In three-dimensional Euclidean space, when integrating over a spherical region whose density follows the inverse-square law ρ(r) = 1/r², the Jacobian r² of the volume element and the density 1/r² cancel. As a result, the total mass within the unit sphere coincides with the total solid angle of 3D space, 4π (the full measure in steradians). We formulate this as the “4π Mass Cancellation Theorem.”
We further show that for a general exponent n, m(n)=(4π)/(3-n), and that n=2 is the unique exponent making m coincide with the solid angle itself (Section 5). Writing m=4π· S, reading S=(1)/(3-n) as the “number of overlapping sheets of 4π” yields a module staircase 2π, 4π, 8π.
Positioning of this paper: the 4π appearing here is the angular integral ∫_(S²)dΩ itself, the same solid angle as the 4π in Gauss's law, Coulomb's law and Poisson's equation. This paper claims no new physical law; it shows what becomes visible when the known structure — that the geometry of 3D space requires the inverse-square law — is rewritten in the form m=4π.
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 $4\pi$ Mass Cancellation Theorem [Paper 2]. Zenodo.
DOI: 10.5281/zenodo.21403931
ORCID: 0009-0006-8587-3450