Reconciling fractional entropy and black hole entropy compositions

Manosh T. Manoharan; N. Shaji

REVIEW 4 major objections 5 minor 2 cited by

Reconciling fractional entropy and black hole entropy compositions

T0 review · 4 major / 5 minor · reviewed 2026-08-10 · deepseek-v4-flash

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Pith's one-line read A fractional entropy with $q=2$ exactly reproduces black hole merger entropy and yields a universal horizon temperature.

desk verdict A technically competent but conditional paper: the headline reconciliation of fractional entropy with black-hole composition rests on an equal-prior assumption the authors themselves concede may not hold. read the letter →

arxiv 2501.18151 v1 pith:34USTODL submitted 2025-01-30 gr-qc

Manosh T. Manoharan , N. Shaji This is my paper · ORCID

classification gr-qc

keywords fractionalentropyblackholecompositionnon-additiveempiricaltemperatureZerothlawofthermodynamicsmassquantizationinformationfluctuationhorizon

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

The paper argues that a fractional generalization of Shannon entropy, $S^U_q = \sum_i p_i (-\ln p_i)^q$, provides the missing statistical composition rule for black hole mergers. For $q=2$, this entropy composes as $S^U_2(A+B)=S^U_2(A)+S^U_2(B)+2S^U_1(A)S^U_1(B)$, and when the Shannon part satisfies $S^U_1=\sqrt{S^U_2}$ — which holds exactly for equal microstate probabilities $p_i=1/W$ — it matches the ideal merger formula $S_f = S(M_1)+S(M_2)+2\sqrt{S(M_1)S(M_2)}$ implied by energy conservation. From that match the paper derives a Zeroth-law-compatible empirical temperature $T=T_P/(2\sqrt{\pi})$ that is the same for every horizon, and an equidistant mass spectrum $M=\mathcal{M}\,n$ instead of the usual area quantization. A sympathetic reader would care because the result recasts black hole composition as an isothermal statistical process and points to Planck-scale degrees of freedom, while tying the second law to an information-fluctuation bound $\sigma^2=S^U_2-(S^U_1)^2$.

What carries the argument

The carrier of the argument is the fractional entropy $S^U_q=\sum_i p_i(-\ln p_i)^q$, built from a fractional derivative acting on the Shannon generating function; $q=1$ recovers Shannon entropy. Its composition rule for $q=2$ contains both $S^U_2$ and $S^U_1$, and the identity $S^U_1=\sqrt{S^U_2}$ for equal priors is the bridge to black hole entropy composition. The variance identity $\sigma^2=S^U_2-(S^U_1)^2$ (information fluctuation) supplies the second-law bound, and the additive functional form $S^U_1=\sqrt{S^U_2}$ defines the Zeroth-law temperature. The concavity condition for thermal stability, $W>\exp(q-1)$, is what rules out bits and qubits for $q=2$.

What would settle it

Compute $S^U_1$ and $S^U_2$ for any explicit horizon microstate model with non-uniform probabilities: if $S^U_1\neq\sqrt{S^U_2}$, the fractional-entropy composition rule no longer matches the black hole merger formula. Alternatively, observations showing that black hole masses are not equidistant, or that a horizon has a two-level state space, would contradict the paper's derived spectrum and stability bound.

Watch

Extended reading notes

Core claim

On the paper's own terms, the central discovery is that fractional entropy $S^U_q=\sum_i p_i(-\ln p_i)^q$, with $q=2$, is the statistical entropy whose composition rule coincides with black hole entropy composition under energy conservation. The composition rule for two independent systems is $S^U_2(A+B)=S^U_2(A)+S^U_2(B)+2S^U_1(A)S^U_1(B)$, so the cross term is controlled by ordinary Shannon entropy $S^U_1$. With equal priors $p_i=1/W$, $S^U_1=\ln W$ and $S^U_2=(\ln W)^2$, so $S^U_1=\sqrt{S^U_2}$ and the rule becomes the black hole merger formula $S_f=S(M_1)+S(M_2)+2\sqrt{S(M_1)S(M_2)}$. Taking the additive functional $S^U_1$ to define empirical temperature gives $T=T_P/(2\sqrt{\pi})$, independent of surface gravity, with Hawking temperature related by $T_H=T/(2S^U_1)$. The same counting produces a mass spectrum $M=\mathcal{M}\,n$, and stability for $q=2$ requires microstate dimension $W>e$, excluding two-level bits or qubits.

Load-bearing premise

The load-bearing premise is that horizon microstates are equally likely ($p_i=1/W$), so that $S^U_1=\sqrt{S^U_2}$; the authors state this may not generally hold, and if it fails the cross term in the composition rule, the universal temperature, and the mass spectrum all change.

Editorial extensions

If this is right

Black hole mergers become isothermal in the empirical temperature: every horizon carries $T=T_P/(2\sqrt{\pi})$, even though Hawking temperatures differ.
Mass, rather than horizon area, is quantized in equal steps $M=\mathcal{M}\,n$, a spectrum derived from $S^U_1$.
The generalized second law is tied to information fluctuation: it holds when $S^U_2 \ge \sigma^2$, and would require an unphysical imaginary $S^U_1$ to fail.
Horizon microstate spaces for $q=2$ must have dimension $W>e$; two-level bits or qubits are thermally unstable for this composition.
The equilibrium distribution for the $q=2$ entropy is $p_i=\exp(-1-\sqrt{1+\phi_i})$, a stretched exponential that deviates from the Boltzmann distribution.

Reading between the lines

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

If the universal empirical temperature is real, it provides a horizon-scale thermometer independent of mass, which could be probed through gravitational-wave ringdown spectra or Hawking-radiation analogues; the paper does not develop this observable signature.
The requirement $W>e$ suggests testing candidate quantum-gravity microstate models: any model built from two-level systems would be incompatible with the $q=2$ composition, while qutrit or higher-dimensional models would be preferred.
The paper's mass quantization is connected to, but derived differently from, recent black-hole 'atom' proposals; a direct comparison of the predicted fundamental mass $\mathcal{M}$ with observational remnants would test whether the spectra coincide.
The stretched-exponential equilibrium distribution might be observable in analogue gravity or trapped-ion experiments simulating fractional entropy; this is an inference, not a claim of the paper.

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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

Two things to know. First, the paper is honest about its own load-bearing weakness: matching the fractional-entropy composition rule to the black-hole merger formula requires S1 = sqrt(S2), and the authors explicitly say this holds only for equal priors and "may not generally hold." Second, the universal empirical temperature and the equidistant mass spectrum are not independent outputs; they are direct consequences of that same ansatz. The stress-test note is correct: from the variance identity S2 = S1^2 + sigma^2, equality holds iff sigma=0, i.e. uniform microstate probabilities. Section V's discussion of the second law does not establish zero variance; it only uses non-negativity of variance, which is true regardless.

There is real content here. The q=2 composition rule for Ubriaco's fractional entropy is derived cleanly, and the appearance of S1 in the composition is a genuine observation. The stability analysis, giving W > e for q=2, is a concrete and checkable result. The paper also credits Ubriaco, Tsallis-Cirto, and Volovik appropriately, and it distinguishes its approach from Volovik's. In terms of scholarly honesty, this is not a paper that hides its assumptions.

The soft spots are the central one plus a few smaller ones. The paper never supplies a physical argument for why horizon microstates would be exactly equally probable, so the match with Eq. (11) remains a special case, not a reconciliation. The dimensional handling of k_B in Eq. (30) is sloppy; the cross term should have k_B in the denominator if S1 and S2 both carry units of k_B. That is probably fixable but should be flagged. The "universal empirical temperature" is, as the reader says, a normalization artifact: once you posit S1 proportional to M, the coefficient is fixed by the area law.

Who is this for? Someone working on non-additive entropies in quantum-gravity phenomenology will find the composition algebra and the stability condition useful. The paper deserves a serious referee who can press on the equal-prior question and ask whether any horizon-state model satisfies it. As written, the central claim is conditional, but the paper is not incoherent and the mathematics is mostly sound. I would send it to review rather than desk-reject, with the expectation that the authors either justify the uniform measure or substantially soften the claim.

For my own work, I would not cite it in the next year. But I would bring it to a reading group as an example of how non-additive entropy frameworks can be self-consistent yet physically underdetermined.

Referee Report

4 major / 5 minor

Summary. The paper proposes that the fractional (Ubriaco) entropy S^U_q = k_B sum_i p_i (-ln p_i)^q, for q=2, reproduces the black hole merger entropy composition S_f = S(M1)+S(M2)+2 sqrt(S(M1)S(M2)). For q=2 the fractional composition rule introduces a cross term involving S^U_1, and the match to the black hole formula is made by imposing S^U_1 = sqrt(S^U_2). The authors show this relation holds for equal priors p_i = 1/W, and they then use it to derive a universal empirical temperature T = T_P/(2 sqrt(pi)) for all horizons, an equidistant mass spectrum M = M n, and various stability and distribution-function results. The paper also discusses the second law in terms of information fluctuation variance.

Significance. If the central reconciliation were established, the framework would offer a novel statistical interpretation of black hole entropy composition and a concrete prediction of a universal horizon temperature. The paper does contain useful observations: the composition rule for q=2 naturally brings in the Shannon entropy; the identity sigma^2 = S^U_2 - (S^U_1)^2 correctly connects the matching condition to vanishing information fluctuation; and the stability analysis for q>1 is clearly presented. However, the central claim depends entirely on the assumption S^U_1 = sqrt(S^U_2), which the authors themselves state may not generally hold. Because the universal temperature and mass spectrum follow directly from that unproven special case, the main results are conditional rather than robust. The manuscript would need a physical argument for equal prior horizon microstates, or a reformulation of the claims as conditional, before the results could be accepted.

major comments (4)

[Section IV, Eqs. (30)-(32)] The claimed match between fractional entropy composition and black hole composition is established only under the assumption S^U_1 = sqrt(S^U_2). Section IV.A proves this relation solely for the equal-prior case p_i = 1/W, and the text immediately concedes that this assumption 'may not generally hold'. Section V shows that the relation is equivalent to vanishing information fluctuation variance but does not prove that horizon microstates have zero variance; it only uses the non-negativity of variance. Thus the central reconciliation is conditional on an unjustified special case.
[Section VI.B, Eq. (50)] The universal empirical temperature T = 1/(2 sqrt(pi)) sqrt(c^5 hbar/(G k_B^2)) is derived from S^U_1 = sqrt(S^U_2) combined with the Schwarzschild area law S^U_2 proportional to M^2. Since S^U_1 = sqrt(S^U_2) is exactly the equal-prior assumption, the temperature is not an independent result but a restatement of that assumption. Moreover, for a Kerr black hole with J != 0, sqrt(Area) is not proportional to M, so the claimed universality for 'all black hole horizons' is not supported by the derivation.
[Section VI.C, Eqs. (56)-(61)] The equidistant mass spectrum M = M n follows from identifying ln W = sqrt(Area/(4 l_p^2)) and then imposing the condition sqrt(4 pi/alpha)(R/l_p) = integer. This condition is an additional quantization assumption, equivalent to requiring that the exponent of W is an integer; it is not derived from the fractional entropy framework itself. The derivation therefore does not establish a mass spectrum from fractional entropy alone.
[Section IV, Eq. (30)] Equation (29) gives S^U_2(A+B)/k_B = S^U_2(A)/k_B + S^U_2(B)/k_B + 2 S^U_1(A) S^U_1(B)/k_B^2. Multiplying through by k_B gives a cross term 2 S^U_1(A) S^U_1(B)/k_B, not 2 S^U_1(A) S^U_1(B) as written in Eq. (30). As written, Eq. (30) is dimensionally inconsistent unless k_B is set to 1. This missing factor propagates into the matching condition in Eq. (31) and into the restoration of physical constants in Eq. (50).

minor comments (5)

[Section III.A, Eq. (26)] The proof of independence of n would benefit from an explicit statement of the domain of p_i (0 < p_i <= 1) and the choice of branch for the logarithm, since the integral representation involves powers of -ln p_i.
[Section VII, Eq. (71)] The selection of the branch p_i = exp(-1 - sqrt(1+phi_i)) is made on physical grounds, but the normalization condition in Eq. (73) should be used to show that this branch, rather than the other, yields a normalizable probability distribution over the full energy range.
[Section VIII, Eqs. (75)-(81)] The stability analysis is carried out for equal probabilities p_i = 1/W, but the composition rule in Eq. (30) is defined for general distributions; the restriction to the equal-prior case should be stated explicitly in this section.
[Section VI.B, Eq. (49)] The definition 1/T = dS^U_1/dM is dimensionally inconsistent in SI units unless c=1 is assumed; the standard relation is 1/T = partial S/partial E. The text should state the unit convention before restoring constants in Eq. (50).
[Throughout] There are several typographical and presentation issues: 'Fadeev' should be 'Faddeev', the R'enyi accent is inconsistent, Eq. (15) has a garbled operator expression, and the captions of Figures 1 and 3 could specify the branch choices and parameter values more fully.

Circularity Check

1 steps flagged · score 6.0 of 10

The universal empirical temperature and the equidistant mass spectrum reduce by construction to the assumed S1=√S2 relation, which the authors themselves concede may not generally hold.

fitted input called prediction [Section VI.C, Eqs. (55)–(61)]
"S_U2 ∝ R². This suggests that it is not ln W that is proportional to the area, but rather its square. Therefore, we have: S_U1=√S_U2=k_B√(Area/4ℓ_p²), ... =⇒ ln W=√(Area/4ℓ_p²). ... By setting √(α/16π) m_p=M =⇒ M=M·n."

The equidistant mass spectrum M=M·n is obtained by substituting S_U1=√S_U2 (the equal-prior/zero-variance case) into the state-counting relation ln W=√(Area/4ℓ_p²) and then applying the conventional area-quantization rule. Since S_U1=√S_U2 was introduced solely to make the composition rule match Eq. (11), the mass quantization is the same assumption expressed in mass units, not an independent prediction of the fractional-entropy framework.

full rationale

The genuinely independent content in this paper includes the fractional-entropy composition rule Eq. (30), the variance identity Eq. (39), and the non-negativity inequality Eq. (40), none of which requires the black-hole matching. However, every black-hole-specific conclusion—the exact matching to Eq. (11), the universal empirical temperature Eq. (50), the relation to Hawking temperature Eq. (51), and the mass spectrum Eq. (61)—uses S_U1=√S_U2, a relation the authors introduce as an assumption and explicitly concede is not general. This is not a hidden logical circle of the self-citation type: there is no load-bearing self-citation (refs. [35, 36] are cosmology applications, and ref. [34] is only an end-of-paper agreement remark). Nevertheless, under the circularity rubric, the main 'predictions' reduce by construction to the assumed square-root relation, while the paper presents them as derived results. The reader's take is therefore substantially correct, though the flaw is better described as an unproved, nongeneric ansatz called a prediction than as a formal logical tautology. Score 6 reflects partial circularity: the core conditional result is forced by an input assumption, while other parts of the paper (stability analysis, equilibrium distribution, variance inequality) retain independent content.

Assumptions & free parameters 1 free parameters · 4 assumptions · 2 invented entities

The central construction relies on four stated assumptions: the area law for S^U_2, equal-prior horizon microstates, energy-conserving mergers, and probabilistic independence of the two components. The equal-prior assumption is the most fragile because the entire composition match and the universal temperature depend on it, and the paper itself flags it as unproven. The only hand-set parameter is q=2, fixed by the desired composition rule.

free parameters (1)

q = 2
Chosen so that the fractional entropy composition matches the idealized black hole merger formula; no independent first-principles derivation is given.

assumptions (4)

domain assumption S^U_2 obeys the Bekenstein-Hawking area law S^U_2 = k_B A/(4 l_p^2).
Invoked in Section IV-C and in the Conclusion ('we assume that S^U_2 is the one that is responsible for the area law'), and needed for the S1 proportional to mass relation.
ad hoc to paper Horizon microstates are equally probable, or information fluctuations vanish, so that S^U_1 = sqrt(S^U_2).
Section IV, Eqs. (31)-(32), is the load-bearing step that maps fractional entropy composition to black hole composition; the authors admit this may not hold generally.
domain assumption The merger conserves mass with negligible gravitational wave emission: M_f = M1+M2.
Section II uses this idealized Schwarzschild case to write S_f = S1+S2+2√(S1S2), while noting that real mergers can emit up to about 29% of energy.
domain assumption The two subsystems A and B have independent probability distributions.
Section III.B derives the composition rule under p_{ij}(A+B)=p_i(A)p_j(B); the paper acknowledges gravity may make this independence questionable.

invented entities (2)

Empirical temperature T = T_P/(2√π)
purpose: Zeroth-law-compatible temperature assigned to every black hole horizon, making composition isothermal.
Derived from the assumed S1 proportional to M relation; no external observable is predicted beyond the Hawking-temperature connection.
Equidistant mass spectrum M = M·n
purpose: Replaces the area spectrum with a mass spectrum from fractional entropy state counting.
Eq. (61) presents the spectrum but no experimental handle is proposed; the paper notes Volovik [34] obtained a similar result.

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Pith. "Pith review of Reconciling fractional entropy and black hole entropy compositions." pith.science (2026). https://pith.science/paper/34USTODL

@misc{pith2026250118151,
  author       = {Pith},
  title        = {Pith review of: Reconciling fractional entropy and black hole entropy compositions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/34USTODL}},
  note         = {Machine review of arXiv:2501.18151}
}

abstract

This study investigates the implications of adopting fractional entropy in the area law framework and demonstrates its natural alignment with an isothermal description of black hole composition. We discuss the Zeroth law compatibility of the fractional entropy and define an empirical temperature for the horizon. We highlight the distinction between the empirical and conventional Hawking temperatures associated with the black holes. Unlike the Hawking temperature, this empirical temperature appears universal, and its proximity to the Planck temperature suggests a possible quantum gravity origin. We also establish the connection between these temperatures. Furthermore, extending the conventional fractional parameter $q$, constrained between 0 and 1, we establish that any positive real number can bound $q$ under the concavity condition, provided the log of micro-state dimensionality exceeds $q-1$. Specifically, for black holes, $q = 2$, necessitating micro-state dimensionality greater than $e$, thereby excluding the construction of black hole horizon states with two level bits or qubits. We also identify the connection between the validity of the second law and information fluctuation complexity. The second law requires that the variance of information content remain smaller than the area of the black hole horizon. This constraint naturally gives rise to a Boltzmann-Gibbs-like entropy for the black hole, which, in contrast to the canonical formulation, is associated with its mass rather than its area. Equilibrium distribution analysis uncovers multiple configurations, in which the one satisfying the prerequisites of probability distribution exhibits an exponent stretched form, revealing apparent deviation from the Boltzmann distribution.

Figures

Figures reproduced from arXiv: 2501.18151 by the authors.

**Figure 2.** FIG. 2. The entropy as a function of the binary probability distribution [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** FIG. 3. Entropy [PITH_FULL_IMAGE:figures/full_fig_p011_3.png] view at source ↗

Figures from the paper (3 more)

**Figure 4.** Figure 4: FIG. 4. Logarithm of the negative second derivative of entropy [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]

**Figure 5.** Figure 5: FIG. 5. Stability criterion [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]

**Figure 6.** Figure 6: FIG. 6. Stability criterion [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

Black Hole Thermodynamics via Tsallis Statistical Mechanics
gr-qc 2025-02 conditional novelty 6.0 of 10

The authors derive a q-modified black hole entropy from Tsallis statistics applied to a near-horizon gas and show that a negative non-extensive parameter can stabilize a Schwarzschild black hole.
Black hole thermodynamics and topology
gr-qc 2025-05 reject novelty 4.0 of 10

The author derives S_RN = 4πM^2 for Reissner-Nordström black holes by adding the inverse temperatures of both horizons, contradicting the standard area law.

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Reviewed August 10, 2026 · model on record in the stance chip above.

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Based on the definition provided earlier, for 𝑞 = 2, we can express the entropy as 𝑆𝑈 2(𝐴+𝐵)/𝑘𝐵=(−1)2 ∑︁ 𝑖=1 ∑︁ 𝑗=1 𝑝𝑖,𝑗(𝐴+𝐵) ln𝑝𝑖,𝑗(𝐴+𝐵) 2 = ∑︁ 𝑖=1 ∑︁ 𝑗=1 𝑝𝑖,𝑗(𝐴+𝐵) ln𝑝𝑖(𝐴)+ ln𝑝𝑗(𝐵) 2 = ∑︁ 𝑖=1 ∑︁ 𝑗=1 𝑝𝑖,𝑗(𝐴+𝐵)[ln𝑝𝑖(𝐴)]2 + ∑︁ 𝑖=1 ∑︁ 𝑗=1 𝑝𝑖,𝑗(𝐴+𝐵) ln𝑝𝑗(𝐵) 2 +2 ∑︁ 𝑖=1 ∑︁ 𝑗=1 𝑝𝑖,𝑗(𝐴+𝐵) ln𝑝𝑖(𝐴) ln𝑝𝑗(𝐵). (28) By summing over the probabilities of individual sys...

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This also indicates that the underlying fundamental degrees of freedom can simul- taneously give rise to both𝑆𝑈 1 and𝑆𝑈 2 like entropy, suggesting a more in-depth duality

(32) While this assumption may not generally hold, proving its va- lidity would lead to a consistent composition rule for black hole entropy under energy conservation. This also indicates that the underlying fundamental degrees of freedom can simul- taneously give rise to both𝑆𝑈 1 and𝑆𝑈 2 like entropy, suggesting a more in-depth duality. Can we consider t...

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(42) However, the formal logarithm of 𝑆 exhibits additivity. To demonstrate this, let us multiply both sides by(1−𝑞) and then add 1, (1−𝑞)𝑆𝑇 𝑓 =(1−𝑞)𝑆𝑇 1+( 1−𝑞)𝑆𝑇 2+( 1−𝑞)2𝑆𝑇 1𝑆𝑇 2 1+( 1−𝑞)𝑆𝑇 𝑓 = 1+( 1−𝑞)𝑆𝑇 1+( 1−𝑞)𝑆𝑇 2+( 1−𝑞)2𝑆𝑇 1𝑆𝑇 2 1+( 1−𝑞)𝑆𝑇 𝑓 = 1+( 1−𝑞)𝑆𝑇 1 1+( 1−𝑞)𝑆𝑇 2 . Taking the natural logarithm on both sides yields, ln h 1+( 1−𝑞)𝑆𝑇 𝑓 i = ln 1+...

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