Tolman-R: Mathematician review (Step 2)

Reviewer: PhD mathematician. Subject: R. C. Tolman, The Theory of the Relativity of Motion (University of California Press, 1917). Source tags: [read] from the documents, [web] with URL, [memory] with confidence.

Source note (repeated from claims.md): the project folder holds one OCR scan of the 1917 book. Its text layer garbles nearly every equation. Chapters III-IV and X-XI were deep-read from page images by Step 1. For this review I also read Chapters II, VI, VIII, IX and XII from the Project Gutenberg re-typesetting that the Investigator downloaded (research/files/Tolman-1917-Gutenberg-32857.pdf), and I checked one disputed equation against the scan image. Where a conclusion rests on [memory] alone, it is marked. My web-check budget was only partly usable: the search tool hit its session cap after the first fetches, so most Kaufmann/Bucherer/Fox/Brecher/Alvager dates stay [memory] with the confidence stated.

This is a mainstream textbook, not an outsider's theory. "Mathematical consistency" here means: do the derivations follow from the stated assumptions, are the units right, and what is assumed without being stated.

Corrections to Step 1

1. Gutenberg equation (229) is mistyped; the scan is right. The Investigator (sites.md lead 1, research/2026-10-04-gutenberg.md) says the Gutenberg edition gives a clean copy of every equation. For (229), the Gutenberg text layer reads rho' = rhokappa(1 - u_z V/c^2). The 1917 scan, page 186 (printed p.169), rendered at 300 dpi, reads u_x V/c^2 [read: page image p.186]. Only u_x can be right: charge density transforms as rho' = gamma(rho - V j_x/c^2), and the book's own next step, the charge-invariance argument via eq. (17) (Gutenberg section 157), needs u_x. So the Gutenberg copy is useful, but any equation used for a verdict must still be checked against the scan, as the Investigator said.
2. claims.md claim 6 overstates what the book says about emission theories. The book (Gutenberg Section 22, scan Chapter II) says the Stark-effect canal-ray measurements and Tolman's own 1910 sun-limb experiment are incompatible with emission variants (1) "mirror re-emits at c" and (2) Stewart's mirror-image rule. For Ritz's variant (3) it says: "unfortunately optical experiments for deciding between the Einstein and Ritz relativity theories have never been performed, although such experiments are entirely possible." Against Ritz it offers only the Comstock-de Sitter binary-star argument [read: Gutenberg PDF, Section 22]. So "ruled out by astronomical and Stark-effect evidence" should read: variants (1) and (2) are argued against by Stark-effect and the sun-limb test; Ritz's variant rests only on the binary-star argument, and Tolman says a direct test was never done.
3. Mass-law evidence (claims 2 and 6). facts.json records the Kaufmann/Bucherer/Hupka appeal as a triumph with no data. Section 64 (Gutenberg) adds a logical step that the Investigator's lead (Tolman 1910 caveat) bears on: Tolman writes that, because the mass derivation "makes no reference to any electrical charge", one may "reject the possibility of explaining the Kaufmann experiment by assuming that the charge of the electron decreases with its velocity". That is circular if the experiment is meant to test the mass law: the derivation is valid only if the postulates hold, and the experiments are being used to support the postulates. In 1910 Tolman himself wrote that the experiment "may not really indicate an increase in the mass" [read: Wikisource, Investigator note]. The two positions differ; both are Tolman's.
4. Chapter VIII, XII were digest-only in Step 1; I have now read them. Results are in claims 2, 4, 5 below. No errors found in the printed equations I checked. One rounding: "1 gram = 10^21 ergs" is used again at the end of Section 105; with Tolman's own c it is 8.99 x 10^20 erg, so his 7.47 x 10^-11 g for a mole of massless particles at 300 K is really 8.3 x 10^-11 g (my calculation).
5. Claim 5 wording. claims.md says Maxwell-Lorentz theory "follows from the same least-action principle". More exactly (Gutenberg Section 151): Tolman chooses the known Lorentz-theory Lagrangian (eq. 209) and shows it is already compatible with the H/sqrt(1-u^2/c^2) rule; the transformation laws for e, h, rho (227)-(229) are stated, with the claim they "fulfil all the requirements", not derived [read]. I checked that e^2 - h^2 is invariant under (227)-(228) (it is, with the cross terms cancelling). I also checked that (209) reduces to (e^2 - h^2)/2 (230) once curl h = (e-dot + rho u)/c is used, with the parenthesis in (209) being -phi . (e-dot + rho u)/c. The signs are consistent.
6. The temperature law is not purely a "choice of H". claims.md says (201) "follows from Tolman's chosen kinetic potential". Correct, and it can be said more sharply (claim 4 below): it is equivalent to the assumption that the heat term T dS dt transforms like work, so that T dt is invariant. That is the exact point where the Ott/Arzelies law departs.
7. The Investigator's note naming "Ott-Eddington-Moller" is consistent with Pouryazdan and Vakili 2026 [read: research notes]; I add no correction, only note that facts.json cites "Ott and Arzelies" and the 2026 paper adds Eddington and Moller.

Items I checked and confirm: the Step 1 deep-read equations in Chapters III-IV and X-XI. I re-derived by hand or in SymPy: time dilation, length contraction, the clock offset l'V/c^2, the velocity law and the 4c/5 example, the mass law, Delta E = c^2 Delta m (the integral of u d(mu) equals c^2 dm), the Lorentz transformation and its invariance, (167), (172), (173), (175), the lever result, (200), (201), (204), (205), (208). All agree with Tolman. The Step 1 corrections of digest errors stand.

Claim 1: postulates, time dilation, contraction, Lorentz transformation, velocity addition

a. Internal consistency. Consistent. The Chapter III derivations (light clock, longitudinal round trip, clock offset, velocity composition) are correct and each step reproduces the Lorentz-transformation result. Chapter IV obtains the Lorentz transformation "by trial", which is the weak spot as logic, not as arithmetic. Hidden assumptions, none stated by Tolman except homogeneity: - Linearity of x', t' in x, t. It is never stated. It follows from homogeneity of space and time in standard treatments [memory, high]. - Isotropy (implicit in y' = y, z' = z). - The light-sphere condition only says x^2 + y^2 + z^2 - c^2 t^2 = 0 is preserved. Tolman states the stronger "this expression is invariant". For a linear map, preserving the zero set gives invariance up to a constant factor, and the V -> -V symmetry condition fixes the factor to 1. Not shown in the book. - Without the symmetry condition the three conditions leave a family. My SymPy check: for a general linear map in (x, t), requiring x'^2 - t'^2 = x^2 - t^2 gives a^2 - d^2 = 1, b^2 - e^2 = -1, ab = de, which still admits a free rapidity parameter. The velocity V enters only when the Newtonian limit and the origin condition are applied. So "sufficient to determine the solution" (p.62) is asserted, not shown. The book's answer is the right one. - Newtonian limit: to first order, eq. (12) gives t' = t - Vx/c^2, not t' = t. The Vx/c^2 term is first order in V/c. "Reduces to Galileo when V is small" needs c -> infinity (or |x| << c|t|), not only small V. This is the relativity of simultaneity showing up. - Group statement: two boosts along one line compose to a boost. Two non-parallel boosts do not. My numerical check: boosting 0.6c along y after 0.6c along x gives a non-symmetric 4 x 4 matrix, which is a boost times a rotation (the Thomas-Wigner rotation) [memory, high on the fact; matrix computed]. Tolman's "suitable magnitudes and directions" is true only if rotations are allowed.

Worked numbers (mine): V = 0.6c gives sqrt(1 - 0.36) = 0.8, so a 1 microsecond tick lasts 1.25 microseconds in the other frame. The velocity law gives c/2 + c/2 = (c)/(1 + 1/4) = 0.8c. Units are consistent throughout.

b. Consistency with experiment. Tolman cites a long list of null ether-drift experiments (Michelson-Morley, Trouton-Noble, etc.) [read: p.38]. Later tests he could not cite: Ives-Stilwell 1938 (time dilation via the transverse Doppler effect; modern storage-ring versions limit deviation to about 1e-8) and modern Kennedy-Thorndike versions (light-speed anisotropy limited to about 1e-17) [web] https://en.wikipedia.org/wiki/Tests_of_special_relativity . Muon lifetime and GPS timing agree with time dilation [memory, high].

c. Tested? Yes, many times, by independent groups, over more than a century (above). Replicated.

d. If it were true. Nothing would change: this is the mainstream.

Verdict: Established (kinematics and the Lorentz transformation). The book's Chapter III proofs are heuristic by Tolman's own statement ("no stress on mathematical elegance or logical exactness", p.45); the logical gaps above are expository, not errors.

Claim 2: mass m0/sqrt(1-u^2/c^2), Delta E = c^2 Delta m, relativistic Lagrangian

a. Internal consistency. The Lewis-Tolman colliding-spheres argument (Section 31) is consistent given its assumptions: first postulate (symmetry of the two experiments), transverse lengths equal, time dilation, momentum conserved with momentum = mass x velocity and mass allowed to depend on speed, transverse speeds negligible compared with V. The jump from "B's ball moving at V" to "any speed u" is an extrapolation. Note that this "mass" is momentum/velocity (what older books call transverse mass). The longitudinal mass (force/acceleration along the motion) is gamma^3 m0; the book treats this in Chapter VI [memory, high on the fact; I did not re-derive from the book].

Delta E = c^2 Delta m: the derivation is exact for kinetic energy: u d(mu) = c^2 dm when m = m0 gamma (SymPy confirmed). That gives Delta E = c^2 Delta m, i.e. E = mc^2 + const. The extension to "all energy has mass, all mass is energy" (rest energy m0 c^2) is by an inductive argument (stopping a body by collision or friction), which Tolman signals with "every reason to believe" and "pragmatic" [read: p.57]. I read the later electromagnetic version (Gutenberg Sections 166-167): from the transformation of the field and the work done by external fields he gets Delta E = kappa Delta E0' for a force-free system, hence E = gamma(m0 c^2 + E0). The identification of the leftover function phi(u) with the kinetic energy of the "mechanical parts" is an assumption (the field contributes only through E0). The result is standard. The famous 4/3 difficulty of the classical electron is avoided here because Tolman wraps the system in an impermeable shell whose stresses are part of the system; he does not discuss 4/3 in this place [read, Gutenberg 166].

Relativistic Lagrangian (Chapters VII, IX): the rule H/sqrt(1-u^2/c^2) = H'/sqrt(1-u'^2/c^2) comes from requiring the variation of the action integral to be invariant, with dt'/dt = kappa(1 - xdot V/c^2) [read: Gutenberg Sections 109-111, eqs. 140-146]. The step from "the integral of delta H dt is invariant" to "H dt is invariant" assumes the invariance holds as a local statement. That loses nothing for the physics (the action being a scalar), but it means H is fixed only up to additions that do not change the equations of motion. Tolman admits (Section 139, p.172) that H "was not absolutely uniquely determined". For a free particle, H = -m0 c^2 sqrt(1-u^2/c^2) is the standard Lagrangian [memory, high].

Units check: kinetic potential has units of energy; E0*sqrt(...) is energy; the footnote on p.184 (Gutenberg) honestly admits the electromagnetic H omits epsilon and mu to hide a units problem. Fine in Gaussian-type units.

Numbers: the "1 gram = 10^21 erg" statement is a rounding of 8.99 x 10^20 erg (11% high), and it is repeated in the book (Sections 105, 167). Not a derivation error.

b. Experiment. Electron momentum/velocity data: Kaufmann (c. 1901-1906), Bucherer 1908, Hupka 1910, Neumann 1914 [memory, medium]. History matters: Kaufmann's 1906 analysis was first read as favouring Abraham's rigid-electron model, and the precision needed to distinguish the models was disputed for a long time [web] https://en.wikipedia.org/wiki/Tests_of_special_relativity . Later, cleaner tests: Bertozzi 1964 (electrons 0.5 to 0.99c, kinetic energy against time of flight) [memory, medium; not web-checked], and the daily operation of particle accelerators, where Delta E = Delta (mc^2) is routine [memory, high]. Mass-energy: nuclear mass defects and binding energies agree [memory, high].

c. Tested? Yes. Independent groups, replicated. In 1917 the support was thinner than the book's word "triumph" (see Correction 3).

d. If true. Nothing; mainstream. The "relativistic mass" terminology is replaced in modern texts by invariant mass with p = gamma m0 u and E = gamma m0 c^2 [memory, high]. Historically that bookkeeping is superseded but the content is the same.

Verdict: Established (the physics). Split: Tolman's pedagogic derivation of the general mass-energy equivalence is partly inductive ("Plausible" as an argument); the modern "relativistic mass" language is Historically superseded but instructive.

Claim 3: momentum density = energy flow / c^2, unsymmetrical stress tensor, lever torque

a. Consistency. Consistent, and I checked it against the modern formalism, which the deep reader flagged as unverified. In the rest frame take a body with energy density w and a shear stress sigma (symmetric, Cauchy stress). Boosting the stress-energy tensor along x with speed u (c = 1 units) gives T^{0y} = gammabetasigma and T^{xy} = T^{yx} = gamma*sigma [my calculation from the standard tensor boost]. Tolman's unsymmetrical t has t_xy = gamma sigma (agrees with eq. 167: t_xy = t0_xy/sqrt(1-beta^2)), t_yx = (1-beta^2) t_xy (175), g_y = t_xy u/c^2 (172). Then Tolman's symmetric tensor p = t + g u gives p_yx = (1-beta^2) t_xy + beta^2 t_xy = t_xy = p_xy, so p is symmetric and equals the modern T^{ij}. Equations (172), (175), (176) therefore agree with the modern stress-energy tensor. The unsymmetrical t is just the stress tensor with the convective momentum flux g u removed.

The lever (Section 136): F1 l1 - F2 l2 = F1 l1 - F2 l2 (1 - V^2/c^2) = F1 l1 V^2/c^2 follows from l2 = l2' sqrt(1-V^2/c^2), F2 = F2' sqrt(1-V^2/c^2). Cross-check by the torque integral: (u x g)_z = u g_y = u^2 t_xy/c^2, and the volume integral of t_xy over the arm that carries force F1 is F1 l1 (the virial relation for a body in equilibrium), giving dM/dt = F1 l1 V^2/c^2. Same value from two routes. Units: stress x velocity / c^2 = kg m^-2 s^-1 = momentum per volume. Good.

Numbers (mine, from Tolman's formulas): a shear stress of 10^9 Pa in a body moving at 1000 m/s gives a transverse momentum density of 1.1 x 10^-5 kg m^-2 s^-1. For the lever with F1 l1 = 100 N m at Earth's orbital speed 30 km/s, V^2/c^2 = 1.0 x 10^-8 and the "unbalanced" torque is 1.0 x 10^-6 N m, which is absorbed as angular-momentum growth, not rotation.

Hidden assumption: the lever result holds only for a body in static rest-frame equilibrium and treats the arms as rigid in the sense that the rest-frame forces are the whole story.

b. Experiment. The Trouton-Noble experiment (1903) found no torque on a moving charged capacitor, consistent with no rotation [memory, high; deep read also cites it; Investigator: https://en.wikipedia.org/wiki/Trouton-Noble_experiment is named as a source, not opened]. No experiment tests the small stress-induced momentum: Tolman himself writes "we have no experiments of sufficient accuracy... and hence have found no experimental contradiction" [read: p.172].

c. Tested? The prediction (no rotation) is tested by Trouton-Noble and similar null results. The explanation is debated in the literature: Shapiro 2012 (arXiv:1206.4487; Investigator lead) calls it a paradox with "extensive discussion and indeed controversy" and resolves it through angular momentum and simultaneity [read: p.1 of the downloaded PDF per the Investigator]. Laue's 1911 resolution is the same energy-flow account as Tolman's [memory, high]. Some authors stress "hidden momentum" or simultaneity instead [memory, medium; Jackson, Am. J. Phys. 2004, not web-checked]. All agree the lever does not rotate.

d. If true. Nothing changes; the modern symmetric stress-energy tensor already encodes it.

Verdict: Established (momentum density = energy flux/c^2, lever outcome). The interpretation of which mechanism "explains" the lever is a mild open pedagogical dispute, not a dispute about outcome.

Claim 4: entropy and pressure invariant, T = T0 sqrt(1-u^2/c^2), G = (E+pv)u/c^2, hohlraum

a. Consistency. The algebra is right. The pieces: - Pressure (200) and entropy (197) are invariant; my check: the energy of a moving system from (204), E = (E0 + p0 v0)/sqrt(1-beta^2) - p v, equals the volume-integrated modern T^{00}, gamma(E_rest + beta^2 P v0) [my calculation]. Same for momentum (205)-(206). The hohlraum results (207)-(208) follow: with p0 = E0/(3 v0), E = E0 (3 + beta^2)/(3 sqrt(1-beta^2)) and G = (4/3) E0 u/(c^2 sqrt(1-beta^2)) (SymPy). The 4/3 is for a cavity held by external pressure; for an isolated cavity the wall stresses cancel it. - Chapter VIII kinetic theory: the equipartition quantity = 3kT. I re-derived the integration by parts: with energy E = c sqrt(psi^2 + m^2 c^2) the identity (hc/3) = 1 gives the book's (126) (verified). This is the known Juttner result = 3kT [memory, high]. The pressure law pV = (N/3) (127) is the standard p = n/3. The ultra-relativistic limit E = 3kT per particle is correct. Cycle-based temperature (Section 103): heat = work for a gas whose energy is volume-independent, a valid step.

The temperature step is a definition, and the book does not say so. From (204)-(206), with S, v0 and u held fixed, E - uG = E0 sqrt(1-beta^2) exactly (SymPy: difference zero). Raise the rest-frame entropy by dS, so dE0 = T0 dS: - the energy increase of the moving system is dE = T0 dS / sqrt(1-beta^2) (this is the Ott/Arzelies value T = gamma T0 if the heat is defined as the energy increase at fixed u); - the energy increase less the work of the external force that keeps u constant, dE - u dG, equals T0 dS sqrt(1-beta^2) (this is Tolman's value, the Planck-Einstein T = T0/gamma). So the two classic laws differ by whether the extra momentum G carried in by the heat is counted as "work". Tolman's route: the generalised force conjugate to S in the Lagrange equations is -dH/dS with H = -E0 sqrt(...), and eq. (144)-(145) make the work term transform with dt'/dt, i.e. T dt is invariant. The statement "T/sqrt(1-u^2/c^2) is invariant" (201) is therefore a consequence of assigning heat to the work (action) side of the ledger, not a measurement result.

A third answer, Landsberg's T' = T (1966-67), comes from defining temperature by the rest-frame state only. A fourth route: the radiation from a body moving at speed u seen at angle theta has a Planck shape with Doppler temperature T0/(gamma(1 - beta cos theta)), which varies with direction from 0.5 T0 to 2 T0 at beta = 0.6 (my calculation). So there is no unique "temperature of a moving body": the answer depends on the operational definition. Numerical illustration, T0 = 300 K:

(The last column is T0 [gamma^2 (1 + beta^2/3)]^(1/4) for a photon gas, my calculation.) This agrees with the Investigator's lead: Pouryazdan and Vakili 2026 (arXiv:2606.00521; Phys. Lett. A 590, 131860) define an effective temperature from the transformed energy density, find it rises with speed for a photon gas, an ideal relativistic gas and an electron gas, and conclude temperature is observer dependent; Heras and Osorno 2022 (arXiv:2204.12572) say the Planck-Einstein form is "supported by many authors" beside the Ott-Arzelies and Landsberg laws and argue none satisfies group closure [read: per the Investigator's notes of the downloaded PDFs; I did not re-read the full PDFs]. Both are theoretical. Neither reports a laboratory measurement.

The deduction in the book is internally consistent; the claim "temperature transforms as T0 sqrt(1-u^2/c^2)" is one of several self-consistent conventions.

b. Experiment. Entropy and pressure invariance: no dispute [memory, high]. Temperature of moving bodies: no direct experiment exists that I know of; Cubero et al., PRL 2007, simulations support the rest-frame (invariant) view for the Juttner gas [memory, medium; not web-checked]. The dispute is about definition, so an experiment would first have to say what a moving thermometer measures.

c. Tested? No neutral, controlled test is on record for the temperature law (I know of none; the literature is theoretical). Not dismissed without testing either; there is nothing practical to test (it needs bulk matter at relativistic speed with a thermometer comoving or not).

d. If true. Nothing in mainstream physics breaks; the debate is open inside mainstream physics. If the Planck-Einstein form were adopted, the Tolman-Ehrenfest relation T sqrt(g00) = const (gravitational redshift of temperature) would be its general-relativity partner [read: Investigator's note of Pouryazdan-Vakili's discussion].

Verdict (split): - Entropy and pressure invariance, E, G, hohlraum momentum: Established (agrees with the modern stress-energy tensor). - T = T0 sqrt(1-u^2/c^2) as a law: Not testable as stated (convention-dependent: it equals Planck-Einstein only if heat counts as work; competing forms are equally self-consistent) and never properly tested. Not internally inconsistent.

Claim 5: Maxwell-Lorentz electrodynamics, field transformation, charge invariance, EM mass, no ether

a. Consistency. Consistent, with the caveats in Correction 5. The variation of H = integral [e^2/2 + (curl phi)^2/2 - phi.(e-dot + rho u)/c] dV with the constraint rho = div e (Lagrange multiplier psi) yields e = -(1/c) dphi/dt - grad psi and curl curl phi = (e-dot + rho u)/c, i.e. the four field equations (221)-(224); I checked the signs in the chain (215)-(219). The force law (226) is left to the reader ("complicated transformation ... not presented"). The transformation equations (227)-(229) agree with the standard ones (E'_y = gamma(E_y - vB_z), E'_z = gamma(E_z + vB_y), B'_y = gamma(B_y + vE_z/c^2), B'_z = gamma(B_z - vE_y/c^2), rho' = gamma(rho - v j_x/c^2)) [memory, high]; but (229) must read u_x (Correction 1). Charge invariance: rho'/rho = sqrt(1-u^2/c^2)/sqrt(1-u'^2/c^2), the same ratio as the volume contraction, so Q is invariant (my check of the algebra with eq. 17). The supporting argument, "count the electrons", is a plausibility argument, not a proof.

b. Experiment. Fields and charge: relativistic electrodynamics is confirmed by everything from motors to accelerators [memory, high]; neutrality of atoms and matter to extreme precision supports velocity-independence of charge [memory, medium; not web-checked]. The "ether is unnecessary" statement: no experiment has found an ether; see claim 1.

c. Tested? Yes, by essentially all electromagnetic engineering; the book's Wilson-experiment section (172, moving dielectric) was an open experimental suggestion in 1917 ("a matter of great interest" per the Investigator's note). I did not read Section 172 in detail, and the Investigator has not searched it yet (sites.md lead, related work). Later moving-dielectric experiments agree with relativity [memory, medium; not checked].

d. If true. Nothing; mainstream.

Verdict: Established. Digest-level caution is lifted for Sections 151-157 and 166-167 (read), not for Sections 168-172 (moving media, Wilson), which I did not read in full.

Claim 6: Kaufmann-Bucherer-Hupka support the mass law; emission (Ritz) theories refuted by Comstock/de Sitter and Stark-effect

a. Consistency of the book's argument. The binary-star argument (Gutenberg Section 21) is arithmetically correct: if light speed is c plus u at emission, light from the approaching side arrives with delay l/(c+u) and from the receding side with l/(c-u). The apparent half period changes by 2ul/(c^2 - u^2), about 2ul/c^2. My worked number: u = 100 km/s, l = 10 light-years (9.5 x 10^16 m) gives 2.4 days, comparable with spectroscopic-binary periods of days, so Kepler's laws would visibly fail [my calculation]. The argument assumes the light travels the whole distance at its emission speed. The Investigator's lead is that J. G. Fox (1962-65), as reported by Martinez 2004, argued from the Ewald-Oseen extinction theorem that light passing through a medium is replaced by light at speed c relative to the medium after one extinction length, so the binary-star argument and interferometer tests with sunlight or starlight (Tomaschek, Miller, and "those using Tolman's apparatus") are not decisive; Martinez himself says the early evidence was "misconstrued as telling against it more conclusively than actually was the case" [read: research/2026-10-04-martinez-ritz.md, quoting the downloaded PDF]. The 1910 Tolman paper notes the same possibility himself, that a speed difference may be destroyed near the earth, "not entirely excluded" [read: Wikisource, Investigator]. The mathematics of the 2ul/c^2 shift is unaffected; the physical premise is what Fox disputed. Whether the extinction length in space is short enough to matter I could not check numerically (the quantity depends on interstellar density; I have no reliable number [memory, low]).

b. Experiment. Tests designed to avoid the extinction objection came later: Alvager et al. 1964, where photons from fast-moving pi-zero decay did not take the speed of their source [web] https://en.wikipedia.org/wiki/Tests_of_special_relativity ; Brecher 1977, X-ray binaries [memory, medium; not web-checked]. Mass law: see claim 2. Tolman's own 1910 sun-limb null result was against variant (1); by his own account, a decision between Einstein and Ritz needed a new test (1912) [read: Wikisource, Investigator].

c. Tested? The Ritz theory in 1917: never properly tested by a neutral, controlled experiment that avoids extinction; it was argued against by an astronomical inference and not dismissed outright (Tolman, Ehrenfest and others called for a decisive test, per Martinez; Ritz died in 1909). It was not "dismissed without testing" either, since the sun-limb and starlight tests were run (1910, 1924), but those were later contested by Fox. After 1964, direct tests exist. The mass law: tested, with a mixed early record (Kaufmann 1906 read at first as favouring Abraham [web, Wikipedia above]).

d. If true. If an emission theory were true, the second postulate would fail and relativity as a theory of light propagation would have to be rebuilt (a Ritz-type theory keeps the first postulate but would give different Doppler, aberration and Michelson-Morley results). If the Kaufmann data had been explained by a velocity-dependent charge instead of mass, the Lewis-Tolman derivation and everything in Chapters VI-IX built on it would need a separate justification. Both are now excluded by later data.

Verdict (split): - Mass law: Established (modern accelerator and Bertozzi-type data), though Tolman's 1917 support was thinner than the word "triumph". - Emission theories ruled out: in 1917, Never properly tested for Ritz's version (the book says so itself); the binary-star argument is mathematically sound but its premise was disputed (Fox). Now Tested, not supported (Alvager 1964 and later [web/memory above]).

Investigator findings kept in the record (with my weighing)

• Gutenberg edition (priority 1): useful for equations; contains at least one transcription error, (229) (Correction 1). Verify against the scan.
• Tolman 1910 and 1912 papers (priority 1): show Tolman first treating emission as live, running a sun-limb test, saying the Kaufmann experiment might not show a mass increase, and calling in 1912 for a decisive Einstein-Ritz test. Pertinent to claims 2 and 6; I agree with the Investigator's reading. The 1917 book is more confident than the 1910-1912 papers.
• Caltech/OAC papers and NAS memoir (priority 1 and 4): lecture notes and manuscripts may exist but the folder list was not read; the memoir was not read. No mathematics to review there. The record around Tolman is continuous (no absence-of-evidence anomaly).
• Temperature papers (priority 2/3): Heras-Osorno 2022 and Pouryazdan-Vakili 2026; they show the dispute is alive, and my analysis shows why (definition of heat).
• Shapiro 2012 (priority 2): agrees on the no-rotation outcome; supports claim 3.
• Martinez 2004 / Fox (priority 3): the strongest dissent against the book's emission-theory argument. It is a historian's secondary account of Fox's papers; the primary Fox papers (Am. J. Phys. 1962, 1965 [memory, medium]) have not been read. Single account, no second source in the record yet; it is evidence and stays in the record.
• Not yet searched by the Investigator: H. A. Wilson moving-dielectric experiment and the 1909 Lewis-Tolman paper (colliding-sphere and lever origins).

Summary table

What would settle the open items

• Temperature: state the operational definition (what a comoving or lab thermometer exchanges, and whether the exchanged heat carries momentum); the Planck-Einstein and Ott values then both follow by the calculation above. No new physics is needed.
• Gutenberg errors: compare every equation used in a verdict with the scan.
• Fox and extinction: obtain Fox 1962/1965 and compute the extinction length for the binary-star sight lines.