In 3D Euclidean space R³, we prove that the necessary and sufficient condition for the spherical-shell mass growth rate dm/dr of a spherically symmetric power-law density distribution ρ(r) = r⁻ⁿ (n ∈ R) to be independent of the radius r is exactly n = 2. Beyond the purely analytical route of differentiation and integration, we confirm the necessity of n = 2 from four different academic approaches: (1) Gauss's flux conservation (spatial geometry), (2) Bertrand's theorem on closed-orbit stability (classical mechanics), (3) the fundamental solution of the Laplace equation (potential theory), and (4) data-density culling in the Universe OS (information theory). Furthermore, by carrying out the volume integral over a sphere of radius 1 under n = 2, we show the mechanism by which the spatial expansion (r²) and the density attenuation (1/r²) cancel completely, yielding the limit mass m = 4π, the pure total solid angle of 3D space.
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 Uniqueness Theorem of $n = 2$ for Spherically Symmetric Density Distributions [Paper 1]. Zenodo.
DOI: 10.5281/zenodo.21403186
ORCID: 0009-0006-8587-3450