Whittaker-ET: Arbiter (STEP 3)

Inputs weighed: facts.json, claims.md, physicist.md, mathematician.md, contrarian.md, deep/, sites.md, research/ (incl. Q07 follow-up and A07). Tags: [read] / [web] URL / [memory] confidence / [calc] my own computation (project venv, mpmath/SymPy).

Source gap (carried over): the 1903 Math. Annalen paper is a scan with no text layer; equations were read from page images. Its section 5 parts 2-4 (journal pp. 347-353) rest on a digest alone. The 1910 book is digest only. History Vol. 2 (1953) was added to the folder after this file was first written and is now read (Chapter V deep read; see "QE revision 2"). Not obtained by anyone: the 1927 paper beyond section 1, the 1928 lecture, the 1944 Endeavour article "Old and new ideas of gravitation", From Euclid to Eddington, Whittaker's reply to Stoney (if any), the 1903 press coverage, the Fisher Library papers (MS COLL 00219). Where a conclusion below rests on [memory] or on A07/Q07 web notes alone, it says so. The gravity passage (journal pp. 354-355) was read from page images by the deep reader, physicist, mathematician and contrarian; those readings agree.

What kind of project this is

Whittaker (1873-1956) was a first-rank mathematical physicist, FRS, RAS Secretary 1901-06 [web: research/2026-10-07-mactutor.md]. There is no anomalous device or experiment. The project has (i) theorems, which are checked, not tested; (ii) one physical guess (gravity as longitudinal waves in the medium of light, 1902-03); (iii) the "longitudinal standing wave" hypothesis put to the review (Q07), which is a later descendant of (ii) and not Whittaker's wording.

Where the three reviewers agree

• Claims 1-3 mathematics: Established. All three verified the identities independently.
• The 1903 gravity sum is sgn(Vt - r)/r, not a bare 1/r: +1/r inside the front, -1/r outside; the source changes sign from -1 to +1 at t = 0 (a static -1/r plus a switched-on 2/r); the "flickering" parts of Stoney's critique collect at the wave front. Physicist, mathematician and contrarian each reached this. Earlier files (claims.md, facts.json, A07) are incomplete here (see Corrections).
• "Longitudinal" follows from using a scalar potential (force = -grad V lies along the wave vector); it is not a discovery about a medium.
• The speed is unconstrained by static fields (any k works).
• "Standing" is not in the 1902 or 1903 text; it comes from Barrett (1993), Bearden, Titleman.
• The 1904 paper has no gravity and no longitudinal waves; each scalar plane-wave term gives a transverse EM wave, and Whittaker's 1951 History p.212 (Ch. XIII) calls F and G "two plane-polarised components" [read].

Recomputed numbers (where reviewers differ or only one computed)

1. Cutoff sum (contrarian): (2/pi) Si(mu_max (Vt - r)) at mu_max = 10, r = 1 gives 1.0557 (Vt-r = 1), 0.9878 (5), 0.9985 (20), 0.9996 (100). [calc] Matches the contrarian's 1.056 / 0.988 / 0.9985 / 0.9996 exactly.
2. Gibbs overshoot (mathematician): (2/pi) Si(pi) = 1.1790, i.e. 18 percent. [calc] Confirmed.
3. 1/r integral on journal p.334: at (0.3, 0.4, +1.2) = +0.769231, at z = -1.2 = -0.769231. [calc] Confirmed.
4. Standing-wave test (mathematician): cos(kappa r)/r gives r^2 F = 1 + (kappa r)^2/2 - (kappa r)^4/8. [calc] Confirmed. Correction to the worked example: kappa r = 1e-4 gives a deviation of 5e-9, not 1e-8. For a deviation of exactly 1e-8, kappa r = 1.414e-4, so at 1 AU kappa = 9.45e-16 per metre, wavelength 6.6e15 m (0.70 light year), omega = c kappa = 2.8e-7 rad/s, period 2.2e7 s (257 days). The original statement ("below 1 part in 1e8" needs kappa < 6.7e-16, period about 363 days) is a conservative bound and not wrong; both say the period must be months to a year or longer if the wave travels at c. This is the mathematician's own illustration, not from any source, and assumes wave speed c.
5. Force sign: V = sgn(Vt - r)/r gives force -dV/dr = sgn(Vt-r)/r^2 + 2 delta(Vt-r)/r (physicist). The mathematician wrote the front spike as "-(2/r) delta", which is the derivative dV/dr, not the force. The physicist's sign is right for the force; the mathematician's is a labelling slip and nothing depends on it.

Web checks (budget 8; used 3)

• Carlip, "Aberration and the Speed of Gravity" (Phys. Lett. A, 2000): abstract confirms Newtonian aberration needs speeds above 2e10 c, and that "aberration in general relativity is almost exactly canceled by velocity-dependent interactions, permitting c_g = c" [web: https://arxiv.org/abs/gr-qc/9909087]. Retags the physicist's and contrarian's [memory, medium] items as [web].
• Kopeikin-Fomalont 2003 Jupiter measurement: reported the speed of gravity equal to c within 20 percent; Clifford Will and Stuart Samuel dispute that it measured the speed of gravity at all [web: https://www.phys.ufl.edu/~cmw/SpeedofGravity.html; https://www2.lbl.gov/Science-Articles/Archive/Phys-speed-of-gravity.html; https://arxiv.org/abs/astro-ph/0311063]. The contrarian's "disputed" is confirmed; the figure (1.06 +/- 0.21) c is not re-checked. No verdict depends on it.
• Short-range inverse-square tests: Eot-Wash reached 56 micrometres in 2007 (Kapner et al., PRL 98, 021101) and 52 micrometres in a later test [web: https://arxiv.org/pdf/2002.11761]. The reviewers' "about 50 micrometres" is right.
• Left as [memory]: Khmelnitsky-Rubakov 2014 oscillating potential from scalar dark matter (medium); preferred-frame bounds from lunar laser ranging (medium); optical-metric/analogue-gravity statement (medium); the Eotvos-test sensitivity of 1e-13 by the 2000s (medium). None is load-bearing; every verdict rests on the [web] items cited by the reviewers (GW170817 arXiv:1710.05834, GWTC-3 arXiv:2112.06861, PSR B1913+16 arXiv:1606.02744, MICROSCOPE arXiv:2209.15487, Ibison astro-ph/0302273, Cassini). I did not re-fetch these; they stand as retrieved by the Q07 follow-up and the physicist.

Corrected claim list and final verdicts

Claim 1. Whittaker's integral solutions of Laplace's and the wave equation (1903)

Agreement: all three. Dispute: only the scope of "general". The mathematician's analysis is the most careful: proved per polynomial degree; summing over degrees, convergence and interchange are asserted; local near a point where V is analytic. "Periodic in u, v" is vacuous for the integral (only the range [0, 2 pi] x [0, pi] enters), so Munera's objection (as relayed by Titleman 2023 [web: https://www.qeios.com/read/Y3D1HM]) bites only if "general" means global. The 1/r identity on p.334 holds only for z > c. Final verdict: Established (local, real-analytic). The global "every solution everywhere" form: Not proved by the paper and not needed by any use made of it. Settles it: nothing needed; a proof of convergence of the sum over degrees would close the one assertion.

Claim 2. Finite wave solutions decompose into plane waves in all directions

Agreement: all three. Final verdict: Established. Qualify as "finite, continuous on a range, locally" (Whittaker's wording).

Claim 3. Two scalar potentials F, G for the field of moving electrons (1904)

Agreement on the mathematics (Lienard-Wiechert reproduced to 12 digits by the deep reader; plane-wave terms transverse, checked analytically by the mathematician). The physicist and contrarian differ in wording on the attached "hidden longitudinal scalar waves" reading: physicist says contradicted where made specific; contrarian says not testable as stated; mathematician says it contradicts Whittaker's own gloss. I side with the combination: it is not in the paper, the paper's own plane-wave terms are transverse, Whittaker's 1951 gloss (p.212) calls F and G polarisation components, and no one has stated a measurable difference from Maxwell-Lorentz theory. Titleman 2023 (Qeios, arXiv:2205.08309) and Bearden hold the longitudinal reading; their preprint has open reviews on Qeios (mean 3.25 per the physicist) [web]. That is a view by living authors and stays in the record. Final verdict: Established (as a reformulation of Maxwell-Lorentz theory). The "longitudinal scalar wave" reading: Not testable as stated (potentials are gauge-dependent; no distinct prediction given), and at odds with the paper's text (not a separate scale verdict). Settles it: a stated, measurable difference between a longitudinal-potential-wave model and Maxwell's equations; none exists.

Claim 4. Gravity as longitudinal waves in the medium of light (1902 MNRAS 62, 617; 1903 section 5 part 5), and the "standing wave" form put to the review (Q07)

The reviewers agree on every fact of what Whittaker wrote and computed. They differ on how to split the verdicts: - Physicist splits 4A static math, 4B scalar/longitudinal gravity waves with speed above c, 4C shared medium, 4D standing wave. - Contrarian splits similarly but gives more weight to "dismissed by argument, not by experiment". - Mathematician gives the mathematical statement Established and the physical model Never properly tested. The differences are over what "tested" means. The speed, polarisation and radiation-rate tests (GW170817, GWTC-3, PSR B1913+16) were not designed against Whittaker, but they are controlled observations of exactly the properties the idea would need, so for the specific forms they count as tests. Where the idea is stated generically (a shared medium with tensor response) there is no test, because GR's shared light cone gives the same speed. I accept the physicist's and contrarian's treatment of tests, and the mathematician's point that Whittaker never wrote a moving-source field equation, so nobody has tested "Whittaker's model" by name.

Disagreement on 4C: the physicist says "Refuted by calculation" for the scalar polarizable-vacuum version; the contrarian says "Tested, not supported", noting that it rests on one paper (Ibison 2003, 2/3 of GR's radiated power, from inside the Puthoff group) and is not replicated. The scale defines "Refuted by calculation" as accepted laws applied correctly, no experiment needed. Ibison's calculation is a calculation within one specific model (Dicke-Puthoff), compared with a measured decay (0.9983 +/- 0.0016 of quadrupole prediction). I side with the contrarian's wording: it is a model-specific calculation compared against data, by a single analysis, so "Tested, not supported" for that model is the safer label. It does not touch tensor-response media or Whittaker's own undeveloped note.

Stoney 1903 (MNRAS 63, 424): praised the mathematics, rejected the physics as requiring an "overwhelmingly improbable" 1/mu spectrum. The arbiter's weighing: his arithmetic is right and so is the physicist's reply that the spectrum is the Fourier transform of a step, so the objection is correct about a finite-band set of waves and answered for the full set. What it leaves is a physical picture in which the source changes sign from -1 to +1 at t = 0 (a static -1/r plus a switched-on 2/r), with an expanding front at speed V. Whittaker's reply, if any, is not found.

Final verdicts for claim 4, split: - 4A. Static 1/r as a superposition of waves whose time dependence cancels inside the front: Established (mathematics; steady only for r < Vt, -1/r outside, source a step; Gibbs ringing of 18 percent overshoot under a hard cutoff). - 4B. The constituent waves are physically real with a 1/mu spectrum (Stoney's objection): Never properly tested (argued against in 1903, not measured; not blocked or suppressed). - 4C. Gravitational radiation is longitudinal/scalar: Tested, not supported (LVK finds tensor modes and no evidence of non-tensor modes; strictly longitudinal modes are only weakly constrained by interferometers, and a small scalar admixture remains allowed [web: Q07 note on arXiv:2610.08588, preprint]). - 4D. Speed of gravitational waves enormously above c: Tested, not supported (GW170817: -3e-15 <= (v_gw - c)/c <= +7e-16 [web: https://arxiv.org/abs/1710.05834]). For the static field the speed is not testable. - 4E. Shared medium for light and gravity: generic form Never properly tested in a form that differs from GR (equal speed fits both); the scalar polarizable-vacuum form Tested, not supported (Ibison; one analysis). - 4F. Q07 hypothesis, "longitudinal standing wave": Not testable as stated (no frequency, amplitude, coupling or medium properties given). Each concrete version is Never properly tested as a named model, though generic searches (oscillating g, short-range inverse-square down to 52 micrometres, pulsar timing for scalar dark matter) found nothing. It is not what Whittaker wrote. Physicist's point kept: a travelling spherical wave is the sum of two standing patterns in time quadrature, so "standing vs travelling" is partly a presentation matter; but a literal standing wave would oscillate or have nodes, and Whittaker's field has omega = 0. Where might the Q07 hypothesis live best? Whittaker's section 5 part 5 and the 1902 note are the right origin: they contain the "gravity as waves in the light medium" suggestion in its first form. The most defensible way to state the Q07 hypothesis is the mathematician's: a genuine standing wave obeying the scalar wave equation gives r^2 F = 1 + (kappa r)^2/2, so it can mimic Newton only if its period is months or longer (for wave speed c), and then g should oscillate everywhere in step. That is a definite prediction that can be searched for. Settles it: (1) a measured departure of a moving source's gravitational response from the GR retarded form that a scalar longitudinal channel fits better; (2) detection or tight exclusion of scalar/breathing/longitudinal polarisation with a network able to see it; (3) for standing-wave variants, a gravimeter, atom-interferometer or pulsar-timing line at a fixed frequency that correlates with an optical-clock modulation at the same frequency; (4) Whittaker's own statement of the moving-source field, if he ever wrote one.

Claim 5. The 1951 aether remark and the history of aether gravity

Agreement: not a derivation; the history (Riemann 1853, Maxwell 1864, Kelvin 1896 contested by FitzGerald in a private letter) is accurately reported [read: Chapter VIII deep read]. The mathematician adds one sharp point: Maxwell's energy argument (the field energy of Newtonian gravity is negative, -g^2/(8 pi G) [memory, high], where electrostatic energy is positive) is an obstacle any medium-gravity model must pass and Whittaker's waves do not address. The vacuum is not empty (Lamb shift, Casimir force [memory, high]); but it is Lorentz-invariant, and the vacuum-energy mismatch with gravity is the cosmological constant problem [memory, high]. Navarro (2020, 2025) on Vol. 2: Whittaker's commitment to the aether and a gravity-EM link, and a priority dispute crediting Poincare and Lorentz [web: research/2026-10-07-sagepub.md, abstract only; Vol. 2 now read, see QE revision 2]. That is one historian's account; antisemitism is left open by him; Navarro is not an interested party in the physics. Final verdict: Historically superseded but instructive (material aether as the seat of fields); vacuum-as-aether proposal: Not testable as stated (terminology). Underlying vacuum physics: Established.

Absence of evidence: where the record goes dark

1. Whittaker's reply to Stoney, and any 1903-1927 development of the wave-gravity idea. Competing explanations: (a) he considered it answered or merely a representation (supported by "this investigation does not explain the cause of gravity"); (b) he gave it up as relativity arrived (supported by his 1927 work inside GR, 1928 lecture calling the medium analogy "not quite perfect" [web: research/2026-10-07-cambridge-synge.md]); (c) he kept it privately or in later writings (supported by Navarro: "convinced that gravitation and electromagnetism were ultimately connected"; the 1943-44 Endeavour articles exist but are unread; Fisher Library letters and an autobiography exist). Against (b): his 1951 preface still keeps the aether name. The record fits all three. Deciding records: the 1944 Endeavour article; Vol. 2 (now read: see QE revision 2); Fisher Library MS COLL 00219 Box 11; the 1903 daily-press articles McCrea mentions.
2. Brush's Bureau of Standards check (1911-29): the result is not located. Competing explanations: never completed; completed with a null; lost. Later composition limits (MICROSCOPE 1.5e-15 scale, Eotvos tests) cover the same effect with vastly greater sensitivity, but are not a replication of Brush's design. Deciding record: US Bureau of Standards files, 1911-1929.
3. No funding or publication was blocked: the idea was published in Math. Annalen and MNRAS and received reasoned criticism. The record is thin, not hostile; there is no evidence here for suppression and none against it beyond that.

Open questions

• Did Whittaker ever write the field of a moving source in his wave picture? (Would show whether he met the aberration problem Laplace raised.)
• What is the best tensor-valued "medium" that reproduces GR's wave speed, polarisations and pulsar decay, and does it differ observationally from GR at all?
• Is there an observable signature (frequency line) of a months-to-years-period scalar standing wave in superconducting gravimeters or pulsar timing, and does the published dark-matter literature already exclude it?
• Does Munera's composition-dependence result survive Darmos-response and MICROSCOPE?
• What did Whittaker's Vol. 2 and the 1944 Endeavour article say about gravity and the aether? (Vol. 2 answered in QE revision 2; the 1944 article is unread.)
• (QE revision) What do Hicks (1879-80), Korn (1898) and Larmor (1895) actually derive for the retarded pulsation force, and how is common phase justified?

Corrections to earlier documents

1. claims.md claim 4, facts.json (1903 entry) and the A07 summary: they say the sum is 1/r "once Vt > r" and call it a steady field. The sum is sgn(Vt - r)/r: -1/r outside the front, the source is a step at t = 0, and the force carries a delta shell at r = Vt. [calc, three reviewers agree]. Steady only inside r < Vt. A07's inferred line should be amended; this is an addition, not a retraction (A07 table row "1/r for r < Vt" is correct).
2. claims.md claim 4 should cite the 1902 MNRAS note (62, 617-620) as the first statement: the "identical property of the medium" and "enormously greater" sentences appear word for word there. claims.md Source gap lists the 1902 note as "not obtained"; the Investigator has since read it.
3. claims.md claim 4 / facts.json: add that "standing wave" wording is from Barrett (1993), Bearden and Titleman, not Whittaker.
4. claims.md claim 2 should be qualified ("finite solution", continuous on a range, locally). Claim 1: "periodic without loss of generality" is vacuous for the integral; real restriction is analyticity and local validity.
5. facts.json 1951 entry should list History Vol. 1 p.212 (Chapter XIII, "The Two Scalar Potentials": F and G are "two plane-polarised components") beside the Chapter VIII deep read.
6. A07's "the scalar version of that medium fails on radiation" overstates; it rests on Ibison's analysis of the Dicke-Puthoff model (2/3 of GR power). Tensor-response media are not excluded.
7. Mathematician's front-spike sign ("-(2/r) delta" labelled as force) is the derivative; the force is +2 delta/r. Standing-wave worked example: kappa r = 1e-4 gives 5e-9, not 1e-8 (see Recomputed numbers 4).
8. Carlip 2000 and Kopeikin-Fomalont items in the physicist and contrarian were [memory]; now [web] as above.
9. Stale-copy check for the orchestrator: search facts.json, claims.md, followups.json (A07), entries and the primer for "1/r once Vt > r", "steady superposition" and "standing wave" attributed to Whittaker.

QE revision (2026-10-07)

Inputs: qe.md (verdict MINOR), and the new deep reads in deep/: the 1951 History Ch. IX (PDF pp. 146-152, section 12 onward: sections 14-22), Ch. V (pp. 71-81), the gravity passages (pp. 16, 21-23, 32-33, 56, 110-111, 154; section 12), Ch. XIII (pp. 200-212); the 1910 comparison deep read; and the 1903 addendum (journal pp. 347-353). Tags: [read] project documents, [web] URL, [calc] computed. Web checks: 4 of 8 used in total (3 earlier, 1 now).

What qe.md found and what the new deep reads say

QE's MINOR verdict stands. No central verdict changes. Four additions are material; the rest are corrections.

R1. Pulsation models of gravity (qe.md M1-M2). Confirmed and sharpened by the deep read [read: 1951 PDF p.149-150, bk 284-286; page images]. - C. A. Bjerknes (1870s): two spheres in an incompressible fluid that change volume together attract by the inverse-square law if their pulsations are concordant, repel if half a period apart, and exert "no action" at a quarter period (the quarter-period sentence is added in 1951; absent in 1910). Whittaker: "It is necessary to suppose that the medium is incompressible, so that all pulsations are propagated instantaneously; otherwise attractions would change to repulsions and vice versa at distances greater than a quarter wave-length." Hicks (1879-80) did the mathematics (cited, not read). - Leahy (1884): for an elastic medium, "the wave-length of the disturbance being supposed large in comparison with the distance between the spheres", the results are reversed: like phases repel, unlike attract; still inverse square. QE's M1 omitted the long-wavelength condition. Leahy's reversal is a property of an elastic medium at short range, not the same effect as the quarter-wave sign change. - Korn (1898, footnote): gravitating particles as slightly compressible spheres pulsating in an incompressible perfect fluid, intensity proportional to mass, giving "forces of the Newtonian kind". Larmor (1895): gravitation between electrons from vacuous electrons with free periods in the aether, following Bjerknes (1951 only). Whittaker gives all of these as other people's hypotheses with no test and no number. - [calc, deep read section 19] For a medium whose compressional waves travel at c, attraction at distance d needs period T > 4d/c: Sun-Earth 1996 s; Sun-Neptune 16.7 h; Sun-Proxima 17 yr. This is only the sign bound. The stronger bound is fidelity to the inverse square: the mathematician's r^2 F = 1 + (kappa r)^2/2 result means about 1 part in 1e8 at 1 AU needs a period of order a year (the deep read states the (kd)^2 scaling for the retarded Bjerknes force as [memory, medium]; the exact law is in Hicks, not read). For particle-scale pulsation frequencies the medium would have to carry the disturbance at 1e23-1e26 c, which is why Korn kept the fluid incompressible [inferred in the deep read]. A c-speed medium needs periods of 1e3-1e8 s, unlike any known atomic frequency. - Effect on 4F: the Q07 hypothesis has a named 19th-century ancestor with explicit conditions (common phase for every gravitating body, mass-proportional amplitude, near-field regime d < lambda/4, medium constitution that fixes the sign). It is closer to an oscillating near-field than to a textbook standing wave with fixed nodes: nodes between the bodies would reverse the force. The modern acoustic counterpart, the secondary Bjerknes force between pulsating bubbles, is laboratory physics [memory, high, per the deep read] and shows a real, testable "bodies pushed by a longitudinal field" model; it is an analogy, not evidence for gravity.

R2. Riemann 1861: the origin of the "density-wave" reading of Whittaker's potentials (qe.md M3, claim 3). [read: 1951 PDF p.152, bk 291; 1910 PDF p.345, bk 324; page images]. Whittaker: Riemann noted that his potentials satisfy dphi/dt + c div a = 0 (1910: without c), the same condition as Lorenz's, "This appeared to Riemann to indicate that phi might represent the density of an aether, of which ca represents the velocity." Whittaker adds the limits: the electric and magnetic forces then correspond to second derivatives of the displacement, so the field energy is hard to match to the model's energy. A modern limit: the condition is a gauge choice [memory, high], so a "density" read off phi depends on the gauge. This is the 19th-century root of the reading in Barrett, Bearden and Titleman, held by living authors, and stays in the record. It is a different object from Whittaker's 1904 F and G, which his 1951 text (p.212) calls plane-polarised components. No change to the claim 3 verdicts.

R3. How the elastic-solid aether treated the longitudinal mode (qe.md M4). [read: 1951 Ch. V, PDF pp. 71-81]. Verdict: the QE reading is partly corrected. In every elastic-solid scheme, longitudinal and transverse waves are two modes of one medium set by two constants (Poisson: transverse sqrt(n/rho), longitudinal sqrt((k+4n/3)/rho)). Theorists kept the longitudinal mode quiet by making it infinitely fast and uncoupled (Fresnel "indefinitely great", Stokes; Green's jelly-like aether with a very large ratio), zero-speed and energy-free (Cauchy/Kelvin contractile aether, k + 4n/3 = 0, read on page image p.79), or absent by construction (MacCullagh's rotational aether). Navier's sqrt(3n/rho) mode is not suppressed. Cauchy alone expected it might be found, perhaps as heat. Two corrections follow. (i) The deep read did not find the phrases "identical property of the medium" or "enormously greater" in pp. 71-81; the nearest are Fresnel's "indefinitely great" (bk 128) and Green's "very large". These phrases are Whittaker's own 1902-03 wording and are not borrowed from the History. (ii) Whittaker's 1903 closing sentence ties the property to the electromagnetic transmission of light, not to the elastic-solid mode; the elastic-solid history is context for why a fast longitudinal mode was thinkable, and is not something Whittaker states as his source. Maxwell's medium has no longitudinal mode (mathematician, gradient versus curl). Net: a "very fast longitudinal mode" in the light medium was a respectable 19th-century option, deliberately made harmless; it was never observed.

R4. The speed-of-gravity record in the same volume (qe.md M5). [read: 1951 PDF pp. 16, 21-23, 32-33, 56, 110-111, 154; page images; deep read section 12]. Newton: conjectured aether density gradient, no speed. Cotes (1713): no mechanism, instantaneous in his day. Le Sage (1782): screening of ultramundane corpuscles; inverse square from shadowing; no speed. Aepinus (1759): gravity perhaps a small residual of nearly balanced electric forces. Euler (1760-61): aether pressure "Constant - 1/r", static, so weight proportional to atomic volume. Laplace: gravific fluid, aberration-type tangent force, speed "at least a hundred million times greater than that of light" (1e8 c) from the Moon's secular motion; Whittaker: "evidently in the highest degree questionable". Tisserand (1872): Weber's law with h = c gives 14 arcsec per century at Mercury, 3/8 of the 38 arcsec excess (3/8 x 38 = 14.25 [calc]); Levy (1890): Riemann's law gives 3/4 (28.5 arcsec), and "a combination" gives all of it; 3/8 + 3/4 = 9/8, so the combination must be a weighted mix, not a sum [calc]. Kelvin's vortex atoms make gravity instantaneous. Two cautions: Whittaker's 38 arcsec is his figure (the modern excess is about 43 arcsec [memory, high], not used in the History); and his own comment that gravity "had been believed to be instantaneous" since Laplace sits uneasily with Laplace's lower bound, consistent only as "so large as to be treated as instantaneous". Bearing on claim 4: Whittaker's "enormously greater than light" is the same order of speed as Laplace's bound, an old, reasoned position that avoided aberration. Modern observation (Carlip; GW170817) rules out speeds far above c for gravitational waves; the Laplace-type aberration problem is why.

R5. Whittaker's History is silent on his own gravity idea (qe.md M6). [read: deep read 2, section A]. Ch. XIII (PDF pp. 200-212) mentions Whittaker once: the 1904 F and G result (p.211, bk 409; footnote Proc. LMS (2) i, p.367), in the third person, with no mention of the 1903 paper or the 1902-03 undulatory gravity. A whole-volume text search finds "Whittaker" only on the title page and preface, in one citation of his Analytical Dynamics, and in that sentence; "gravit" has 47 hits, those inspected relate to Newton, Maxwell, Riemann, Planck and others, but not all were read in context. So "never mentioned in Vol. 1" is confirmed for Ch. XIII and strongly indicated, not exhaustively proved, for the volume; Vol. 2 is unread. Added to "Absence of evidence" item 1 as a further datum: in 1951 Whittaker, writing the history of exactly this subject, does not mention his earlier gravity-wave note. This fits all three explanations (considered it a representation only; gave it up; kept it elsewhere); it is mild support for (a) or (b) over (c) in Vol. 1 but cannot decide. Also in that chapter: Planck's 1899 compressible aether "subject to gravity", compressed round the earth like an atmosphere (p.200), and Michelson's 1897 null result for height dependence of the speed of light (p.202) [read]; background for any idea in which gravity changes the medium near the Earth.

R6. Whittaker's 1951 definition of the aether. [read: 1951 Preface, PDF p.2]. "It seems absurd to retain the name 'vacuum' for an entity so rich in physical properties, and the historical word 'aether' may fitly be retained." The 1910 edition has nothing like it; the 1910 closing passages on the aether's future are dropped in 1951 (1910 deep read). Supports claim 5's "vacuum-as-aether" reading. No change in verdict.

R7. The 1903 paper, parts 2-4 (journal pp. 347-353). [read: page images; the deep-read addendum]. The source gap is closed: every page of the paper has now been read as an image. Parts 2-4 are pure mathematics: the Helmholtz equation (nabla^2 V + V = 0, from V e^{it}), the expansion in spherical Bessel functions, and the generalised Bessel functions J_{m,n} with an integral representation and an addition theorem, and they make no physical claim. Misprints found: the constant (pi/2)^{1/2} should be (2 pi)^{1/2} (p.349); P_n^2 should be P_n^1, A_n for A_n^n (p.350); prefactors 1/(4 pi^2) and 1/(4 pi) on p.352 should be -1/(4 pi^2) and 1/(4 pi^2) [calc]; the exponent signs on p.353. The claim that every solution has the integral form is asserted by reference to section 4, true locally; growing solutions such as cos(sqrt2 x) e^z are not represented by bounded integrals [calc]. This strengthens the claim 1 verdict (local, real-analytic) and leaves it. Relevant to the Q07 hypothesis: part 2 is the step from a general field to a single-frequency field V cos t built from plane waves in all directions, which is literally a standing-wave pattern; Whittaker does not use it for gravity (his gravity argument is the zero-frequency limit, part 5 section 5) [deep-read reading, not Whittaker's]. The section heading says gravitation is "explained as a mode of wave-disturbance" while the text on p.24 says "this investigation does not explain the cause of gravity"; the heading claims more than the text.

R8. The oscillating-g / pulsar-timing statement (qe.md process point). Retagged. [web: https://arxiv.org/pdf/1810.03227] The Parkes Pulsar Timing Array searched 26 pulsars (2004-2016) for the periodic arrival-time signal from an oscillating gravitational potential due to ultralight scalar-field dark matter, found no statistically significant detection, and set a 95 percent upper limit of 6 GeV cm^-3 on the local density for boson masses up to 1e-23 eV (nanohertz frequencies). So the "found nothing" statement is sourced for that band and for that particular scalar-dark-matter model only; it is not a search for a Whittaker-type standing wave, and says nothing about other frequencies or couplings. The superconducting-gravimeter and atom-interferometer statements remain [memory, medium] and are dropped as support; the verdict on 4F does not depend on them. Short-range inverse-square limit (52 micrometres [web: https://arxiv.org/pdf/2002.11761]) is unchanged.

Retained material (qe.md M7)

Kept in the record next to their verdicts, not dropped: Riemann 1853 (gravity and electrostatics as the aether's resistance to compression) and Kelvin 1896 condensational waves, contested by FitzGerald; Wangerin's 1903 summary ("longitudinal waves ... with very great speed"); Munera's periodicity critique; Trovon de Carvalho and Rodrigues (2001), a critique of the AIAS/Bearden use of Whittaker, listed beside Titleman under claim 3 (known from research/2026-10-07-qeios.md; not read here).

Changed verdicts

None of the verdict names change. The wording of three entries changes, shown in the table below: - 4F: the concrete versions are now named: (i) in-phase near-field pulsation (Bjerknes/Korn/Larmor type) and (ii) a genuine standing wave obeying the scalar wave equation (mathematician's r^2 F result). Both: Never properly tested as a named gravity model. The generic oscillating-potential searches (PPTA) found nothing in the nanohertz band for one scalar-dark-matter model. The general 4F verdict, Not testable as stated, stands. - 3: the "hidden longitudinal wave" reading acquires a documented 19th-century origin (Riemann 1861). Verdict unchanged: Not testable as stated, absent from the paper's text. - 5: Chapter V and Chapter IX evidence added. Historically superseded but instructive unchanged.

Where the Q07 hypothesis lives best (revised)

Two homes, to be kept apart. (1) Whittaker 1902-03 as the origin of "gravity as waves in the light medium", with the stated static result 1/r inside the front only. (2) Bjerknes-Korn-Larmor as the origin of "gravity from in-phase longitudinal oscillation of a medium", which Whittaker's own History records. The Q07 phrase "longitudinal standing wave" is closer to (2) than to (1): an oscillating near-field with common phase, not a travelling wave. Honest statement of the question: what common-phase pulsation, of what amplitude, period and coupling, would give the inverse square to 1 part in 1e8 at 1 AU (period of order a year or longer if the medium's speed is c), and where would it show as a line in gravimeter, pulsar-timing or clock data? Settles it: the same experiments listed under claim 4 settle (3) of "Settles it"; add the targeted searches for a common-phase periodic signal at periods from 1e3 s to years, correlated across bodies.

Corrections to earlier documents (from this revision)

1. arbiter.md, "Where the three reviewers agree", claim 4 (Stoney paragraph) and Corrections item 1: "switch-on at t = 0" should read: the source changes sign from -1 to +1 at t = 0, that is, a static -1/r plus a switched-on 2/r (qe.md correction 3). The sum is sgn(Vt - r)/r as stated; nothing else changes.
2. primer.md section 3.5: Laplace's bound is "at least a hundred million times" the speed of light (1e8 c; 1951 PDF p.110, bk 206-207), not "millions of times" (qe.md correction 1). The "7e6 c" figure sometimes quoted is [memory, medium], not checked, and not Whittaker's number.
3. primer.md section 4.2 and contrarian.md claim 4: the 1902 and 1903 closing sentences differ. The "identical property" and "enormously greater" sentences match; the closing sentence of 1903 reads "with a definite though large velocity ... similar to those whose propagation ... constitutes, according to the electromagnetic theory, the transmission of light" (journal p.355, page image), and says "exactly similar" where 1902 has "of the same type" (qe.md correction 2).
4. qe.md M4 implied that the elastic-solid history supplies Whittaker's phrases. The Ch. V deep read finds the phrases are not there (R3).
5. qe.md M1 omitted Leahy's long-wavelength condition (R1; the deep read already records this).
6. A07, physicist.md and contrarian.md cite "History Vol. 1 pp. 206-207" in book pages; Laplace is on bk 206-207 (PDF p.110), Tisserand on bk 207-208 (PDF p.111). State the convention (qe.md correction 4). The earlier deep read spelled these as bk 206-207 and 208-209 in places; the page-image check gives PDF 110 = bk 206-207 and PDF 111 = bk 208-209.
7. facts.json 1903 entry: how_read is out of date (all PDF pp. 2-24 now read as page images; parts 2-4 closed; qe.md correction 5). The digest of the 1903 paper garbles the wave equation (qe.md correction 6); superseded by the deep read.
8. Corrections 1, 2 and 3 of "Corrections to earlier documents" above still stand and are not withdrawn.

QE revision 2 (2026-10-07)

Inputs: qe.md run 2 (verdict MINOR) and the new deep read deep/1953-A-History-...-Vol-2-...-264407.md (Chapter V "Gravitation", PDF pp. 155-207, bk 144-196, text layer plus page images; PDF = bk + 11; Whittaker's own cross-references are book pages). A sixth document, A History of the Theories of Aether and Electricity, Vol. 2 (1953), entered the folder at 13:03 after the reviews and the first arbiter were written. Every earlier "Vol. 2 not obtained / unread" statement in this file (Source gap, claim 5, "Absence of evidence" item 1, "Open questions") is out of date; see the corrections below. Tags: [read] project document, [web] URL, [calc] computed, [memory] with confidence. No new web calls this round (4 of 8 used in total).

New evidence: what Whittaker wrote in 1953 (the inventor's own later, published view)

V1. Speed of gravity. [read: Vol. 2 PDF pp. 157, 160, 162, 194; bk 146, 149, 151, 183; page images at 160, 162, 194] - On Laplace and Lorentz's 1900 paper, bk 149: "the assumptions of Laplace's investigation, which led him to conclude that the velocity of propagation of gravitation must be vastly greater than that of light, do not to twentieth-century minds seem very plausible" (footnote: Vol. I, pp. 207-8). Lehmann-Filhés (1895) is filed with Laplace (bk 149, fn 6). - Bk 151: relativity (Poincare, from 1904) requires that gravity "must be propagated with the speed of light", and "this supposition was not contradicted by the results of observation, as Laplace had supposed it to be". - Bk 183: "gravitational waves are generated, which are propagated outwards with the speed of light". Bk 159-160: no signal exceeds c (group velocity "never greater than c"). - The chapter never mentions his 1902 MNRAS note, his 1903 Math. Ann. section 5 or Stoney. A full-chapter read plus a whole-volume text search found no "Stoney", no "lvii", no MNRAS "lxii (1902)", no "Endeavour", no undulatory, longitudinal or standing-wave gravity. His self-citations in the chapter are all other subjects (bk 146 small divisors 1917; 165 null geodesics 1928; 170 variational paper 1927; 184/186 de Sitter world 1931; 194 Gauss's theorem 1935). The deep read weighed three passages as possible allusions (the Laplace sentence, Lorentz's penetrating waves, the FitzGerald medium passage) and judged that reading them as allusions would go beyond the text. The record is silence, not denial.

V2. Scalar potential abandoned. [read: bk 153, 156-157; PDF pp. 164, 167-168] Abraham (1912) let the light speed depend on the gravitational potential; Nordstroem is named with references only (the label "scalar" is QE's and the primer's [web], not Whittaker's). Einstein and Grossmann's ten g_pq "implied the abandonment of the time-honoured belief that a gravitational field can be specified by a single scalar potential-function". Einstein's stated reason: for radiation in a box a single scalar gives "only one-third of the work", "a conclusion which was obviously wrong"; Whittaker adds that his "strongest reason" was general covariance. This is the step that the 1902-03 picture (one scalar potential V) never had to face.

V3. Medium to curvature, in Whittaker's words. [read: bk 157-158; PDF pp. 168-169, page image 169] FitzGerald (1894): gravity "is probably due to a change of structure of the aether, produced by the presence of matter". Whittaker: Einstein's "space" or "space-time" is FitzGerald's aether, and "structure" became "the more precise 'curvature'". In a static gravitational field, weak electromagnetic phenomena follow Maxwell's theory in a medium whose permittivity and permeability vary from point to point; for a point mass the law is (r+1)^3/[r^2(r-1)], credited to E. T. Copson (Proc. R.S. A 118, 1928, p. 184), with the dielectric reading of FitzGerald credited to Wiechert (Ann. d. Phys. 63, 1920, p. 301). The deep read reconstructs r [calc] as the isotropic radius in units of GM/2c^2, giving weak-field epsilon = mu = 1 + 2GM/(c^2 rho) and the full 1.75 arcsec deflection. Whittaker does not use the phrase "not quite perfect"; that is from the 1928 lecture report [web: research/2026-10-07-cambridge-synge.md]. Both attributions stand, and Copson was Whittaker's colleague at Edinburgh [memory, medium].

V4. Lorentz 1900. [read: bk 149-150; PDF pp. 160-161, page image 160] Le Sage's corpuscles replaced by "trains of electric waves of very high frequency" crossing space; a net force between ions "could exist only if in some way or other electromagnetic energy were continually disappearing", so Lorentz rejected it. He then took up the Mossotti type (unlike charges attract slightly more than like charges repel; (gamma - alpha)/alpha of order 1e-35), for which, Whittaker says, no strong physical reason could be given in 1900 (the deep read corrects QE: Whittaker does not say Lorentz "rejected" the Mossotti model). This is a wave-gravity model in the light medium three years before Whittaker's, and its energy-budget objection is the same kind of obstacle the mathematician raised against any medium gravity (negative gravitational field energy, claim 5).

V5. The aether after relativity. [read: bk 248, 103-104, 40; PDF pp. 259, 114-115, 51] Cunningham (1914) gives the aether velocity c at every point; "The scheme is relativistically invariant: so that an objective aether is not necessarily foreign to the point of view of the principle of relativity" (bk 248). Planck's 1911-12 zero-point energy and Nernst (1916), "the aether everywhere might be occupied by zero-point energy" (bk 103-104): the historical root of the 1951 preface's "so rich in physical properties". Einstein "published a paper which set forth the relativity theory of Poincare and Lorentz with some amplifications"; constancy of light speed was "widely accepted, but has been severely criticised by later writers" (bk 40). The Ch. II framing is read on the passages cited; the rest of Ch. II rests on digests.

V6. Caution about general relativity (GR), and alternatives he records. [read: bk 146, 168, 175, 180-182] Light deflection: "there is a discordance"; Mercury: "there are alternative explanations"; the red shift "does not, properly speaking, constitute a test". He reports with respect Whitehead (1922, flat-space action at a distance at c; Synge 1952), A. G. Walker's 1951 flat-space model and Rosen's flat-space programme (offered as a possible account of D. C. Miller's 1925 ether-drift residuals). He writes that the truth of a law of gravitation "cannot be tested by means of controlled experiments in a laboratory" (bk 146), and rejects Mach's principle as needing undetected "enormous quantities of matter" (bk 168). Gerber (1898, 1917) is cited for a retarded-law fit to Mercury (bk 148). These keep the medium-versus-geometry question (4E) open in his own text; they are reported, not endorsed by him or by us.

V7. Vol. 1, Faraday 1851. [read: Vol. 1 PDF p.104; digest, not deep-read] "It is not at all unlikely that if there be an aether, it should have other uses than simply the conveyance of radiations"; Whittaker: this "may be regarded as the origin of the electromagnetic theory of light". The earliest statement in the folder of one medium serving several roles, the idea of Whittaker's 1903 "identical property of the medium". Minor; the QE-requested deep read of Faraday (and of Ch. II, the index and the preface) was not run, so these rest on the digest and a text search.

How this bears on the verdicts

Claims 1-3 (mathematics): untouched; Vol. 2 does not discuss the 1903 or 1904 papers.

Claim 4. The mathematics (4A) is untouched. The physical suggestion meets four statements Whittaker himself printed in 1953: a superluminal gravity speed rests on implausible assumptions; gravity travels at c and observation allows it; gravitational waves travel at c and are produced by moving matter; a single scalar potential was rightly dropped. This is the inventor's own published view, in agreement with the modern tests that already put 4D and 4C at Tested, not supported (GW170817). It does not turn those tests into a test of his 1903 model, and it is not a recantation: he neither retracts nor repeats the 1903 statement, and the 1903 mathematics is not withdrawn. One difference between the pictures: the 1953 gravitational wave is radiated by moving matter; the 1903 superposition is time-independent and its source is left unexplained. His remarks on medium (V3), flat-space alternatives (V6) and "aether" (V5) leave 4E open in his own text.

Claim 5. Strengthened and clarified. The aether he keeps in 1953 is space-time with medium-like properties (epsilon, mu varying from point to point; Cunningham's invariant scheme; zero-point energy), not a mechanical substrate with its own longitudinal waves. This fits the verdict Historically superseded but instructive (material aether) with the vacuum-as-aether proposal Not testable as stated (terminology). Navarro (2020, 2025) characterizes the History's relativity chapter as shaped by Whittaker's aether commitment and priority view [web: research/2026-10-07-sagepub.md, abstract only]. The primary text can now be set beside it: "with some amplifications" and "severely criticised by later writers" (bk 40) and the Ch. II title "The Relativity Theory of Poincare and Lorentz" are confirmed on the page [read]. Whittaker's framing is the author's own and Navarro's is one historian's reading of it; the reader weighs both. This round did not read all of Ch. II.

"Absence of evidence" item 1, updated

Vol. 2 is no longer a missing record. It contains no reply to Stoney and no development of the wave idea, so those records stay dark. - (b) he gave it up as relativity arrived: more strongly supported, for the speed and scalar parts, by his own 1953 statements (gravity at c, gravitational waves at c, end of the single scalar potential). - (a) he regarded it as a representation only: consistent (he never presents the 1903 result as physics in the History, which fits "does not explain the cause of gravity"). - (c) he kept a private or later conviction: not excluded, and for the medium part still alive. He keeps the aether word, identifies it with space-time, gives gravity a medium (epsilon, mu) reading, reports flat-space alternatives and calls the 1952 deflection evidence discordant. This fits Navarro's "convinced that gravitation and electromagnetism were ultimately connected" ([web], a historian's characterization, not Whittaker's words); it equally fits plain historical thoroughness. - Net: the weight moves toward (b) for the speed and the scalar potential, and (c) survives for the medium. Silence in both volumes about his own note is a datum, not a verdict (SOURCE HONESTY: silence in a history written by the man is an anomaly to explain, and it fits all three). - Deciding records, minus Vol. 2: the 1944 Endeavour article, Fisher Library MS COLL 00219 Box 11, the 1903 daily-press articles, the 1927 Proc. R.S. A 116 paper "On electric phenomena in gravitational fields" ([memory, medium] citation, not cited in Ch. V and not obtained), and Copson (1928). Investigator queue (next run): Copson, Proc. R.S. A 118 (1928) 184; Wiechert, Ann. d. Phys. 63 (1920) 301; Lorentz, Proc. Amst. Acad. 2 (1900) 559; plus Korn, Larmor, Bjerknes and Leahy from QE run 1.

Open question answered in part

"What did Vol. 2 and the 1944 Endeavour article say about gravity and the aether?" Vol. 2 (1953) is now answered as above. The 1944 article remains unread.

Changed verdicts

None of the verdict names change; the table below is amended only in the wording noted. Rows touched: 4C, 4D (add Whittaker's own 1953 view as concordant, not as a test), 4E (add that his text records the medium reading of GR and flat-space alternatives, which keeps the generic form Never properly tested), and 5 (add the 1953 evidence). The borderline condition QE set (a Vol. 2 passage discussing his 1902-03 gravity or a 1903-1926 test of it) is not met: none exists, so the result is MINOR, not MAJOR for claim 4.

Corrections to earlier documents (from this revision)

1. This file, line 5 (Source gap) and "Open questions": "Not obtained ... History Vol. 2 (1953)" is wrong as of 13:03 today. Evidence: _text/manifest.json (330 pages, 15 chunks) and the deep read. The 1944 Endeavour article, 1927 paper beyond section 1, the 1928 lecture, From Euclid to Eddington, a Stoney reply and the Fisher papers remain not obtained.
2. This file, "Claim 4" Stoney paragraph (formerly line 55): 'a particle is "switched on" at t = 0' replaced by 'the source changes sign from -1 to +1 at t = 0 (a static -1/r plus a switched-on 2/r)'. The first QE revision recorded this correction but did not apply it at that line; now applied.
3. This file, claim 5 and "Absence of evidence" item 1: "Vol. 2 not read" and "(apparently open on archive.org ... needs owner approval)" are out of date (above). Item 1(b) cites the 1928 lecture's "not quite perfect"; incomplete, since Vol. 2 credits the medium result to Copson (1928) and Wiechert (1920).
4. QE run 2 itself, three details corrected by the deep read: Chapter V ends at PDF p.207 (bk 196), not p.200; Nordstroem is not called "scalar" by Whittaker; Whittaker does not say Lorentz rejected the Mossotti model. QE's self-citation list also omits bk 186 (1931) (the earlier QE list gave 146, 165, 170, 184, 194 plus 269 and 279 outside this chapter).
5. For the orchestrator: claims.md already carries the Vol. 2 update; stale "Vol. 2 not obtained / unread" copies remain in physicist.md, mathematician.md and contrarian.md (original reviewer text, left as written), primer.md (Source note, section 4.8, section 7, section 9, section 1 "as far as we know"), facts.json 1951 summary, and followups.json A07 ("1944 Endeavour ... unread"; add V3 to "known"; label "pp. 206-207" as book pages).

Verdict summary

Files: review/Whittaker-ET/arbiter.md