We give a geometric reformulation of the plasma frequency _p = √((n e²)/(m ₀)) by applying the Yamagishi patch m=4π (from the n=2 uniqueness theorem, dm/dr = 4π) together with the Yamagishi sync patch ₀ = r (Paper 14). Stage 1: Substituting m=4π and ₀ = r into _p² = (n e²)/(m ₀) gives _p² = (n e²)/(4π r). Stage 2: Incorporating the Bohr–Yamagishi equation (4π v)· r = e (Paper 15) yields the form 16π² r² _p² = n e². Stage 3: We check mode consistency against the criteria of Paper 14, 2nd edition.
[Withdrawals in the 2nd edition]: The first edition called Stage 2 a "Plasma Zero-Tension Confinement Theorem" with "cancellation ratio 1". This is withdrawn. That a ratio equals 1 asserts nothing — it holds for any nonzero quantity. What remains is only the algebraic fact 16π²r²_p²=n_ee², which explains no confinement mechanism (Section 4.1). The Stage 3 claims of an "isomorphism" with tokamak geometry and of "topological control of MHD instabilities" are likewise withdrawn, no quantitative correspondence having been exhibited (Section 5).
[What the 2nd edition does establish]: The adoption of ₀=r (normal rendering mode) is consistent with the criteria of Paper 14. The Debye scale sits clearly on the near-field side, _D/(c/_p)=4.4×10⁻³, and this holds generally for non-relativistic plasmas (Section 3).
(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 Zero-Tension Plasma Equation [Paper 22]. Zenodo.
DOI: 10.5281/zenodo.21601236
ORCID: 0009-0006-8587-3450