c as the Light Sphere — The Causal Sphere r=ct, Volume Restoration, and the Origin of the Inverse-Square Law.
Light spreads over a sphere of radius r = ct, whose radius equals c at unit time. Since this light sphere is a Euclidean sphere in real space, E = 4πc² (Paper 12) reads as a genuine metric area — the coordinate-area concern of velocity space (Paper 60) does not arise. Volume restoration ∫₀^c 4πr² dr = (4/3)πc³ equals the causal-ball volume light fills, exactly the n = 2 shell integral. The inverse-square law is realized literally as dilution over the light sphere (I ∝ 1/(ct)² = 1/r²), grounding the r of n = 2 as the causal radius ct. This light sphere corresponds to the boundary v = c of velocity space, so A (boundary light, flat) and B (interior curvature, hyperbolic) are complementary faces of the same c. Note: for indefinite integrals a constant +C represents a baseline, where in gravitation the cosmological constant Λ can enter (unimodular gravity).
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. 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). $c$ as the Light Sphere [Paper 61]. Zenodo.
DOI: 10.5281/zenodo.21598127
ORCID: 0009-0006-8587-3450