The Yamagishi Patch within Emergent Gravity: Reading G=4π as the Elastic Constant of Spacetime — The Umbrella over the Thermodynamic Arc (Papers 62, 65, 72, 73).
Is gravity fundamental, or emergent — a coarse-grained, collective description like the elasticity of a solid or the hydrodynamics of a fluid? The emergent/thermodynamic-gravity programme argues the latter: Sakharov (1967) showed the Einstein–Hilbert term can be INDUCED by matter vacuum fluctuations, so G is not fundamental but an induced elastic modulus of spacetime; Jacobson (1995) derived the Einstein equations as an equation of state; Padmanabhan and Verlinde read gravity as equipartition and entropy. This paper PLACES the Yamagishi system in that (established, non-string) family. It reads G=4π (Paper 35, the “defrosting” of G) as the elastic constant of an emergent spacetime — the coupling 1/(16πG) being the rigidity of spacetime — rather than a fundamental input. It then exhibits the system’s thermodynamic arc under one umbrella: Λ as an integration constant (Papers 62, 72), horizon entropy 1/4 as equipartition (Paper 73), the causal-sphere entropy on a Jacobson horizon (Paper 65) are one stance applied several times, NOT four separate solutions. IMPORTANT: this is a positioning (“address registration”), not a proof that gravity is emergent (a debated programme), not a derivation of G, and not a quantisation of gravity. The frame houses the system’s constants without melting their distinct roots (Papers 48/58).
A note on the aim of this work. This is a structural interpretation — a placement of the system within an established, non-string research programme — not a new physical law and not a proof that gravity is emergent. 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 Yamagishi Patch within Emergent Gravity: Reading $G=4\pi$ as the Elastic Constant of Spacetime. The Umbrella over the Thermodynamic Arc (Papers 62, 65, 72, 73) [Paper 74]. Zenodo.
DOI: 10.5281/zenodo.21670102
ORCID: 0009-0006-8587-3450