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arxiv: 2605.31211 · v1 · pith:WRMYDLACnew · submitted 2026-05-29 · 🌀 gr-qc

Classical Corrections to Black Hole Entropy II

Daulet Berkimbayev , Martin Blaschke This is my paper

Pith reviewed 2026-06-28 21:31 UTC · model grok-4.3

classification 🌀 gr-qc
keywords black hole entropylogarithmic correctiondiscrete recursionSchwarzschild-TangherliniReissner-Nordströmone-bit absorptioncharge dependencehistory ensemble
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The pith

Discrete recursion in black-hole absorption produces the area law plus a logarithmic correction whose coefficient is fixed by the dimensional dependence of the mass step.

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

The paper treats the classical one-bit absorption model of black-hole growth as a discrete recursion rather than a continuum equation, with each absorption unit measured in natural logarithmic units. For Schwarzschild-Tangherlini black holes the recursion recovers the expected area scaling of entropy and generates a logarithmic correction whose coefficient is set solely by how the mass increment depends on spacetime dimension. The same recursion applied to Reissner-Nordström black holes at fixed charge makes the logarithmic coefficient depend explicitly on charge. Ordered histories built from positive, negative, and neutral charge labels are shown to define formal ensembles in which different histories can reach the same macroscopic state; the construction separates dynamical entropy from thermodynamic, history, and hidden conditional entropies. A reader would care because the corrections arise from the discrete character of absorption without reference to quantum gravity and therefore apply to both neutral and weakly charged final states.

Core claim

Modeling black-hole growth as a discrete recursion with one-bit absorption units measured in natural logarithmic units yields the area-law scaling of entropy together with a logarithmic correction for Schwarzschild-Tangherlini black holes, the coefficient of the correction being fixed by the dimensional dependence of the mass step; the identical construction applied to fixed-charge Reissner-Nordström black holes produces a logarithmic coefficient that depends explicitly on charge. Ordered histories labeled by charge sign define formal ensembles rather than direct microstate counts, and the framework distinguishes dynamical entropy, thermodynamic state entropy, history entropy, and hidden con

What carries the argument

The discrete recursion that replaces the continuum absorption equation and incorporates the dimensional dependence of the mass step to determine the logarithmic correction coefficient.

If this is right

  • The logarithmic correction coefficient for Schwarzschild-Tangherlini black holes is fixed by the mass step's dimensional dependence without additional fitting.
  • In the fixed-charge Reissner-Nordström sector the logarithmic coefficient acquires an explicit dependence on charge.
  • The fixed-charge recursion corresponds to the large-mass neutral growth stage after a charged seed has formed.
  • Ordered charge histories define a formal ensemble in which two histories may reach the same macroscopic state while differing in hidden degrees of freedom.
  • Adding a mass or energy label reduces endpoint degeneracy but does not eliminate it within the ensembles considered.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • If the discrete recursion is a faithful model of accretion, the same logarithmic correction should appear in numerical simulations that enforce a minimal absorption unit.
  • The charge dependence of the correction coefficient offers a potential observational signature when entropy is compared across black holes of different charge-to-mass ratios at large mass.
  • Treating histories as ensembles rather than microstate counts suggests that unitary evolution can preserve distinctions invisible at the macroscopic level.
  • The separation of entropy types can be applied to other classical growth models to check whether analogous corrections arise.

Load-bearing premise

Black-hole growth can be represented accurately as a discrete recursion whose absorption units are measured in natural logarithmic units and whose mass-step dimensional dependence alone fixes the logarithmic entropy correction without extra parameters.

What would settle it

A numerical evaluation of the discrete recursion across several spacetime dimensions that produces a logarithmic coefficient different from the one predicted by the mass-step dimensional dependence would falsify the central claim.

Figures

Figures reproduced from arXiv: 2605.31211 by Daulet Berkimbayev, Martin Blaschke.

Figure 1
Figure 1. Figure 1: FIG. 1. Numerical verification of the corrected logarithmic [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. RN leading coefficient [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Relative deviation between the fitted RN logarithmic [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Fraction of total history entropy stored in the macro [PITH_FULL_IMAGE:figures/full_fig_p007_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Charge distribution from formal sign-history counting at [PITH_FULL_IMAGE:figures/full_fig_p008_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Endpoint degeneracy in a toy ensemble with both [PITH_FULL_IMAGE:figures/full_fig_p008_7.png] view at source ↗
Figure 9
Figure 9. Figure 9: FIG. 9. Linear calibration between the discrete step count and [PITH_FULL_IMAGE:figures/full_fig_p009_9.png] view at source ↗
Figure 11
Figure 11. Figure 11: FIG. 11. Window stability of the logarithmic coefficient [PITH_FULL_IMAGE:figures/full_fig_p009_11.png] view at source ↗
read the original abstract

We reconsider the classical one bit absorption model of black hole growth as a discrete recursion rather than a continuum equation. In this paper, one bit means one elementary absorption unit measured in natural logarithmic units. For a Schwarzschild Tangherlini black hole, the discrete treatment gives the expected area scaling and also produces a logarithmic correction. The coefficient of this correction is fixed by the dimensional dependence of the mass step. We then extend the construction to the Reissner Nordstr\"om case at fixed charge. In this sector, the logarithmic coefficient gains an explicit charge dependence. The fixed charge calculation can be interpreted as the large mass neutral growth stage after a charged seed has already formed. In that interpretation, realistic charged absorption affects the initial cutoff, while the large mass recursion controls the dominant entropy for weakly charged final states. We also study ordered histories built from positively charged, negatively charged, and neutral labels. These histories define a formal ensemble rather than a direct count of black hole microstates. In a unitary description, two different histories may lead to the same reduced macroscopic state, described by mass, charge, and entropy, while remaining different in hidden or environmental degrees of freedom. Adding a mass or energy label reduces the endpoint degeneracy, but it does not remove it in the ensembles considered here. The corrected interpretation separates dynamical entropy, thermodynamic state entropy, history entropy, and hidden conditional entropy. It also clarifies that the multinomial formula is a restricted history counting result, not a proof that different histories correspond to unique or identical microscopic black hole states.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript reconsiders the classical one-bit absorption model of black hole growth as a discrete recursion in natural logarithmic units rather than a continuum equation. For Schwarzschild-Tangherlini black holes, the discrete treatment is claimed to produce the expected area scaling for entropy along with a logarithmic correction whose coefficient is fixed by the dimensional dependence of the mass step. The construction is extended to the Reissner-Nordström case at fixed charge, where the logarithmic coefficient acquires an explicit charge dependence. The paper also studies ordered histories with charged and neutral labels, interpreting them as a formal ensemble, and separates dynamical entropy, thermodynamic state entropy, history entropy, and hidden conditional entropy.

Significance. If the central derivation is free of circularity in the definition of the mass step, this work could provide a classical discrete model that naturally generates logarithmic corrections to black hole entropy with explicit dependence on dimension and charge. The separation of different entropy concepts and the interpretation of histories as ensembles rather than microstate counts is a positive clarification. However, the significance is tempered by the need to verify that the mass step rule is independently derived from geometry without encoding the logarithmic term.

major comments (2)
  1. [Abstract] Abstract: The claim that the logarithmic coefficient is fixed by the dimensional dependence of the mass step requires explicit demonstration that the recursion relation and the rule for ΔM (in natural log units) are defined using only the horizon geometry or mass in d dimensions, without reference to the target entropy form or the log term itself. If the step definition already incorporates logarithmic scaling to match the expected correction, the coefficient is imposed by construction rather than derived.
  2. [Abstract] The extension to Reissner-Nordström at fixed charge (abstract): the charge dependence of the log coefficient must be shown to follow solely from the same independent mass-step rule applied to the RN geometry, without additional modeling choices that encode the desired correction.
minor comments (1)
  1. The abstract refers to 'one bit means one elementary absorption unit measured in natural logarithmic units' without a precise definition of the unit or its relation to the horizon area; this should be clarified with an equation in the main text.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive feedback on our manuscript. We address the major comments point by point below, providing clarification on the independence of the mass-step definition while agreeing to strengthen the presentation for explicitness.

read point-by-point responses

  1. Referee: [Abstract] Abstract: The claim that the logarithmic coefficient is fixed by the dimensional dependence of the mass step requires explicit demonstration that the recursion relation and the rule for ΔM (in natural log units) are defined using only the horizon geometry or mass in d dimensions, without reference to the target entropy form or the log term itself. If the step definition already incorporates logarithmic scaling to match the expected correction, the coefficient is imposed by construction rather than derived.

    Authors: The mass increment ΔM per one-bit absorption is fixed by the d-dimensional horizon geometry alone: for the Schwarzschild-Tangherlini solution the relation M ∝ r^{d-3} together with the area-radius scaling determines the discrete change in mass that corresponds to a unit increment in the leading (area-law) entropy term. The recursion is then written directly in terms of this geometrically determined ΔM; the logarithmic correction appears only after summation of the discrete steps and is therefore a derived consequence rather than an input. We acknowledge that the abstract does not spell out this derivation in full detail and will revise the manuscript by adding an explicit subsection that derives ΔM from the metric without reference to the target entropy expression. revision: yes

  2. Referee: [Abstract] The extension to Reissner-Nordström at fixed charge (abstract): the charge dependence of the log coefficient must be shown to follow solely from the same independent mass-step rule applied to the RN geometry, without additional modeling choices that encode the desired correction.

    Authors: In the fixed-charge RN sector the identical geometric rule for ΔM is applied, now using the outer-horizon radius of the d-dimensional RN metric at constant Q. The charge dependence of the resulting logarithmic coefficient arises automatically from the modified M-r relation at fixed Q; no extra terms or ad-hoc adjustments are introduced. The fixed-charge regime is interpreted as the late-time neutral accretion phase after an initial charged seed, but this interpretation does not alter the mass-step definition itself. We will revise the text to display the explicit RN mass-step formula and its insertion into the recursion, thereby demonstrating that the charge dependence follows solely from the geometry. revision: yes

Circularity Check

1 steps flagged

Log coefficient fixed by input choice of mass-step dimensional dependence, reducing central claim to model assumption by construction

specific steps
  1. fitted input called prediction [Abstract]
    "For a Schwarzschild Tangherlini black hole, the discrete treatment gives the expected area scaling and also produces a logarithmic correction. The coefficient of this correction is fixed by the dimensional dependence of the mass step."

    The dimensional dependence of ΔM is an explicit modeling input. Inserting a chosen d-scaling for the step size into the recursion algebra immediately yields the log term with a specific coefficient; the coefficient is therefore not an output but the direct algebraic consequence of the chosen input scaling.

full rationale

The paper's core result is that the discrete recursion produces S ~ A/4 + c(d) log(A), with c fixed solely by the d-dependence chosen for the elementary mass step ΔM (in natural-log units). This dependence is introduced as part of the model definition rather than derived from horizon geometry alone or from an independent principle; once the d-scaling of ΔM is set, the logarithmic term and its coefficient follow directly from the recursion algebra. No external benchmark or parameter-free derivation is shown to constrain that d-dependence independently of the target correction. Consequently the claimed 'prediction' of the coefficient is equivalent to the modeling choice that was inserted to produce it.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 1 invented entities

The central claim depends on the discrete recursion assumption and the interpretation of histories as formal ensembles rather than microstate counts; the mass step serves as the key input that sets the correction coefficient.

free parameters (1)
  • mass step
    Dimensional dependence of the mass step is used to fix the logarithmic correction coefficient in the discrete recursion.
axioms (1)
  • domain assumption Black hole growth proceeds via discrete one-bit absorptions in natural log units
    The paper reconsiders the model as a discrete recursion rather than continuum equation.
invented entities (1)
  • history entropy no independent evidence
    purpose: To account for entropy from different absorption sequences in the ensemble
    Defined to separate from thermodynamic and dynamical entropy in unitary descriptions of black hole states.

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discussion (0)

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Reference graph

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