We examine the geometric expression of the temperature kT. The argument proceeds in three stages. Stage 1: Substituting the Yamagishi kinetic energy E_k = 2π v² (Paper 18) into the equipartition theorem ⟨ E_k⟩ = (3)/(2) k_B T yields kT = (4)/(3)π v² in natural units. Stage 2: Combining this with the pressure P = (4)/(3)π n v² (Paper 19) through the standard ideal gas law P = nkT, we show the two agree exactly. Stage 3: We state the Universe OS clock frequency model.
[Substantive correction in the 2nd edition]: The first edition started from 3/2 kT = E_k· v and obtained kT = 4/3 π v³. This is an error and is corrected here. The standard equipartition theorem reads ⟨ E_k⟩=3/2 k_BT; there is no operation multiplying the right-hand side by the speed v. The first edition introduced one extra factor of v.
The correction improves the internal consistency of the theory. With kT=4/3 π v², the standard form P=nkT=4/3 π nv² holds directly and agrees exactly with the pressure of Paper 19. The non-standard form P=nkT/v required by the first edition becomes unnecessary (Section 2.3).
The geometric reading "temperature = the volume of a ball in velocity space" is likewise withdrawn. Temperature must carry the dimension of energy (∝ v²), not of volume (∝ v³) (Section 2.4).
(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. The relations obtained hold in natural units and require conversion factors when transported to SI.
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 Overclock Temperature Equation: Derivation of $kT = \frac{4}{3}\pi v^2$ and Consistency with the Standard Equation of State [Paper 20]. Zenodo.
DOI: 10.5281/zenodo.21601035
ORCID: 0009-0006-8587-3450