Substituting the Yamagishi patches M = 4π and G = 4π into Kepler’s third law T² = (4π²/GM)a³ cancels the 4π² of the numerator against the 16π² of the denominator and gives 4T² = a³, that is, T = a^(3/2)/2. The cancellation is correct. It is, however, an expression in natural units and does not hold as such in SI: setting M = 4π and G = 4π already constrains the units of mass, length and time against one another.
[What the 3rd edition treats] — the one thing of substance that survives every choice of units: the exponent 3/2.
First, 3/2 is not a dimension. Paper 94 separated fractal dimension (non-integer, Hausdorff), topological dimension and special integer dimensions as distinct roots. The exponent 3/2 is none of these; it is a scaling exponent of dynamics, a fourth distinct root. The error in the first edition’s reading — that the square root of a³ reduces three dimensions to two — lay, before any arithmetic (a square root halves an exponent rather than lowering it by one), in a category mistake: taking an exponent for a dimension. Reading a^1.5 as “1.5 dimensions” is the same kind of error as inferring, from the fact that velocity is measured in metres per second, that there is a space of dimension metres-per-second.
Second, 3/2 is a member of a family. For a power-law central force F ~ r^n one has in general T ~ a^((1-n)/2): the inverse-square law gives 3/2, the 1/r law gives 1, and Hooke’s law gives 0 — the last corresponding to the known isochronism of the harmonic oscillator, whose period does not depend on amplitude. This paper verifies the exponent without recourse to observational data, by integrating the two-body problem directly: for eight bound orbits of eccentricity 0 to 0.6, the measured periods agree with 2πa^(3/2)/sqrt(GM) to within 5×10⁻¹², and the log–log regression of period against semi-major axis returns 1.50000000, differing from 3/2 by 8×10⁻¹³.
Third, in d dimensions the exponent becomes d/2. Gauss’s flux theorem in d dimensions gives F ~ r^-(d-1) — the same circumstance by which 4π becomes Vol(S^(d-1)) — and substitution yields T ~ a^(d/2). Hence 3/2 is merely the d = 3 member of a family with one member per dimension, and by the measure of Paper 114 it belongs on the dimension-blind side: it is not 3/2 because three dimensions are special, but because d/2 at d = 3 is 3/2. This is the same verdict Paper 35 (3rd edition) returned for the 4π of Poisson’s equation.
Fourth, the root of 3/2 is the same as that of Poisson’s 4π. Since the inverse-square law is forced by Gauss’s flux in three dimensions, the exponent belongs to what Paper 35 called Root 1 (fact, not chosen), whereas the G = 4π in the same equation is Root 2 (convention, may be chosen). The single equation 4T² = a³ contains at once something that could not be chosen — the exponent — and something that was — the coefficient 1/4. Relatedly, that π vanishes entirely from the Kepler row is only a forced π² divided by an inserted π²; no meaning should be read into the disappearance.
Fifth, what makes the Kepler problem special is not its exponent but the fact that its orbits close. By Bertrand’s theorem (1873), the only central-force potentials for which all bounded orbits are closed are the inverse-square law and the harmonic oscillator; the reason orbits close — that the perihelion does not precess — is the hidden SO(4) symmetry, the conservation of the Laplace–Runge–Lenz vector (Paper 79). Rotational symmetry SO(3) alone fixes only the plane of the orbit; the enlargement to SO(4) fixes its orientation as well. But “three dimensions, therefore inverse square” (Gauss’s flux, geometry) and “inverse square, therefore closed orbits” (Bertrand, dynamics) are two independent theorems. Their meeting at the inverse-square law is a signature, not a derivation, and must not be read as “orbits close because space is three-dimensional” — indeed the other solution Bertrand allows, the harmonic oscillator, does not arise from Gauss’s flux in three dimensions at all.
The 2π appearing in T = 2πa/v is the periodicity of phase — Root B in the roll call of Paper 18 (3rd edition) — and 4π² is its square. For a circular orbit it happens to point at the same circle as the circumference factor Vol(S¹) (Root C), but that is a special circumstance of the circle and not a general identification of the two roots.
Finally, the intuition behind the first edition — that for a distant body the universe does not handle its interior detail — is itself correct as physics; what it corresponds to is Newton’s shell theorem, by which a spherically symmetric body exerts exactly the gravity of its total mass at its centre, the internal density distribution having no effect whatsoever from outside. In the multipole expansion the corrections enter as (R/r)^n, and for nearby stars r/R ~ 10⁷ makes the quadrupole term ~10⁻¹⁴, unmeasurable. The correction is only that this applies to the shell theorem rather than to a^(3/2).
Scope: a structural interpretation of established physical law (Layer 2). No new physical law and no prediction is claimed; the exponent is not shown to select three dimensions; Gauss’s flux and Bertrand’s theorem are not merged; time is not claimed to be a “rendering lag”, which is a name and not a derivation. The main result 4T² = a³ and the shell-theorem section are unchanged. The Universe OS is a non-scientific Layer-3 lens.
(scope note) Positioning of this paper: This paper offers a structural interpretation of established physical law. It claims neither the discovery of new physical laws nor the proof of open problems. The algebraic relations presented are substitutions into known expressions; the aim is to propose a geometric reading of them. No claim contradicting established theory is made.
Note on units (important): The substitutions m=4π, G=4π and the like employed here constitute a choice of natural units. G and ħ are dimensionful quantities whose numerical values depend entirely on the unit system. Setting G=4π is possible in exactly the sense that G=1 (geometrised units) or G=6.674×10⁻¹¹ (SI) is possible; physics does not determine which is “correct.”
What this paper exhibits is therefore how the equations look once this unit system is adopted, not a new fact about nature. Moreover the relations obtained hold in natural units, and conversion factors are required to carry them into SI. Whatever value this paper has lies in making geometric structure visible — rendering the provenance of π and the role of solid angle easier to read off the equations.
A note on the aim of this work. This is a structural reinterpretation of well-established physics, not a proposal for a new physical law. Every relation shown is a substitution into, or a geometric re-reading of, equations that are already known and that remain unchanged. The aim is a modest one: to make a few recurring geometric structures — in particular the solid angle 4π and the exponent n = 2 — easier to see within a single picture. Where a particular unit convention is adopted (for example G = 4π), this is a choice of notation, not a claim about nature. Nothing here is intended to contradict existing results.
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.
Reference
Yamagishi, Y. (2026). The Yamagishi-Kepler Equation [Paper 28]. Zenodo.
DOI: 10.5281/zenodo.21430531
ORCID: 0009-0006-8587-3450