Finite geometry and black hole stability: Embedding discrete space into classical manifolds

Arkady Bolotin

REVIEW 4 major objections 5 minor 32 references

Finite geometry and black hole stability: Embedding discrete space into classical manifolds

T0 review · 4 major / 5 minor · reviewed 2026-08-15 · deepseek-v4-flash

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Pith's one-line read A Schwarzschild black hole has a nonzero minimum volume, so evaporation stops at a Planck-scale radius and leaves stable remnants.

desk verdict A clear, readable speculative paper whose Planck-scale remnant conclusion is effectively put in by hand; novel construction but load-bearing assumptions remain ungrounded, and it deserves a serious referee. read the letter →

arxiv 2505.11585 v1 pith:ZKDGAECX submitted 2025-05-16 gr-qc

Arkady Bolotin This is my paper · ORCID

classification gr-qc PACS 04.70.-s04.60.-m

keywords blackholeremnantsfinitegeometrySchwarzschildvolumePlancklengthBekenstein-HawkingentropyboundinformationdensityPoissonpointprocessfaithfulembedding

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

General relativity makes the volume of a black hole slice-dependent, yet entropy bounds require a volume with a boundary. To fix this, the paper models physical space as a finite set of points in which each point is one bit of information, and embeds that set faithfully into a Riemannian manifold so that information density can be measured by ordinary volume and area. The Bekenstein-Hawking entropy fixes the areal density, the observed cosmic scale bounds the volumetric density, and the result is a definite, observer-independent interior volume for a Schwarzschild black hole. The minimum such volume is nonzero, about $4\ell_H\ell_P^2$, which forces evaporation to stop at an event-horizon radius $r_S=\ell_P/\sqrt{\pi}$. Stable black hole remnants are therefore a consequence of the finite character of geometry, not of a particular quantum-gravity model.

What carries the argument

The central object is the faithful embedding of a finite point set $\{M\}$ into a Riemannian manifold $\mathcal{R}$, with each point treated as one bit of information. Faithful means the count of points in any region and on its boundary is proportional to volume and area, respectively: $H(V)=k_B\rho_V V$ and $H(A)=k_B\rho_A A$. The paper models the sprinkling as a homogeneous Poisson process, so the probability of finding $n$ bits in a volume is $P(n)=(\rho_V V)^n e^{-\rho_V V}/n!$. From the Bekenstein-Hawking area law it reads off $\rho_A=1/(4\ell_P^2)$; from the entropy bound and the cosmic ratio $A/V\approx H_0/c$ it obtains $\max\rho_V\approx(1/4\ell_P^2)(H_0/c)$. The minimum volume then follows from setting the Poisson parameter $\lambda=\rho_V V_{\min}$ equal to 1, giving $V_{\min}=(\max\rho_V)^{-1}\approx4\ell_H\ell_P^2$, and comparing with the saturated entropy bound fixes the terminal horizon radius $r_S=\ell_P/\sqrt{\pi}$.

What would settle it

Recompute the argument with the Poisson expectation set to $\lambda=2$: the remnant radius becomes $\ell_P\sqrt{2/\pi}$, showing that the Planck-scale conclusion is directly controlled by the one-bit normalization. Observationally, a confirmed observation of a black hole evaporating completely, with no stable remnant and no cutoff in the emitted graviton spectrum, would refute the claim.

Watch

Extended reading notes

Core claim

The paper's discovery is that a discrete, finite model of space can be embedded so faithfully into a classical continuum that black hole volume becomes a well-defined physical quantity. In this embedding the entropy in a region is $H(V)=k_B\rho_V V$, the entropy on its boundary is $H(A)=k_B\rho_A A$, and the Bekenstein-Hawking formula $\rho_A=1/(4\ell_P^2)$ fixes the areal information density. The entropy bound then requires $\rho_V\le A/(4\ell_P^2 V)$; using the observed flatness of the universe, $A/V\approx H_0/c$ for the observable region, so $\max\rho_V\approx(1/4\ell_P^2)(H_0/c)$. Requiring the smallest information-bearing volume to contain one bit gives $V_{\min}=(\max\rho_V)^{-1}\approx4\ell_H\ell_P^2$. Saturation of the entropy bound at this volume yields $r_S=\ell_P/\sqrt{\pi}$: a Schwarzschild black hole cannot evaporate below about the Planck length, leaving a stable remnant.

Load-bearing premise

The entire numerical result hinges on the choice that the smallest information-bearing volume contains exactly one bit ($\lambda=1$); any other normalization would rescale the remnant radius by $\sqrt{\lambda}$.

Editorial extensions

If this is right

A Schwarzschild black hole has an observer-independent volume bounded above by $\max V\approx 4\pi\ell_H r_S^2$, so the notion of black hole volume is meaningful after all.
Evaporation must stop at $r_S\approx \ell_P/\sqrt{\pi}$, leaving a stable remnant; total collapse is avoided by the finiteness of geometry.
Because $V_{\min}$ depends only on $c$, $\ell_P$, and $H_0$, remnant stability is a model-independent prediction that any candidate quantum theory of gravity should reproduce.
The entropy bound is preserved: with $\rho_V\le (1/4\ell_P^2)(A/V)$, volumetric entropy cannot exceed areal entropy in faithful embeddings.
A cutoff in the graviton spectrum emitted during black hole evaporation could be an observational signature of discrete space, and stable remnants remain a dark matter candidate.

Reading between the lines

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

Beyond the paper: the choice $\lambda=1$ is a normalization, not a derivation; a different filling number would rescale the remnant radius by $\sqrt{\lambda}$ and would test how much of the Planck-scale remnant is forced by geometry rather than by convention.
Beyond the paper: because $V_{\min}$ depends on today's Hubble length, the remnant mass is tied to cosmology; if Hubble-tension measurements revise $H_0$, the predicted remnant scale shifts in a way that could be compared with observations.
Beyond the paper: the isotropy argument excludes rotating black holes; extending the Poisson-embedding construction to Kerr geometries would require an axis-dependent density and could predict a different remnant shape or spin cutoff.

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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

The one thing to know: this paper argues for stable Planck-scale black hole remnants from a finite-geometry embedding of information bits. The conclusion is a familiar expectation in quantum gravity; the specific route through a volumetric information density is new. The paper is clearly written, engages the right literature, and takes the Weyl tile problem and the identification problem seriously. The derivation of the areal density ρ_A = 1/(4ℓ_P²) is clean, and the construction leading to Eqs. (12) and (15) is not in the prior literature. Credit where due: the motivation is legitimate, since black hole volume is slicing-dependent and the Bekenstein-Hawking bound implicitly needs a volume, and the sprinkling picture is a coherent way to connect finite geometry to a Riemannian manifold.

The soft spots are load-bearing. The central step equates the entropy-bound upper limit on volume, maxV ≈ 4πℓ_H r_S², with the one-bit minimum volume V_min ≈ 4ℓ_Hℓ_P², and from that equality reads off r_S = ℓ_P/√π. But maxV is an upper bound on the volume of any region satisfying the entropy bound, while V_min is a lower bound on the region needed to hold one bit. The consistency requirement is V_min ≤ V_actual ≤ maxV, not V_actual = maxV. The paper supplies no model for the actual interior volume, which in GR is slicing-dependent. If you instead took a Euclidean ball interior V_actual = (4/3)πr_S³, the cutoff would not be the Planck length. Second, the λ=1 condition is introduced as "reasonable to propose" rather than derived; choosing λ ≠ 1 would shift the remnant radius by √λ. That is an ad hoc normalization doing real work. Third, the universal density bound rests on taking a cosmological area-to-volume ratio and applying it inside a black hole; the isotropy argument is hand-wavy near a singularity and does not justify the same bound where the geometry is very different.

So the central claim, as stated, is not established. The math is consistent but the physical assumptions are underdetermined. That said, the paper is not confused; it is a speculative proposal with clear steps, and the author honestly engages with alternatives. The citation pattern is fine (the one self-citation is to prior related work). This paper is for readers interested in quantum gravity phenomenology and black hole volume debates — it will provoke discussion, but I would not rely on it yet. I would not cite it in my own work, but I would bring it to a reading group. For peer review: yes, send it to referees. A competent referee will quickly locate the two unforced choices, and the paper is concrete enough that the report can be constructive rather than dismissive.

Referee Report

4 major / 5 minor

Summary. The paper proposes that physical space is a finite geometry whose points carry one bit of information, and that such a discrete space can be faithfully embedded into a Riemannian manifold through a Poisson sprinkling process. Using the Bekenstein-Hawking entropy and the holographic entropy bound, the author derives an upper limit on the interior volume of a Schwarzschild black hole, maxV(RR) ≈ 4πℓ_H r_S^2 (Eq. 12), and a minimum volume V_min(RR) ≈ 4ℓ_Hℓ_P^2 containing one bit (Eq. 15). Equating these two expressions yields r_S = ℓ_P/√π, which is interpreted as the smallest possible Schwarzschild black hole and as evidence that black hole evaporation halts at the Planck scale, leaving stable remnants. The paper also discusses the Weyl tile problem, the identification problem in discrete geometry, and potential observational signatures in the gravitational-wave spectrum.

Significance. If the central derivation were valid, the paper would offer a simple, quantum-gravity-free argument for stable Planck-scale black hole remnants and would connect the idea of discrete spacetime to an observable cutoff in the gravitational-wave spectrum from black hole evaporation. The paper is clearly written and gives a useful discussion of the identification and distance-function problems in finite geometry, proposing a concrete embedding picture. However, the logical path from the entropy bounds to the remnant claim has several load-bearing gaps: an undetermined normalization λ=1 is introduced by hand, the cosmological information density is assumed without justification to saturate inside a black hole, and an upper bound on volume is compared with a lower bound to infer the Planck-scale horizon radius. The result is therefore not a robust derivation but a dimensional argument whose specific numerical prediction depends on arbitrary choices.

major comments (4)

[Section 3.1, Eqs. (12) and (15)] The paper derives maxV(RR) as an upper bound on the interior volume of a Schwarzschild black hole and V_min(RR) as a minimum volume containing one bit, and then compares them to conclude r_S = ℓ_P/√π. This is logically invalid as stated: Eq. (12) is only an upper limit, while the actual interior volume is not specified and is slicing-dependent in general relativity (see Christodoulou and Rovelli, Ref. [6]). The most that follows from the entropy bound and the one-bit requirement is the consistency inequality V_min ≤ V_actual(RR) ≤ maxV(RR), which yields only r_S ≥ ℓ_P/√π. To claim that evaporation stops at this radius, the paper must show that V_actual cannot remain above V_min for smaller r_S; no such relation between the actual interior volume and the horizon radius is provided.
[Section 3.1, Eq. (14)] The choice λ=1, namely that the minimum information-bearing volume contains exactly one expected bit, is an ad hoc normalization and not a consequence of the preceding physics. Since V_min = λ/maxρV and the resulting horizon radius scales as r_S = √λ ℓ_P/√π, the precise Planck-length prediction is directly controlled by this arbitrary input. The paper states only that it is 'reasonable to propose' λ=1, but provides no principle from finite geometry or quantum theory that selects this value over, say, λ=1/2 or λ=2.
[Section 2.3, Eqs. (7)–(10)] The upper limit maxρV is derived from the cosmological region R_U using the Euclidean area-to-volume ratio A/V ≈ H0/c, and then assumed to hold inside a black hole because the Poisson sprinkling is homogeneous and isotropic. This extrapolation is not justified: the interior of a Schwarzschild black hole is not a Euclidean region at rest, and near the singularity the notions of isotropy and spatial volume themselves break down. Moreover, the paper assumes that the informational density actually attains the upper bound inside the black hole; if the true density is lower, then V_min = 1/ρV is larger and the remnant radius is correspondingly larger. No mechanism enforcing saturation is given.
[Section 3.1, Eq. (15)] The predicted minimum volume depends on the Hubble length ℓ_H through V_min(RR) ≈ 4ℓ_Hℓ_P^2. The Hubble constant H0 is not a fundamental constant; it is the present-day value of a time-dependent cosmological parameter. As a result, the supposed remnant scale and remnant mass would depend on the cosmological epoch, which is incompatible with the claim that stable black hole remnants are characterized by the fundamental constants c and ℓ_P alone. The paper does not address how V_min should be defined or evaluated at other times.

minor comments (5)

[Eq. (13)] The numerical value r_sphere ≈ 3.32×10^10 m is not consistent with Eq. (12). For r_S = 3 km and ℓ_H ≈ 1.32×10^26 m, the sphere with volume 4πℓ_H r_S^2 has radius (3ℓ_H r_S^2)^{1/3} ≈ 1.5×10^11 m, not 3.32×10^10 m. Please recheck the expression and the numerical evaluation.
[Section 2.2] The notation is confusing where the paper writes 'Let {M} denote the set of points representing a space M'; the symbol M is used for both the set and the space it represents, making statements such as |{M}| ambiguous.
[Eq. (9)] The approximation A(δR_U)/V(R_U) ≈ H0/c corresponds to r_U ≈ 3c/H0, i.e., about three Hubble radii, but the paper does not explain why the observable universe radius should be three Hubble radii rather than the Hubble radius itself or the standard particle horizon radius.
[Section 3.1] The statement that λ→0 'eliminating quantum fluctuations' is an unsupported physical assertion; the Poisson model only describes the probability of bit counts, and the requirement that a region contain at least one bit deserves a more detailed justification than the brief remark given.
[Section 3.2, Discussion] The claim that the approach is 'model-independent' is overstated: the faithful-embedding picture, the identification of points with bits, the Poisson sprinkling, and the λ=1 normalization are substantive model assumptions rather than consequences of general principles alone.

Circularity Check

2 steps flagged · score 8.0 of 10

Planck-scale remnant follows from stipulated λ=1 and saturation of the entropy bound, not from a derivation.

fitted input called prediction [Section 3.1, Eqs. (14)-(15)]
"Thus, it is reasonable to propose that the volume Vmin(RR) must be such that the expected number of bits of information it contains is equal to 1. This leads to the definition of Vmin(RR) as: Vmin (RR) = (max ρV )−1≈ 4ℓHℓ2P"

Vmin is presented as a found minimum volume, but its value is fixed by setting the Poisson parameter λ=1 in Eq. (14). This is a stipulated normalization, not a consequence of finite geometry: any λ would give Vmin=λ/maxρV and would change the inferred remnant radius by √λ. The nonzero Planck-scale minimum volume is therefore the arbitrarily chosen one-bit condition restated as a result, i.e. a fitted input renamed as a prediction.
self definitional [Section 3.1, after Eq. (15), comparing with Eq. (12)]
"Comparing Vmin(RR) with Eq. (12) suggests that a Schwarzschild black hole with the smallest possible interior volume would have a Schwarzschild radius on the order of the Planck length. Specifically, rS = ℓP√π≈ 0.9× 10−35 meters."

Eq. (12) is only an upper bound maxV≈4πℓHrS^2 and Eq. (15) is only a lower bound Vmin=4ℓHℓP^2. The claimed radius rS=ℓP/√π is obtained by equating these two bounds, i.e. by assuming the actual black-hole volume saturates both the entropy bound and the one-bit minimum simultaneously. That saturation is not derived; without it the consistency condition is Vmin≤V_actual≤maxV, and a different model of V_actual (e.g. Euclidean 4/3πrS^3) gives an entirely different cutoff. Thus the central Planck-remnant result is the saturation assumption expressed as an algebraic identity, not a prediction.

full rationale

The central claim of the paper—that a Schwarzschild black hole stops evaporating at rS=ℓP/√π—reduces by construction to two stipulated inputs. First, Vmin is defined as (maxρV)^{-1} by choosing λ=1 ('reasonable to propose'), so the nonzero minimum volume is a restatement of that normalization. Second, the Planck radius is obtained by equating the upper bound Eq. (12) with the lower bound Eq. (15); this saturation of the entropy bound is assumed, not derived. Neither step involves a load-bearing self-citation: the only self-citation ([9] in Section 2.1) supports the general motivation of finite degrees of freedom and is accompanied by an external reference and the entropy bound, so it is not load-bearing. Because the conclusion follows from definitions and arbitrary equalities rather than from independent physics, the circularity score is high; however, the paper's cosmological input maxρV≈(1/4ℓP^2)(H0/c) is derived from standard flat-universe geometry, so the derivation is not wholly empty. Score 8 reflects that the headline remnant prediction is forced by the definitions and assumptions, without a full equivalence of the entire framework to its conclusion.

Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The model rests on several strong assumptions: identifying points with bits, uniform faithful embedding, the holographic bound, and the critical λ=1 normalization. The free parameters λ and H0 directly determine the predicted remnant scale, so the ledger shows that the central conclusion is largely built in.

free parameters (2)

lambda (expected number of bits in minimum volume) = 1
Chosen by hand in Section 3.1 so that the minimum volume contains exactly one bit; this directly sets the remnant radius r_S = sqrt(lambda) * l_P / sqrt(pi).
Hubble constant H0 = ~67.4 km/s/Mpc (Planck 2015)
Used in Eq. (9)-(10) to set the maximum volumetric information density via A/V ~ H0/c; an external empirical input that controls the magnitude of V_min.

assumptions (7)

domain assumption Entropy equals the number of points times the Boltzmann constant (Eq. 2)
The paper identifies each point in finite geometry with one bit of information, making entropy proportional to point count. This is a modeling choice, not derived.
domain assumption Faithful embedding: point count is proportional to volume and area (Section 2.2)
The paper assumes points can be embedded uniformly into a Riemannian manifold, leading to H(V) = k_B rho_V V and H(A) = k_B rho_A A.
standard math Bekenstein-Hawking entropy equals the information content of the event horizon (Eq. 5)
Standard result used to fix rho_A = 1/(4 l_P^2).
standard math Holographic entropy bound holds (Eq. 1 and Eq. 7)
The paper relies on Bousso's holographic principle to constrain rho_V V <= rho_A A.
domain assumption The universe is a flat 3D Euclidean ball with A/V ~ H0/c (Eq. 9)
The paper identifies the observable universe's radius as r_U ~ 3c/H0 and uses the Euclidean sphere ratio A/V = 3/r_U to fix max rho_V.
domain assumption The maximum volumetric density rho_V is location-independent and applies inside black holes (Section 2.3)
The paper argues that faithful embedding forces rho_V to be uniform, and assumes this uniform density holds in the black hole interior, despite the highly curved geometry.
domain assumption A black hole interior has a well-defined Euclidean volume V(R_R) and surface area A(delta R_R)
Implicit in applying Eq. (7) and the area formula A = 4 pi r_S^2 to a Schwarzschild black hole, without addressing coordinate-dependence of interior volume.

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Cite this review

Pith. "Pith review of Finite geometry and black hole stability: Embedding discrete space into classical manifolds." pith.science (2026). https://pith.science/paper/ZKDGAECX

@misc{pith2026250511585,
  author       = {Pith},
  title        = {Pith review of: Finite geometry and black hole stability: Embedding discrete space into classical manifolds},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZKDGAECX}},
  note         = {Machine review of arXiv:2505.11585}
}

read the original abstract

The issue of defining the volume of black holes has significant implications for quantum gravity. Drawing on concepts from quantum theory and general relativity, several motivations for introducing discreteness in geometry can be proposed. However, to seriously consider any proposal for a discrete geometry, the identification problem and the challenge of defining a distance function within such a geometry must be addressed. This paper proposes the faithful embedding of sets representing spaces in finite geometry -- a specific type of discrete geometry characterized by a finite set of points -- into Riemannian manifolds as a solution to these problems. Similar to a classical measuring apparatus that interprets and understands quantum results in classical terms, classical geometry serves as a bridge between the discreteness of the physical world and our continuous understanding of the properties of space. In this framework, the volumetric density of information contained within a black hole is established, providing a consistent volume for the Schwarzschild black hole observed by all observers. Furthermore, the study finds that the minimum volume of the Schwarzschild black hole is non-zero. This fact implies that a black hole can only evaporate until its event horizon radius reaches the Planck length, signifying that black hole remnants are stable. Consequently, the total collapse of a black hole is prevented by the finite nature of the geometry describing physical space.

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Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

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