We substitute the Yamagishi mass patch m=4π into the basic equations of Newtonian mechanics. First: Newton's second law becomes F=4π a. Second: momentum and kinetic energy become p=4π v and E_k=(1)/(2)mv²=2π v². Third: substituting G=4π and m=4π into U=-(GMm)/(r) gives U=-(16π² M)/(r). Fourth: the work integral returns the change in kinetic energy.
Position of the 2nd edition: All of the algebra in this paper is correct — each result is a substitution into a known expression, and no error was found on verification. The second edition corrects four inaccuracies in the geometric labels (Section 6). It also shows that E_k=2π v² agrees exactly with the corrected temperature of Paper 20, kT=4/3 π v² (Section 5), providing independent support for that correction.
(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 Classical Mechanics Reformulation [Paper 26]. Zenodo.
DOI: 10.5281/zenodo.21601339
ORCID: 0009-0006-8587-3450