Substituting the Yamagishi patch m = 4π = Vol(S²), supplied by the n = 2 uniqueness theorem, into the classical kinetic energy E_k = ½ m v² gives
E_k = ½ (4π) v² = 2π v² .
The substitution is correct. The coefficient ½ is the result of the integration ∫₀^v v' dv' = v²/2 and is not an open puzzle.
[What the 3rd edition adds] — the identification of the 2π that appears in this equation. The number 2π occurs in several places in this series, but it is the same number, not the same root. This edition calls the roll of three distinct roots.
Root A — the solid angle of a hemisphere. This is the 2π of the present paper: Ω_hemi = 2π, carrying the same dimension of solid angle as 4π, so no length is confused with a solid angle.
Root B — the periodicity of phase, sin(x + 2π) = sin(x). The 2π in h = 2πħ, and in the phase-space cell d³x d³p / (2πħ)³, belongs to this Root B and is not a new fourth root, since it descends from the periodicity of e^(iθ). Not multiplying roots carelessly is itself part of the roll call: what is to be counted is not the places where 2π appears, but the independent mechanisms that produce it.
Root C — Vol(S¹), a value on the odd staircase (Paper 115).
Root C yields a sharper diagnosis of the first edition's error. The first edition identified 2π as “the circumference of S¹”; the second edition withdrew that on dimensional grounds (a length against a solid angle). Paper 115 adds a second and more structural reason: ½ · Vol(S²) = Vol(S¹) is numerically correct, but Vol(S²) lives on the even staircase and Vol(S¹) on the odd one, and 4π → 2π is not a step of the recursion — the step after 4π is 8π²/3, and the step before 2π² is 2π. The coincidence is therefore a signature, and not a link by generation rule (Paper 100). A signature must not be mistaken for a root.
The 3rd edition further makes explicit: that the relativistic expansion E = 4πc² + 2πv² + (3/2)π v⁴/c² + ⋯ is a power series in the dimensionless quantity v/c, which is where the “module boundary” of Paper 113 lives; that the v² here is a quantity in the flat approximation of velocity space, the hyperbolic geometry of Papers 60 and 63 showing itself only as v approaches c; and that this equation is merely the d = 3 member of a family with one member per dimension (the measure of Paper 114 — for d = 4 one obtains E_k = π² v²).
This version of the record carries five languages: Japanese, English, German, French and Italian.
Scope: a structural interpretation of established physical law (Layer 2). No new physical law and no prediction is claimed. The main result E_k = 2π v² and the algebraic compatibility with Papers 17, 19 and 20 are unchanged from the second edition.
(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 substitution m = 4π employed here constitutes a choice of natural units. G and ħ are dimensionful quantities whose numerical values depend entirely on the unit system. Setting m = 4π is possible in exactly the sense that G = 1 (geometrised units) or SI units are 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, 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 Kinetic Energy Equation — 3rd Edition: $E_k = 2\pi v^2$, and the Three Distinct Roots of $2\pi$ — [Paper 18]. Zenodo.
DOI: 10.5281/zenodo.21430706
ORCID: 0009-0006-8587-3450