The “1/4” in Black-Hole Entropy: Anchoring It via the 4π Solid Angle, the Euler Characteristic, and Equipartition — and Placing the Causal Sphere on Jacobson’s Horizon.
The Bekenstein–Hawking entropy S = A/(4 l_P²) carries a famous coefficient, 1/4. This paper anchors it structurally (it does NOT give a microscopic derivation; the area law is assumed). In the first-law derivation for a Schwarzschild horizon, the Hawking temperature T = ħκ/(2πck_B) contributes a 2π — the period of Euclidean time (an S¹) — while the horizon area A = 4πr_s² contributes a 4π — the solid angle of the spatial 2-sphere, which by Paper 67 is 2π·χ(S²), a topological quantity. With the surface gravity κ = c⁴/(4GM) these fix the coefficient of A/l_P² to exactly 1/4. So the 1/4 is not one mysterious number but a composite of a thermal 2π and a spatial 4π that — by the discipline of Papers 48/58 — are DISTINCT roots (imaginary-time circle vs. spatial sphere). Equipartition (Padmanabhan): S = N/4 with N = A/l_P² — one bit per four Planck areas. Finally, the causal sphere r=ct of Paper 61 is a Jacobson local horizon, so the causal-sphere entropy of Paper 65 (S = πc⁵, exponent 5=2+3) is its Bekenstein–Hawking entropy, its 1/4 anchored by the topological 4π of Paper 67. This is a structural anchoring in the emergent-gravity neighbourhood, not a microscopic derivation.
A note on the aim of this work. This is a structural interpretation (an anchoring of a known coefficient using established gravitational thermodynamics), not a microscopic derivation and not a new physical law. 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 “$1/4$” in Black-Hole Entropy: Anchoring It via the $4\pi$ Solid Angle, the Euler Characteristic, and Equipartition (and Placing the Causal Sphere on Jacobson’s Horizon) [Paper 73]. Zenodo.
DOI: 10.5281/zenodo.21666086
ORCID: 0009-0006-8587-3450