Generalized Entropies and Black Hole Area Quantization from Landauer's Principle

Jorge Ananias Neto; Ronaldo Thibes

REVIEW 3 major objections 6 minor 1 cited by

Generalized Entropies and Black Hole Area Quantization from Landauer's Principle

T0 review · 3 major / 6 minor · reviewed 2026-08-02 · deepseek-v4-flash

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Pith's one-line read This paper argues that Landauer's principle, the thermodynamic cost of erasing one bit of information, uniquely fixes the spacing parameter of the black hole area spectrum, reproducing the Bekenstein–Mukhanov value for standard entropy and

desk verdict The Barrow and modified Rényi sections are fine, but the modified Kaniadakis conclusion relies on a small-κ expansion used outside its domain; the exact Landauer equation actually has no solution for large n, so that headline result does not hold. read the letter →

arxiv 2605.26386 v2 pith:7JZNW3U4 submitted 2026-05-25 gr-qc

Jorge Ananias Neto , Ronaldo Thibes This is my paper · ORCID

classification gr-qc

keywords blackholeareaquantizationLandauer'sprincipleBekenstein–HawkingentropyBarrowRényiKaniadakisinformationerasurespectrum

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper argues that Landauer's principle, the thermodynamic cost of erasing one bit of information, uniquely fixes the spacing parameter of the black hole area spectrum. For standard Bekenstein-Hawking entropy the constraint ΔS = k_B ln 2 reproduces the Bekenstein–Mukhanov value γ = 4 ln 2. Applied to Barrow, modified Rényi, and modified Kaniadakis entropies, the same constraint yields a level-dependent γ that distinguishes regular branches from singular ones and controls whether adjacent area levels become densely packed at large n. The paper claims this provides a direct way to determine γ without assuming a specific microscopic degeneracy for horizon states.

What carries the argument

The load-bearing device is the discrete Landauer condition ΔS(n) = S(n+1) − S(n) = k_B ln 2, imposed on entropy functions evaluated on an evenly spaced area lattice A_n = γ ℓ_P² n. Solving this one-bit condition for γ replaces the usual degeneracy-counting W = k^n and produces, for each entropy functional, a specific γ or γ(n). The second diagnostic is the relative spacing ΔA_n/A_n, whose large-n limit tells whether the discrete spectrum becomes effectively continuous.

What would settle it

An exact quantum-gravity computation of the area operator spectrum that yields a transition entropy different from k_B ln 2 between neighboring levels, or an observed spectral spacing not equal to 4ℓ_P² ln 2 in the Bekenstein-Hawking limit, would falsify the Landauer-fixed spectra.

Watch

Extended reading notes

Core claim

The central claim is that replacing degeneracy-counting arguments with Landauer's erasure cost as the physical criterion for transitions between adjacent area levels determines the area-spectrum parameter γ for a whole family of entropy functionals. Imposing ΔS = S(n+1) − S(n) = k_B ln 2 on the spectrum A_n = γ ℓ_P² n yields γ_B = 4 [ln 2 / ((n+1)^{1+Δ/2} − n^{1+Δ/2})]^{2/(2+Δ)} for Barrow entropy, γ_R = 4(2^λ−1)/[λ(1−n(2^λ−1))] for modified Rényi entropy, and γ_κ ≈ 4 ln 2 [1 + κ²(ln 2)²(3n²+3n+1)/6] in the small-κ modified Kaniadakis expansion. The relative level spacing ΔA_n/A_n then distinguishes the models: it vanishes like 1/n for Barrow, like 1/n² for the nonsingular Rényi branch, but

Load-bearing premise

The entire derivation rests on identifying the entropy difference between adjacent quantized area levels with exactly one bit of erased information, ΔS = k_B ln 2, and this equality is not derived from black hole microphysics.

Editorial extensions

If this is right

If correct, the area-spectrum parameter of a black hole is fixed by the thermodynamics of information erasure rather than by counting horizon microstates, recovering the Bekenstein–Mukhanov spacing for standard entropy.
For Barrow entropy, the Landauer constraint makes γ run with level number n, so the area spectrum is not exactly evenly spaced even though it becomes quasi-continuous at large n.
For modified Rényi entropy, only the negative-deformation branch yields a positive, nonsingular area spectrum; the positive branch's pole at n_c = 1/(2^λ−1) signals that Landauer's principle selects admissible entropic deformations.
For modified Kaniadakis entropy, a fixed deformation parameter prevents the spectrum from becoming continuous at large n; recovering a macroscopic regime requires κ(n) to decrease faster than 1/√n.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Extending the same Landauer-fix procedure to other entropic proposals (for instance logarithmic or Tsallis-corrected entropies) would immediately separate their regular from singular branches, without new assumptions.
The level-dependent γ discovered here can be read as the value needed to keep each single-area-quantum transition at exactly one bit of information cost, giving an operational meaning to 'running' of the area-quantum parameter that analogue black hole experiments could probe.
If the Landauer constraint is universal, the large-n spacing fingerprint distinguishes the underlying quantum-gravity microstructure: Barrow and nonsingular Rényi spectra become effectively continuous, while a fixed Kaniadakis deformation does not.
A testable extension would be to derive the Hawking line spectrum from each Landauer-fixed area spectrum, since the elementary transition must satisfy ΔM = k_B T ln 2, and compare the predicted line spacings.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Quick take: the paper applies a single idea—imposing Landauer's ΔS = k_B ln2 between adjacent area levels—to four entropy functionals. The Bekenstein–Hawking and Barrow sections are clean and correct; the modified Rényi section is correct apart from an obvious typo (the printed formulas use 2λ instead of 2^λ, which changes the coefficient but not the behavior). The modified Kaniadakis section has a load-bearing problem: the small-κ expansion is used to conclude a large-n result, and that step is invalid.

What's new: applying the Landauer condition to Barrow, modified Rényi, and modified Kaniadakis entropies is not in the cited literature, and the BH result is honestly attributed to Ref. [11]. The Barrow computation yields a level-dependent γ and 1/n spacing in the large-n limit; that's correct. The MRE branch analysis, including the pole for λ>0 and the finite-asymptotic area for λ<0, is also correct in substance.

The soft spot is MKE. The expansion of S_kappa(n) in powers of κ requires κγn/4 ≪ 1. But the solution for γ grows like n² for large n, so the expansion parameter is not small in the regime where the paper claims the relative spacing does not vanish. Worse, the exact equation for the entropy difference has a maximum as γ→∞, equal to (k_B/κ) ln(1+1/n). Landauer's condition can be met only if this maximum ≥ k_B ln2, i.e., n ≲ 1/(κ ln2). Beyond that, no real positive γ exists. So the MKE construction does not merely fail to make the spacing vanish; it stops having a solution. The paper's caveat about the expansion remaining controlled does not address this.

The overall approach is what it is: the Landauer identification is an assumption, not a derivation, and the level-dependent γ is a consequence of that assumption, not a prediction. Within that limited scope, the correct sections are a useful template.

Who this is for: people working on Bekenstein–Mukhanov spectra or generalized entropy models. They should read the BH and Barrow sections and the MRE analysis, and ignore the MKE conclusions until the exact solvability issue is sorted out. The paper deserves a serious referee, not a desk reject, because the flaw is specific and fixable and the other sections are usable.

Referee Report

3 major / 6 minor

Summary. This paper imposes Landauer's principle, ΔS = k_B ln 2, as the entropy difference between consecutive area levels A_n = γℓ_P² n, and solves for the spectrum parameter γ for four entropy functionals. For Bekenstein–Hawking entropy it recovers γ = 4 ln 2 (Sec. 2). For Barrow entropy it obtains a level-dependent γ_B(n) and relative spacing ΔA/A ~ [2/(2+Δ)](1/n) (Sec. 3). For the modified Rényi entropy it solves Eq. (38) to get γ_R = 4(2^λ−1)/[λ(1−n(2^λ−1))]; the λ>0 branch has a pole, while the λ<0 branch has a finite asymptotic area and ΔA/A ~ 1/n² (Sec. 4). For the modified Kaniadakis entropy, an O(κ²) expansion yields γ_κ(n) and, the paper claims, a non-vanishing relative spacing at large n for fixed κ; a running κ(n) is suggested to restore the macroscopic limit (Sec. 5).

Significance. If the Landauer criterion is accepted as a quantization principle, the paper provides a coherent and largely checkable derivation of area-spacing parameters across several generalized entropy models. All three generalized cases correctly reduce to γ = 4 ln 2 in the κ, λ, Δ → 0 limits, and the algebraic steps in Sections 2–4 are transparent and easily reproduced. The paper is honest about the assumed nature of Eq. (5) — though, as noted below, it overstates the independence of the resulting γ(n). The MKE section contains a load-bearing validity gap; my exact large-argument analysis suggests the qualitative MKE conclusion may survive in modified form (ΔA/A → 2^κ − 1 rather than κ²(ln 2)²n), so the section is repairable. Strengths include explicit closed-form formulas, a clear branch analysis for the modified Rényi case, and appropriate credit to Ref. [11].

major comments (3)

[Sec. 5, Eqs. (52)–(66)] The central MKE conclusion is drawn from the O(κ²) expansion of asinh(κγn/4), which is controlled only when κγn/4 ≪ 1, i.e., n ≪ 1/(κ ln 2) for fixed κ. Eq. (66) is then invoked at n→∞, precisely outside that domain; within it, the κ²n term is O(κ) relative to the 1/n term, which still vanishes. The fixed-κ claim is therefore not established. A concrete test: the exact equation asinh(κγ_{n+1}(n+1)/4) − asinh(κγ_n n/4) = κ ln 2, in the large-argument limit asinh z ≈ ln(2z), gives A_{n+1}/A_n → 2^κ, hence ΔA/A → 2^κ − 1 > 0 for fixed κ; this supports the qualitative claim but requires γ_n ~ 2^{κn}/n (exponentially running), not the polynomial correction of Eq. (59). The authors should add this exact/numerical analysis and amend or qualify the large-n statement.
[Sec. 4, Eqs. (41)–(45)] These equations are displayed with a factor that reads as 1 − 2λ, whereas the solution of Eq. (38) — and Eq. (39) itself — requires 1 − 2^λ. With the linear reading, Eq. (41) fails to reduce to the Bekenstein–Mukhanov spectrum in the limit λ→0⁻, so the section is internally inconsistent as printed. If the exponential form is intended, the typesetting must be corrected throughout, including Eqs. (41)–(45) and the surrounding discussion. The qualitative conclusions of Sec. 4 (finite asymptotic area, vanishing relative spacing) are otherwise unaffected.
[Secs. 1 and 6] The claim that the procedure provides 'a direct way to determine the parameter γ ... without assuming, from the beginning, a specific microscopic degeneracy' overstates the status of the results. The Landauer condition (5) is itself an assumed physical input; each section simply solves it for γ(n) within a chosen entropy functional, so the level dependence and the spacing asymptotics are consequences of that constraint, not predictions independent of it. The paper is explicit about Eq. (5), so this is not an internal inconsistency, but Sec. 6 should acknowledge the conditional character of the determination and note that for the generalized entropies there is no external benchmark beyond the κ, λ, Δ → 0 limits.

minor comments (6)

[Abstract and Sec. 5] The phrase 'the small κ expansion shows that a fixed deformation parameter prevents the relative spacing from vanishing' should carry the regime qualification from Major Comment 1; as written it invites the large-n reading that the calculation cannot support.
[Sec. 4, before Eq. (41)] 'The branch selected by Landauer's principle, λ<0' — the Landauer condition alone does not select a branch; the choice λ<0 follows from the additional requirement that γ_R and A_n remain positive for all n. Please reword to reflect the actual selection criterion.
[Eq. (67)] The sufficiency condition κ(n)√n ≪ 1 is fine, but if the exact MKE analysis of Major Comment 1 is added, the authors should check whether a running κ(n) reproduces it; κ(n) ~ 1/√n is only one possibility.
[Figs. 1 and 2] The figures were not available in the version I reviewed; the curves should be verified against Eqs. (23) and (39), especially the Δ=0 and λ=0 limits where the curves must coincide.
[Eqs. (26)–(27)] The exact Barrow expression for ΔA_n is unusually cumbersome; since only the large-n result (29) is used, consider omitting (27) or moving it to an appendix.
[Sec. 2, Eq. (14)] It would be useful to state explicitly at Eq. (14) that this reproduces the result of Ref. [11]; the current text makes this point only in the Introduction.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the derivation chain is explicit algebra from the stated Landauer premise and entropy functionals.

full rationale

The paper's chain is explicit: assume ΔS = k_B ln 2 (Eq. 5); use S(A) for each entropy functional; impose the equal-n area ansatz A_n = γ l_P^2 n (Eq. 1); solve for γ (Eqs. 14, 23, 39, 59). Each γ is the direct algebraic solution of the Landauer constraint, and the relative-spacing formulas (Eqs. 17, 29, 45, 64-66) are derived consequences. No step silently feeds the result back as an input. The claim that this 'determines γ without assuming a specific microscopic degeneracy' is rhetorically overstated: since S = k_B ln W, ΔS = k_B ln 2 is equivalent to W(n+1)/W(n) = 2. But this is a framing issue, not a circular derivation, because Landauer's principle is an explicit premise and the derivation does not use γ to justify it. The MRE and MKE functionals come from Refs. [13,14] involving the present co-author, but the paper re-derives both forms in Eqs. (33)-(35) and (48)-(50), so the self-citation is not load-bearing. The MKE large-n conclusion is governed by the paper's own caveat that the small-κ expansion must remain controlled (after Eq. 67); this is a validity limitation, not circularity. The BH case reproduces an external benchmark, and the generalized results are conditional on stated assumptions. Score 0.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The paper's central output is gamma(n), solved from the Landauer constraint for each entropy; thus the free parameter of the area spectrum is converted into a function of n. The physical input is the Landauer identification, which is assumed. The generalized entropy functionals come from prior work (including the authors' own Refs [13,14]). No new particles or forces are introduced.

free parameters (4)

gamma (area spectrum parameter) = 4 ln 2 for BH; level-dependent gamma_B(n), gamma_R(n), gamma_kappa(n) for Barrow/MRE/MKE
Introduced as free in Eq. (1), then fixed by imposing Delta S = k_B ln 2. It is the output of the prescription, not an independent prediction.
Delta (Barrow deformation parameter) = 0 <= Delta <= 1, not fitted here; input from Barrow 2020
Controls the Barrow entropy exponent; assumed from the literature, with Delta = 0 recovering BH.
lambda (modified Rényi parameter) = lambda < 0 branch; not fitted
Deformation parameter in Eq. (35); the negative branch is chosen for regularity. Landauer alone does not fix its value.
kappa (modified Kaniadakis deformation) = small, fixed; alternatively scale-dependent kappa(n) with kappa(n) sqrt(n) << 1
Deformation parameter; perturbative expansion in kappa; the paper suggests making it scale-dependent to restore vanishing relative spacing.

assumptions (6)

domain assumption Landauer's principle applies to black hole area transitions with Delta S = k_B ln 2
Stated in Sec. 2, Eq. (5); load-bearing for all derivations. No microscopic derivation is provided.
domain assumption Area spectrum is evenly spaced, A_n = gamma l_P^2 n
Eq. (1), inherited from the Bekenstein–Mukhanov approach; the generalized cases keep this form with level-dependent gamma.
standard math Bekenstein–Hawking entropy S_BH = k_B A / (4 l_P^2)
Standard black hole thermodynamics, used in Eqs. (3), (12), etc.
domain assumption Barrow, modified Rényi, and modified Kaniadakis entropy functionals are valid black hole horizon entropies
Taken from Refs [12–14]; not derived here. The MRE and MKE definitions come from the authors' own prior papers.
domain assumption Small-kappa expansion of S*_kappa is valid up to O(kappa^2)
Used in Sec. 5; the paper itself later notes the expansion is restricted to a controlled regime, weakening the large-n conclusion.
ad hoc to paper lambda < 0 branch is 'selected' by Landauer's principle
Landauer only fixes gamma given lambda; the choice of negative lambda is made to avoid a pole and negative gamma, not forced by the principle.

invented entities (1)

scale-dependent kappa(n)
purpose: To prevent the MKE relative spacing from growing, allowing Delta A/A -> 0 at large n
Proposed in Sec. 5, Eq. (67), without a physical mechanism or independent prediction.

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Pith. "Pith review of Generalized Entropies and Black Hole Area Quantization from Landauer's Principle." pith.science (2026). https://pith.science/paper/7JZNW3U4

@misc{pith2026260526386,
  author       = {Pith},
  title        = {Pith review of: Generalized Entropies and Black Hole Area Quantization from Landauer's Principle},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/7JZNW3U4}},
  note         = {Machine review of arXiv:2605.26386}
}

abstract

We investigate black hole area quantization by imposing Landauer's principle on the discrete entropy change between consecutive area levels. The elementary transition is identified with the entropy cost of erasing one bit of information, \(\Delta S=k_B\ln 2\). For the Bekenstein--Hawking entropy, this gives the standard Bekenstein--Mukhanov value of the area spectrum parameter, which is used as the reference limit. The same discrete construction is then applied to generalized entropy functionals. For Barrow entropy, the parameter \(\gamma\) becomes level dependent, while the relative separation between adjacent area levels still vanishes for large \(n\). For the modified R\'enyi entropy, the nonsingular branch has vanishing relative spacing at large \(n\), whereas the singular branch develops a finite-level pole. For the modified Kaniadakis entropy, the small \(\kappa\) expansion shows that a fixed deformation parameter prevents the relative area spacing from vanishing in the large \(n\) limit. Overall, the results suggest that Landauer's principle provides a useful way to analyze generalized entropic extensions of the Bekenstein--Mukhanov approach.

Figures

Figures reproduced from arXiv: 2605.26386 by the authors.

**Figure 2.** presents γR as a function of n for different negative values of λ. As n increases, γR decreases toward zero, indicating an effective running of γR induced by the Landauer criterion. 10 20 30 40 50 n 0.0 0.5 1.0 1.5 2.0 2.5 R = 1.0 = 0.5 = 0.2 = 0 FIG. 2: Parameter γR, given by Eq. (39), as a function of n for selected values of λ ≤ 0 in the MRE framework. Substituting Eq. (39) into Eq. (1), for the branch selected b… view at source ↗

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Landauer entropy of spacetime
gr-qc 2026-05 unverdicted novelty 5.0 of 10

Defines Landauer entropy for static spherically symmetric spacetimes as a surface integral over geodesic entropies, establishes a second law under mild assumptions, and relates the result to Bekenstein-Hawking entropy.

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As emphasized in Ref

INTRODUCTION Black hole area quantization is rooted in the idea that, for a non-extremal black hole, the horizon area has the character of a classical adiabatic invariant [1–5]. As emphasized in Ref. [5], the Ehrenfest principle [6] then suggests that the corresponding quantum observable should have a discrete spectrum. A minimal implementation of this id...
[2]

LANDAUER PRINCIPLE Landauer’s principle provides a direct link between information theory and thermody- namics [8–10]. It states that erasure of one bit of information requires a minimum energy cost, according to the inequality ∆E≥k BTln 2,(6) wherek B is the Boltzmann constant andTis the temperature. The factor ln 2 reflects the entropy change associated...
[3]

BARROW ENTROPY The possibility that quantum gravitational effects give rise to deformations in the black hole horizon with fractal features has been considered by Barrow in [12]. In this framework, the entropy associated with the horizon is modified and takes the form SB =k B A 4ℓ2 P 1+ ∆ 2 ,(19) 4 whereAis the usual horizon area and ∆ denotes the Barrow ...
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MODIFIED R ´ENYI ENTROPY To introduce the modified R´ enyi entropy, we first briefly review Tsallis statistics. Tsal- lis statistics [29, 34, 35] provides a nonextensive extension of the usual Boltzmann–Gibbs (BG) framework, in which the standard definition of entropy is generalized by means of an entropic index parameterq. The corresponding entropy is gi...
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MODIFIED KANIADAKIS ENTROPY Kaniadakis statistics defines a nonextensive extension of the standard Boltzmann– Gibbs (BG) framework through a real deformation parameterκ[59–62]. In this formalism, the entropy is written as Sκ =−k B WX i=1 p1+κ i −p 1−κ i 2κ ,(46) wherep i denotes the probability of theith microstate andWis the total number of accessible st...
[6]

The analysis was performed at the exact discrete level by identifying the entropy variation between two neighboring area levels with the elementary one-bit entropy increment

CONCLUSIONS In this paper, we examined the implications of Landauer’s principle for the quantization of black hole horizon area in different entropic frameworks. The analysis was performed at the exact discrete level by identifying the entropy variation between two neighboring area levels with the elementary one-bit entropy increment. This procedure provi...
[7]

ACKNOWLEDGMENTS Jorge Ananias Neto would like to acknowledge CNPq (Conselho Nacional de Desen- volvimento Cient ´ ıfico e Tecnol´ ogico), Brazilian scientific support federal agency, for partial financial support, CNPq-PQ, Grant number 305984/2023-3

2023

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Reviewed August 2, 2026 · model on record in the stance chip above.

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As emphasized in Ref

INTRODUCTION Black hole area quantization is rooted in the idea that, for a non-extremal black hole, the horizon area has the character of a classical adiabatic invariant [1–5]. As emphasized in Ref. [5], the Ehrenfest principle [6] then suggests that the corresponding quantum observable should have a discrete spectrum. A minimal implementation of this id...

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LANDAUER PRINCIPLE Landauer’s principle provides a direct link between information theory and thermody- namics [8–10]. It states that erasure of one bit of information requires a minimum energy cost, according to the inequality ∆E≥k BTln 2,(6) wherek B is the Boltzmann constant andTis the temperature. The factor ln 2 reflects the entropy change associated...

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BARROW ENTROPY The possibility that quantum gravitational effects give rise to deformations in the black hole horizon with fractal features has been considered by Barrow in [12]. In this framework, the entropy associated with the horizon is modified and takes the form SB =k B A 4ℓ2 P 1+ ∆ 2 ,(19) 4 whereAis the usual horizon area and ∆ denotes the Barrow ...

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MODIFIED R ´ENYI ENTROPY To introduce the modified R´ enyi entropy, we first briefly review Tsallis statistics. Tsal- lis statistics [29, 34, 35] provides a nonextensive extension of the usual Boltzmann–Gibbs (BG) framework, in which the standard definition of entropy is generalized by means of an entropic index parameterq. The corresponding entropy is gi...

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MODIFIED KANIADAKIS ENTROPY Kaniadakis statistics defines a nonextensive extension of the standard Boltzmann– Gibbs (BG) framework through a real deformation parameterκ[59–62]. In this formalism, the entropy is written as Sκ =−k B WX i=1 p1+κ i −p 1−κ i 2κ ,(46) wherep i denotes the probability of theith microstate andWis the total number of accessible st...

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The analysis was performed at the exact discrete level by identifying the entropy variation between two neighboring area levels with the elementary one-bit entropy increment

CONCLUSIONS In this paper, we examined the implications of Landauer’s principle for the quantization of black hole horizon area in different entropic frameworks. The analysis was performed at the exact discrete level by identifying the entropy variation between two neighboring area levels with the elementary one-bit entropy increment. This procedure provi...

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ACKNOWLEDGMENTS Jorge Ananias Neto would like to acknowledge CNPq (Conselho Nacional de Desen- volvimento Cient ´ ıfico e Tecnol´ ogico), Brazilian scientific support federal agency, for partial financial support, CNPq-PQ, Grant number 305984/2023-3

2023