from 1896 [read: Hunt p.7]. Pupin's loading patent (1899) was sold to AT&T for $500,000 [read: p.5]. - Poynting (Jan 1884 abstract) preceded Heaviside; FitzGerald called Hertz's work the confirmation of Maxwell in an address at the 1888 Bath meeting (Hunt says "presidential address"; not web-checked) [read: p.6]. - Hunt is a popular-history source in Physics Today; the single-source caveat on claim 7 stands.

Round 2 (after the second deep read of Dynamical Theory PDF pp.53-60 on page images)

Source: deep/dynamicaltheoryo00maxw-471f79.md, section "QE round 2 deep read" (printed pp.505, 509, 511, 512 rendered at 150 dpi) [read: page image pp.53, 57, 59, 60]. No new web checks; unit conversions stay [memory: medium], as tagged.

Result: no verdict changes. Every verdict and the verdict table below stand. The deep read settles the OCR doubt left open in Round 1 and adds four findings, all bearing on the worked examples and the primer, none on a verdict.

1. Sunlight formula and numbers confirmed on the page image (claim 4 context). The printed formula is P = sqrt(8 pi mu V W), with mu (magnetic coefficient, 1 in air or ether). Round 1 and the QE task text had dropped the mu (OCR loss); with mu = 1 the forms agree. The printed result is "P = 60,000,000, or about 600 Daniell's cells per metre", so the digits flagged as OCR-garbled in Round 1 are now read correctly. Printed magnetic force 0.193 (Earth), 4.13 (Sun); Kew horizontal field 1.76 in the same units [read: page image p.53].
2. P is a peak value and W a time average. Maxwell's W = P^2/(8 pi mu V) equals the modern S = E0^2/(2 mu0 c) for peak E0. So a sqrt(2) RMS/peak correction is not the explanation of his gap (this confirms my Round 1 finding that 1/sqrt(2) does not fit). [read + calc]
3. The 1.6 shortfall is real and unexplained. With his V = 3.1074e8 m/s and 1217 W/m^2 (83.4 ft.lb/s/ft^2), his formula gives 9.75e7 units = 975 V/m; he prints 6.0e7 = 600 V/m. Ratio 1.62, i.e. 0.62 in the other direction (the same number as Round 1's "0.62"). The magnetic value 0.193 = 6e7/3.1074e8 agrees with the printed 6e7, so the gap is not an OCR or typesetting slip in one number; it is in the calculation or the inputs. The modern peak 958 V/m differs from 975 only by sqrt(3.1074/2.998) = 1.018 (his V against modern c). Cause not identified; the primer must say so. [read: page image p.53; calc]
4. The Sun-surface values look a factor of ten too small. About 215 solar radii give P about 215 times larger (about 129,000 cells per metre); he prints about 13,000 (magnetic 4.13 against 0.193, ratio 21.4). Both Sun numbers are mutually consistent but off by about 10 from inverse-square scaling. The 215 figure is [memory: high]. Not verdict-bearing; a worked-exercise item. [calc]
5. Gold-leaf anomaly, stated by Maxwell himself. Top of printed p.505: transmission of green light through gold "cannot be reconciled with the electromagnetic theory of light, unless we suppose that there is less loss of energy when the electromotive forces are reversed with the rapidity of the vibrations of light than when they act for sensible times" [read: page image p.53]. This is an open anomaly Maxwell conceded in 1865, relevant to the contrarian-style question "what did the theory not explain". Whether it was resolved by frequency-dependent conductivity (Drude model) is [memory: medium], not checked, and no verdict depends on it.
6. British Association coil, confirmed (claim 3 and 4 yardstick for experimental scatter). Dimensions (mean radius 0.158194 m, n = 313, wire 0.00126 m, and so on) and all sums were re-checked: 437,440 - 7,345 + 70 = 430,165 m [read: page image p.59; calc]. Least-squares value 456,748 m (+6.18 %, his factor 1.0618, from 16 runs), electric-balance "rough experiment" 410,000 m (-4.69 %); spread between highest and lowest 10.9 % of the calculation [read: page image pp.59-60; calc]. Maxwell's only stated reason for trusting the calculation is one sentence, "probably much more accurate than either of the other determinations"; he gives no uncertainties, so it is an opinion, not a measurement. SI equivalent about 43 mH is [memory: high]. This strengthens Round 1's reading that 5 % scatter was normal, so the 5 % light-speed agreement was suggestive rather than decisive.

Small tag change: the Round 1 note "digits partly OCR-garbled, not checked on the page image" for para 108 is superseded; it is now [read: page image p.53].

Open items closed or changed: "view the p.53 image and redo Maxwell's conversion" is done (image viewed; factor 1.6 persists; comparison with Thomson's 1854 foot-pound convention still not done). New open item: check the factor-of-ten in the Sun values against the original printing or another edition.

What would settle the open items (additions)

• The sunlight factor of 0.62: view the p.53 image and redo Maxwell's conversion; compare with Thomson's 1854 paper for his foot-pound convention.
• Whether anyone other than Hertz repeated his air experiments in 1888-89 with a documented source (Lodge, Lecher, others): web search or Hunt's book "The Maxwellians" (1991), cited by Belrose.

Verdict table (final, after QE revision and Round 2; no verdict changed in either round)