The central observation of this paper is a change of reading, not a calculation. Adopting the geometric mass parameter m=4π (from the n=2 Uniqueness Theorem), Einstein's relation becomes
E = mc² = 4π c² ,
and the right-hand side is literally the surface area of a sphere of radius c. That is, once m is read as a solid angle, E=mc² has the form of an area — the boundary of a region, not its interior. This is consistent with the dimensional structure of c², whose numerator carries the dimension of area.
Once the expression is seen as a surface area, the corresponding volume follows immediately, since integrating a sphere's surface area over its radius returns its volume:
∫₀^c 4π r² dr = (4)/(3)π c³ + C .
We refer to this as the "Einstein–Yamagishi Volume Restoration Equation". We stress that the content lies in the geometric reading, not in the integration, which is elementary calculus; the integration merely records what the area–volume relation already implies. Furthermore, we expand this into dynamic systems by reinterpreting the general relation E² = (mc²)² + (pc)² as the "Yamagishi Overclock Pythagoras Equation"—the universe OS's task manager. Through this lens, the uncertainty relation can be read as a bound keeping the momentum leg pc — and hence the total energy — finite.
Two clarifications added in revision. (i) On the integration variable: the integration is carried out over the radius r, with c entering only as the upper limit. This is the standard fact that integrating a sphere's surface area over its radius returns its volume. It is not an integration with respect to c itself, which — c being a constant — would not be a standard operation. (ii) On the uncertainty relation: the reading above is an interpretation, not a derivation. In standard quantum mechanics the uncertainty relation follows from the Robertson inequality together with [x,p]=iħ; the commutator comes first. This paper does not claim to explain why that commutator holds.
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). From Boundary Energy to Volume Energy: The Einstein-Yamagishi Volume Restoration Equation and the Overclock Pythagoras Theorem [Paper 12]. Zenodo.
DOI: 10.5281/zenodo.21404703
ORCID: 0009-0006-8587-3450