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 ${\Omega }_1={\sum }_{n}1/n!=e$, one nat. The $N=A/{\ell }_P^2$ independent cells give $\Omega =e^N$, hence $S=N$. Multiplied by the kinematic factor $\frac{1}{4}$ (the causally accessible fraction of the null phase space, established separately), this returns $S_{BH}=A/4{\ell }_P^2$. The derivation invokes no field equations, no microscopic theory, and no posited Hilbert space.

3 pages, published version. EPL 156 (2026) 49002