The single phrase equivalence principle names three claims that differ in what they assume and in how they are tested.
This paper calls them by name and shows in numbers that the third is not a refined version of the first. NO NEW PHYSICAL LAW IS CLAIMED, and nothing here contradicts general relativity. The distinction among the three principles, the Nordtvedt effect and the Brans-Dicke parametrisation are all standard. Nor does this paper assert experimental values: what is shown is the conversion, if the Eotvos parameter falls below such a value then the Nordtvedt parameter falls below such another, and every conversion here was computed in this paper.
First, in the weak equivalence principle the mass cancels only because inertial and gravitational mass were identified. The mass drops out of Newton’s equation of motion only when the two are set equal; write the ratio of gravitational to inertial mass as one plus a small excess and that excess returns to the acceleration. For two bodies the Eotvos parameter equals the difference of the two excesses to first order. A test of the weak principle therefore measures the failure of that identification and nothing else, and what must be moved to move it is the composition of the bodies, not their place or their speed.
Second, Einstein’s equivalence principle answers a different question. The gravitational redshift over a tower of 22.5 metres is 2.45506 times ten to the minus fifteen, and a GPS clock, at a semi-major axis of 26561 kilometres, differs from one on the ground by plus 45.719 microseconds gravitationally and minus 7.21337 microseconds from velocity, for a net of plus 38.5056 microseconds per day, all computed here. None of these numbers depends on what the clock is made of. What they test is that the laws do not depend on where the laboratory is or how fast it moves, that is local position invariance and local Lorentz invariance, so refining an experiment that varies composition leaves part of the Einstein principle untested. The same distinction is needed as in Paper 123, where what moves is not the clock but the worldline.
Third, and this is the core. The strong equivalence principle requires self-gravity. A violation appears as the ratio of gravitational to inertial mass equal to one plus the Nordtvedt parameter times the binding-energy fraction. Computing that fraction for uniform spheres gives 1.4365 times ten to the minus twenty-six for a one kilogram laboratory sphere of density eight thousand, 1.8829 times ten to the minus eleven for the Moon, 4.1768 times ten to the minus ten for the Earth, 1.2735 times ten to the minus six for the Sun, and 0.10337 for a neutron star of 1.4 solar masses and twelve kilometres. The Earth exceeds a laboratory mass by 2.9076 times ten to the sixteen, which is sixteen orders of magnitude. As far as self-gravity is concerned, a body on a bench barely exists.
Fourth, a laboratory test of the weak principle therefore barely constrains the strong one. Converting a bound on the Eotvos parameter into a bound on the Nordtvedt parameter, a precision of ten to the minus eleven gives 6.961 times ten to the fourteen with laboratory masses and 0.02507 with the Earth-Moon pair. Pushing to ten to the minus fifteen still gives only 6.961 times ten to the ten in the laboratory, which is no constraint at all, while the same precision on the Earth-Moon pair gives 2.507 times ten to the minus six. An experiment testing the strong principle is not a refined version of one testing the weak principle. It is a different experiment on different bodies, and seventeen orders of magnitude are not closed by better apparatus.
Fifth, general relativity satisfies all three and Brans-Dicke only the first two. In Brans-Dicke theory gamma equals one plus omega over two plus omega, beta equals one, and the Nordtvedt parameter is four beta minus gamma minus three, which is one over two plus omega. At omega equal to five this gives gamma minus one equal to minus 0.1429 and a Nordtvedt parameter of 0.1429; at five hundred, minus 1.992 times ten to the minus three and 1.992 times ten to the minus three; at forty thousand, minus 2.5 times ten to the minus five and 2.5 times ten to the minus five, all computed here. Only omega tending to infinity, that is general relativity, gives a Nordtvedt parameter of exactly zero. Since Brans-Dicke satisfies the weak and Einstein principles for every omega, the strong principle is the only one of the three that separates general relativity from its nearest rivals, and the sentence the equivalence principle has been confirmed says nothing about choosing between theories until it says which one. The Nordtvedt parameter stands where Paper 134 placed gamma.
A census. The weak principle says that falling does not depend on composition, composition is what moves it, and a laboratory suffices. The Einstein principle says that special relativity holds locally, the laboratory’s place and speed are what move it, and a laboratory suffices. The strong principle says that it holds for self-gravitating bodies too, the binding-energy fraction is what moves it, and astronomical bodies are needed. These are not one principle at three strengths. The third is not reached from the first by improving precision: it needs another body.
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; any assertion of measured values, none is made, since what is given is the conversion alone. What is added: computing the binding-energy fraction for five bodies and exhibiting the sixteen orders between them, setting side by side what one precision buys in a laboratory and on the Earth-Moon pair, and writing, for each of the three, what must be moved for it to move. What remains: three principles under one name appear to lie along a single line. They do not. No refinement of the first arrives at the third, and reaching the third means leaving the bench and borrowing a planet.
Reference
Yamagishi, Y. (2026). There Are Three Equivalence Principles [Paper 165]. Zenodo.
DOI: 10.5281/zenodo.22092957
ORCID: 0009-0006-8587-3450