A New Way to Write the Holographic Entropy Bound

In the paper's words

The holographic entropy bound is conventionally stated in bits, a unit that embeds two human conventions: integer counting of states and the choice of logarithm base 2. This paper asks whether the bound can be restated entirely in observer-independent quantities, and it situates the question in the current holographic literature. Working in Planck natural units (c=ℏ=G=k_B=1), area is a dimensionless number A in Planck areas and physical entropy is the natural-logarithm entropy S=A/4 (nats).

The abstract, verbatim, and its opening sentence is the claim. Every line below is quoted from the text.

Numbers it reports

1. (iv) A classification table separating base-free, derived, and conventional ingredients of the bound, and a corrected attribution of the Ostrowski-theoretic framing to [14] — Sections 4 and 6.

   from 1. Introduction

2. The parallel is exact in structure: where [3] removes a scheme choice from the definition of energy, this paper removes a base choice from the definition of entropy. [4] studies the action growth rate $\dot{I}_{\rm WDW}$ of the Wheeler-DeWitt patch for $D\ge4$ AdS black holes with spherical, toric, and hyperbolic horizons ($k=1,0,-1$), proposes bounds on holographic complexity under the complexity=action conjecture, and introduces a modified complexity=volume conjecture with growth rate $\dot{\cal C}_{V}=2P_{\rm th}\,\Delta V_{\rm th}$.

   from 2. Background and Related Work

3. This is the established practice, within a subfield, of expressing physical law in dimensionless constant form — the practice this paper extends to information measures. [7] supplies the historical and conceptual anchor: it recounts that Planck introduced the natural units $h$, $c$, $G$, $k$ in 1897–1899, that only $c$, $\hbar$, $G$ retained status in the "cube of theories," and it argues from the definition $T=dU/dS$ that $k$ is not a natural unit at all but "a ratio of 'hand-crafted' units of temperature and energy," i.e. a conversion factor from degrees to joules.

   from 2. Background and Related Work

4. The same argument applies verbatim to the bit: $\ln 2$ is a conversion factor between the natural-log entropy and a unit chosen for engineering convenience. [9] gives the mathematical formalism behind such conversions: it constructs the vector space of physical values, in which different unit systems correspond to different bases and a transformation between unit systems is a matrix-vector multiplication, and it demonstrates the formalism on the subspace spanned by $\{10,\mathrm{m},\mathrm{s},\mathrm{kg}\}$.

   from 2. Background and Related Work

5. We adopt Planck natural units throughout: $$ c=\hbar=G=k_{B}=1. $$ In these units the Planck length $\ell_{P}$ is the unit of length, the Planck area $\ell_{P}^{2}$ is the unit of area, and any physical area $A_{\text{phys}}$ is represented by the dimensionless number $$ A=\frac{A_{\text{phys}}}{\ell_{P}^{2}}. $$ This is the move that [14] describes generally — in Planck units every physical quantity becomes a dimensionless number — applied to the one geometric quantity that enters the holographic bound.

   from 3.1 Natural-unit setting

6. The bit is defined as the entropy of a two-outcome event: $$ b=\frac{S}{\ln 2}=\frac{A}{4\ln 2}. $$ Decimal digits are $d=S/\ln 10$.

   from 3.3 Recovery of derived units

How it is done

1. The holographic entropy bound is conventionally stated in bits, a unit that embeds two human conventions: integer counting of states and the choice of logarithm base 2.

   from Abstract

2. Anyone who specifies information-capacity limits for quantum-gravity experiments — for example, tabletop analog black-hole simulators or quantum memories used as entropy reservoirs in analog-gravitation platforms — must translate the holographic entropy bound into an operational unit.

   from 1. Introduction

3. Holographic bounds in cosmology. [1] compares two holographic arguments that limit inflation — the de Sitter Equilibrium (dSE) cosmology and the approach of Banks and Fischler (BF) — and finds that simple versions yield the same bound on the number of e-foldings.

   from 2. Background and Related Work

4. We adopt Planck natural units throughout: $$ c=\hbar=G=k_{B}=1. $$ In these units the Planck length $\ell_{P}$ is the unit of length, the Planck area $\ell_{P}^{2}$ is the unit of area, and any physical area $A_{\text{phys}}$ is represented by the dimensionless number $$ A=\frac{A_{\text{phys}}}{\ell_{P}^{2}}. $$ This is the move that [14] describes generally — in Planck units every physical quantity becomes a dimensionless number — applied to the one geometric quantity that enters the holographic bound.

   from 3.1 Natural-unit setting

What would falsify it

1. The empirical-equivalence result (R5) would be falsified by exhibiting a well-defined observable $O$ that depends on the logarithm base while all invariant quantities ($A$, $S$, $D$) are held fixed.

   from Falsification condition

2. The rounding analysis (R4) would be falsified if a quantum-gravity framework predicted a different integer rounding rule for $D$ at small areas, for example a rule giving $D_{\text{int}}\neq\lfloor e^{A/4}\rfloor$ at $A=4$.

   from Falsification condition

3. Section 6 discusses limitations, failure modes, and falsification conditions.

   from 1. Introduction

Where the claim stops

1. Assumed Bekenstein–Hawking relation.

   from Limitations

2. The bound assumes a well-defined Hilbert-space dimension for a sub-region.

   from Limitations

3. The argument that the natural logarithm is singled out does not prove that entropy must be logarithmic; it inherits that assumption from the Bekenstein–Hawking framework.

   from Limitations

Where to read

1. Abstract187 words · 1 min
2. 1. Introduction439 words · 2 min
3. 2. Background and Related Work1,135 words · 5 min
4. 3. Methods362 words · 2 min
5. 4. Analysis771 words · 3 min
6. 5. Results178 words · 1 min
7. 6. Discussion579 words · 3 min
8. 7. Practical Implications and Engineering Pathways814 words · 4 min
9. 8. Conclusion241 words · 1 min
10. Provenance and AI Involvement240 words · 1 min