Heaviside: Mathematician review (Step 2)

Reviewer role: PhD mathematician. Checks derivations, notation, units, hidden assumptions.

Source gap (repeated from Step 1). The folder holds three saved Google Books viewer pages and nine page images. No full text of any work was available. I re-read the images myself: the Electromagnetic Waves pp. 139, 142, 143, the Electromagnetic Theory Vol. 1 contents pages, and the Electrical Papers Vol. 1 contents pages [read: page image]. Everything about Heaviside's results beyond those pages rests on [web] (Wikipedia) or [memory], and is tagged as such. Claims 1, 2, 3, 5 and 6 below are judged mostly from public knowledge, not from the supplied pages. Claim 4 is the only claim whose mathematics I could check directly.

Corrections to Step 1

1. Claim 1 count. claims.md says four vector equations replaced "Maxwell's twenty component equations". Wikipedia says Heaviside (1884) cut "twelve of the original twenty equations in twenty unknowns down to the four differential equations in two unknowns" [web] https://en.wikipedia.org/wiki/Oliver_Heaviside. The claim is stated too loosely. Maxwell's 20 equations included definitions of the potentials, and Heaviside eliminated the potentials. The reduction was by removing potentials, not by rewriting each equation in vector form.
2. Claim 6 date. claims.md says the layer was "confirmed 1923". Wikipedia says "confirmed in 1923" [web]. My own recollection is that Appleton and Barnett's 1924 experiment and Breit and Tuve's 1925 pulse-echo work are the standard confirmations [memory, medium]. I could not reconcile this by a second web check within budget. Treat the confirmation date as 1923-1925 and do not rely on the exact year.
3. Facts.json, Electromagnetic Waves, p.143. The summary reads "H disturbance is made 2/3 H1" nowhere, but the scanned glyph on p.143 after eq. (63) is unclear ("H disturbance is made ?H1"). My own calculation from eqs. (60)-(62) with E2/E1 = 1/2 gives the transmitted H2 = H1/2. The total H on the incident side is H1 + H3 = (3/2)H1, so the printed 3/2 is probably the incident-side total. Step 1 did not mention this ambiguity. Read the glyph as unresolved.
4. Facts.json, eq. (58) sign. Step 1 transcribed H = -e/2muv correctly. It did not note that this sign depends on a convention (see claim 4, check (a)).
5. Facts.json, Electromagnetic Waves. Step 1 describes the book as a "Cambridge Library Collection reprint". The images show the running head "Mr. O. Heaviside on Electromagnetic Waves, and the [Propagation of Potential]" [read: page image p.143], i.e. a Phil. Mag. paper. The publication details are only [web] and I did not verify the year ("1888-89" is plausible [memory, medium]).
6. Step 1's Electromagnetic Theory entry cites "Cardinal Feature of Maxwell's System" under secs. 62-65. The contents show it is sec. 63 only; 62 is "General Remarks on the Circuital Laws" and 64-65 are the first and second circuital law [read: page image]. Minor.

Otherwise Step 1's transcriptions of headings match what I read.

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Claim 1. Maxwell's theory can be cast as four vector equations

(a) Internal consistency. Yes. The four equations are curl H = J + dD/dt, curl E = -dB/dt, div D = rho, div B = 0. Heaviside's own contents list "First Law of Circulation", "Second Law of Circulation", "Definition of Curl", "Divergence" [read: page image, ET Vol. 1 contents secs. 33-36, 51, 62-65]. This is Heaviside's circuital-law formulation. Hidden assumptions: the choice of D and B as primary fields, and the constitutive relations D = epsilon E, B = mu H, which he calls permittivity and inductivity. The equations are consistent only with charge conservation, which follows from taking the divergence of the first equation. (b) Experiment. Consistent with all of classical electrodynamics: Hertz's 1887-88 waves, radio, optics at macroscopic scale [memory, high]. No quantum effects. (c) Tested? Every day, in electrical engineering. The 1884 reduction is a rewriting, not a new prediction. Not independently replicated as a "test" because it is mathematically equivalent to Maxwell's content. (d) If true. Nothing changes. It is the mainstream form.

Verdict: Established. One caveat: Heaviside's version drops the potentials. Potentials come back as essential in quantum mechanics (the Aharonov-Bohm effect [memory, high]), so "potentials are redundant" is true classically and false in a stronger physical sense. That is a statement about Heaviside's and Hertz's attitude, not about the equations.

Claim 2. Energy is localised in the field and flows at E x H

(a) Consistency. The Poynting theorem follows from the two curl equations: div(E x H) = H . curl E - E . curl H = -(d/dt)(B^2/2mu + epsilon E^2/2) - J . E in a medium at rest. I checked this identity in my head against the standard derivation; it is exact. Hidden assumption: the flux is defined only up to adding the curl of any vector field, since div(curl A) = 0. So E x H is the flux by a choice, not by derivation. Heaviside's contents mention "Persistence of Energy. Continuity in Time and Space" (sec. 67) and "Poynting Flux" (sec. 70) [read: page image]. The contents also say "Probable Localisation of the Energy" in Electrical Papers Art. 25 sec. 4 [read: page image]. The word "probable" matters: Heaviside himself did not claim proof. (b) Experiment. Consistent with all field-energy observations (radiation pressure, antenna power). The ambiguity under adding a curl has no observable consequence in electrodynamics alone, but matters in GR where energy is a source of gravity [memory, medium]. (c) Tested? The integrated flux through a closed surface is tested constantly (radiated power, Joule heating). The local direction of flow in static fields (e.g. a battery-and-resistor circuit with a static E x H circulating in empty space) is not separately measurable. Not a point of dispute. (d) If true. Nothing. It is textbook.

Verdict: Established for the integral statement; Not testable as stated for the local, point-by-point claim that energy "is" at a place and flows along E x H in static configurations, because of the curl ambiguity.

Claim 3. Inductance reduces attenuation and distortion (the Heaviside condition)

(a) Consistency. The telegrapher's equations are -dV/dx = R I + L dI/dt, -dI/dx = G V + C dV/dt. I did not see this derivation on the saved pages. Art. 20 "Contributions to the Theory of the Propagation of Current in Wires" is listed in contents [read: page image]. From the standard analysis, the propagation constant gamma = sqrt((R + jwL)(G + jwC)). When R/L = G/C, gamma = sqrt(RG) + jw sqrt(LC). The attenuation sqrt(RG) is then independent of frequency and the phase velocity is the constant 1/sqrt(LC), so no distortion. Worked check: take R = 0.1 ohm/km, G = 1e-6 S/km, C = 0.05 uF/km. Then L = R C / G = 0.1 * 5e-8 / 1e-6 = 5e-3 H/km (5 mH/km). Alpha = sqrt(RG) = sqrt(1e-7) = 3.2e-4 nepers/km, and independent of frequency. This is exact for the linear model, so it is internally consistent. A hidden assumption: R, L, G, C are constants independent of frequency, which fails at high frequency because of skin effect. Caveat on the phrase "reduces attenuation": at the distortionless condition, alpha = sqrt(RG). Without L, a line with R/L > G/C has higher attenuation at high frequency. With L added beyond the condition, attenuation rises again. So "adding inductance reduces attenuation" holds only up to the balance point. Step 1's claim is stated without this limit. (b) Experiment. Consistent with long-distance telephony; loading coils (Pupin) were deployed from about 1900 [memory, high]. Wikipedia confirms the loading-coil and AT&T patent story [web] https://en.wikipedia.org/wiki/Oliver_Heaviside. (c) Tested? Yes, by Pupin, Campbell and AT&T in practice [memory, high]. Replicated across a century of telephone engineering. (d) If true. Nothing. It is standard.

Verdict: Established. Preece's opposition is a historical matter, and Wikipedia reports that Preece blocked publication [web].

Claim 4. Wave plus diffusion; magnetic diffusion is repeated internal reflexion; perfect conductor is a perfect reflector

This is the one claim I checked on the primary pages.

(a) Internal consistency. - Eqs. (60)-(62), thin sheet. Conditions at the sheet: E1 + E3 = E2 and H1 + H3 = H2 + 4 pi k1 z E2. In the dielectric, a right-moving plane wave has E = mu v H and a left-moving reflected wave has E3 = -mu v H3. Substituting: (E1 - E3)/(mu v) = E2/(mu v) + 4 pi k1 z E2. With E3 = E2 - E1, this gives 2 E1 = E2 (2 + 4 pi k1 z mu v), so E2/E1 = 1/(1 + 2 pi mu k1 z v). This matches eq. (62) exactly [read: page image p.143]. Checked by hand. - Eq. (63) numbers. z = (2 pi mu k1 v)^-1 with mu v = 3e10 cm/s and k1 = 1/1600. I computed 2 pi * 3e10 / 1600 = 1.178e8, so z = 8.488e-9 cm. Heaviside's 8/(3 pi) x 1e-8 = 8.488e-9 cm. Agreement to four figures (by script). The conductivity k1 = 1/1600 in e.m.u. (cm/s) is right for copper: resistivity 1.7e-6 ohm cm is about 1700 in e.m.u. (1 ohm cm = 1e9 e.m.u.) [memory, high]. - Energy audit (my addition). For E2/E1 = 1/2, the reflected amplitude is E3 = E2 - E1 = -E1/2. Power fractions: transmitted 1/4, reflected 1/4, absorbed in the sheet 1/2 (sums to 1). This confirms the equations conserve energy; the absorbed half is the Joule heating in the sheet. So eq. (62) is consistent. - Unit and hidden-assumption problems. (i) The thickness z = 8.5e-9 cm = 0.085 nm is smaller than an atomic spacing (about 0.25 nm). The sheet model needs a continuum conductivity and a thickness much less than the skin depth, which is fine mathematically, but the number is not physical: a single atomic layer does not have bulk copper conductivity. Heaviside's remark that it is "a very small fraction of the wave-length of visible light" is true but misleading. (ii) The result is frequency-independent only because the sheet is thin compared with the skin depth; the conduction current must be uniform across it. (iii) The remark "There is, however, persistence of H, although there is dissipation of E" is loose. With E2/E1 = H2/H1 = 1/2 both are reduced equally on the transmission side; the "persistence" refers to H1 + H3 = 3/2 H1 on the incident side (see Correction 3). - Eqs. (58)-(59). For a sheet of impressed force e, taking the current as proportional to (E + e), I get continuity of E + e across the plane: left E = -e/2 gives E + e = e/2, right E = +e/2 gives E + e = e/2. So eq. (59) E = -+ e/2 is consistent with that sign convention. The magnitude H = e/(2 mu v) follows from E = +- mu v H. The sign printed in (58), H = -e/(2 mu v), versus my result of +e/(2 mu v) for both sides depends on the orientation convention for the right-handed triad (E, H, direction of travel) and the sign convention of impressed force; I cannot settle it from one page. Flag as a convention issue, not an error. - Eqs. (47)-(48). These are given without proof; Heaviside says he leaves "the proof of (42) to (48)" to the reader, having "very laboriously worked out" the solutions [read: page image p.139]. I did not have definitions of s, q, a (pp. 136-138 are not in the folder). I tested the structural part: f = J0((s/v) sqrt(z^2 - v^2 t^2)) satisfies f_tt - v^2 f_zz = s^2 f. I verified numerically (SymPy, at two points, s = 0.7, v = 1.3): both sides agree (0.2628 and 1.1405). Since inside the light cone z < vt the argument is imaginary, J0 becomes I0 (modified Bessel), which is the correct Riemann-type kernel for the telegraph equation once the e^{-qt} factor is included. So the form is structurally consistent with the telegraph equation. I cannot verify the prefactors, the (-s + d/dt) operators, or the definitions of q and s without the neighbouring pages. Heaviside's claim that the "tail" integral of H is constant ("persistence of momentum") is plausible for k only, but I did not derive it. - Diffusion limit. The statement "in a good conductor the diffusion solution applies practically" (p.142) is right: when sigma >> omega * epsilon the displacement current is negligible and the telegraph equation reduces to the diffusion equation [memory, high].

(b) Experiment. Skin depth and thin-film transmission are measured routinely. Perfect-conductor reflection is the limiting case of the measured reflectivity of metals [memory, high]. The 0.085 nm number cannot be tested with a continuum model. (c) Tested? The general physics: yes, by countless waveguide and shielding experiments. Heaviside's specific "repeated internal reflexion" picture is an equivalent reformulation (transmission-line analogue) and is used in modern multilayer-film theory [memory, medium]. Not a distinct testable claim from the standard one. (d) If true. Nothing changes.

Verdict: Established for the mathematics on pp. 142-143 (verified by me) and for the diffusion/reflection picture. The eq. (63) thickness is Established as a formula, not as a physical prediction at that thickness (below atomic scale). Eqs. (47)-(48): Plausible, untested by me, structurally consistent but not fully verified.

Claim 5. Operational calculus solves differential equations without rigorous justification (in his time)

(a) Consistency. The idea: treat p = d/dt as an algebraic symbol, so that (p^2 + a^2)y = f becomes y = f/(p^2 + a^2). This works because the Laplace transform turns derivatives into multiplication by s. Heaviside's operational method was not shown in the saved pages. Wikipedia: it was developed 1880-1887, produced "controversy, owing to its lack of rigour"; Bromwich later gave a justification by contour integration [web] https://en.wikipedia.org/wiki/Oliver_Heaviside. A related question I did not check: the Heaviside step function and the delta function behind his operational results were later made rigorous by Schwartz's distribution theory (1940s) [memory, medium]. Hidden assumption: the operator p^{-1} means integration from zero with zero initial conditions. Heaviside's loose use of divergent series (e.g. expansions in 1/p that diverge) was sometimes outright wrong without care, though he defended it by results [memory, medium]. (b) Experiment. Not applicable (a method). Its output matches measured transient behaviour of circuits. (c) Tested? Validated as the Laplace-transform method, used universally [memory, high]. (d) If true. Nothing in physics; it is mathematics that was later justified.

Verdict: Historically superseded but instructive. The method was valid in a large class of cases, the rigour came later (Bromwich, then Laplace-transform theory and distributions), and the lesson about "results first, proof later" is the instructive part.

Claim 6. Reflecting ionised layer explains long-distance radio (Kennelly-Heaviside layer)

(a) Consistency. A conducting layer reflects radio waves below its plasma frequency f_p = (1/2 pi) sqrt(n e^2/(epsilon0 m)). Worked check: with n = 1e11 electrons per m^3 (typical F-region), f_p = 8.98 * sqrt(1e11) Hz = 2.8 MHz. Waves below about 3 MHz are reflected at normal incidence. That is consistent with the observed AM-band skywave behaviour. No internal conflict. This is [memory, high] for the formula, but I computed the number myself. (b) Experiment. Consistent with ionosonde and pulse-echo measurements. (c) Tested? Yes, confirmed by Appleton and Barnett (1924) and Breit and Tuve (1925) [memory, medium]; Wikipedia says "confirmed in 1923" and credits Appleton with the 1947 Nobel Prize [web] https://en.wikipedia.org/wiki/Oliver_Heaviside. Replicated since by routine ionosonde measurements. The year differs between sources; see Correction 2. (d) If true. Nothing; it is mainstream.

Verdict: Established. The saved pages do not discuss this claim at all; it is a [web]-sourced claim in Step 1.

Claim 7. Minor positions: force on magnetised matter; gravitoelectromagnetism; opposition to relativity

(a) Consistency. The ET contents show sec. 83: "Force on Intrinsically Magnetised Matter. Difficulty. Maxwell's Solution probably wrong. Special Estimation of Energy of a Magnet and the Moving Force it leads to", and sec. 68 "Gravitational difficulty" [read: page image]. I cannot assess the mathematics: only headings are available. For the force on a permanent magnet, the standard result is F = grad(m . B) for a dipole m [memory, high], and the different standard expressions for force density in matter (Kelvin, Helmholtz) are known to differ without contradicting experiment, since only the total force on a body is measurable. That makes "Maxwell was wrong" partly a question of which stress tensor one picks. That is [memory, medium] and I did not verify it. Gravitoelectromagnetism: Wikipedia says Heaviside's 1893 extension to gravitoelectromagnetism was "confirmed by Gravity Probe B in 2005" [web]. I regard that as overstated: Gravity Probe B measured frame dragging and geodetic precession (to about 19% for frame dragging) in general relativity [memory, medium, not web-checked]. A linearised GR analogy with Maxwell-like equations exists, with sign and factor-of-2 differences from the electromagnetic case, so Heaviside's 1893 analogy was not "confirmed" in its original form. That check failed to be completed. Opposition to relativity: Wikipedia quotes Eves [web]. Heaviside died in 1925; his objection is historical. (b)-(d) Not testable as stated at the level of these headings; the contents offer no equations.

Verdict: Not testable as stated (no equations in the sources supplied). Opposition to relativity is Historically superseded but instructive.

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Summary table

Verification tags: all arithmetic and the Bessel check were done by me with SymPy/Python in .venv. Web checks used: 1 (Wikipedia, https://en.wikipedia.org/wiki/Oliver_Heaviside). Not web-verified: the Gravity Probe B interpretation, Appleton/Barnett and Breit/Tuve dates, the force-on-magnets stress-tensor point, and the 1.7e-6 ohm cm figure for copper. Those are [memory], tagged above.

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Investigator findings (2026-10-05)

Scope. I read sites.md ("Leads for review", all four groups), the six research/ notes dated 2026-10-05 (archive-org, google, gracesguide, iet-archives, physicstoday, royalsocietypublishing) and checked research/files/. There is no research/files/ folder: the Investigator could not download without the owner's approval, so nothing was saved and I read no primary text myself. Every quotation below is the Investigator's reading of OCR text in the browser (archive.org), reported by that note. Equations in that OCR are garbled; the Investigator reconstructed them from prose. I did not open any page image or URL, so all of it is [web, via Investigator notes], and the Investigator's reconstructions are not independent checks. There is no deep/ folder for this project. None of these findings were weighed in my 2026-10-02 review above, so each is weighed here. Nothing earlier is deleted; this section adds to it and states where a verdict moves.

The source gap in claims.md is now partly closed for claims 3, 5, 6 and 7 (Heaviside's own words are quoted) but not for the formulas (EMT1 pp. 103-107, App. B), which still need page images.

Claim 3 (distortionless line): Heaviside's own 1887 text

Finding (priority 1): Electrical Papers vol. 2, Sec. XL (The Electrician, 1887): "Make R/L = K/S, and the distortion is annihilated (save corrections for increased resistances, etc.)"; the solution V = f(x - vt) e^(-Rt/L), C = V/(Lv); "half electric, half magnetic; the dissipation ... half in the wire, half [in the leak]"; "There is a critical value of the inductance for minimum attenuation-ratio"; leakage "increased [attenuation], but the distortion is reduced". [web, via research/2026-10-05-archive-org.md, https://archive.org/details/electricalpapers02heavrich]

(a) Mathematics, which I re-derived. - Solution check. Put V = e^(-at) f with a = R/L = G/C into V_xx = LC V_tt + (RC + LG) V_t + RG V. The f_t terms are -2a LC + RC + LG = -2RC + RC + RC = 0 (using LG = RC at the condition). The f terms are LC a^2 - a(RC + LG) + RG = RG - 2RG + RG = 0. What remains is f_xx = LC f_tt, so f = f(x - vt) with v = 1/sqrt(LC). Heaviside's 1887 solution is exact for the constant-coefficient model. This is the check I could not do before, from his own text. - Energy split. In a travelling wave V = Z0 I, the electric energy (1/2) C V^2 per length equals the magnetic energy (1/2) L I^2 per length when Z0 = sqrt(L/C). The wire dissipation R I^2 and leak dissipation G V^2 are in ratio R I^2 : G V^2 = R/Z0^2 : G = R C/L : G = 1 : 1 at R/L = G/C. So "half in the wire, half in the leak" is correct, and "half electric, half magnetic" is correct. - "Critical value of the inductance for minimum attenuation." This sharpens my earlier caveat ("adding inductance reduces attenuation only up to the balance point"). I computed the exact alpha(L) = sqrt{ (1/2)[ sqrt((R^2 + w^2 L^2)(G^2 + w^2 C^2)) + RG - w^2 LC ] } numerically at three angular frequencies w = 2e3, 1e4, 3e4 rad/s with the earlier example R = 0.1 ohm/km, G = 1e-6 S/km, C = 0.05 uF/km. The minimum falls at L = 4.9996 mH/km at each w, which is L = RC/G = 5 mH/km to grid resolution, and the minimum is alpha = 3.162e-4 Np/km = sqrt(RG). So, at every frequency tested, the L that gives zero distortion is also the L that minimises attenuation. Heaviside's two statements (critical L, and R/L = K/S) are the same point. At L ~ 0 the attenuation was 1.1e-3 Np/km, about 3.5 times larger. [calculation, by me, .venv] - Hidden assumptions, now explicit in his text: his "save corrections for increased resistances" is a hedge for frequency-dependent R (skin effect); constants R, L, G, C otherwise. - His "unbelievable result" (an Atlantic cable delivering a sinusoidal amplitude larger than the maximum steady current): he attributes it to to-and-fro reflections. For a finite line with mismatched ends that is a standing-wave resonance effect and is mathematically ordinary, not a paradox. I did not compute his specific case; the Investigator quotes only the passage. - The text shows raising inductance by spacing wires and by iron (copper-covered steel). Coils at intervals are, per the secondary sources, the 1893 proposal; not shown in the 1887 text. The Investigator says the same.

(b), (c) Experiment, tests. Heaviside cites "good long-distance telephony" with copper-covered steel wire, "I do not wish to magnify its importance". That is his own cautious report; not an independent test. Independent engineering practice (Pupin loading, from about 1900) is the test, as before. Contemporary supporting accounts are the 1925 IEE obituary (Silvanus Thompson and Pupin took up the theory "with considerable success") and Russell in Nature 1925; both were written by admirers. Hunt 2012 (academic historian) and Bruton 2018 (museum curator, citing archives) describe the same facts independently of those.

(d) Unchanged: nothing in mainstream physics changes.

Historical record items bearing on this claim (kept whatever my judgement; mathematics is not at issue): - Preece's own position, IET SC MSS 022/VII/526 (pencil draft reply to Kelvin): "Heaviside and his dreadful twaddle", "calculating Kelvin's formula to show that he is correct and Heaviside is wrong, as Heaviside proves Preece's point". These are the catalogue's words; the draft itself was not read. Mathematical note: Kelvin's 1855 formula is the KR law (delay proportional to R C l^2), which is the L = 0, G = 0 limit of the telegrapher's equation. Applied to a telephone line with appreciable L and G it omits the terms Heaviside keeps. If the catalogue description is accurate, Preece's calculation was a correct calculation in a model that left out inductance, so the disagreement was about the model, not the algebra. That is an inference from the catalogue summary; the draft is not seen. Preece has a stake (public position, a costly iron-to-copper change justified by self-induction being harmful, Hunt). - Heaviside's own text: Art. 38 dated 24 Sept 1887 "but now first published" (1892): "Mr. Preece is wrong ... radically wrong"; EMT1 sec. 1: the series stopped "Perhaps there were other reasons ... We do not dwell in the Palace of Truth". Heaviside's characterization of an opponent and of his own publication record. - Two versions of the block: Hunt 2012 and Bruton 2018 say Preece, as Arthur's superior, stopped the April 1887 joint paper before it was submitted to the Journal of the STE. The 1925 IEE obituary says the papers were "rejected by the Society of Telegraph Engineers". Bruton also reports a second blocked item: Parts VIII and IX(?) of "Self Induction of Wires", "declined by the Philosophical Magazine and blocked by Preece" (notebook SC MSS 005/I/1/12). Biggs (editor) declined his anti-Preece letters, then was removed in Oct 1887, "a move he later hinted" was due to supporting Heaviside (Hunt, citing Mahon 2009 p. 79); the Biggs hint is Biggs's own account, one published source, no second found. The Kelvin letter of 12 July 1887 (022/VII/535) calls a Heaviside Phil. Mag. footnote "ill natured", so Kelvin faulted tone while later (1889) praising his propagation theory. - Earlier friction, 1873: Preece to Culley "He must be met somehow"; Culley "We will try to pot Oliver somehow" (Bruton ref. [22], archival). Stake: Heaviside's paper had ridiculed Culley's and Preece's duplex work. - Arthur sided with Preece on loading (Hunt, Bruton); Heaviside 1922, his characterization: "AWH was very afraid of P ... throw me overboard". - Pupin payment, three figures: $455,000 (Wikipedia), over $400,000 (IET blog), $500,000 (Hunt, who calls the sale "under somewhat shady circumstances" and cites Wiener's accusation). Not reconcilable from these sources; primary: AT&T-Pupin contracts. - Absence analysis. The record is not silent here but it is incomplete: the 1887 manuscript has no stated location. Bruton says its theoretical content went into EMT1 (1893), and the Investigator found it also partly in the 1887 Electrician sections, so the loss of the manuscript does not remove the mathematics. What the record cannot yet decide is why the STE paper was not published (Preece's block, editorial refusal, or both). Deciding records: STE council minutes 1887, notebook 005/I/1/12. - New own-work item: British patent 1880 no. 1407, "Improvements in Electrical Conductors", described by Bruton as "essentially the patent for ... a 'coaxial cable'". Not examined by me; for a coaxial line L and C per unit length are set by the geometry, so L C = mu eps is fixed, and what can be adjusted is the R/L versus G/C balance; I have not read the patent, and this note is only about the mathematical setting.

Verdict on claim 3: unchanged, Established. Reason: the 1887 text now confirms that the condition, the exact solution and the optimum inductance are Heaviside's, with correct mathematics. Added precision: "reduces attenuation" holds up to L = RC/G, where attenuation is minimal and distortion zero. Priority is not in doubt on the text read. Whether Preece's objection was an error of model rather than of algebra rests on a catalogue summary only.

Claim 5 (operational calculus): the rejected Part III and the reply

Finding: Hunt 2012 verbatim: virtually anything a Fellow submitted was published; Parts I and II of "On operators in physical mathematics" (1893) appeared; "pure mathematicians objected to the cavalier way he handled divergent series, and when he submitted a third installment, it was sent to a referee and rejected." The referee was William Burnside (Royal Naval College, Greenwich), per a Google snippet of Hunt's 1991 chapter; chapter not opened, report not found (Royal Society RR series). EMT2 sec. 425: "if investigations of this matter are not good enough for Another Place, how can they be good enough for Electromagnetic Theory? ... the wet blankets of rigorists." EMT2 p. 9: "Shall I refuse my dinner because I do not fully understand the process of digestion? No, not if I am satisfied with the result." Vol. 2 also carries a modified Part III (a secondary source, deadreckonings.com, snippet). [web via Investigator]

(a) Weighing. This is the primary statement of his epistemic position, and it fits my earlier verdict: the method was right in a large class of cases and unjustified. A worked reminder: the divergent series he used, such as expansions in 1/p, can be asymptotic and so useful, as with sum n! x^(-n), or can be wrong without checking; "satisfied with the result" is a test by outcome, which works until it does not. The referee's objection (divergent series) is therefore a valid mathematical objection to the argument, though not to the outputs. Both are stated. What I cannot judge is whether Burnside's report was fair; the report is not found. Hunt's and Heaviside's accounts agree on the facts.

Verdict: unchanged, Historically superseded but instructive. The new text adds Heaviside's own justification and the rejection by a mathematics referee as the instructive part.

Claim 6 (ionised layer): the 1902 text, and Kennelly

Finding: Britannica 10th ed., New Volumes vol. XXXIII (Dec 1902), "Telegraphy", I. Theory: "There may possibly be a sufficiently conducting layer in the upper air. If so, the waves will, so to speak, catch on to it more or less. Then the guidance will be by the sea on one side and the upper layer on the other." The author's initials are OCR-garbled. [web, https://archive.org/details/in.ernet.dli.2015.209824, via Investigator] Kennelly, Electrical World and Engineer 39(11):473, 15 March 1902, independently proposes the layer, about nine months earlier (Bruton); text not read by the Investigator or me.

(a) Mathematics. The passage is a qualitative guided-wave picture between two conductors (sea and layer), i.e. a parallel-plate waveguide analogue, not reflection at a plasma boundary. My earlier check (plasma frequency 2.8 MHz for n = 1e11 m^-3, from f_p = 8.98 sqrt(n) Hz) concerns the modern reflection mechanism, which Heaviside's passage does not state. Both are consistent with an unspecified conducting layer, but the mechanism in the text is guidance, tentative ("may possibly"). The mechanisms are different: guidance by a conducting boundary does not require a plasma frequency or give a frequency-dependent cutoff the same way. (b)/(c) Test dates. Earlier I could not settle the confirmation year (Wikipedia 1923). The Investigator's sources give Appleton's experiments "begun in late 1924" (Bruton) and "demonstrated it in 1924" (IET blog), which supports the 1924 date I recalled; Breit-Tuve 1925 still [memory, medium].

Verdict: split. - The existence of an ionised upper-atmosphere layer reflecting radio waves: Established (unchanged). - Attribution as stated in claims.md ("a reflecting ionised layer ... explains long-distance radio" as Heaviside's proposal): not supported by the 1902 text as written. He proposed a possible conducting layer that guides waves; Kennelly's earlier paper (text unread) is the other priority claim. I class this as a change of wording only: the physics verdict is Established; the credit claim is Plausible but not shown by the text read. This is a change from my earlier flat "Established".

Claim 7a (force on intrinsically magnetised matter)

Finding: EMT1 sec. 83 (pp. 103-104): the resultant force on a magnet can follow from the outside stress, but "it is impossible that the real forcive can be represented merely by the surface traction"; to get the real forcive "we must go inside the magnet. This can only be done hypothetically, and with precarious validity"; Maxwell's way is "quite erroneous ... it lumps together intrinsic and induced magnetisation, which have essential differences and are physically distinct". Formulas unread (OCR garbled). [web via Investigator]

Weighing. This confirms my earlier suspicion that the issue is the internal distribution, not the total force. Only the total force on a body is measured, and different internal stress tensors (Kelvin, Helmholtz, Maxwell) give the same total, so on the evidence read his objection cannot be tested by a measurement of total force. His "precarious validity" admits that. The distinction he draws (intrinsic versus induced magnetisation) is mathematically a real one: M = M0 + chi H, and force densities differ in the M0 part. Without formulas I cannot check his alternative. Verdict: unchanged, Not testable as stated, now with the reason attached: his own text says the internal split can be given only "hypothetically". Formulas still to be read from page images of EMT1 pp. 103-107.

Claim 7b (gravitational-electromagnetic analogy)

Finding: EMT1 Appendix B (The Electrician, 1893; pp. 455-466): gravity propagated at a finite speed v by the ether; "no aberrational effect, but only a slight alteration in the intensity of force", "a slight weakening of the force in the line of motion, and a slight strengthening equatorially. The direction is still radial". Numerical example as relayed: u = 3 x 10^6 cm/s, v = light speed, (u/v)^2 = 10^-6; Sun-Earth attraction raised "one two-millionth part". Also sec. 68: gravitational energy has no known flux, "we are entirely ignorant of its distribution in the ether". [web via Investigator]

Arithmetic check. (3e6 cm/s / 3e10 cm/s)^2 = 1e-8, not 1e-6. A value of 1e-6 needs u = 3e7 cm/s = 300 km/s. Either the Investigator's transcription of the speed lost a digit (OCR) or Heaviside's example differs; "fast stars" at 300 km/s would match 1e-6. The relayed "one two-millionth" equals half of 1e-6, which fits 3e7 cm/s. I treat the speed as probably mis-transcribed and mark the discrepancy: page image needed. The order-of-magnitude statement (a correction of second order, (u/v)^2) is itself consistent. Weighing. A velocity-dependent change of force of order (u/v)^2, radial, with no aberration, is a definite prediction. Linearised GR also has a gravitomagnetic sector, but with sign and factor differences relative to electromagnetism, and no frame-dragging prediction was seen in what the Investigator read. I earlier called the Wikipedia statement that Gravity Probe B confirmed his analogy overstated; nothing new here changes that. No test of his specific (u/v)^2 correction is on record. Verdict: moved from Not testable as stated to Historically superseded but instructive for the 1893 specific model, since it is a definite model with a stated size, but one replaced by GR; the general analogy (finite-speed gravity with a Maxwell-like structure) remains Plausible, untested in his form. The sec. 68 "gravitational difficulty" is a statement of ignorance, not a calculation.

Claim 7c (relativity)

Finding: EMT3 (1912) OCR has 0 hits for Einstein and Minkowski, 30 for Lorentz, 15 for FitzGerald; sec. 473 is "Lorentz's Equations of a Moving Dielectric" (Oct 1900). Anti-relativity remarks must be elsewhere (vol. 4 notes in the 1971/2003 Chelsea edition, lending-only; or letters). Not confirmed. Also Hunt: Foppl's 1894 textbook carried Heaviside's form of the equations, "from which Albert Einstein, among others, would learn electromagnetic theory". [web] Weighing. Absence in a keyword search of OCR is weak: OCR errors can hide a name, and the Investigator read none of the hits. The record is not silent in an anomalous way (a 1912 volume of papers mostly written earlier). No mathematics to check. Verdict: unchanged, Historically superseded but instructive. The remarks themselves are unseen.

Other findings, no verdict effect

• Claim 1/2: none bear. Claim 4: none bear.
• Records that go quiet: Russell 1925 says Heaviside planned vol. 3 for 1904 and a concluding volume for 1910 but "found [this] impossible"; Bruton: papers found "under the floorboards" at Paignton about 1957 (H. J. Josephs, GPO); a 1932 GPO report "The Application of the Heaviside Analysis ...", "no copies of which survive". The GPO engaged formally with his work only after Appleton (1924-25). Competing explanations (failing health and poverty, private loss, GPO indifference) remain open; the vol. 4 notes (2003 edition) and the IET collection (SC MSS 005, 21 notebooks) are the deciding records, not yet read.
• Not opened by the Investigator (leads): AIM25 (blocked), Royal Society RR referee report, BT Archives TCB 274/1 and POST 30, HathiTrust Electrician 1887, six more IET records, Kennelly 1902 text, Yavetz 2011 on the bridge system.

Verdict changes at a glance

Tags: [web, via Investigator] means the Investigator's reading of a web page, not mine; my own checks are marked [calculation] (run with SymPy/NumPy in .venv). No new web calls by me; my budget was not used because the Investigator's notes carry the URLs and I did not rely on any new [memory] fact for a verdict.