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Ives: Mathematician review (STEP 2, redo after QE round 2)

Reviewer role: PhD mathematician. I check derivations, notation, units and hidden assumptions, and I answer questions a-d for each claim in claims.md (claims 1-7, with 7a-7f taken separately). Tags: [read] = from a document (page given), [web] = from a web page (URL given), [memory] = recalled, with confidence, [calc] = my own arithmetic or SymPy/mpmath/NumPy check in this review, assumptions stated.

This file replaces the first Mathematician review in full. The first review judged claim 4 (gravitation) from one abstract and called claim 3 "limited by the unread 1945 and 1951 papers". Both limits are gone: the 31 Ives papers are in the manifest, and deep reads exist for 1941 (Ives-Stilwell II), 1945, 1947, 1948, 1949, 1939 and the 1937-38 papers. Claim 4 changes verdict. Several other statements change. See "Corrections to earlier documents" at the end.

Source gap (what this review rests on)

Read by me from deep reads (which read page images for equations): Ives-Stilwell 1938 and 1941 (Part II), Ives 1937 (JOSA 27:263, 27:310, 27:305), 1938 (JOSA 28:296), 1939 (JOSA 29:183), 1945 (Phil. Mag. 36:392), 1947 (Phys. Rev. 72:229), 1948 (JOSA 38:413), 1949 (JOSA 39:757), the Pobedonostsev-Parshin paper (6 pages), the 1952 and 1953 mass-energy papers.
I read myself from the extracted text: the 1942 radiation-pressure paper (JOSA 32:32) and the 1942 ripple-tank paper (JOSA 32:25), at the lines quoted below.
Digest only: 1924-26 photoelectric papers, 1933 Ives-Fry, 1937 JOSA 27:177 and 27:389 (the latter I read earlier as OCR), 1939 chronotopic interval and double Fizeau wheel, 1940 moving mirror, 1943-44 wave-packet papers, 1947 historical note (read earlier as OCR), 1948 JOSA 38:879, 1950, 1951 papers, 1951 Rumford lecture. Claims resting on these are marked.
Ruderfer's appendix in Moray's book is quoted as QE and the deep reads quote it (review/Morray-TH/_text, Sea of Energy 5th ed. pp.271-274); I did not re-read it.
Still not available: JOSA 42:540 in a legal copy, Rev. Sci. Instr. 9:283 (1938), Nature 168:246 (1951), Hazelett and Turner (1979), R. C. Jones JOSA 29:337, Otting.

Symbols and two ideas the reader needs

c: speed of light. beta = v/c. gamma = 1/sqrt(1 - beta^2) (the Lorentz factor of special relativity). lambda0: wavelength of the line from an atom at rest (H-beta, 4861.3 Angstrom; 1 Angstrom = 10^-10 m).
Second-order Doppler shift: the part of a wavelength change proportional to beta^2, which does not change sign when the direction of motion reverses. Ives-Stilwell looked for it.
Ives's y in the gravitation papers (he writes gamma, but it is NOT the Lorentz factor): y = 1 - 2GM/(R c^2), where G is Newton's constant, M the attracting mass and R the distance from its centre. It is the number by which a clock's rate and a ruler's length (squared) change at distance R. For the Sun's surface 2GM/(R c^2) = 4.25 x 10^-6 [calc]. The Schwarzschild solution of general relativity (GR) is the statement that near a mass, clocks tick at the rate sqrt(y) and radial rulers are shortened by sqrt(y). I introduce it in this plain way because the reader profile does not assume GR.
Poincare group, in plain terms: the set of coordinate changes (rotations, shifts, and Lorentz boosts) under which the laws of special relativity keep their form. "Lorentz-covariant" means "keeps its form under a boost".

Claim 1. Moving clocks run slow (second-order Doppler shift lambda0 (1 + beta^2/2))

a. Is the mathematics internally consistent?

Yes, with one misprint in the 1938 paper itself. The derivation is unchanged from my first review, so I summarise it. For the two viewing directions the relativistic Doppler formula gives lambda_B = lambda0 (1 - beta cos theta)/sqrt(1 - beta^2) and lambda_R = lambda0 (1 + beta cos theta)/sqrt(1 - beta^2). Their average is lambda0/sqrt(1 - beta^2) = lambda0 (1 + beta^2/2 + ...) exactly in theta, so the experiment does not need an exact right-angle view [calc]. The p.216 formula "lambda0 (1 - V^2/c^2)^(1/2)" is a frequency factor attached to a wavelength; the sign of the exponent is wrong for a wavelength. It is in the printed 1938 paper (page image) and is not a fault in the physics. Table III's computed column reproduces to about 1% [calc].

New in this round, Part II (JOSA 31:369, 1941): - The tube ran to about 43,000 V. My own check: for H2+ (mass 2 protons) beta^2 = 2eV/(M c^2) = 4.583e-5 at 43,000 V, so the centre shift lambda0 beta^2/2 = 0.1114 Angstrom; at 42,260 V it is 0.1095 Angstrom [calc]. The paper's observed 0.113 and 0.1145 Angstrom at 42,260 and 42,280 V (computed 0.1098 and 0.1073 from the measured first-order line separations 32.50 and 32.29 Angstrom) match this [read: deep/Ives-Stilwell-1941, Table I; my check: 32.29^2/(2 x 4861.3) = 0.1072 Angstrom, 32.50^2/(2 x 4861.3) = 0.1086 Angstrom, so the first computed entry is about 1% above what its own separation gives, the second exact] [calc]. - That shift is 1.24 times the 0.09 Angstrom H-beta doublet separation [calc]. The 1938 caveat was that an intensity change between the two doublet components could fake a shift smaller than their separation. A shift larger than the separation cannot be produced by moving intensity from one component to the other (the centre of gravity of two lines cannot move outside the interval between them). So the 1938 caveat is answered by arithmetic, not only by the authors' statement. - The Part II sign pattern is worth a mathematician's remark. In 1938 seven of eight observed shifts lay below the computed one (mean ratio 0.964). In Part II nine H2 points sit on average about 3% above the computed curve (six above, three at or below) [read: deep/Ives-Stilwell-1941], and the highest two are +2.9% and +6.7% [calc from the table rows above]. A systematic error with one sign would not flip between the two papers; it looks like scatter plus plate-to-plate differences, as the authors' mirror-reflectivity explanation allows. I treat the two sets as consistent with coefficient 1/2 at the few-percent level, which is what the earlier review concluded for 1938 alone. - One H3O table entry looks misprinted (0.0012 where about 0.0118 is expected) [read: deep]. It is a misprint of a decimal place, not a data problem.

b. Consistent with established experiment?

Yes. It is the relativistic Doppler factor tested at 10^-9 in storage rings (Botermann 2014, 7Li+ at beta = 0.338; the relation gamma sqrt(1 - beta^2) = 1 holds to within 2.3 x 10^-9) [read: earlier review of arXiv 1409.7951, abstract and p.2].

c. Tested? By whom? Replicated?

Ives and Stilwell 1938, 1941; Otting 1939; Mandelberg and Witten 1962 (exponent 0.498 +/- 0.025, second hand through Giuliani); modern storage-ring tests as above. All support the coefficient.
The 1938 data scatter more tightly about the prediction than the paper's quoted line-separation error allows (chi-square 1.5 to 3.2 for 8 points depending on the assumed error; calc and assumptions in my first review, unchanged). This agrees with Mandelberg and Witten's remark that the fit is better than the real uncertainty. It is a reason to take the quoted 3-4% precision as optimistic; I make no stronger claim.
Dissent kept in the record: Pobedonostsev and Parshin, SPIE 1121 (1989), a repeat at 150-300 keV and 1.5-2.0 MeV. Deep read of the 6 pages: observed second-order shift 40 to 82 per cent smaller than special relativity at 150-300 keV (observed/SR from 0.17 to 0.60), within about 10 per cent either side at 1.5-2.0 MeV; only statistical error stated (0.04 Angstrom per line position); no systematic-error budget; the two viewing arms are separate optical paths, one through a fibre [read: deep/Pobedonostsev]. My arithmetic on their 150-300 keV numbers (from the deep read's [calc]): the deficit is 0.31 to 0.52 Angstrom, many times the stated 0.04 Angstrom, but a misalignment of the two arms of about 0.5 degrees would produce a deficit of that size, and the authors' own SR columns imply viewing angles differing between series (78.2 versus 77.2 degrees) [calc, reported in the deep read]. I add one check of my own on the voltage-to-speed step, since I could not see it in the first review: at 150 keV for an H2+ beam the speed per atom is beta about 0.013, where relativistic corrections to kinetic energy are of order 10^-4 relative, far too small to account for deficits of 40-80% [calc: kinetic energy 75 keV per proton, rest energy 938 MeV, ratio 8e-5]. So the voltage conversion is not where the deficit comes from. This is one group; I found no replication; their own classical model (a Marinov-type angular dependence) predicts a shift 5% above SR at their angle, so it does not explain their deficit either [read: deep/Pobedonostsev].
Faraj (emission-theory advocate) claims the experiment shows only an asymmetry in the longitudinal Doppler shift; the deep read finds the attribution to Ives wrong and Faraj's supporting algebra invalid [read: deep/gsjournal-727]. I accept that finding; I have no independent check of the algebra beyond the deep read.
R. C. Jones's objection (JOSA 29:337, 1939) that the undisplaced reference atoms might themselves be carried along: Part II reports an imparted velocity of "about 750 meters per second" with the mirror covered at 14,000 V and no central-line displacement within error with it in, and a reversed double-ended tube giving 0.0744 against 0.0717 Angstrom computed at 28,300 V (one point, no error bar). My check: at 28,300 V the computed centre shift is 0.0733 Angstrom, 2% above the paper's 0.0717 [calc]. A speed of 750 m/s is beta = 2.5e-6, whose first-order Doppler shift is 0.012 Angstrom, about 10% of the 0.11 Angstrom signal at the top and 25% of the 0.047 Angstrom signal at the old maximum [calc]. That is why the covered-mirror result matters, and why "no displacement within error" with the mirror in is the number to look for. The paper's answer rests on that comparison; I did not see an error bar for it.

d. If true, what would have to change?

Nothing; it is standard physics.

Verdict, claim 1

Established. The answer to the 1938 caveat is in hand (Part II, shift above the doublet separation). Dissent (Pobedonostsev and Parshin) is stated above and is not a verdict-changer against 10^-9 storage-ring tests, but it is an unreplicated report, not an explained one.

Claim 2. The result shows a real slowing in a fixed ether and favours Larmor-Lorentz over Einstein

a. Is the mathematics internally consistent?

The logic is consistent; the conclusion exceeds the data. Ives's argument (1938 p.226, 1937 JOSA 27:177 and 27:389, 1947 Note): Michelson-Morley fixes only the ratio of lengths along and across the motion; Kennedy-Thorndike with the Ives-Stilwell clock rate then fixes the absolute contraction. His 1937 papers say Kennedy-Thorndike alone "cannot by itself test the Larmor-Lorentz hypothesis" without the contraction exponent, and propose the canal-ray measurement to fix it [read: digest; qe.md]. The 1947 Note's family of contractions and clock periods I re-derived earlier [calc]; at 18,350 V the four cases give 0.095, 0.0475, 0.063 and 0 Angstrom against 0.047 observed.

Hidden assumption, unchanged: "stationary in the ether" is not what the experiment measures; it measures atom speed relative to the laboratory. Ives's own 1937 algebra (JOSA 27:389, eta = 1 case) shows that with the Larmor-Lorentz clock factor the laboratory's ether velocity drops out of the centre shift, which then depends only on the relative velocity of source and spectrograph. In Ives's theory the experiment therefore cannot locate the ether frame. Part II adds a small, relevant fact: its six pages credit the prediction to Larmor and Lorentz and use neither "ether" nor "relativity" [read: claims.md claim 2]. Ives's 1937 clock-paradox papers (JOSA 27:305 and Nature 168:246, the latter not obtained) say that the ether frame decides which clock lags; I judge below (claim 3) that this is a convention-dependent statement.

b. Consistent with established experiment?

The measured number, yes. The strong reading (real slowing relative to a fixed ether, so special relativity is not needed) makes the same predictions as special relativity in every experiment that the Robertson-Mansouri-Sexl framework covers ([web] https://en.wikipedia.org/wiki/Test_theories_of_special_relativity, secondary). Jones 1939 derived the same predictions from special relativity.

c. Tested?

The measurement, yes. The ether-versus-relativity reading, no experiment can separate the two because every observable is the same function of the same invariants. Searches for a preferred frame (Ruderfer 1960 proposal; Preikschat's rotor limit 2.0 +/- 5.2 cm/s; storage-ring limits in the RMS framework) have not found one; those are physicist matters and I only record them. The 1938 press reception, with the NYT of 27 April 1938 quoting Einstein ("the most direct proof yet brought forth in support of the theory of relativity") and the report that it verified "the existence of ether" [web: nytimes.com/1938/04/27/archives/einstein-hails-proof-dr-ives-provides-best-support-for-relativity.html, from the Investigator's note] shows both camps claiming it.

d. If true, what would have to change?

If "a real slowing relative to a privileged frame" only means that such a frame exists and cannot be detected, nothing in the mathematics changes and nothing is tested. If it meant a velocity-dependent effect on the laboratory's motion through that frame, an experiment would have to show it; none has (RMS limits of order 10^-8 on the clock-rate parameter alpha and 10^-6 on light-speed anisotropy from satellite clocks, as quoted by Botermann).

Verdict, claim 2 (split)

Weak form (Ives-Stilwell fixes the clock-rate coefficient left open by Michelson-Morley and Kennedy-Thorndike): Established.
Strong form (the experiment shows a real slowing relative to a fixed ether and so decides for Larmor-Lorentz over special relativity): Not testable as stated.
As a position in 1938: Historically superseded but instructive. Robertson, who wrote the relativity assessment in the 1956 memoir, was a referee and critic of Ives; that stake is stated.

Claim 3. Ives's alternative theory can be built from classical ether physics and is special relativity in different clock-setting conventions

I now judge the three routes separately.

3(a) The 1937 transported-rods-and-clocks route

a. Internally consistent. My first review only skimmed JOSA 27:263. The deep read and the 40-digit check recorded in claims.md show that for finite transport speeds the 1937 equations agree with special relativity applied to the same transport procedure, and reduce to the Lorentz transformations as the transport speed goes to zero. The 1937 equations carry misprints (eqs. 14, 25, 33, 34, 38-40) that do not change the conclusions [read: claims.md; deep]. The difference between "transport-synchronised" and "Einstein-synchronised" times is a first-order effect in the transport speed, about 1.7 ps over 1 km at 300 m/s [read: claims.md; deep]. That agrees with my own estimate of a 5.6e-17 s offset for walking pace over 10 m in the 1949 case below (same formula). b. Consistent with experiment: yes, because it reproduces the Lorentz transformations. c. Tested: the shared predictions, yes. No measurable difference from special relativity is named by Ives. d. If true: a privileged frame and a mechanism for the contractions; no change in equations.

3(b) The 1943-45 conservation-law route (JOSA 33:163, 34:222; Phil. Mag. 36:392)

This is the item QE asked me to check for hidden assumptions, and I have the deep read of the 1945 paper.

a. Consistent as far as written, but not a derivation from conservation laws alone. The deep read lists these inputs: Maxwell radiation pressure on a moving mirror P = 2E(c - f)/(c + f), momentum E/c for a wave train, the mass equivalent E/c^2 for radiation, Newtonian space and time, and conservation of mass, momentum and energy. From these Ives obtains the mass law M = M0/sqrt(1 - v^2/c^2) (eq. 18), then length and clock factors D2, D3, D1 from a light-clock argument. - My answer to QE's question "is the E/c^2 mass already relativistic input?": no. Radiation mass equivalent E/c^2 follows from radiation pressure and momentum in Maxwell's theory (Poincare 1900) and is not taken from special relativity [read: deep/1945 Step B; memory: medium for the attribution to Poincare 1900]. But the next step does assume something. Ives assumes that "conservation of mass" holds for all the masses and mass equivalents involved; the particle's mass must grow by the absorbed energy over c^2. That is the inertia of energy applied to matter, which is the physical core of relativistic dynamics, though not a formula copied from it. - The "extra term must drop out" step (Step F) is where I put the largest hidden assumption. When Ives repeats the impact on a platform moving through the medium, there are two unknowns and the requirement that a quantity depend on the platform speed V alone leaves a free function. The deep read gives a counterexample: with c = 1, V = 0.5 and rest-case final speed 0.3, choosing a bracket 1 + h with h = 0.1 satisfies energy and momentum conservation exactly, with w = 0.709 instead of 0.696 [calc in deep read]. Setting h = 0 gives the exact relativistic Doppler factor and Einstein's velocity addition. That is the relativity principle for this one process, brought in as an assertion. - The light-clock step requires all clocks, in both orientations, to share the rate D1. Equal rates in both orientations is the Michelson-Morley null result; equal rates for different clock types is the content of Kennedy-Thorndike. Ives names these experiments himself (1945 p.11), so the closure of the argument uses what the experiments established. - Misprint: eq. 19 prints Mc^2 for M0 c^2 [read: deep]. b. Consistent with experiment: yes, since it reproduces the Lorentz transformations and the mass law. c. Tested: Ives cites Kaufmann (mass), Michelson-Morley (length), Ives-Stilwell (clocks) as supporting "that these changes of dimensions are real" [read: deep]. These are tests of the results, not of the route. d. If true: Ives says the "Special Principle of Relativity" is a "superfluous hypothesis" [read: _text 1945]. The mathematics shows it is not removed: it moves into the two premises above, where it is not stated as a principle. So if Ives were right, mainstream physics would have to accept a different axiom set, not different equations.

3(c) The 1948-51 operational route and the 1949 "internal inconsistency" charge

The QE flagged that the earlier "SR internally contradictory: Refuted by calculation" verdict was reached from the 1953 summary. I have the 1949 deep read, which reached its conclusion by symbolic algebra, and I checked the key identity.

a. The charge: if clocks are admitted for distant time-setting, special relativity's derivation "leads to an internal inconsistency", and the Lorentz transformations are "an approximation, owing their simplicity of form to the exclusion of clocks in the determination of epoch" [read: text 1949]. The deep read finds: Ives's equations (19)-(20) are term for term the Lorentz transformation written with clock readings that each lag Einstein time by k r' (r' the distance from the place where the setting clock started); SymPy confirms this identically. Special relativity predicts exactly Ives's one-way result c/(1 - ck) for a clock carried at finite speed q, which tends to c as q goes to 0. My own algebraic comment, which makes the point sharper: a one-way speed c/(1 - ck) in one direction and c/(1 + ck) in the other is the Reichenbach synchronisation family c+ = c/(2 epsilon), c_- = c/(2 (1 - epsilon)) with epsilon = (1 - ck)/2. The round-trip speed 2/(1/c_+ + 1/c_-) = c exactly for every k [calc by hand]. So the only freedom is the label epsilon (a convention about how distant clocks are set); the measured round-trip speed does not move. Ives's "approximation" is a statement about which epsilon the transport procedure produces, not a failure of the Lorentz group. Size of the offsets: to first order k = q/(2 c^2), so the measured one-way speed is about c + q/2 and the clock offset r' q/(2 c^2). For a clock walked at 1 m/s across 10 m, 5.6e-17 s and a fractional speed change 1.7e-9; for a jet at 250 m/s over 10,000 km, 14 ns [read: deep/1949; I reproduced the formula arithmetic: 10 x 1/(2 x 8.988e16) = 5.56e-17 s] [calc]. Ives rejects the q goes to 0 limit because it "could not be performed" (an operationalist point, not a mathematical one). The same limit is accepted as an experimental fact in his 1950 paper [read: deep]. So: no inconsistency. What survives is the conventionality of one-way simultaneity, a recognised point of the mainstream literature. b. Consistent with experiment: yes, because it predicts the same measurable quantities. c. Tested: Ives names no experiment; his 1950 and 1951 papers cite only Michelson-Morley and Ives-Stilwell, which special relativity predicts identically [read: deep/1949]. The 1948 JOSA 38:879 paper, in which "rod-clock-quotient" is first defined (the earlier files say 1953), and the 1950 paper are digests only to me. d. If true: Ives writes that abandoning the constancy principle "removes the basis for geometric theories claiming fusion of space and time" [read: digest 1951 via claims.md]. Mathematically, nothing of that follows: the Minkowski interval is invariant under the Lorentz group whichever way clocks are set.

Verdict, claim 3 (split)

Ives's 1937 transported-rod-and-clock theory as a re-description of special relativity: Not testable as stated (equal predictions), mathematics consistent; also Historically superseded but instructive.
The 1943-45 claim that the Lorentz transformations follow from conservation laws and Newtonian space-time with the relativity principle "superfluous": Not testable as stated as an explanation; in strict logic the relativity principle is carried by the premises (the free function h, clock universality), so "superfluous" does not hold as a logical claim. I do not call it Internally inconsistent, since nothing in the derivation contradicts itself; it is incomplete.
The 1949 charge of internal inconsistency in special relativity: Refuted by calculation (his own equations (19)-(20) are the Lorentz transformation in a non-Einstein synchronisation, round-trip speed c for all k). This now rests on the 1949 paper itself, not only the 1953 summary.

Claim 4. Ives's gravitation theory is the Schwarzschild solution of GR expressed as a fixed ether (replaces the earlier version, which rested on an abstract)

Background for the reader (plain language)

General relativity (GR) says that near a mass M clocks run slow and radial rulers are shortened by the factor sqrt(y) (y = 1 - 2GM/(R c^2)); a planet moves along the path that a free body takes in that distorted arrangement. The Schwarzschild solution is that statement written exactly for a single, non-rotating mass. Ives's route is different in words only: he says a body at rest in the field has the dimensions, clock rate and mass of a body moving with the speed it would acquire falling from infinity under Newton's law, V^2 = 2GM/R. Then 1 - V^2/c^2 = y automatically.

a. Is the mathematics internally consistent?

I check the three papers in turn.

1939, JOSA 29:183 (deep read, reproduced by the deep reader with SymPy). From A1 (the equivalence as motion), A2 (null reading of an interferometer preserved) and A3 (Fitzgerald contraction relative to the local light speed) Ives obtains the local light speeds radial c_R = c y and transverse c_T = c y^(1/2) (eq. 2), c_theta^2 = c^2 y^2/(cos^2 theta + y sin^2 theta) (eq. 3), and the interval ds^2 = c^2 y dT^2 - R^2 dtheta^2 - dR^2/y, which is exactly Schwarzschild's in Schwarzschild coordinates, not a 1/c^2 approximation [read: deep/1939]. I confirm the identification through a numerical check of my own: Fermat's principle with the light-speed field above gives, for the null-ray orbit, (dR/dphi)^2 = R^4/b^2 - R^2 + 2mR (m = GM/c^2, b the impact parameter), which is the standard Schwarzschild null-geodesic equation. Integrating it for a ray grazing the Sun gives a deflection of 1.7512 arcseconds [calc, mpmath; the closed form 4GM/(b c^2) = 1.7512 arcseconds]. So Ives's light-speed field gives the full GR deflection, not the "half" value 0.88 arcseconds [calc]. He states the red shift and light bending in words only, with no numbers [read: deep/1939]. Note the dependence: A1 uses the exact Newtonian escape speed V^2 = 2GM/R with no further correction. That makes y have no (GM/Rc^2)^2 term. I examine that choice below.

1947, Phys. Rev. 72:229. A reinterpretation of Birkhoff's equations, not a new force. Rewritten in ordinary time and force, they contain a force normal to the motion of size v^2/c^2 times the normal part of gravity; Ives says an infinite number of energy-neutral extra terms could be written and that "an independent physical derivation of these transverse forces would be welcome", so he does not endorse Birkhoff's theory as physics [read: deep/1947]. For Mercury v is about 48 km/s so v^2/c^2 is 2.6e-8 [calc: (4.8e4)^2/(9.0e16) = 2.6e-8].

1948, JOSA 38:413 (deep read from page images). The postulate extended to mass gives gravitational mass m_g = m0/S with S = (y - rdot^2/(c^2 y) - r^2 thetadot^2/c^2)^(1/2). At low speed and weak field, m_g = m0 (1 + GM/(r c^2) + v^2/(2 c^2) + ...) [read: deep; calc by SymPy series].

Correction to my first review: I wrote that the abstract's mass factor expands to m0 (1 + v^2/(4 c^2)). That is an arithmetic error on my part. The expression under the square root is about 1 - 2GM/(rc^2) - v^2/c^2, whose square root is 1 - GM/(rc^2) - v^2/(2c^2), so the velocity part is the ordinary special-relativistic v^2/(2c^2), and the new part depends on position. My statement that the theory "appears to ascribe a velocity-dependent factor to the gravitating mass different from the inertial one" was wrong.

Reproduction of the numbers [calc]. The perihelion advance per orbit from Ives's orbit equation d^2u/dtheta^2 + u (1 - 6 G^2M^2 b^2/(c^2 h^2)) = K-bar is pi times epsilon, epsilon = 6 G^2 M^2/(c^2 h^2). With h^2 = GM a (1 - e^2), the advance is 6 pi GM/(c^2 a (1 - e^2)). For Mercury (GM = 1.32712e20 m^3/s^2, a = 5.7909e10 m, e = 0.20563, 415.2 orbits per century) that is 5.019e-7 radian per orbit, 42.98 arcseconds per century. Sommerfeld's special-relativistic mass increase with a fixed Newtonian force gives epsilon = G^2M^2/(c^2h^2), one sixth: 7.16 arcseconds per century [calc; deep read's integration confirms both ratios]. The "one sixth" holds for the fixed-force reading; if the force also scales with the moving mass the deep read's integration gives one third [read: deep/1948]. Both statements are conditional on how force is defined.

Is anything fitted? No numerical constant is tuned; the 6 comes out of the algebra. But there are choices, and the first Mathematician and arbiter reading ("chosen to reach 43") was too hasty in the one direction and too generous in the other. The choices are:

F_theta = 0 (constant moment of momentum), justified by "the general validity of the conservation laws" [read: deep/1948 p.1-2], and splitting the transverse force (11) into the pair (12)-(13) by analogy with Birkhoff. Ives's final footnote concedes that on the GR relations alone "there is no ground for favoring (22) and (23) over the simple and symmetrical equations (12) and (13)" [read: deep/1948 p.4]. The deep read shows the two sets give identical accelerations, so these choices do not move the orbit.
The exact form y = 1 - 2GM/(R c^2), with no term in (GM/(R c^2))^2. I did not find this examined in the earlier reviews. Ives writes that departures from the Newtonian law "would enter as additional terms in 1/c^4 in (7)", so one may use (7) "provided we restrict ourselves to terms in 1/c^2" [read: deep/1948 p.2]. That bookkeeping is not right for forces. The Newtonian pull is the gradient of the 1/c^2 term of y (times c^2); a (GM/(R c^2))^2 term in y would therefore contribute a force of relative order GM/(R c^2), which is the same order as the perihelion effect. I tested what that does. Take y = 1 - 2m/r + kappa m^2/r^2 (m = GM/c^2) in the same metric form ds^2 = y dt^2 - dr^2/y - r^2 dtheta^2 and linearise the orbit equation u'' = m/h^2 - u + 3 m u^2 - kappa m^2 u/h^2 - 2 kappa m^2 u^3 about the circular orbit. SymPy gives the oscillation frequency squared omega^2 = 1 - (6 - kappa) m^2/h^2 + O(m^3) [calc], so the perihelion advance is (6 - kappa)/6 times the Ives/GR value. For kappa = 0 it is 43 arcseconds per century; for kappa = 2 (for example y = exp(-2m/r) to second order) it would be two thirds of that. So the 43 arcseconds is a consequence of the specific choice "the Newtonian escape speed, exactly", not of the ether idea. It is an input, but one Ives states openly ("by the demands of internal consistency, also exact" in the note added in proof [read: deep/1948 p.4]) and it coincides with GR's Schwarzschild coordinates, where Mercury's measured value is matched. I flag it as the one place where the freedom the earlier reviewers feared is real: the "no free parameter" description is true only given kappa = 0, which is a postulate about the 1/c^4 structure of the field. Whether nature picks kappa = 0 is, of course, what the observed 43 arcseconds (with other tests of GR) says it does. [Parametrised post-Newtonian statement for context: the factor (2 + 2 gamma_PPN - beta_PPN)/3 multiplies the GR perihelion advance and the tests give it as 1 within about 10^-4 to 10^-5; memory: medium, not checked on the web. I use it only as context, not for a verdict.]
Exactness. The orbit equation (25) and force equation (23) are first order in 1/c^2, as the paper says in the text ("Agreeing to discard terms in 1/c^4"). In the note added in proof Ives claims more, that the relation S = gamma/b^(1/2) and gamma = 1 - 2GM/(rc^2) are "also exact". The deep read confirms the three governing integrals (the interval, the angular-momentum constant h, the energy constant) are exactly Schwarzschild's, but finds that the printed force equation (23) differs from the exact Schwarzschild radial equation by terms that give no perihelion advance (a constant times GM/r^2, which only rescales GM, and a radial force proportional to rdot^2/r^2, whose perihelion shift integrates to zero over an orbit; for circular orbits it implies thetadot^2 = (GM/r^3)(1 - 2GM/(rc^2))/(1 - 3GM/(rc^2)) against exactly GM/r^3, a difference of about 2.5 parts in 10^8 for Mercury, absorbed in the fitted GM of the Sun) [read: deep/1948; calc]. So: exact for the integrals; first order for the force equation; Ives's claim "also exact" holds in the first sense only. A misprint (a dot for a minus sign under the root) is in Eq. (23) [read: deep/1948].

Internal consistency verdict for (a): the mathematics is internally consistent. The one weakness is the bookkeeping remark about 1/c^4 terms above; the author resolves it by stating the exact form as a postulate.

b. Consistent with established experiment?

Mercury's perihelion: yes, 42.98 arcseconds per century matches the GR value [calc]. Light deflection: the full GR value follows from his own light-speed field (1.7512 arcseconds at the limb) [calc]. Gravitational red shift: the clock-rate factor sqrt(y) is the GR one [read: deep/1939]. Shapiro delay: Ives does not discuss it, but because his light-speed field is Schwarzschild's, the delay follows from the same field; I computed nothing here beyond the identification of the metric. Binary pulsars and gravitational waves are not addressed, and could not have been in 1948 [read: deep/1948]. His own limit statements: the 1939 result is "of purely optical significance", has "no non-Euclidean geometry, nor new law of gravitation", and "calls for anisotropy of the ether ... but makes no suggestion as to how this anisotropy affects other than optical phenomena" [read: deep/1939]. The single-central-mass, static, non-rotating case is all that is covered. No frame dragging, no multi-body, no radiation reaction.

Two further mathematical observations.

Ruderfer's statement, as quoted ("any Lorentz-ether theory of gravitation which provides the static Newtonian gravitational force is necessarily equivalent observationally to general relativity"), is stronger than the mathematics supports. A Lorentz-covariant flat-space theory with the Newtonian static limit does not by itself fix the 1/c^2 terms. Nordstrom's scalar theory (1913) has the Newtonian limit and a different (zero) light deflection, and Brans-Dicke and Whitehead's theory differ from GR in other terms [memory: high for Nordstrom's zero light deflection and Brans-Dicke; not web-checked, and not used as a verdict]. What gives Ives the GR values is the equivalence postulate with the exact y, not the presence of an ether. Birkhoff's own flat-space theory also gets the classic tests; it is criticised for requiring a super-hard equation of state for matter and for lacking a derivation from more general propositions [web: https://arxiv.org/pdf/gr-qc/9912003, search summary of "Field Theory of Gravitation: Desire and Reality"; I read only a search-result summary and did not open the paper]. That answers the web check QE asked for on Birkhoff, at that level.
In GR the coordinate R is a convention; in Ives's ether it is a physical distance measured by uncontracted ("Lorentz stationary") rulers. The two are consistent: a radially contracted ruler counts more rulers over the same ether distance, so proper radial distance = ether distance / sqrt(y) = the Schwarzschild proper distance dR/sqrt(y) [calc, by comparing his A1 with the metric]. So Ives's flat Euclidean ether space with coordinate R maps onto the Schwarzschild solution without remainder.

c. Tested?

The tests of GR (perihelion, deflection, red shift, radar delay) are tests of every theory with these formulas, which includes Ives's. No test separates his statement from GR's, because it is the same set of predictions. His one open door is a conjecture, which he expects to close in GR's favour: if invariance of the interferometer null does not hold for large interferometers, "a sufficiently large and sensitive interferometer could be used to detect motion in a gravitational field" [read: deep/1939]. No such experiment is on record. In the 1939 chronotopic-interval paper (JOSA 29:294, digest) he gives a table of combinations of contraction and clock rate that all give a null interferometer result, and says choosing among them needs experiments where the gravitational parameter varies, such as near the Sun [read: digest; qe.md]. So Ives himself names where his framework is underdetermined.

Reception: Robertson refereed the 1947 paper, called the "Lorentzian" framing "a bit (ca. 40 yrs) out of date", and it was published [read: Investigator's note, Lalli 2016]; Ruderfer's call that "the continued neglect of Ives' gravitational theory is unwarranted" is quoted in claims.md. The papers were not tested and refuted; they were not tested at all because they predict the same numbers as GR, which means there was nothing to test.

d. If true, what would have to change?

Mathematically, nothing in predictions for one central mass. Conceptually, the geometric curvature of GR would be replaced by a state of the ether (anisotropic light speeds, position-dependent mass, clock rate and length) with Newtonian absolute space and time. That is an interpretation. What would change is the answer to "what is gravity", not "what happens". The reading in the question ("only a difference of explanation") holds with the refinement recorded in the deep read: relative to special relativity alone, gravity needs position-dependent mass, clock rate and length, not only speed-dependent ones (speed-dependent mass gives one sixth of the advance).

Verdict, claim 4 (split)

Predictions for one static central mass (red shift, light bending, Mercury's perihelion advance): Established (they are the GR values, which are established; Ives derives them, with no fitted constant, from an equivalence postulate whose one hidden input is y exact, kappa = 0).
The claim that a Birkhoff-type transverse force or Ives's version of it is a separate physical force beyond GR: Not testable as stated (Ives himself says the transverse force and the position-dependent mass factor are the same effect seen two ways, and the force pair is not preferred by GR; any energy-conserving transverse term could be written).
The ether reading ("anisotropy of the ether", absolute space and time): Not testable as stated.
Extensions beyond one static mass (Shapiro delay, pulsars, gravitational waves, rotation): Never properly tested for Ives's theory as such, because he did not work them out. Shapiro delay would follow from the same light-speed field.
Replaces the earlier verdict "Never properly tested (low confidence)" and the statement that the velocity terms were "presumably chosen to reach 43".

Claim 5. Einstein did not derive E = mc^2 in 1905

Unchanged from my first review except for one point. Part A (the circularity charge): Einstein's chain gives K0 - K1 = L (gamma - 1), and L (gamma - 1) = L beta^2/2 + O(beta^4), and K = (1/2) m v^2 at low speed gives m - m' = L/c^2 with no use of the result. The factor L/((m - m') c^2) appears only after the particle formula K = m c^2 (gamma - 1) is inserted into Ives's Table I. Ives's remark that subtracting the two observers' equations "should be obvious a priori" to give 0 = 0 is wrong: the difference is L (gamma - 1), nonzero [calc, SymPy, first review]. Ives's 1953 Note narrows the charge to "the two-observer data alone do not fix the constant" and concedes that "something must be added" [read: 1953 Note]. Part B (momentum conservation): net momentum of the two pulses (L/2c)[(1 + beta) - (1 - beta)]/sqrt(1 - beta^2) = L v/(c^2 sqrt(1 - beta^2)); conservation gives m - m' = L/c^2 [calc]. The algebra is correct; the input m gamma v for the massive body is a relativistic momentum.

New points from the Ives-papers update: - The 1945 paper (claim 3b) gives Ives's earlier route to the same relation, taking the radiation mass equivalent E/c^2 from Maxwell's radiation pressure and then using it as a premise. So Ives's own derivations use the relation as an input (the 1945 inertia-of-energy premise). This does not make the 1952 charge wrong or right; it shows the 1945 and 1952 papers argue from different starting points. - Journal: JOSA 42(8):540-543, August 1952 (Ohanian's "Am. J. Phys." is a slip). The abstract says Einstein's reasoning, "questioned by Planck [1907], is defective" [read: first review]. - Ohanian's own critique (non sequitur outside slow internal motion) stands, and Planck 1907 and Laue 1911 are the history ([web] https://en.wikipedia.org/wiki/Mass%E2%80%93energy_equivalence, secondary).

Verdict, claim 5 (split)

Strong form (Einstein's 1905 derivation is circular): Refuted by calculation. Ives partly withdrew it in 1953.
Weak form (Einstein's 1905 paper does not give a fully general derivation): Established as a statement about the proof (Planck, Laue, Ohanian). E = mc^2 itself is not in dispute.

Claim 6. Atoms are not clocks (Bonizzoni and Giuliani, Giuliani; not Ives's claim)

Unchanged. a. Giuliani's version of Schrodinger's 1922 argument obtains the photon energy E_p = E_p0 sqrt(1 - beta^2)/(1 - beta cos theta1), formally identical to the wave Doppler formula; the recoil correction E_p0 = Delta E (1 - Delta E/(2 E1)) is about 1.4e-9 for H-beta (Delta E = 2.55 eV [memory: high], E1 = 938 MeV) [calc]. b. Same numbers for this experiment. c. Only Giuliani and Bonizzoni-Giuliani; A. I. Miller holds the opposite view. d. Pedagogy, not physics.

Verdict, claim 6

Not testable as stated (an interpretation with no measurable difference for this experiment); the mathematics is consistent.

Claim 7. Ives's separate physical claims (items a-f of claims.md)

Not all are "same prediction, different words". I take each in turn, and say which kind it is.

7(a) The photon will "go the way of the caloric" (1951 Rumford lecture)

a. Not a derivation. Ives's argument is that photons would have to "exist and not exist alternately" in the loops and nodes of a standing wave [read: digest; QE section 3.9], and he concedes "the difficulty of arriving at an explanation of all optical phenomena, such for instance as the Compton effect". Mathematically, the standing-wave argument is a misreading of what the quantum theory predicts. In quantum optics the one-photon state of a standing-wave mode is a superposition; the probability of detecting the photon at a point is proportional to the squared mode amplitude |E(x)|^2, so detection is largest at the antinodes and zero at the nodes. That is exactly the standing-wave photoelectric result that Ives himself measured (emission proportional to standing-wave intensity). The photon picture and the wave picture predict the same pattern. [memory: high, standard quantum optics.] So the experimental support for a standing-wave response does not discriminate between them. b. Consistent with established experiment: no. The experiments that distinguish a classical field from photons exist and post-date the lecture: Kimble, Dagenais and Mandel 1977 (photon antibunching in resonance fluorescence; a result understandable only with a quantised field), Grangier, Roger and Aspect 1986 (single-photon anticorrelation at a beamsplitter, violating a classical inequality by 13 standard deviations) [web: https://resolver.caltech.edu/CaltechAUTHORS:KIMprl77 and https://courses.physics.illinois.edu/phys513/sp2019/reading/week1/GrangierSinglePhoton1986.pdf, from a search; I read search summaries, not the papers]. The Compton effect (1923) is a prior example that Ives acknowledged. c. Tested: by others, after 1951, and not supported (the above). Beck and Sommerfeld bipartition-angle data that Ives cites as favouring waves are not examined here (digest only). d. If true, quantised electromagnetic field theory would have to be reworked, including the Compton and photoelectric analyses. It is a different prediction, not an interpretation. Kept in the record next to the judgement as the SOURCE HONESTY rule requires: Ives's reasoning rests on standing-wave and coherence arguments (Lorentz 1910) that were respectable in 1951 and he states the Compton difficulty himself.

Verdict 7(a): Tested, not supported. (Tests are after 1951; the 1951 prediction was a testable claim.)

7(b) Radiation pressure on a moving absorber (JOSA 32:32, 1942)

a. I read the paper's text. His eq. (7): P = E_a (c + v)/(c - v) for a completely absorbing surface moving toward a point source; and eq. (3): P = 2 E_i (c + v)/(c - v) on a moving perfect reflector (the standard moving-mirror result, one factor of the Doppler-squared change of energy density, (c + v)^2/(c - v)^2, times (c - v)/(c + v) [read: _text 1942 p.1-2; calc]). He then argues that Lorentz's P = E_i and Poynting-Abraham's P = E_i (1 + v/c) would make the pressure differ between a stationary and a co-moving source-absorber system, and that his result removes this: with E_mv = E_s (c - v)/(c + v) (eq. 12, from the Larmor-Lorentz frequency change) the pressure P = E_mv (c + v)/(c - v) = E_s, the stationary value [read: _text 1942 p.2]. The algebra is consistent. The OCR makes the definition of E_a in (5)-(7) ambiguous (in (5)-(6) it is a "rate of absorption of energy"; in (7)-(11) it is used where an energy density is needed), so I cannot fully check the units of (7) from the text alone. Reading E_a as the lab-frame incident energy density, (7) equals the energy density seen in the absorber's own rest frame, u' = u (1 + beta)/(1 - beta) for a plane wave. That is the standard relativistic transformation [memory: high]. In that reading Ives's result is what the relativity principle gives (a co-moving source and absorber behave as if stationary), and so "Lorentz's and Poynting-Abraham's values are wrong" would be a statement about incomplete older calculations, not about a prediction that differs from special relativity. I cannot confirm that E_a has that meaning in his eq. (7) (the OCR is ambiguous) and I have not checked textbook relativistic radiation pressure on a moving absorber in detail. Marked as an open point, not a verdict-bearing conclusion. b. Consistent with established experiment: no experimental claim. c. Tested: not by Ives (the effect is first order in v/c but "practically impossible to detect", per Abraham's remark that Ives quotes [read: _text]). I found no test. d. If true: older textbook formulas would be corrected; for relativistic physics nothing changes.

Verdict 7(b): Not testable as stated (as read, it restates a Lorentz-covariance requirement; the measurable difference from older formulas is first order in v/c and unmeasurable in 1942).

7(c) The rotating disk "dishes" (JOSA 29:472, 1939)

a. Digest and QE's quoted lines only. Ives argues that Wood and Hackett's "twist" in the double Fizeau toothed wheel follows from the Fitzgerald contraction alone, and for the Ehrenfest paradox (the rim of a rotating disk contracts, the radius does not) the disk dishes like a saw blade with a paraboloidal profile at small angular velocity [read: _text lines 197, 234-242; qe.md]. I did not rederive the profile. The Ehrenfest problem in the mainstream treatment is resolved by the elastic stresses of a rigidly built disk: the rim wants to contract, the radius cannot freely follow, and the actual shape comes from the elasticity equations. Dishing is one possible response of a thin disk; whether it appears for a thin disk with real elastic constants is a matter of its stress state. Mathematically I cannot tell from the digest whether the paraboloidal profile is derived or assumed. b/c. No test is cited in the material I have. d. If true, nothing in special relativity changes; this is a statement about elasticity of a spinning disk.

Verdict 7(c): Never properly tested (a stated deformation with no experiment on record; the digest gives no derivation I could check).

7(d) Ripple-tank analogues with Stilwell (JOSA 31:14, 32:25)

a. Consistent as stated, with two arithmetic points from my reading of the 1942 text. The paper prints v/c = 0.585 for v = 11.28 cm/s and c = 19.20 cm/s; the quotient is 0.5875, not 0.585 (a 0.4% rounding difference) [read: _text 1942 lines 38-41; calc: 11.28/19.20 = 0.5875]. The comparison it makes is with the factor (1 - v^2/c^2) = 0.658 (at v/c = 0.585) against a measured wavelength ratio 0.852 [read: _text lines 216-218; calc: 1 - 0.585^2 = 0.6578, 1 - 0.5875^2 = 0.655]; the digest's exponent 1/2 (which would give 0.811) is wrong, as QE noted. The authors attribute the difference to capillary-wave dispersion (speed depends on wavelength). A discrepancy of 0.852 against 0.658 is a 30% difference; the authors' dispersion explanation is plausible but not quantified in the lines I read. One OCR line of the same paper reads "v/c = 0.535" in the text beside the 0.658 comparison; 0.658 corresponds to 0.585, so I treat 0.535 as an OCR slip. b/c. This is an experiment on a classical medium, not light. It shows: for a wave medium with constant wave speed, interference patterns of moving sources look like the stationary pattern with lengths scaled by the Fitzgerald factor and periods by the Larmor-Lorentz factor, and Fresnel drag and aberration appear [read: digests; QE section 3.7]. Mathematically that is the statement that the wave equation is invariant under a Lorentz transformation of coordinates with the medium's wave speed in place of c. That is true for any medium with a linear dispersion-free wave equation, and it is a clean physical picture of why a Lorentz ether reproduces special relativity. It does not show light has such a medium. d. Nothing in physics changes; it supports the existence of the mathematical structure that Ives claims.

Verdict 7(d): Established (as a statement about waves in a non-dispersive classical medium; it is not a claim about light).

7(e) Measured-velocity statements and the "rod-clock-quotient" (1948-51)

A difference of explanation, not of prediction (claim 3c). Verdict: Not testable as stated.

7(f) Interferometer in a gravitational field: the open door for large interferometers

Ives's one conjecture beyond GR. a. Consistent. b/c. No experiment on record; the conjecture is that a large interferometer in a gravitational field might not preserve its null. The question is empirical, and the weak-field gravitational-redshift and Sagnac-type tests of GR are not tests of this. d. If a null were not preserved, the equivalence postulate would be violated.

Verdict 7(f): Never properly tested (a testable conjecture; Ives expected it to fail).

What the documents establish on the owner's two questions (mathematician's reading)

Question 1: Why is Ives interesting to Moray? The file chain is Ruderfer's appendix "Neutrino Structure of the Ether" (Lett. Nuovo Cimento 13:1, 1975) in Moray's The Sea of Energy, 5th ed. pp.271-274, as quoted by QE and claims.md. Its three steps, in Ruderfer's words: (1) the rejection of an ether applies "to a Galilean ether, but not to a Lorentz-invariant ether since its description of observation is necessarily identical to that of relativity"; (2) Ives gave "the most direct verification of the validity of a Lorentz ether ... independent of relativity", a classical derivation of the Lorentz transformations (1945) and of "the Schwarzschild metric, hence ... an identical explanation of the three famous tests of relativity"; (3) an admissible ether "focuses attention on its properties", defined as "the totality of all the presently unobservable forms of energy that may exist in the microcosm", proposed as a sea of low-energy neutrinos whose "total unobservable energy ... may far exceed the total energy contained in matter, i.e. the vacuum is then a plenum rather than a void" [read: qe.md section 9; claims.md]. I now know exactly what Ruderfer cites: the 1945 Phil. Mag. paper and the 1939/1947/1948 gravitation papers, all of which I have read.

My mathematical reading of each step: - Step 1 is correct as a statement about equivalence. Adding a preferred frame to a Lorentz-invariant theory adds a unit time-like vector field n^mu that no observable depends on, so it adds nothing to the predictions. - Step 2 is accurate about what Ives did, with the reservations above: the 1945 route carries the relativity principle in its premises (claim 3b), and the gravitation papers give the Schwarzschild metric because the equivalence postulate and the exact y are inputs (claim 4). So Ives's papers show an ether picture is admissible; they do not show an ether exists. - Step 3 is Ruderfer's addition. A Lorentz-invariant vacuum has an energy-momentum tensor proportional to the metric (pressure equal to minus energy density), which defines no preferred frame and contains no energy that can be extracted, since extraction requires a lower state to go to [memory: high; standard]. A sea of neutrinos at rest in a preferred frame has an energy-momentum tensor that is not proportional to the metric; it defines a frame, and it would show up in precisely the frame-dependent experiments (Ruderfer's own 1960 rotor proposal) that look for such effects. So the statement "the ether is Lorentz-invariant and has large energy content that is a tappable resource" has a mathematical tension that Ruderfer does not resolve in the lines I have. None of the 31 Ives papers I could consult assigns the ether an energy content that could be tapped (deep reads of 1939, 1945, 1948 say the ether supplies a frame and a mechanism, not a field, force or energy source) [read: deep/1945 section 8; deep/1948]. No Ives paper mentions Moray.

Question 2: "even if Ives is correct then the aether follows Lorentz and Einstein ... a Lorentz aether contributes no new forces beyond those specified by Lorentz/Einstein, so it is only a difference of explanation." For the core theory that reading is correct, and it can be stated precisely. Lorentz's and Einstein's theories share the same equations (the Lorentz transformations, Maxwell's equations, relativistic mass). They differ by an extra postulate (a preferred frame) that no measurement depends on. In Reichenbach's terms the difference is the synchronisation parameter epsilon, and the round-trip speed is c for every epsilon (claim 3c). So there is no extra force and no extra prediction, only a different story about why rods shrink and clocks slow (Ives's "real changes in instruments" against Einstein's property of space-time). Ives states it in his own words: Robertson's "completely equivalent in substance to the special theory" (as Robertson judged, and which Ives did not dispute); "no ... new law of gravitation" (1939); "the force equations are actually implicit in the general relativity solution" (1948).

Four qualifications make the question's summary exact: 1. Lorentz's explanation carries a hidden assumption that Einstein's does not need. If contractions are real effects of motion through a medium, then every force that holds matter together must itself transform under the Lorentz group, otherwise rods of different materials would contract differently and Michelson-Morley and Kennedy-Thorndike would not be null. Einstein's principle puts this in as an axiom; Lorentz needs it as a separate assumption about molecular forces (tested to high precision by Hughes-Drever-type experiments [memory: high]). This is the real cost of the ether picture: not a new force, but a requirement that all forces be Lorentz-covariant. 2. In gravity, relative to special relativity alone, one needs position-dependent mass, clock rate and length, not only speed-dependent ones. In GR that is the curved metric; in Ives it is the state of the ether. Ives writes this as forces: the 1947 transverse force and the 1948 position factor are, in his paper's own words, the same effect: "the equivalence of a multiplying factor gamma^(1/2) in the denominator of the local mass to a force at right angles to the motion". So it is a rewriting of the same physics as forces, not a new force. 3. The claim that "any" Lorentz-ether theory of gravity with Newtonian statics is equivalent to GR (Ruderfer) is stronger than the mathematics shows (claim 4b): it is the equivalence postulate with the exact y that gives Ives the GR values. 4. Ives's other physical claims (7a photon, 7b radiation pressure, 7c disk) are not rewritings. The photon prediction is a different prediction, and later experiments did not support it.

Investigator's findings that bear on these claims (kept in the record whatever the verdict)

The 1938 paper with Tables I-III and Part II (1941): supply the numbers and answer the doublet caveat (claim 1).
1952 mass-energy paper and 1953 Note: claim 5. The 1953 "Genesis" paper argues Sagnac and Michelson-Gale give "positive first-order results exactly in conformity with the idea that the light signals travel in a fixed ether". The Sagnac phase shift 4 A Omega/(c lambda) (A enclosed area, Omega rotation rate) is derived in special relativity for a rotating frame, and its first-order form coincides with the fixed-ether form; it does not single out an ether [memory: high, standard result; not web-checked]. Ives's 1938 closed-path paper derives the standard first-order Sagnac difference (4 A w/c^2) himself and agrees that transported "local time" clocks give c [read: claims.md].
Pobedonostsev and Parshin versus Botermann 2014: claim 1.
Einstein in the NYT (27 Apr 1938), the Bell Laboratories Record (1947), the NRL bibliography (Benton, 1959): record of reception.
Robertson's referee report on the 1947 paper and his authorship of the 1956 memoir's relativity assessment: stake stated (claims 2, 4).
Lalli on Library of Congress MSS27508: Ives to Butterfield, 5 Oct 1951 ("the classical English school of physical thought ... against the paradox of the Semitic school"); Lalli's reading "seems to imply" an "implicit anti-Semitism". Outside the mathematics; kept next to the equations. The 1950 Ives letter on hostile Princeton reception in 1938 (Pauli reporting "my work didn't amount to anything") is documented in one published source (Hazelett and Turner 1979, via Bonizzoni-Giuliani) and is Ives's own account; no second source found yet.
Absence of evidence: the saved Bell Laboratories Record issues are from 1947; nothing on the experiment in the 1938 issues was found by text search [read: Investigator's note]. Not a mathematical question.
New: the Ives standing-wave photoelectric numbers (claims.md, "Also in the record") are mainstream and uncontested; they matter for Moray only through his interest in standing waves (Q10). One mathematician's note: a measured ratio of selective to normal emission of 15-25 against at most about 4 from absorption differences (ApJ 60:209, 1924) is a statement that the thin-film emission is controlled by the standing-wave field in the film, which the photon picture reproduces with detection probability proportional to |E|^2 (see 7a).

Corrections to earlier documents

mathematician.md (my first review), claim 4a. I wrote that the mass factor expands to m0 (1 + v^2/(4c^2)) and differs from the inertial mass. This is an arithmetic error. Correct: m_g is about m0 (1 + GM/(r c^2) + v^2/(2c^2)); the velocity part is the ordinary special-relativistic one. Evidence: 1948 paper Eq. (2), expansion by SymPy in the deep read, and my reading above.
mathematician.md, claim 4 verdict ("Never properly tested; low confidence; the extra velocity terms ... a velocity-dependent force with free structure can be made to match one number"). Overtaken. The perihelion result is not matched to one number; it follows from the same three integrals as Schwarzschild's, and the full GR light deflection follows from the 1939 light-speed field. The real input is that y is exactly 1 - 2GM/(R c^2); changing the second-order coefficient kappa changes the advance by (6 - kappa)/6 [calc].
mathematician.md, source gap. The statement that the 1945, 1947, 1948 and 1941 papers are "not obtained" is out of date; they are in the manifest and were read in the deep reads.
mathematician.md, claim 3 ("Unread 1945 and 1951 papers limit confidence"; "never coincide" for finite q). Replaced by the three-route analysis above. The "never coincide" remark by Bonizzoni and Giuliani is accurate for the formulas but not for the predictions: the transformations differ only in synchronisation convention; the same Lorentz-group invariants appear in both.
claims.md and arbiter.md, "SR internally contradictory: Refuted by calculation" (reached from the 1953 summary). Now supported from the 1949 paper itself: his own equations (19)-(20) are the Lorentz transformation in non-Einstein synchronisation; the round-trip speed is c for all k.
mathematician.md, claim 1c on the Pobedonostsev voltage-to-speed step. The relativistic correction to kinetic energy at 150 keV per H2+ is about 8e-5, so the voltage conversion is not the source of a 40-80% deficit [calc].
claims.md, claim 7(d), the v/c value. The paper prints 0.585; 11.28/19.20 = 0.5875 [calc]. This is a small rounding or printing slip in the 1942 paper, not a defect in the review; the comparison factor 0.658 corresponds to 0.585.
arbiter.md line 15 repeats the m0 (1 + v^2/(4c^2)) figure; it must be corrected per item 1.
QE round 2, Ruderfer quote. QE quotes Ruderfer; I did not re-read his appendix. His statement "any Lorentz-ether theory of gravitation which provides the static Newtonian gravitational force is necessarily equivalent observationally to general relativity" is quoted accurately as far as I can tell, but it is not a theorem (claim 4b).

What would settle what is open

Claim 1: a replication of Pobedonostsev and Parshin's 150-300 keV points with an error budget for the angle and arm alignment, and the reference atoms' imparted velocity measured with the mirror covered, would show whether the 40-80% deficit is instrumental.
Claim 3(b): a derivation of the exact free function h in Ives's moving-platform step from conservation laws alone, or a proof it cannot be fixed without the relativity principle.
Claim 4: a second-order (m^2/r^2) test of the field: a deflection or perihelion measurement precise enough to fix kappa independently of other parameters (the existing tests already bound it near the GR value).
Claim 7(b): the definition of E_a in the 1942 equation (7) from a clean copy, and the textbook relativistic pressure on a moving absorber.
Claim 7(c): Ives's profile for the dished disk compared with the elasticity solution of a thin spinning disk.

Summary of verdicts

Moving clocks run slow (second-order Doppler): Established (the 1938 doublet caveat is answered by Part II: shift 0.11 Angstrom above the 0.09 Angstrom separation; Pobedonostsev-Parshin dissent unreplicated).
Real slowing in a fixed ether: weak form Established; strong form Not testable as stated; as a 1938 position Historically superseded but instructive.
Alternative theory from classical ether physics: 1937 route Not testable as stated (equal predictions) and Historically superseded but instructive; 1943-45 route Not testable as stated, with the relativity principle moved into the premises (the free function h; clock universality); 1949 "inconsistency" charge Refuted by calculation.
Gravitation (changed from Never properly tested): predictions for one static mass Established (they are the GR values, derived without a fitted constant; one hidden input: y exact, kappa = 0); transverse force as a separate force and the ether reading Not testable as stated; extensions (Shapiro delay, pulsars, waves) Never properly tested for Ives's theory.
Einstein did not derive E = mc^2: circularity charge Refuted by calculation; "no fully general derivation in 1905" Established.
Atoms are not clocks: Not testable as stated.
Ives's separate claims: (a) photon abandoned Tested, not supported; (b) radiation pressure Not testable as stated; (c) dishing disk Never properly tested; (d) ripple-tank analogue Established (for non-dispersive classical waves; not a test of light); (e) rod-clock-quotient Not testable as stated; (f) large interferometer door Never properly tested.