Substituting m=4π into the kinetic-theory pressure P = (1)/(3) n m v² gives
P = (4)/(3)π n v² ,
in which the coefficient (4)/(3)π equals the volume of the 3D unit ball, Vol(B³). This connects to the kinetic energy of Paper 18 via P = (2)/(3) n E_k with E_k = 2π v². All of these are elementary substitutions.
Correction carried over in the 2nd edition: The first edition combined this with the temperature of Paper 20 in its first-edition form kT = (4)/(3)π v³, which required the non-standard equation of state P = n kT/v. The second edition of Paper 20 corrected the temperature to kT = (4)/(3)π v² (the first edition had misapplied equipartition by an extra factor of v). This correction works in favour of the present paper: with the corrected temperature, the standard ideal gas law
P = n kT = n·(4)/(3)π v² = (4)/(3)π n v²
holds directly, and the non-standard form is no longer needed.
(scope note) Scope of this paper: This paper offers a structural interpretation of existing physical law. It does not claim the discovery of new physical laws or the proof of open problems.
Note on units: Substitutions such as m=4π are a choice of natural units. G and ħ are dimensionful; their numerical values depend entirely on the unit system. Setting G=4π is legitimate in exactly the same sense as setting G=1 (geometric units) or G=6.674×10⁻¹¹ (SI); physics does not single out one of these as "correct." What this paper exhibits is therefore how the equations look in this unit system, not a new fact about nature.
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 Spherical Compression Pressure Equation [Paper 19]. Zenodo.
DOI: 10.5281/zenodo.21600912
ORCID: 0009-0006-8587-3450