The Bekenstein–Hawking Entropy of the Causal Sphere and c⁵: The Dimensional Ladder of Information Capacity — An Exponent-Structure Observation, Not a Derivation.
Paper 44 assigned the exponent c⁵ to the “capacity” rung of its dimensional ladder (shared with gravitational-wave luminosity and bulk density). This paper gives that rung an information-theoretic reading. Combining three established ingredients — the Bekenstein–Hawking entropy S = A/(4 l_P²); the reciprocal-patch scaling l_P² = c^(−3) under ħG = 1 (Paper 44); and the causal (light) sphere A = 4πc² at t=1 (Papers 60, 61) — gives S = πc⁵, with the exponent decomposing exactly as 5 = 2 + 3 (area c² times inverse Planck area c³). This is the sibling of the Planck-pressure result of Paper 50 (c⁷, 7 = 5/2 + 3·3/2) and is consistent with the c⁵ rung of Paper 44. Stated as an exponent-structure observation: the “c⁵ = capacity” intuition acquires a Bekenstein–Hawking reading as the information capacity of the causal sphere. IMPORTANT SCOPE: the paper does NOT derive the Bekenstein–Hawking entropy, does not claim a literal SI identity (S = πc⁵ is an exponent-convention statement, dimensionless in SI), depends on the unit-time choice t=1, and — per the discipline of Paper 48 — reports the exponent match as a resonance, not a shared origin.
A note on the aim of this work. This is a structural interpretation (an exponent-structure observation) of well-established physics, not a proposal for a new physical law and not a derivation of the entropy 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 Bekenstein–Hawking Entropy of the Causal Sphere and $c^5$: The Dimensional Ladder of Information Capacity (An Exponent-Structure Observation, Not a Derivation) [Paper 65]. Zenodo.
DOI: 10.5281/zenodo.21616466
ORCID: 0009-0006-8587-3450