We rewrite the self-gravitational potential energy of a uniform sphere, U = -(3GM²)/(5R), using M = (4)/(3)π R³ ρ and the Yamagishi mass patch ρ = 4π n, and examine the origin of the resulting constant 256π⁴ = (16π²)².
Principal withdrawal in the 2nd edition: The first edition claimed to have explained hydrostatic equilibrium. This is withdrawn. The paper merely rewrites the self-gravitational energy U and never writes down a pressure equation at all. Hydrostatic equilibrium is dP/dr = -Gm(r)ρ/r², and stability follows from the virial theorem as the condition γ>4/3. From the expression for U alone, nothing can be said about stability (Section 5).
The reading of 256π⁴ as a "double toroidal barrier" is likewise withdrawn: 256π⁴ is a consequence of choosing the mass unit m=4π, and setting m=7 would produce a different constant with equal justification (Section 4).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.
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 Double-Torus Stellar Fortress Equation [Paper 30]. Zenodo.
DOI: 10.5281/zenodo.21601732
ORCID: 0009-0006-8587-3450