Two Entropies, Not to Be Conflated: A Structural Contrast of 3D Coarse-Graining Entropy (the Arrow of Time) and the Black-Hole Horizon Area Entropy (the Holographic Bound).
Under the single word “entropy,” two different things are often conflated. This paper separates them and states their one legitimate relation. (A) Coarse-graining entropy: the thermodynamic / Gibbs entropy of a 3D bulk, as when ink dropped in water spreads. The microscopic laws are reversible (Liouville’s theorem, Paper 77), and the irreversible “arrow of time” lives in the observer’s resolution / coarse-graining (Papers 77/78). (B) Horizon area entropy: the Bekenstein–Hawking entropy of a black hole, S = A/(4 lP^2). It scales with AREA, not volume (A proportional to M^2), and is where Paper 73’s “1/4” lives. (A) and (B) are distinct quantities (distinct roots, Papers 58/48) — one lives in resolution, the other in dimension/boundary. Their one legitimate relation is the holographic bound: the coarse-graining entropy packable into a region cannot exceed a ceiling set by the area of its boundary, S_bulk ≤ A/(4 lP^2) (the Bekenstein bound; ’t Hooft / Susskind). So the true nature of “subsumption” is neither identity nor a sum but a ceiling — area bounding the bulk’s information from above. We organize this as a double structure of “resolution (coarse-graining) × dimension (boundary ceiling)” and warn against double-counting. IMPORTANT (scope): this is a structural arrangement (Layer 2); it produces no new mathematics and no new prediction. It does NOT solve “why information is encoded on a 2D boundary” (the holographic principle itself) — that is an open research question, and this paper does not answer it; it only maps how not to conflate the two entropies. The Universe OS’s “Zip compression / task-kill / singularity crash” is a Layer-3 non-science lens, and the paper notes that an apparent answer to “why 2D” is a signature, not a proof.
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). Two Entropies, Not to Be Conflated [Paper 88]. Zenodo.
DOI: 10.5281/zenodo.21747809
ORCID: 0009-0006-8587-3450