When a number agrees exactly, the first thing to suspect is not nature but the definitions.
The single phrase Mach’s principle names several claims that do not imply one another. This paper calls them by name and shows in numbers that a much-quoted numerical coincidence is in fact an identity following from the definitions. NO NEW PHYSICAL LAW IS CLAIMED and nothing here contradicts general relativity. That Mach’s principle is not one sentence, the Lense-Thirring effect, Thirring’s shell coefficient, Godel’s universe and Sciama’s argument are all standard. Nor does this paper rule on whether Mach’s principle is right: what can be written is what each reading demands.
First, and this is the core. The claim often quoted runs thus: summing the ratio of gravitational potential to the square of the speed of light over the visible universe gives something near one, as if distant matter were paying for inertia. So the number was actually computed, from the Hubble radius at the critical density. At Hubble constants of fifty, sixty-seven point four, seventy, one hundred and one thousand kilometres per second per megaparsec, the answer is one half, to every digit computed. The reason is visible on substitution alone: the enclosed mass equals the cube of the speed of light divided by twice the gravitational constant times the Hubble constant, so the ratio is one half and the Hubble constant cancels. This is not a number that was measured and found to agree. It is an identity following from the definitions, and in the six roots of the equals sign of Paper 16 it is a consequence of substitution rather than an empirical law.
Second, then what does the coincidence record? Computing away from the critical density gives 0.05, 0.15, 0.5 and 1.5 at density parameters of 0.1, 0.3, 1 and 3, that is simply half the density parameter. So what the observation that it comes out near one records is that the density parameter is near one, which is flatness. That is a fact about the geometry of the universe and says nothing about where inertia comes from. What must be moved for the number to move is the density parameter, not the amount of matter, not the age of the universe and not the Hubble constant. Multiplying the Hubble constant by twenty did not move it, and a quantity that does not move cannot be offered as evidence.
Third, geodetic precession and frame dragging have distinct roots. For a gyroscope orbiting the Earth, whose angular momentum computes to 5.84571 times ten to the thirty-three in SI units, the two characteristic rates are 6.6048684 against 0.081436822 arcseconds per year at a radius of 7027 kilometres, a ratio of 81.1; the ratio is 96.8 at ten thousand kilometres, 157.7 at 26561 and 198.7 at 42164. The ratio moves, because the geodetic rate falls as the radius to the minus five halves and the dragging rate as the radius to the minus three. Having different dependence on the radius, these are not one effect at two strengths. Stopping the Earth’s rotation and recomputing leaves the geodetic rate at 6.6048684 and sends the dragging to zero: the angular momentum does not appear in the geodetic rate at all, which survives unchanged around a body that does not rotate. One experiment holds two distinct roots, mass curvature on one side and rotation on the other, by the same discipline with which Paper 165 separated the three equivalence principles. And the Machian half is the small half.
Fourth, dragging is always partial, and extrapolation does not help. Thirring’s weak-field result for the dragging fraction inside a slowly rotating thin shell computes to 9.28175 times ten to the minus ten for the Earth, 2.83009 times ten to the minus six for the Sun, and 0.229704 for a neutron star of 1.4 solar masses and twelve kilometres. Machian talk wants complete dragging, that is a fraction of one. Solving the formula for that gives a radius of four thirds of the gravitational radius, and comparing it with the Schwarzschild radius gives a ratio of exactly two thirds, the same for every mass since the mass cancels. So the weak-field formula reports that complete dragging is reached inside the horizon. That is not a discovery about black holes; it is a proof that the formula was carried outside its domain. Whether complete dragging occurs is therefore not settled by extrapolating a weak-field coefficient, and this paper claims no such answer, writing only how far its own tools reach, as Paper 159 did when separating the readings of ill posed.
Fifth, there is a counterexample from the other side. Minkowski space has well-defined inertial frames and no matter at all; the Schwarzschild exterior has none locally; Godel’s universe of 1949 has rotating dust and no global rest frame; the Kerr exterior has frames that are dragged and still defined. General relativity has vacuum solutions carrying a perfectly good inertial structure, so the strong reading, that inertia is produced entirely by distant matter, is false inside the very theory usually credited with vindicating it. The reason is structural: the field equations relate matter to curvature, and picking out one solution needs boundary conditions as well, with asymptotic flatness a choice rather than a consequence of the matter. Read as an inverse problem, Mach asks whether the local inertial frame determines the distribution of matter, and it does not, because the same matter admits different boundary data. Where Paper 89 assembled tools for reading a hidden cause from a perfect null, here stands an example on the unreadable side.
A census. The dynamical reading, that the universe is not static, is satisfied. The strong reading, that an isolated body has no inertia, is not satisfied, vacuum solutions being counterexamples. The local reading, that matter fixes local inertial frames, holds only partly and demands boundary conditions. The cosmological reading, that the density parameter equals one, is a question for observation, and what moves it is that parameter. Dragging, that rotation drags inertial frames, is satisfied, is proportional to the angular momentum, and is small. These five are not a staircase of strength: general relativity satisfies some, violates others, and for others demands boundary conditions. The sentence that general relativity realised Mach’s principle has no truth value until it says which reading.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
⚠️ Scope ── what is not claimed
a census of established standard facts (Layer 2). What is not shown: any new physical law, none is claimed; any contradiction with general relativity, there is none; a verdict on Mach’s principle, none is given; an exact answer about complete dragging, none is claimed. What is added: computing the ratio across values of the Hubble constant and confirming that it is an identity, showing that what it records is half the density parameter, that is flatness, showing that the ratio of geodetic precession to frame dragging moves with the radius, and showing that extrapolating the weak-field coefficient gives two thirds of the Schwarzschild radius for every mass, which proves the extrapolation invalid. What remains: when a number agrees exactly, the first thing to suspect is not nature but the definitions. The ratio was one half before anyone measured it. Mach’s question does not vanish with it, but this number is no longer evidence for it, since only a quantity that can be moved can serve as evidence.
Reference
Yamagishi, Y. (2026). Sciama’s Famous Coincidence Is an Identity [Paper 169]. Zenodo.
DOI: 10.5281/zenodo.22093563
ORCID: 0009-0006-8587-3450