Hilgenberg-OC: Mathematician review (Step 2)

Reviewer: PhD mathematician. Scope: derivations, notation, units, hidden assumptions. Physics-versus-experiment judgements belong chiefly to the physicist; I answer questions b-d only as far as the mathematics reaches them.

Tags: [read] from the project documents or the Investigator's notes of documents they read; [web] with URL; [memory] with confidence; [calculation] my own arithmetic (Python, checked).

Source gap (repeated from claims.md)

The project folder holds one document: Scalera and Braun (2003), a secondary summary by Earth-expansion proponents. It contains no equations [read: deep/ p.6-15]. Hilgenberg's own mathematics is available to this review only through (i) the authors' paraphrase and (ii) the Investigator's reading of two page images of his 1933 book Vom wachsenden Erdball (archive.org Hilgenberg1933, scan n20-n21), which is the one place where numbers and an equation of his appear [web: https://archive.org/details/Hilgenberg1933]. I did not read the 1931 booklet, the 1933 Magnus reports, the 1939 sink/source booklet, the 1965 paper or the 1974 paper. Where I say "his law" below I mean the growth law of the 1933 book as the Investigator reports it. Conclusions about his sink-flow gravity rest on that summary, on the authors' paraphrase, and on my own reconstruction of the mathematics a sink-flow model must contain (marked as reconstruction, not as his).

Nothing was downloaded to research/files/ (empty); the Investigator listed the 1933 OCR text as a candidate. A mathematician's check of the whole book is the first thing to obtain.

Corrections to Step 1

1. claims.md claim 3 says the three numbers "do not agree with each other as printed". True of the folder document, and the Investigator's 1933 reading resolves the first pair: 60% is a diameter ratio (0.62-0.65 from the shelf-sphere construction), so volume and mass ratio are 0.6^3 = 0.216, and 1/0.6 = 1.67 is the "nearly doubling". I confirm [calculation]: 0.6^3 = 0.216; 1/0.216 = 4.63; ln 4.63 = 1.532; /3x10^16 s = 5.1x10^-17 s^-1. Consistent with his printed ~5x10^-17. Small correction to the Investigator: his own text gives the ratio as 0.62-0.65, and 0.62^3 = 0.238, not 0.216; the 0.6 used in the exponential law is a rounding of the lower limit (or of sqrt(0.35) = 0.59, [calculation]). That is a 10% difference in volume ratio, and a change in mu of about 6% (ln 4.2 vs ln 4.63), harmless for a one-significant-figure rate but worth stating.
2. Step 1 (deep read) treats 4590 km and 4 mm/yr as simply "unreconciled". They can be tested against his own 1933 law, and they are mutually inconsistent in a specific way (Q3 below). The 1933 law, applied to the Permian (250 Myr, [memory, high]), gives R = 5570 km, not 4590 km, and a present growth rate of 3.4 mm/yr, not 4. The 4 mm/yr is therefore close to his 1933 law's present rate, but the 4590 km is not. So the printed caption mixes two parameter sets. Source of the figures remains the 1965 paper, not read.
3. Step 1 and the Investigator say "g/sec" is a printed unit slip. Agreed; mu is a relative rate (s^-1). I add: the sentence "1 g hydrogen gives 15 million new molecules per second" is correct only for H2 molecules: 1 g H2 = 3.0x10^23 molecules, times 5x10^-17 = 1.5x10^7 [calculation]. Not an error of his, but the phrase "1 g hydrogen" needs "H2".
4. Step 1 calls the 1974 Riemann/"simple mathematical conversion" a claim with no equations. Correct, and it should stay classed as an unsupported assertion by the authors' paraphrase, not by Hilgenberg's text as seen. The 1933 book (Investigator) shows Hilgenberg's actual view of Riemann: a precursor idea of ether flowing into matter, which Riemann replaced by an abstract four-dimensional picture. That is not the same as the 1974 statement the authors report (zero ether energy density plus a fourth coordinate gives general relativity), and the two should not be conflated.
5. Investigator's Leads, section 3, on Bjerknes. Flagged there as [memory, medium], to check. I did a mathematical check below (Claim 1) and it holds in a specific, useful form: the sign rule Hilgenberg lists (like sinks attract, like sources attract, sink and source repel) is the pattern of Bjerknes' pulsating spheres [memory, medium], and the force law in that theory is inverse square with strength proportional to the product of the two volume fluxes. That is exactly what a sink-flow theory of gravity needs, and it leads to a growth law (below).

Reader's mini-glossary (symbols used below)

• Sink / source: a point where fluid disappears / appears at volume rate m (cubic metres per second). For a steady spherical sink in incompressible fluid the radial speed is v(r) = -m/(4 pi r^2): flux through any sphere is constant.
• rho_e: density of the ether (kg per cubic metre). Unknown; no source gives it.
• M: mass of a body. mu: the relative growth rate dM/dt = mu M (per second). G: Newton's gravitational constant, 6.674x10^-11 N m^2 kg^-2 [memory, high].
• Continuity equation: d(rho)/dt + div(rho v) = s, where s is the rate at which fluid is created (+) or destroyed (-) per unit volume. If s = 0 outside a body, the flow is conserved there.

Claim-by-claim

Claim 1. Gravity is the accelerated flow of space (a Cartesian ether) towards massive bodies, each a sink.

a. Internal consistency. The folder document gives no equation, so consistency cannot be checked on it. I checked what any version of the claim must contain. There are two readings, and the claim as printed ("gravitational field is nothing but the effect of the accelerated flow of space towards the planetary body" [read: p.8]) does not say which it uses.

Reading A: a body is carried along by the flow. Take steady incompressible flow into a sink, v = -k/r^2 (k = m/4 pi). A parcel carried by the flow has acceleration a = v dv/dr = (-k/r^2)(2k/r^3) = -2k^2/r^5. It falls off as r^-5, not r^-2. Worked numbers [calculation]: matching the surface value g = 9.8 m/s^2 at r = R = 6.371x10^6 m needs v(R) = sqrt(g R/2) = 5.6 km/s. At the Moon (r = 60 R) the same flow gives 9.8/60^5 = 1.3x10^-8 m/s^2, against the observed 2.7x10^-3 m/s^2 [memory, high], wrong by five orders of magnitude. So Reading A fails by simple calculus. To make a = -GM/r^2 the flow speed must be v = sqrt(2GM/r) (the free-fall speed), which is not incompressible: its divergence is (3/2) v/r, not zero. Then continuity requires ether to be destroyed everywhere in the space outside the body, at rate (3/2) rho_e v/r per unit volume, which contradicts "conversion inside planets". This dilemma (inverse-square requires v proportional to r^-1/2, sink-in-planet requires v proportional to r^-2) is the central mathematical difficulty of the claim as worded. Reconstruction: nothing in the folder document or the Investigator's pages shows Hilgenberg resolving it. His 1933 text (Investigator) says "a body accelerates in a field because a sink sits in a stream", which is Reading B, not A.

Reading B: a sink is pushed by the stream (Bjerknes/Kelvin force). A point sink of volume strength m_2 placed in an ambient stream of velocity u feels a force F = rho_e m_2 u (momentum balance on the absorbed fluid; magnitude [memory, medium], textbook hydrodynamics, Lamb Hydrodynamics and V. Bjerknes, Fields of Force; I did not recheck on the web). In the field of a second sink, u = m_1/(4 pi r^2), so F = rho_e m_1 m_2/(4 pi r^2): inverse square, attractive for like sinks. This is how a sink-flow theory can give Newton's form. It is consistent with the sign rule Hilgenberg lists in 1933 (Investigator, scan n20). It has a hidden requirement that I now derive.

Hidden requirement. Equate with Newton: rho_e m_1 m_2/(4 pi) = G M_1 M_2 for all bodies, so m must be proportional to M: m = s M, with s the volume of ether swallowed per kg per second. The mass gained by a body is then dM/dt = rho_e m = rho_e s M, i.e. exponential growth, mu = rho_e s, and G = rho_e s^2/(4 pi) = mu^2/(4 pi rho_e) [reconstruction, medium confidence on the textbook force law]. Thus exponential mass growth is not an extra assumption in this model; it is forced by "gravity from sinks" plus "swallowed ether becomes mass". That is a real mathematical point in favour of internal coherence, and it matches Hilgenberg's 1933 phrase (Investigator) that G "in the sense of the ether-stream hypothesis" is the volume of ether swallowed per unit mass per unit time (his s). Using his mu = 5.1x10^-17 s^-1: rho_e = mu^2/(4 pi G) = (2.6x10^-33)/(8.4x10^-10) = 3.1x10^-24 kg/m^3 [calculation]. He gives no ether density, so this is an implied value, not his; it is a number the theory could be asked to defend.

Unresolved inside the model: (i) the same mu for every body, including the Sun and Moon, is required (G universal); (ii) a universal exponential growth means every body gains 1.6x10^-9 of its mass per year [calculation]; (iii) Reading B's force is the force between sinks in an ideal incompressible fluid at rest, whereas gravity acts on a body through any intervening matter without screening; the model gives no account of that (a physics point for the physicist, noted here because it is a statement about the hypothesis, not about data).

b. Consistency with established results. Mathematically derived consequences I can check, treating them as consequences of the model as reconstructed: - Newton's inverse square: reproduced by Reading B, not by Reading A (Moon test above). - Relative mass growth 1.6x10^-9 per year for all bodies. For the Sun this means d(GM_sun)/dt over GM_sun of about +1.6x10^-9 per year [calculation]. Planetary ephemerides bound this at about 10^-13 per year (EPM fits give (-5 to -10) x 10^-14 per year, with uncertainties of a few 10^-14; INPOP limit about 2x10^-13 per year) [web: https://arxiv.org/abs/2201.09804 ; https://arxiv.org/pdf/2111.04499]. That is four orders below the model's rate, if the Sun grows as the Earth does. If the Sun is exempt, universality of G fails (hidden requirement i above). I flag this as a consequence for the physicist to weigh; the arithmetic is mine. - Present rate of Earth radius growth under his 1933 law with constant density: dR/dt = mu R/3 = 3.4 mm/yr [calculation: (5.1x10^-17/3) x 6.371x10^6 m x 3.156x10^7 s/yr]. Space geodesy (VLBI, SLR, GPS, DORIS) gives a mean radius change of 0.1 mm/yr with 1-sigma uncertainty 0.2 mm/yr [web: https://www.sciencedaily.com/releases/2011/08/110817120527.htm, reporting Wu et al., Geophys. Res. Lett. 2011]. The model's 3.4 mm/yr is about 17 times that uncertainty. (The reader should know this tests the current rate; a model with decreasing rate escapes it. His 1933 exponential law has an increasing rate, so it does not escape.) - Mass creation rate today: mu M_earth = 3.1x10^8 kg/s, or 9.6x10^15 kg/yr [calculation]. If the created mass carries rest energy (E = m c^2) from nowhere, that is 2.7x10^25 W, compared with the Sun's power intercepted by Earth, about 1.7x10^17 W, and Earth's heat flow, about 4.7x10^13 W [memory, high]. The model does not say whether the ether supplies this energy; Hilgenberg predicts the Earth's own temperature rises with mass (Investigator, scan n21), and the heat flow constrains that. - The ether inflow needed at the Earth's surface (Reading B, if one assumes all swallowed ether is converted at the surface): volume flux m = s M = mu M/rho_e = 3.1x10^8/3.1x10^-24 = 1x10^32 m^3/s [calculation], an inflow speed at the surface of m/(4 pi R^2) = 2x10^17 m/s, vastly above c. This follows from the implied rho_e above (low, because rho_e is derived from G and mu). So with those two inputs the inflow speed through the Earth's surface exceeds the speed of light by eight orders of magnitude. Hilgenberg's 1974 statement as reported (a star whose ether inflow reaches c is equivalent to a black hole) is a statement about this same quantity, and the numbers show it is violated by the Earth by a huge factor in this model. This calculation depends on a chain of my reconstruction (force law, G = mu^2/(4 pi rho_e)) and should be re-done once the 1931 and 1939 booklets are read; I flag it as a strong red flag, not a settled refutation. A different ether density or a different force law would change it.

c. Tested? The Investigator's reading (1933 book): Hilgenberg describes sink/source flow effects as medium-independent, with his own experiments in the 1931 booklet and the 1939 booklet (72 pp., "flow experiments that explain the nature of gravity" [read: deep p.16]); he credits Westphal with earlier one-dimensional experiments. I could not read these. The mathematical content of any such experiment would be a demonstration of the Bjerknes-type sign rule, which is a known hydrodynamic result [memory, medium] (including the inverse-square force between pulsating spheres), not a test of gravity: it shows that fluids can mimic an attraction, not that space is such a fluid. No independent replication on record in the folder or the Investigator's notes.

d. If true: The mainstream identification of gravity with the curvature of spacetime (general relativity) would have to be an effective description of a conserved-or-created ether. The following is exact, not mere analogy [memory, high]: for the Schwarzschild solution there is a coordinate system (Painleve-Gullstrand) in which space is flat and "flows" inward at v = sqrt(2GM/r), reaching c at the horizon (A. J. S. Hamilton and J. P. Lisle, Am. J. Phys. 74 (2006), "The river model of black holes"; not rechecked). So Hilgenberg's "flowing space" with v = c at the black-hole boundary has a legitimate mathematical image in general relativity: but that flow is a choice of coordinates, makes no measurable difference, and contains no ether density, no created matter and no growth. Adding ether density and mass creation is the departure, and it makes the model testable (and, by the numbers above, in conflict with measured GM_sun stability). Also required to change: conservation of mass-energy (see Claim 2).

Verdict (Claim 1): Split. - Qualitative picture (gravity as flow into sinks): Plausible, untested as a mathematical analogy; the Bjerknes inverse-square force between sinks exists. The reading in which bodies are carried along by an incompressible sink flow is Refuted by calculation (acceleration scales as r^-5; Moon test above). The reading consistent with inverse-square requires m proportional to M, hence exponential growth of every body. - Quantitative form as a competing theory of gravity: Never properly tested (no equation set in the sources I could read; Hilgenberg's own 1931/1939 booklets unread). The solar mass limit above suggests that if it were stated with universal mu, it would be Refuted by calculation, but this rests on my reconstruction and I do not make it the verdict.

Claim 2. Inflowing ether is converted inside planets into new atoms; the Earth gains mass exponentially.

a. Internal consistency. Hilgenberg's 1933 law: V' = V e^(mu t), mass proportional to volume at constant density, mu = (1/t) ln(V'/V). Arithmetic is correct (above). Assumptions he states: constant uniform density ("not true, but simplifying"), age 1 Gyr "since before the Paleozoic" (his figure, from radioactive dating after Born; the Paleozoic began about 540 Myr ago, [memory, high], and ended about 252 Myr ago, so "1 Gyr" does not match either end; he also allows 100 Myr from sediment estimates). Hidden assumptions: - Constant density is inconsistent with the physics of a compressible self-gravitating body: central pressure scales as G M^2/R^4, i.e. as rho^2 R^2 for constant density, so a 0.6-scale Earth at the same density has 0.36 of present central pressure [calculation, scaling argument]; the real Earth's density rises with depth because of that pressure. Not fatal (he flagged it) but the "V/V' = M/M'" step is an approximation of unknown size. - The rate mu is a constant in the 1933 law, but a body's acquisition of mass under Reading B is dM/dt = rho_e s M with s constant: also exponential. These two agree. Good internal coherence. - Choosing t = 1 Gyr as the interval from 0.6 R to R and mu constant: the ratio of Permian to present radius then follows (5570 km), not the 4590 km of the 1965 caption (below), so the 1933 and 1965 numbers cannot both come from one exponential law with t=1 Gyr. - Surface gravity: with constant G and constant density, g proportional to R. At 0.6 R the surface gravity would be 0.6 g, a quantitative consequence a mathematician can state; the claim in the folder document gives none.

b. Consistency with established results. Mass conservation: stated plainly, the claim is the creation of mass (3x10^8 kg/s today in his law, 9.6x10^15 kg per year) with the ether as the reservoir. A critic recorded in the TU Berlin forum (Bruhn, Prof. of Mathematics, TU Darmstadt): expansion "massively violates the law of conservation of mass" [read: Investigator, TU intern 2001 reader responses]. That is true by construction, so the objection is a consistency objection with accepted laws, not a mathematical error in his law. Measured: no excess in GM_earth is on record in the sources here; the ephemeris limit on the Sun (above) is the applicable web-checked limit. I have not web-checked Earth GM stability over decades (satellite laser ranging) and say so.

c. Tested? No test of the mass-creation hypothesis in the folder or the Investigator's notes. Scalera (2021) is the one attempt at numbers for ether density, velocity and flux [read: Investigator, Earth-Prints abstract; Scalera is an interested party]; I read only the Investigator's summary of an Italian abstract. He cites geoneutrino experiments (Borexino, KamLAND) as "corroboration", which is the author's characterisation and which I cannot evaluate from the summary: geoneutrino flux measures radioactive decay in the Earth, which the mainstream explains without mass creation.

d. If true: conservation of mass-energy would not hold for isolated bodies; the equivalence of mass and energy would need an energy source (the ether). General relativity would have to treat the ether as a field with a source term. The implied ether density 3x10^-24 kg/m^3 (above) is of the same order as the mean cosmological density [memory, high that critical density is about 10^-26 kg/m^3; this number is not Hilgenberg's].

Verdict (Claim 2): Never properly tested as a mass-creation hypothesis (no neutral, controlled test on record in these sources; the question of dismissal without testing is the physicist's and contrarian's). The strong form (universal exponential growth at his mu) is Tested, not supported by the space-geodetic radius limit and the solar-mass limit cited, if the numbers I reconstructed are taken as his. The 1933 constant-density relation M/M' = (R/R')^3 is Established only as an approximation, not as a property of real planets.

Claim 3. Earth's radius has nearly doubled since the Paleozoic; 60%; 4590 km; 4 mm/yr.

a. Internal consistency. Fully checked. - 60% as a diameter ratio is explained by the 1933 book (above). It gives R = 0.6 x 6371 = 3823 km at the "shelf-sphere stage" [calculation]. "Nearly doubling" = 1/0.6 = 1.67: not "doubling" (2.0) but "nearly" in a loose sense. - 4590 km is 72% of present (4590/6371 = 0.720) [calculation]. It cannot be the same stage as 60%. - If the shelf sphere is the pre-Paleozoic stage (as the 1933 text says: "since before the Paleozoic"), then Permian at 0.72 is a later stage and the two are reconcilable only if the 1965 reconstruction places the Permian at an intermediate radius; my calculation shows the 1933 exponential law cannot do this: it gives R(250 Myr ago) = 5570 km with t = 1 Gyr [calculation]. For R_Permian = 4590 km at 250 Myr with an exponential law, mu = 1.25x10^-16 s^-1 and the present growth rate would be 8.4 mm/yr [calculation], not 4. A linear law would give (6371-4590)/250 Myr = 7.1 mm/yr [calculation]. So 4590 km, 250 Myr and 4 mm/yr cannot all hold for a constant-rate or exponential law. 4 mm/yr over 250 Myr would give 1000 km growth, leaving a Permian radius of 5370 km (that equals 84% of the present radius), not 4590. Alternative reading not excluded: the Permian was dated differently in his time or the 4 mm/yr is an average over a different stage; the 1965 paper would say. [read: deep p.13 figures; calculations mine] - The map scale: only partly legible; no use.

b. Consistency with established results. Paleomagnetic palaeoradius estimates: McElhinny, Taylor and Stevenson (Nature 271, 316, 1978) limit possible expansion over the past 400 Myr to less than 0.8% [web: https://www.nature.com/articles/271316a0 , via search summary; abstract only, paper not read]. Hilgenberg's Permian radius ratio 0.72 would be a 28% change over 250 Myr: excluded by that limit by a factor of order 30. Note the independence: these limits come from paleomagnetic data, which Hilgenberg himself used (1965 paper) to place poles. Current rate: 3.4-4 mm/yr versus 0.1 +/- 0.2 mm/yr (Wu et al. 2011, above).

c. Tested? By others: McElhinny et al. (1978), Wu et al. (2011). By Hilgenberg: his reconstructions are geometrical constructions (paper outlines on globes, 30 projections in 1965 [read: deep p.13]); the 1933 book invites the reader to repeat the construction ("nachzuprüfen") (Investigator). A construction that fits is not a test of expansion, because he has degrees of freedom: the cut lines (the Mackenzie-Florida megashear of about 2000 km, an Alaska line), the 0.62-0.65 rather than 0.59 (see below), and the reader's own freedom in handling the distortion. Whether a neutral fit test was ever applied to continental outlines beyond Hilgenberg and his supporters is not shown in these sources.

d. If true: mass of Earth 0.216 times present at the pre-Paleozoic stage; mantle and core structure formed under a different pressure history; ocean-floor magnetic anomalies and subduction (Benioff zones) would have to be reinterpreted; seismic tomography showing slabs is not discussed by the sources.

Verdict (Claim 3): Split. Printed numbers (60%, 4590 km, 4 mm/yr, "nearly doubling"): Internally inconsistent as a set (calculation above), partly repairable by the diameter/volume distinction. Radius growth as a physical claim: Tested, not supported (paleomagnetic limit of 0.8% over 400 Myr; geodetic limit of about 0.2 mm/yr at 1 sigma on the current rate), the paleomagnetic limit by McElhinny et al. 1978 being a calculation from independent data, not a Hilgenberg-versus-X experiment. The geometry-only statement that a continent-and-shelf outline can be fitted onto a smaller sphere is Plausible, untested as a geometric exercise (see Claim 5).

Claim 4. Ether flow to the Sun plus the "inverted Magnus effect" spins planets up until an equatorial ring detaches.

a. Internal consistency. The folder document gives no equations; the authors say this part is "less valid today" [read: p.10]. The "inverted Magnus effect" as named is a real fluid-dynamics phenomenon: a spinning sphere or cylinder can experience lift opposite to the usual Magnus force, at certain Reynolds numbers and spin ratios, when the boundary layer on one side goes turbulent [web: https://cambridge.org/core/journals/journal-of-fluid-mechanics/article/inverse-magnus-effect-on-a-rotating-sphere-when-and-why/0E9FEF2880AA45C8B12587374C6F10EE and https://meetings-archive.aps.org/dfd/2011/a7/8; search summary only]. This is a modern, experimentally established effect in viscous fluids, with a boundary-layer explanation. It is a statement about drag and lift on a body in a real (viscous) fluid at specific Reynolds numbers. Transferring it to a planet in an ether requires the ether to be viscous, with a boundary layer on the planet's surface and Reynolds numbers in a window; none is given. Hidden assumption: that a force on a spinning body in a stream gives a torque that increases spin without bound until Roche-type detachment. A force perpendicular to the stream (Magnus lift) is not a torque on the spin axis; spin-up needs a separate argument that is not in the sources read. Angular momentum balance (given a sink flow carrying the momentum) is not stated.

Further, the Investigator's reading of the 1933 book includes the "predicted consequences": distance to the Sun grows, spin speeds up, days shorten, years lengthen (1931 booklet pp.43-50, 68-72, unread). Under Kepler's third law, orbital period T^2 proportional to a^3/(GM): with GM_sun growing at 1.6x10^-9 per year (above) and a growing, the sign of the period change depends on a's growth; no rate is given in the sources. For the Earth's rotation, the paleontological record of daily growth lines in corals (about 400 days per year in the Devonian [memory, medium]) shows days lengthening over time, the opposite of "days shorten". I did not web-check this and flag it as [memory, medium] for the physicist. If correct, Hilgenberg's predicted sign for the length of day is opposite to the observed secular trend.

b. Consistency with established results. Moon-like final state: the authors themselves note planets reach synchronous rotation via tidal friction and that no planet is near the needed spin for ring expulsion [read: page image p.10]. Mathematically: ring detachment needs equatorial centrifugal acceleration omega^2 R to equal surface gravity g, i.e. omega = sqrt(g/R) = 1.24x10^-3 rad/s, period 84.5 minutes for Earth [calculation: sqrt(9.81/6.371e6)=1.241x10^-3 s^-1; T = 2 pi/omega = 5063 s]. Earth rotates at 7.29x10^-5 rad/s, about 17 times slower. Jupiter's rotation (period 9.9 h [memory, high]) is also far below its break-up speed (about 3 hours [memory, medium]). So "none of the planets shows the needed angular velocity" is correct as the authors say.

c. Tested? Hilgenberg reports a laboratory demonstration of one of "four new fluid-dynamic effects ... completely proven by experiment (see the report of 1933)" [read: p.9, quotation from his curriculum]. Which effect, what apparatus, what measurements, are not in the folder. The 1933 papers (Z. Techn. Phys. 14, 211-214) are not in the folder; their existence as a refereed paper in a major journal is [read: reference list]. Replications: none found. The wider literature on inverse Magnus (above) does not cite him in the search summaries; I did not search for earlier work.

d. If true: celestial mechanics would need a viscous, ponderable ether acting on planets, contradicting the very precise ephemeris fits that need no such drag [memory, high]. Note also the sign-of-spin sequence "retrograde to zero to inverted" is an assertion without a torque calculation in the sources.

Verdict (Claim 4): Split. (i) The inverse Magnus effect in laboratory fluids: Established (modern literature above), as a general fluid-dynamics fact; Hilgenberg's 1933 experiment is Never properly tested by others on record in these sources (not shown to have been checked, and not shown to have been dismissed untested). (ii) Planetary spin-up and ring detachment as stated: Not testable as stated (no equation, no torque, no rates) and, against the authors' own caveat and the break-up calculation, Historically superseded but instructive.

Claim 5. Continental outlines move between smaller-radius globes; paleopoles give a consistent paleogeography; no subduction needed.

a. Internal consistency. This is the part where Hilgenberg's mathematics is most concrete and where I can check something. - Area arithmetic (1933 book, Investigator). Shelf areas are about 35% of the Earth's surface; a sphere with area proportional to R^2 closes if (R'/R)^2 = 0.35, R'/R = 0.59 [calculation]. He used 0.62-0.65 "to leave play" (area ratio 0.384-0.423), i.e. 9-21% slack, and at 0.62 "hardly any gap remained" (his words). A fit with 9% area slack plus two cut lines (2000 km shear) is not a closed-form test, because cuts and slack are adjustable. The 35% figure is as stated by him citing Wegener and Born; I have not rechecked it ([memory, medium]: continents plus shelves are about 35-40% of the surface). - Gauss's theorema egregium (basic fact [memory, high]). A piece of a sphere of radius R cannot be mapped to a sphere of a different radius without changing lengths or angles, because Gaussian curvature 1/R^2 changes. So placing a continent on a smaller globe necessarily distorts it. His method (authors' account: paper outlines moved between globes, "eliminating some thin radial slices", equivalent to an equidistant azimuthal projection [read: p.11]) preserves distances along radial lines from a centre and distorts the tangential lengths. Worked number [calculation]: map a point at distance d = 3000 km from the centre from a sphere of radius R' = 3823 km to one of R = 6371 km. The circumference of the circle at that distance changes by [R sin(d/R)]/[R' sin(d/R')] = 2891/2702 = 1.07: a 7% tangential strain at 3000 km, growing with d. Over a continent 6000 km across, that is a gap or overlap on the scale of hundreds of kilometres, which is the size of rift zones. This is a calculation showing that the "slices eliminated" are not small bookkeeping; they absorb a few percent of length, which is comparable to the size of rifts the model explains. So the reconstruction has freedom proportional to the distortion; this limits what a "good fit" proves, and, in the other direction, makes the fit seem natural: any outline fits to within about this tolerance in the narrowing direction. That does not mean the fit is false, only that the criterion of success must be stated. - Paleopole transposition. The authors' account [read: p.12-13]: a single pole seen from different sites on a rigid continent gives a scatter of apparent pole positions when the continent is transferred to a larger sphere "depending on the transformation law". Mathematical check: a paleomagnetic pole at colatitude p from a site is computed from the inclination I via tan I = 2 cot(colatitude) (geocentric axial dipole) and the declination D [memory, high], using angular quantities only. Under pure radial scaling (every point keeps its latitude and longitude) angular distances are unchanged, so no dispersion arises. Dispersion arises only when the continent is treated as a rigid cap of fixed arc length on spheres of different radius (the angular size changes by R/R'), i.e. when a particular transformation law is chosen. That choice is Hilgenberg's, not forced by the data; he says a different law is needed for the paleopoles (distance in degrees maintained) [read: p.13]. Hence the paleomagnetic fit has a free modelling choice (which law) that a reader should know about. The sources do not give the poles themselves or any misfit statistic. - Rift, graben and trench mechanics: the authors say rifts, grabens and crust overlaps arise from "mechanical forces that a spherical continental cap undergoes as it attempts to readjust to a flattening surface curvature" [read: p.11]. This is qualitatively consistent with the Gauss argument above (a cap on an enlarging sphere must stretch tangentially), but no stress calculation is given. A quantitative check would need elasticity of a thin spherical shell; not in the sources.

b. Consistency with established results. The authors themselves say the Pacific solution "is not in agreement with the most up-to-date paleomagnetic data" [read: p.11]. The slab image of seismic tomography is described by them as extending to 700 km; Hilgenberg's interpretation (dunite rising from the deep mantle) is reported without a calculation and the authors only say it "might once again be appraised" [read: p.11-12]. Seafloor-spreading, ocean crust ages and the heat flow are not discussed in the sources. The paleomagnetic palaeoradius limit above (0.8% over 400 Myr) applies directly. The Bruhn/Seiler/Thorweihe criticisms (mass conservation, tomography, no mechanism) are on record in the Investigator's note.

c. Tested? Reconstructions by Hilgenberg (1965: 5 reconstructions, 30 projections) and replicated by supporters (Scalera's four replica globes, workshops in more than ten countries, per Scalera via Jacob); advocates, not independent tests. McElhinny et al. (1978, above) is an independent paleomagnetic test of the expansion size, with a negative result for large expansion. Whether it was applied to Hilgenberg's own reconstructions is not shown. The globe reconstructions can be repeated by a reader with a ball and paper; the sources do not show a blind, neutral comparison.

d. If true: subduction as the main sink of ocean crust would have to be replaced; plate tectonics' quantitative budget (spreading rates matching trench consumption, conserved total area) would not close; the mantle composition and thermal budget would need the new matter.

Verdict (Claim 5): Split. The geometric statement that continent-plus-shelf outlines can be placed with modest gaps on a sphere of about 0.6-0.65 of present diameter: Plausible, untested as geometry (with the caveat that cuts, slack and the transformation law are free parameters, so the fit alone is weak evidence). The statement that this paleogeography is correct, i.e. that Earth was that small: Tested, not supported (paleomagnetic palaeoradius limit, McElhinny et al. 1978; reported by the Investigator's critics also). The claim that trenches, rifts and Benioff zones need no subduction: Never properly tested in the sources reviewed (no calculation or data in the folder; critics cite tomography, per Bruhn; no neutral test of the dunite-uplift idea shown).

Claim 6. The real question is expansion with or without new matter; the hydrodynamic ether view is analogous to, and may converge with, general relativity; a star whose ether inflow reaches c is a black hole.

a. Internal consistency. The reported derivation ("setting the ether energy density to zero, a simple mathematical conversion, adding the fourth time coordinate") is not shown [read: p.14]. Mathematically: general relativity is not obtained from Riemann's ether considerations by that recipe; Einstein's field equations are a specific tensor equation (curvature equals stress-energy) that is not "derived" from a flow by dropping the energy density [memory, high]. The authors' paraphrase is therefore an unsupported assertion in the form given. What can be said exactly is the Painleve-Gullstrand "river" representation of Schwarzschild: space flowing at v = sqrt(2GM/r) with v = c at the horizon (above) [memory, high; Hamilton and Lisle 2006, not rechecked on the web]. This is a genuine mathematical correspondence for static, spherical, uncharged cases; it supports the black-hole statement ("flow speed reaches c") in the narrow sense, with these limits: (1) it is a coordinate choice, so it has no measurable content separate from general relativity; (2) the flow v is not Hilgenberg's incompressible sink flow (that one has v proportional to r^-2, Claim 1); (3) there is no mass creation in it. So "equivalent in every way" (the authors' phrase) is too strong: the equivalence holds only if the ether is the river-model coordinate flow, which then carries none of the features (conversion to matter, growth) that make his theory distinctive.

b. Consistency with established results. General relativity passes precision tests [memory, high] (perihelion precession, light bending, Shapiro delay, gravitational waves). A model that reproduces those via the river representation is consistent; a model with ether growth predicts extra effects (GM change) that the ephemeris limit excludes if universal (Claim 1 b). The authors themselves note Hilgenberg first rejected relativity citing Miller's flawed experiments and later accepted it "with some reserves" [read: p.7].

c. Tested? The black-hole analogy has not been tested as a hydrodynamic claim; the existence of black holes is a general relativity matter. The 1974 open letter drew no reported reply [read: p.13-14]. The sources do not tell whether the addressees received it.

d. If true: general relativity would be an effective, coordinate-level description of an underlying fluid; the cosmological claim "expansion with or without new matter" would become a question about ether supply; Newtonian mass conservation would be violated for isolated bodies.

Verdict (Claim 6): Split. (i) That a black-hole boundary is where a flow of space reaches c: Plausible, untested in the sense that the river model is an exact reformulation, but then Not testable as stated as a distinct theory (no measurable difference from general relativity). (ii) That Hilgenberg's ether-flow theory "converges with" or is "equivalent in every way" to general relativity: Internally inconsistent or at best unsupported as stated: the reported derivation is not shown, and his flow differs from the river-model flow (Claim 1). I use "Never properly tested" for the question as to whether the 1974 argument was examined by anyone, since no reply is reported.

Weighing the Investigator's findings (kept in the record whatever the verdict)

• 1933 book (own work, Priority 1). The most valuable single find for the mathematics: it supplies the growth law, the number 5x10^-17 s^-1, and his candid statement of what is unexplained (how ether enters atoms; how swallowed ether becomes mass). His arithmetic is correct. His admissions are to his credit and should be quoted: he says constant density is "not true but simplifying", and invites readers to repeat the globe construction. Not yet read: the second half of the book and its reference list.
• Yarkovsky (1888) and Riemann. Closest prior parallel to Claims 1-2 (ether inflow, conversion into matter); the Chappell wiki page is an advocacy source (labelled as such). That Hilgenberg reached the idea independently of Riemann (his statement) and that Bruhn named Yarkovsky as the lineage are both on record; whether Hilgenberg knew Yarkovsky is unknown. The Bjerknes lead: confirmed in outline by my calculation above (inverse-square force, like sinks attract), and it is a prior known hydrodynamic analogy.
• Scalera 2021. Only source that tries to put numbers on ether density, velocity and flux; Scalera is an interested party; I saw it only through the Investigator's note of an Italian abstract. My calculation gives an implied ether density under Hilgenberg's mu (3x10^-24 kg/m^3); a comparison with Scalera's numbers is the natural next step, but I cannot make it from the notes.
• Thesis rejection (Priority 4). Three accounts (Scalera-Braun; German Wikipedia: 1938 Fakultaet fuer Maschinenwesen, mainly over relativity; Thielmann 2015 and zillmer.com: submitted 1937, rejected) with the faculty file not yet seen. From the mathematical side, only note that all three place the conflict with relativity as the reason, and that Hilgenberg's 1933 text itself relies on relativity of motion (and on the equality of heavy and inertial mass) as a premise for his ether view, per the Investigator. So the stated mathematical-physical reason for rejection (conflict with relativity) is at least consistent with his own 1933 premises being non-standard; I cannot judge the examiners' reasons without the file. The absence of the faculty record is an anomaly to analyse (Investigator's competing explanations stand); nothing I can add.
• TU Berlin 2001 forum. Critics' arguments recorded (Bruhn: mass conservation, tomography, no quantitative support for decreasing G, meteorite input too small; Seiler; Thorweihe) and the witness account (Fiedler-Volmer: Hilgenberg open to discussion; "sometimes even insulting ignorance"). Both are kept; Bruhn is a mathematician with a stake in his own discipline's standards, Fiedler-Volmer a sympathetic witness. The "ad acta" expert reply (July 2001) was not opened and could add a quantitative critique.
• Honest limits of what the mathematics shows: the authors' own admission that the spin-up part is "less valid today" is evidence from the proponents' side against Claim 4 and is kept as such.

What would settle the points

1. Read the 1931 and 1939 booklets: do they contain the force law (Reading B) and an ether density? The central unknown is whether Hilgenberg derived an inverse-square law and from what.
2. Re-derive G = mu^2/(4 pi rho_e) from his text (not my reconstruction) and test the universality of mu against GM_sun and GM_earth limits (the numbers above).
3. Obtain the 1965 paper: radius law, age scale, and the 4590 km and 4 mm/yr source.
4. A blind reconstruction test of outlines on a 0.62 sphere versus random cuts of equal slack, to quantify how much a "fit" with 9% slack and two cut lines actually shows.
5. Faculty file at the TU Berlin Universitaetsarchiv.

Web verification log (budget 6; used 4)

1. Earth radius change limit: [web] https://www.sciencedaily.com/releases/2011/08/110817120527.htm (Wu et al. 2011, 0.1 mm/yr, 1-sigma 0.2 mm/yr). Search summary only; paper not opened.
2. Inverse Magnus effect: [web] https://cambridge.org/core/journals/journal-of-fluid-mechanics/article/inverse-magnus-effect-on-a-rotating-sphere-when-and-why/0E9FEF2880AA45C8B12587374C6F10EE (established, boundary-layer explanation). Search summary only.
3. Solar GM change limits: [web] https://arxiv.org/abs/2201.09804 and https://arxiv.org/pdf/2111.04499 (EPM about -5 to -10 x 10^-14 per year; INPOP about 2x10^-13 per year). Search summary only.
4. Paleomagnetic palaeoradius limit: [web] https://www.nature.com/articles/271316a0 (McElhinny, Taylor, Stevenson 1978: less than 0.8% over 400 Myr). Search summary of the abstract only. Not web-checked, still [memory]: the Bjerknes/Kelvin force law and its sign (medium; central to Claim 1 Reading B), Painleve-Gullstrand river model (high), coral growth-line day counts (medium), continent plus shelf area share (medium), Earth heat flow and solar constants (high). The physicist should confirm the Bjerknes force-law sign before the Reading B calculations are used.

Summary of verdicts

1. Gravity as sink flow: split. Carried-along incompressible flow: Refuted by calculation (r^-5). Bjerknes-type sink force: Plausible, untested (as analogy); as a full theory of gravity: Never properly tested.
2. Ether to matter, exponential growth: Never properly tested (weak form); universal-rate strong form Tested, not supported (solar GM and Earth-radius limits, on my reconstruction).
3. Radius doubling numbers: printed numbers Internally inconsistent; physical expansion Tested, not supported (paleomagnetic 0.8%, geodetic 0.2 mm/yr).
4. Inverted Magnus spin-up: effect in fluids Established; planetary spin-up Not testable as stated, Historically superseded but instructive.
5. Fit on smaller globes: geometry Plausible, untested (free parameters); correct paleogeography Tested, not supported; no-subduction mechanics Never properly tested.
6. Relativity analogy and black hole: river-model correspondence Plausible, untested but Not testable as stated; "equivalent in every way" Internally inconsistent / unsupported.