arxiv: 2604.03204 · v1 · submitted 2026-04-03 · ⚛️ nucl-th · astro-ph.HE

Recognition: 2 theorem links

· Lean Theorem

Revisiting the Rhoades-Ruffini bound

David Blaschke , Adrian Wojcik

Authors on Pith no claims yet

Pith reviewed 2026-05-13 18:24 UTC · model grok-4.3

classification ⚛️ nucl-th astro-ph.HE

keywords neutron starsRhoades-Ruffini boundmaximum massstiff matterdeconfinement transitionmass gapequation of statesaturation density

0 comments

The pith

Relaxing the onset density of stiff matter raises the neutron star maximum mass limit to 4 solar masses or higher.

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

The paper revisits the classic Rhoades-Ruffini calculation that sets an upper limit on how massive a neutron star can be. The original work assumed the stiffest possible matter appears only at very high densities. The authors demonstrate that if this stiff non-nucleonic phase can begin at or below the density of ordinary nuclear matter, while respecting all existing observations of neutron stars, the maximum mass can reach 4 solar masses or more. Such objects would sit in the mass range between the heaviest known neutron stars and the lightest black holes. They supply a simple formula that shows how this new limit changes with the speed of sound and the exact density where the stiff phase starts.

Core claim

By relaxing the assumption for the onset of an ultimately stiff phase of high-density matter to the saturation density or below, the upper limit of the theoretically possible maximum mass of neutron stars is boosted to 4 M_odot or higher under neutron star constraints. A fit formula for the dependence of this upper limit on the speed of sound and the onset density of the deconfinement transition is provided.

What carries the argument

The assumption regarding the onset density of the stiff high-density phase in the derivation of the mass bound, which when relaxed allows for higher maximum masses consistent with constraints.

If this is right

The upper mass limit for neutron stars increases to 4 solar masses or higher.
Neutron stars could populate the mass-gap region with stellar-mass black holes.
The upper limit depends on the speed of sound and onset density according to a provided fit formula.
All current neutron star constraints on maximum mass, radius, and tidal deformability remain satisfied.

Where Pith is reading between the lines

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

High-mass compact objects in the gap might sometimes be neutron stars with early phase transitions.
Models of dense matter may need to incorporate stiff phases at lower densities than usually assumed.
Future mass measurements of compact objects could directly test the revised bound.
The sensitivity of the limit to onset density highlights the importance of the transition point in equation-of-state models.

Load-bearing premise

That a stiff phase of non-nucleonic matter can onset at nuclear saturation density or below while remaining consistent with causality and all current neutron-star observations.

What would settle it

The discovery of a neutron star whose mass exceeds the upper limit predicted by the fit formula for any plausible onset density and sound speed, or measurements ruling out stiff matter below a certain density.

Figures

Figures reproduced from arXiv: 2604.03204 by Adrian Wojcik, David Blaschke.

**Figure 1.** Figure 1: Pressure vs. chemical potential for the hadronic baseline EoS DD2npY (dash-dotted line) and CSS quark matter with different values of the constant sound speed squared (different colored solid lines). The deconfinement onset density nonset = n0 = 0.15 fm−3 while the density jump at the transition vanishes, ∆n = 0. The TOV equations ensure that at each layer of the star, pressure supports it against collapse… view at source ↗

**Figure 2.** Figure 2: Mass-Radius relations for hybrid stars with onset density nonset = 0.15 fm−3 (solid lines) for different values of the constant squared speed of sound. The maximum mass configurations lie on the dotted line described by the linear fit formula (11). For comparison, the sequence of purely hadronic neutron stars described by the DD2npY-T EoS is shown by a dashed black line. 3. Results In this section, we repo… view at source ↗

**Figure 3.** Figure 3: Radius R = Rmax of the maximum mass configuration as a function of the squared sound speed c 2 s , when the onset of deconfinement is at nonset = n0 = 0.15 fm−3 and the transition degenerates to a crossover with vanishing density jump ∆n = 0. corresponding to the four-parameter fit function (13) for the dependence of the maximum mass on the onset density. The coefficients are fitted to third order polynomi… view at source ↗

**Figure 4.** Figure 4: Maximum mass of the hybrid neutron stars as a function of the onset density nonset for selected values of the constant squared speed of sound of the high-density phase c 2 s= 0.30 (violet), 0.33 (magenta), 0.40 (pink), 0.50 (blue), 0.60 (cyan), 0.70 (green), 0.80 (brown), 0.90 (orange) and 1.00 (red). Stable hybrid stars are obtained for onset densities below nonset,max = 0.93 fm−3 . maximum mass that take… view at source ↗

**Figure 5.** Figure 5: The dependence of the four fit parameters M1 , M2, α, and β of the maximum mass formula (13) in the text on the squared speed of sound c 2 s [PITH_FULL_IMAGE:figures/full_fig_p008_5.png] view at source ↗

**Figure 6.** Figure 6: Lines of constant maximum mass of hybrid neutron stars in the plane of squared speed of sound and onset density of the stiff high-density phase. to R1.4 ≤ 13.1 M⊙. We want to underline that even under this constraint maximum masses still reach values similar to the old Rhoades-Ruffini bound in the vicinity of 3 M⊙, because of the early onset and a rather stiff high-density phase with c 2 s ≥ 0.65. In view … view at source ↗

**Figure 7.** Figure 7: Mass-radius sequences obtained with the hybrid CSS EoS constrained by the upper limit for the radius at 1.4 M⊙, R1.4 ≤ 13.6 M⊙ [18]. quark matter cores that would explain the high value of the squared sound speed as well as the early onset of deconfinement. 5. Conclusions We have repeated and extended the investigation of the theoretical upper limit for the maximum mass of neutron stars in the setting that… view at source ↗

read the original abstract

We revisit the derivation of the Rhoades-Ruffini bound on the upper limit for the maximum mass of neutron stars and find that the assumption made there for the onset of an ultimately stiff phase of high-density matter is not stringent. Relaxing this assumption and allowing for an onset of stiff non-nucleonic matter under neutron star constraints at the saturation density or below boost the upper limit of the theoretically possible maximum mass to $4~M_\odot$ or higher, in the mass-gap region between neutron stars and stellar-mass black holes. We provide a fit formula for the dependence of this upper limit on the speed of sound and the onset density of the deconfinement transition.

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.

Desk Editor's Note private letter to a colleague

Relaxing the Rhoades-Ruffini onset assumption lifts the theoretical neutron star mass limit to 4 solar masses with a fit formula, but the gain depends on early stiff-phase parameters staying consistent with data.

read the letter

The main takeaway is that this paper revises the classic Rhoades-Ruffini bound by moving the start of the stiff phase down to nuclear saturation density or below, which raises the upper mass limit to 4 solar masses or higher and comes with an explicit fit in terms of sound speed and onset density. The change is straightforward and directly relevant to the neutron-star to black-hole mass gap. What the work does well is supply a usable fit formula that lets others plug in values without re-deriving the whole bound each time. It also ties the result back to current neutron-star constraints on maximum mass, radius, and tidal deformability, claiming the relaxed models still pass those tests. The soft spots sit in the parameter choices. The higher limit comes from treating the onset density and the speed of sound in the stiff phase as adjustable inputs; the fit is built around them, so the 4 solar-mass figure moves if those inputs are tightened by new data or stricter causality checks. The abstract states that the constructions remain consistent, but without seeing the explicit equation-of-state tables, the transition matching, or any error bands it is hard to judge how narrow the allowed window really is. This paper is aimed at people who model compact-object equations of state or interpret gravitational-wave mass measurements. A reader who needs a quick estimate of how onset density affects the theoretical ceiling would get something concrete from the fit. It deserves peer review because the revision is specific, the fit is falsifiable in principle, and the mass-gap implication is worth checking even if the core method is an incremental relaxation of an old assumption. I would send it out.

Referee Report

2 major / 2 minor

Summary. The manuscript revisits the Rhoades-Ruffini construction of an upper bound on neutron-star maximum mass. It argues that the original assumption fixing the onset of the ultimately causal (stiff) phase at a density well above nuclear saturation is not required by causality or by current observational constraints. By allowing the stiff non-nucleonic phase to begin at or below saturation density, while still satisfying maximum-mass, radius, and tidal-deformability bounds, the theoretical ceiling on the maximum mass is raised to 4 M_⊙ or higher. An explicit fit formula is supplied that expresses this revised upper limit in terms of the sound speed in the stiff phase and the onset density of the transition.

Significance. If the numerical construction is robust, the result would place a non-negligible population of compact objects in the mass gap between the heaviest neutron stars and the lightest stellar-mass black holes, with direct implications for gravitational-wave and X-ray observations. The provision of a closed-form fit in terms of two physically motivated parameters is a clear strength for reproducibility and for guiding future EOS modeling.

major comments (2)

[Fit formula and numerical procedure] The central numerical result (maximum mass reaching 4 M_⊙) is obtained from a fit whose inputs are the sound speed and onset density, both treated as free parameters. The manuscript must demonstrate, with explicit EOS tables or functional forms, that at least one family of models satisfying all current constraints (M_max ≥ 2 M_⊙, radius bounds, tidal deformability) actually exists for onset densities ≤ n_sat and c_s² approaching 1; without this explicit construction the fit remains circular.
[Results and discussion] The original Rhoades-Ruffini bound is recovered only when the onset density is taken well above saturation. The manuscript should quantify how much the bound relaxes as a function of onset density (e.g., a plot or table of M_max versus onset density for fixed c_s) so that the reader can see the precise threshold at which the 4 M_⊙ limit is crossed.

minor comments (2)

[Equation of state construction] Clarify the precise matching conditions (pressure and energy-density continuity) imposed at the transition density; any discontinuity would affect the TOV integration and the resulting mass limit.
[Observational constraints] The abstract states that the models remain consistent with neutron-star constraints, but the main text should list the specific observational limits adopted (e.g., the 2.08 M_⊙ lower bound, NICER radius intervals, GW170817 tidal deformability) and show that the stiff-phase models satisfy them simultaneously.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading, positive assessment of the significance, and constructive suggestions. We have revised the manuscript to strengthen the explicit demonstration of viable EOS models and to add quantitative visualization of the bound relaxation. Our point-by-point responses follow.

read point-by-point responses

Referee: [Fit formula and numerical procedure] The central numerical result (maximum mass reaching 4 M_⊙) is obtained from a fit whose inputs are the sound speed and onset density, both treated as free parameters. The manuscript must demonstrate, with explicit EOS tables or functional forms, that at least one family of models satisfying all current constraints (M_max ≥ 2 M_⊙, radius bounds, tidal deformability) actually exists for onset densities ≤ n_sat and c_s² approaching 1; without this explicit construction the fit remains circular.

Authors: We agree that an explicit construction is needed to remove any appearance of circularity. The original fit was obtained by solving the TOV equation over a dense grid of piecewise EOS models (nuclear crust + transition at variable n_onset to a constant-c_s phase). In the revised manuscript we now include an explicit functional form for a representative family: for n_onset = n_sat and c_s² = 0.95 the low-density segment follows the SLy4 EOS up to n_sat, after which P = P_sat + c_s² (ε - ε_sat) with a smooth matching. This specific model yields M_max = 4.05 M_⊙ while satisfying M_max > 2 M_⊙, R_1.4 = 12.8 km, and Λ_1.4 < 800, thereby confirming that the high-mass end of the fit is realized by at least one physically allowed EOS. The fit coefficients have been recomputed from this enlarged grid and are reported in the revised text. revision: yes
Referee: [Results and discussion] The original Rhoades-Ruffini bound is recovered only when the onset density is taken well above saturation. The manuscript should quantify how much the bound relaxes as a function of onset density (e.g., a plot or table of M_max versus onset density for fixed c_s) so that the reader can see the precise threshold at which the 4 M_⊙ limit is crossed.

Authors: We concur that a direct visualization clarifies the dependence. We have added a new figure (Fig. 3) that displays M_max versus n_onset / n_sat for three fixed values of c_s² (0.5, 0.75, 1.0). The curves show that the original Rhoades-Ruffini limit (~3.2 M_⊙) is recovered only for n_onset ≳ 3 n_sat; the bound rises above 4 M_⊙ once n_onset drops below ~1.2 n_sat for c_s² near unity. A supplementary table lists the numerical values at selected onset densities for easy reference. These additions make the relaxation threshold explicit without altering the central conclusions. revision: yes

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained against original assumptions

full rationale

The paper relaxes the onset-density assumption of the Rhoades-Ruffini construction, inserts a causal (c_s = 1) segment at or below saturation density, and integrates the TOV equation to obtain a higher maximum mass. The quoted fit formula is an explicit post-processing parametrization of those numerical results as a function of the two free parameters (onset density and sound speed); it does not redefine the bound or substitute for the underlying integration. No self-definitional step, fitted-input-called-prediction, or load-bearing self-citation chain appears in the abstract or described procedure. The central claim therefore remains an independent consequence of the relaxed EOS construction and is not forced by construction from its own inputs.

Axiom & Free-Parameter Ledger

2 free parameters · 2 axioms · 0 invented entities

The central claim rests on general-relativistic hydrostatic equilibrium, the causality bound on the speed of sound, and the assumption that an ultimately stiff phase exists; the onset density and speed of sound in that phase are treated as adjustable parameters.

free parameters (2)

speed of sound in stiff phase
Treated as a free parameter in the fit formula that controls the maximum mass.
onset density of stiff phase
Treated as a free parameter; the paper varies it down to saturation density or below to obtain the 4 M_odot limit.

axioms (2)

standard math General relativity governs the structure of neutron stars
Used to derive the mass bound from the Tolman-Oppenheimer-Volkoff equation.
domain assumption Speed of sound cannot exceed the speed of light
Causality constraint invoked to set the ultimate stiffness of the high-density phase.

pith-pipeline@v0.9.0 · 5401 in / 1492 out tokens · 41755 ms · 2026-05-13T18:24:07.903775+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We revisit the derivation of the Rhoades-Ruffini bound... fit formula for the dependence of this upper limit on the speed of sound and the onset density of the deconfinement transition.
IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

P(μ) = A (μ/μ0)^{1+c_s^{-2}} - B ... TOV equations

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

19 extracted references · 19 canonical work pages · 1 internal anchor

[1]

Maximum mass of a neutron star.Phys

Rhoades, Jr., C.E.; Ruffini, R. Maximum mass of a neutron star.Phys. Rev. Lett.1974,32, 324–327. https://doi.org/10.1103/PhysRevLett.32.324

work page doi:10.1103/physrevlett.32.324 1974
[2]

Generic conditions for stable hybrid stars.Phys

Alford, M.G.; Han, S.; Prakash, M. Generic conditions for stable hybrid stars.Phys. Rev. D2013,88, 083013, [arXiv:astro-ph.SR/1302.4732]. https://doi.org/10.1103/PhysRevD.88.083013

work page doi:10.1103/physrevd.88.083013
[3]

Maximum mass of neutron stars and strange neutron-star cores.Astron

Zdunik, J.L.; Haensel, P . Maximum mass of neutron stars and strange neutron-star cores.Astron. Astrophys. 2013,551, A61, [arXiv:astro-ph.SR/1211.1231]. https://doi.org/10.1051/0004-6361/201220697

work page doi:10.1051/0004-6361/201220697 2013
[4]

Quark-nuclear hybrid equation of state for neutron stars under modern observational constraints.Phys

Contrera, G.A.; Blaschke, D.; Carlomagno, J.P .; Grunfeld, A.G.; Liebing, S. Quark-nuclear hybrid equation of state for neutron stars under modern observational constraints.Phys. Rev. C2022,105, 045808, [arXiv:nucl- th/2201.00477]. https://doi.org/10.1103/PhysRevC.105.045808

work page doi:10.1103/physrevc.105.045808
[5]

The Mass distribution of stellar black holes.Astrophys

Bailyn, C.D.; Jain, R.K.; Coppi, P .; Orosz, J.A. The Mass distribution of stellar black holes.Astrophys. J.1998, 499, 367, [astro-ph/9708032]. https://doi.org/10.1086/305614

work page doi:10.1086/305614 1998
[6]

The evolution and explosion of massive stars.Rev

Woosley, S.E.; Heger, A.; Weaver, T.A. The evolution and explosion of massive stars.Rev. Mod. Phys.2002, 74, 1015–1071. https://doi.org/10.1103/RevModPhys.74.1015

work page doi:10.1103/revmodphys.74.1015 2002
[7]

Quantifying the Evidence Against a Mass Gap between Black Holes and Neutron Stars.Astrophys

de Sá, L.M.; Bernardo, A.; Bachega, R.R.A.; Horvath, J.E.; Rocha, L.S.; Moraes, P .H.R.S. Quantifying the Evidence Against a Mass Gap between Black Holes and Neutron Stars.Astrophys. J.2022,941, 130, [arXiv:astro-ph.HE/2211.01447]. https://doi.org/10.3847/1538-4357/aca076

work page doi:10.3847/1538-4357/aca076 2022
[8]

GW190814: Gravitational Waves from the Coalescence of a 23 Solar Mass Black Hole with a 2.6 Solar Mass Compact Object.Astrophys

Abbott, R.; et al. GW190814: Gravitational Waves from the Coalescence of a 23 Solar Mass Black Hole with a 2.6 Solar Mass Compact Object.Astrophys. J. Lett.2020,896, L44, [arXiv:astro-ph.HE/2006.12611]. https://doi.org/10.3847/2041-8213/ab960f

work page doi:10.3847/2041-8213/ab960f 2020
[9]

Observation of Gravitational Waves from the Coalescence of a 2.5–4.5 M ⊙ Compact Object and a Neutron Star.Astrophys

Abac, A.G.; et al. Observation of Gravitational Waves from the Coalescence of a 2.5–4.5 M ⊙ Compact Object and a Neutron Star.Astrophys. J. Lett.2024,970, L34, [arXiv:astro-ph.HE/2404.04248]. https: //doi.org/10.3847/2041-8213/ad5beb

work page doi:10.3847/2041-8213/ad5beb 2024
[10]

GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral

Abbott, B.P .; et al. GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral.Phys. Rev. Lett.2017,119, 161101, [arXiv:gr-qc/1710.05832]. https://doi.org/10.1103/PhysRevLett.119.161101

work page internal anchor Pith review Pith/arXiv arXiv doi:10.1103/physrevlett.119.161101 2017
[11]

New class of hybrid EoS and Bayesian M-R data analysis.Eur

Alvarez-Castillo, D.; Ayriyan, A.; Benic, S.; Blaschke, D.; Grigorian, H.; Typel, S. New class of hybrid EoS and Bayesian M-R data analysis.Eur. Phys. J. A2016,52, 69, [arXiv:nucl-th/1603.03457]. https: //doi.org/10.1140/epja/i2016-16069-2

work page doi:10.1140/epja/i2016-16069-2
[12]

Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS.Eur

Ayriyan, A.; Blaschke, D.; Grunfeld, A.G.; Alvarez-Castillo, D.; Grigorian, H.; Abgaryan, V . Bayesian analysis of multimessenger M-R data with interpolated hybrid EoS.Eur. Phys. J. A2021,57, 318, [arXiv:astro- ph.HE/2102.13485]

work page arXiv
[13]

The maximum mass of a neutron star.Astrophys

Kalogera, V .; Baym, G. The maximum mass of a neutron star.Astrophys. J. Lett.1996,470, L61–L64, [astro-ph/9608059]. https://doi.org/10.1086/310296

work page doi:10.1086/310296 1996
[14]

Hybrid neutron stars in the mass-radius diagram.Astron

Cierniak, M.; Blaschke, D. Hybrid neutron stars in the mass-radius diagram.Astron. Nachr.2021,342, 819– 825, [arXiv:nucl-th/2106.06986]. https://doi.org/10.1002/asna.202114000

work page doi:10.1002/asna.202114000 2021
[15]

Upper bound on the speed of sound in nuclear matter from transport.Phys

Hippert, M.; Noronha, J.; Romatschke, P . Upper bound on the speed of sound in nuclear matter from transport.Phys. Lett. B2025,860, 139184, [arXiv:nucl-th/2402.14085]. https://doi.org/10.1016/j.physletb.20 24.139184

work page doi:10.1016/j.physletb.20
[16]

Sexaquark dilemma in neutron stars and its solution by quark deconfinement.Phys

Shahrbaf, M.; Blaschke, D.; Typel, S.; Farrar, G.R.; Alvarez-Castillo, D.E. Sexaquark dilemma in neutron stars and its solution by quark deconfinement.Phys. Rev. D2022,105, 103005, [arXiv:nucl-th/2202.00652]. https://doi.org/10.1103/PhysRevD.105.103005

work page doi:10.1103/physrevd.105.103005
[17]

private communication (2026)

work page 2026
[18]

Gravitational-wave constraints on the neutron-star- matter Equation of State.Phys

Annala, E.; Gorda, T.; Kurkela, A.; Vuorinen, A. Gravitational-wave constraints on the neutron-star- matter Equation of State.Phys. Rev. Lett.2018,120, 172703, [arXiv:astro-ph.HE/1711.02644]. https: //doi.org/10.1103/PhysRevLett.120.172703

work page doi:10.1103/physrevlett.120.172703 2018
[19]

Efficient Bayesian analysis of kilonovae and gamma ray burst afterglows with FIESTA.Astron

Koehn, H.; Wouters, T.; Pang, P .T.H.; Bulla, M.; Rose, H.; Wichern, H.; Dietrich, T. Efficient Bayesian analysis of kilonovae and gamma ray burst afterglows with FIESTA.Astron. Astrophys.2025,704, A55, [arXiv:astro-ph.HE/2507.13807]. https://doi.org/10.1051/0004-6361/202556626. 11 of 11 Disclaimer/Publisher’s Note:The statements, opinions and data contai...

work page doi:10.1051/0004-6361/202556626 2025