One nat per cell
Abstract
A black hole horizon carries exactly one nat of information per Planck cell, without naming what the black hole is made of. Two inputs suffice: entropy is irreducible uncertainty (the configurations no observation can resolve at the Planck floor, an objective bound, not ignorance), and the horizon is a complete scrambler, so by the Ehrenfest time its Wigner function has folded below the floor and no cell can be certified empty. A single cell then holds unresolved filament crossings; a cell crossed n times admits n! orderings that scrambling erases, so cardinality n carries weight 1/n! and its count is Ω1=∑n 1/n!=e, one nat. The N=A/ℓP2 independent cells give Ω=eN, hence S=N. Multiplied by the kinematic factor 1/4 (the causally accessible fraction of the null phase space, established separately), this returns SBH=A/4ℓP2. The derivation invokes no field equations, no microscopic theory, and no posited Hilbert space.
Versions
BibTeX
@misc{wolfson2026cell,
title = {{One nat per cell}},
author = {Ira Wolfson},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.32879},
eprint = {2609.32879},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.32879},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - One nat per cell
AU - Ira Wolfson
PY - 2026
DA - 2026/09/26
PB - arXiv
DO - 10.48550/arxiv.2609.32879
UR - https://arxiv.org/abs/2609.32879
AB - A black hole horizon carries exactly one nat of information per Planck cell, without naming what the black hole is made of. Two inputs suffice: entropy is irreducible uncertainty (the configurations no observation can resolve at the Planck floor, an objective bound, not ignorance), and the horizon is a complete scrambler, so by the Ehrenfest time its Wigner function has folded below the floor and no cell can be certified empty. A single cell then holds unresolved filament crossings; a cell crossed n times admits n! orderings that scrambling erases, so cardinality n carries weight 1/n! and its count is Ω_1=∑_n 1/n!=e, one nat. The N=A/ℓ_P^2 independent cells give Ω=e^N, hence S=N. Multiplied by the kinematic factor 1/4 (the causally accessible fraction of the null phase space, established separately), this returns S_BH=A/4ℓ_P^2. The derivation invokes no field equations, no microscopic theory, and no posited Hilbert space.
ER -