{"id":"d300d845-455b-4615-9d11-bdf9c3052a85","arxiv_id":"2505.03061","paper_version":2,"verdict":"REJECT","confidence":"MODERATE","novelty_score":3.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"Modified Renyi entropy produces a modified force with a MOND-like interpolating function, but the force decays as 1/R^4, not MOND's 1/R, and the Bekenstein bound check reduces to a definitional inequality.","lead":"This paper derives a modified gravitational force from a deformed black hole entropy and checks two information theory bounds. The authors claim a MOND-like force law, but the equations show a 1/R^4 decay that does not match MOND's 1/R behavior.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's own Eq. (15) gives a 1/R^4 force at large R, and Section 3 admits this 'does not reproduce MOND either,' so the advertised MOND-like force law is not a MOND force law.","rationale":"The most damaging problem is not the speculative Eq. (3) but the contradiction between the abstract and the paper's own Section 3. Even granting the modified Renyi entropy and the entropic-force formula (13), the derived force (15) has a 1/R^4 tail at large R, whereas MOND's deep-MOND phenomenology requires a 1/R acceleration. The interpolating function (19) satisfying mu(x)->1 and mu(x)->x is necessary but not sufficient; the actual dynamics predicted by Eq. (16) does not satisfy the MOND equation mu(a/a0)*a = a_N. The authors explicitly write that the model 'does not reproduce MOND either.' This alone invalidates the abstract's central claim. I therefore agree with the REJECT verdict, though via a different route than the reader's weakest-assumption choice of Eq. (3). The Bekenstein-bound 'verification' in Eq. (26) is additionally just the elementary inequality e^x >= 1+x for x = lambda*S_R, so it is tautological and carries no independent physical content; this further weakens the paper's set of advertised results but is secondary to the MOND contradiction. A revised version could salvage a modest claim about a deformed Newtonian force with 1/R^4 asymptotics, but not the current abstract's MOND claim.","tokens_in":10406,"tokens_out":7023,"duration_ms":69714,"concrete_test":"Re-derive the force from Eqs. (13)-(15) and compare with MOND's defining relation. With the simple interpolating function mu(x)=x/(1+x), set F = m*a*mu(a/a0) and solve for the actual acceleration a in the deep-MOND limit a_N << a0: the result is a ~ sqrt(a0*a_N) ~ 1/R. Then compare with Eq. (16), which gives a = a_N/(1+a0/a_N) ~ a_N^2/a0 ~ 1/R^4 for large R. The two expressions do not match; this analytic check confirms that the derived force is not a MOND force law despite sharing the formal form of an interpolating function.","verdict_should_be":"REJECT","load_bearing_attack":"The central claim fails on the paper's own algebra. Inserting the modified Renyi entropy into Eq. (13) gives Eq. (15): F_eff = (GMm/R^2)/(1 + lambda*pi*R^2/l_p^2), which at large R behaves as GMm*l_p^2/(lambda*pi*R^4). MOND's deep-MOND scaling is a ~ sqrt(a0*a_N) ~ 1/R. The paper itself acknowledges this in Section 3: 'our model ... leads to a 1/R^4 decay at large distances ... and thus does not reproduce MOND either.' The interpolating function mu(x)=x/(1+x) in Eq. (19) satisfies the formal limits (11)-(12), but that is not sufficient: in MOND the force is F = m*a*mu(a/a0) with the actual acceleration a solving mu(a/a0)*a = a_N. Eq. (16) instead gives a = a_N^2/(a_N+a0), whose small-a_N limit is a_N^2/a0 ~ 1/R^4, not MOND's sqrt(a0*a_N) ~ 1/R. This contradiction is independent of whether the Tsallis-Renyi conjecture (3) is justified, so it is the more basic defect. A corrected abstract would need to rescope the claim to 'a 1/R^4 modified Newtonian force with a MOND-like interpolating function in formal limits' or provide a mechanism restoring 1/R behavior.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a 'modified Rényi entropy' S_R = (k_B/λ) ln(1 + λ S_BH/k_B), obtained from the conjecture that the original Tsallis entropy equals the Bekenstein–Hawking entropy (Eq. 3). Using the entropic-force formula (Eq. 13), the authors derive an effective force law (Eq. 15) and claim that its interpolating function (Eq. 19) makes the model MOND-like. They further claim to verify the Bekenstein bound within this framework (Eq. 26, Fig. 1) and use the Landauer principle to obtain a black-hole mass-loss formula (Eq. 36). The central advertised result is that a MOND-like force law emerges naturally from entropic considerations, together with a verification of the Bekenstein bound and a Landauer-based mass-loss prediction.","tokens_in":10777,"tokens_out":4366,"duration_ms":43950,"significance":"If correct, the paper would connect nonextensive thermodynamics to MOND phenomenology and to information-theoretic bounds for black holes. The algebraic manipulations leading to Eqs. (9), (15), and (16) are internally consistent, and the paper is commendably transparent in Section 3 when it acknowledges that its model does not reproduce MOND. However, the main claims are not supported: the MOND-like force law is contradicted by the model's own large-distance scaling, and the Bekenstein-bound 'verification' is a mathematical identity following from the definition of the modified Rényi entropy. The Landauer result is a straightforward application of ΔS = (dS/dM)ΔM with ΔS = k_B ln 2, rather than a new physical principle. The paper's significance therefore reduces to a formal exercise rather than a substantive contribution.","major_comments":[{"comment":"The advertised MOND-like force law is contradicted by the paper's own algebra. Equation (15) gives F_eff = (GMm/R^2)/(1 + λπR^2/l_p^2), which decays as 1/R^4 for large R, yielding an acceleration a ≈ a_N^2/a_0 ~ 1/R^4 in the deep-MOND regime. MOND requires a ~ sqrt(a_0 a_N) ~ 1/R. The paper itself states in Section 3 that the model 'does not reproduce MOND either.' Moreover, Eq. (19) defines μ(a_N/a_0), while the MOND force law requires μ(a/a_0) with the actual acceleration a satisfying μ(a/a_0) a = a_N. Thus Eq. (19) is not a MOND interpolating function in the dynamical sense. This contradiction undermines the central claim of the abstract and cannot be resolved by rewording alone.","section":"Abstract and Section 3, Eqs. (15)–(19)"},{"comment":"The claimed verification of the Bekenstein bound is an identity, not a physical test. From Eq. (5) in natural units, S_R = (1/λ) ln(1 + λ S_BH), so S_BH = (e^{λ S_R} - 1)/λ. Inequality (26), S_R ≤ (e^{λ S_R} - 1)/λ, is exactly the elementary inequality ln(1+x) ≤ x (equivalently e^u ≥ 1+u), and Fig. 1 merely plots (e^{λ S_R} - 1)/(λ S_R) ≥ 1. Consequently, the Bekenstein bound is satisfied by construction for all λ > 0 and carries no independent physical content.","section":"Section 4, Eqs. (26)–(27) and Fig. 1"},{"comment":"The load-bearing premise is the conjecture that the Tsallis entropy in Eq. (2) exactly equals the Bekenstein–Hawking entropy, as stated in Eq. (3). No independent evidence or physical motivation is provided for this equality, and all subsequent results—the force law, the Bekenstein-bound check, and the Landauer mass loss—follow from it. Since the MOND-like claim fails on its own algebra and the Bekenstein check is tautological, the remaining content reduces to algebraic consequences of this unproven conjecture.","section":"Section 2, Eq. (3)"}],"minor_comments":[{"comment":"The text says the Tsallis entropy 'recovers the Boltzmann-Gibbs entropy when q approaches 0'; the correct limit is q → 1, as the following paragraph correctly states.","section":"Section 2, after Eq. (1)"},{"comment":"Equation (1) is poorly typeset in the manuscript; the denominator should be presented unambiguously as (q − 1) with the numerator k_B(1 − Σ p_i^q), matching the standard Tsallis definition.","section":"Section 2, Eq. (1)"},{"comment":"The sentence following Eq. (26) is grammatically incomplete; it should be completed or merged with the definition of the ratio R_R in Eq. (27).","section":"Section 4, Eq. (26)"},{"comment":"The phrase 'the use of the modified Rényi entropy could be explored in the context of BH area quantization' is vague; the future-work paragraph would benefit from a concrete proposal, such as which area spectrum is expected.","section":"Section 6"}],"recommendation":"reject","confidential_remarks":"The paper's central MOND claim is explicitly contradicted in its own Section 3, and the Bekenstein-bound section verifies an identity. These are load-bearing defects that cannot be fixed by local revision. The Landauer calculation is a trivial manipulation of the entropy definition. I recommend rejection. The authors might consider resubmitting a substantially rewritten manuscript that drops the MOND claim and focuses honestly on the formal properties of their modified entropy, but in its current form the contribution does not meet the standards of the journal."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: this is a clean and transparent substitution exercise, but the headline result is not what it seems. The authors take the modified Renyi entropy of Ref. [13], plug it into their own entropic-force template from Ref. [42], and find F = (GMm/R^2)/(1 + lambda pi R^2/l_p^2). For large R that is 1/R^4, not MOND's 1/R. The paper says this explicitly in Section 3: the model 'does not reproduce MOND either.' So the abstract's claim that a MOND-like force law emerges naturally is contradicted by the body. The interpolating function x/(1+x) satisfies the formal limits (11)-(12), but the actual force law does not have MOND's deep-MOND scaling. That is a load-bearing inconsistency.\n\nWhat is good: the algebra is correct, the derivations are easy to follow, and the authors are honest about the antecedents—the modified entropy is not theirs, the entropic-force formula is from their earlier work, and they cite the Kaniadakis paper [18] that already did the same program. The Landauer part is a routine derivative but correctly done. The Bekenstein bound check, however, is a tautology: since SR = (1/lambda) ln(1 + lambda S_BH) and ln(1+x) < x, SR < S_BH automatically, so verifying SR <= pi R^2 is just plotting e^x >= 1+x. Nothing new there.\n\nNovelty is low: this is a new entropy inserted into an old template, and the MOND connection is formally similar to Ref. [18] but actually comes out worse (1/R^4 vs the Kaniadakis interpolating function). The paper would need a major revision to rescope the claims and delete the vacuous bound check. As it stands, I would not accept it for peer review; the abstract misrepresents the result and the remaining content is minor. If the authors want to salvage it, they should reframe as a modified Newtonian force with a MOND-like interpolation function in a formal sense, and remove the claim that it reproduces MOND.","headline":"The abstract promises MOND, the body delivers a 1/R^4 force law and says so; the Bekenstein check is a tautology.","tokens_in":11317,"tokens_out":4806,"would_cite":false,"duration_ms":47097,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":false},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.Dy","89.70.Cf"],"model":"deepseek-v4-flash","headline":"This paper claims that substituting a modified Rényi entropy for Bekenstein–Hawking entropy in the entropic-force law yields a MOND-like gravitational force, a Bekenstein bound that is always respected, and a Landauer-principle mass loss…","keywords":["Tsallis entropy","Rényi entropy","MOND","Bekenstein bound","Landauer principle","black hole thermodynamics","nonextensive statistics","entropic gravity"],"falsifier":"For a Schwarzschild black hole of mass $M$, compute $W=(1+\\lambda S_{BH}/k_B)^{1/\\lambda}$ with $S_{BH}=4\\pi k_B G M^2/(\\hbar c)$; the conjecture (3) requires $W$ to be an integer microstate count for any $\\lambda\\neq0$, and a non-integer result would falsify it.","tokens_in":10186,"feed_emoji":"🕳️","tokens_out":6730,"duration_ms":61292,"temperature":0.7,"pith_summary":"This paper tries to show that replacing the Bekenstein–Hawking entropy with a modified Rényi entropy produces several black-hole results from one statistical starting point. The derivation rests on the conjecture that the original Tsallis entropy equals the Bekenstein–Hawking entropy, which fixes the microstate count and defines the modified Rényi entropy. Within that framework, the entropic-force formula gives an effective force whose interpolating function is exactly the simple MOND function, the Bekenstein bound holds for typical deformation parameters, and the Landauer principle yields a new mass-loss expression. A sympathetic reader should care because the paper offers a single nonextensive-statistics origin for these separate results, while stating openly that the force law it derives decays as $1/R^4$ and therefore does not actually reproduce MOND's characteristic $1/R$ galactic behavior.","feed_headline":"Black-hole entropy tweak gives a MOND-like force law","feed_subtitle":"A deformed Rényi entropy produces the simple MOND interpolating function, yet the force decays as 1/R^4, not 1/R.","key_machinery":"The carrying object is the modified Rényi entropy $S_R=(k_B/\\lambda)\\ln(1+\\lambda S_{BH}/k_B)$, an entropic deformation with parameter $\\lambda=1-q$, together with the entropic-force derivative $dS/dA$. The entropy provides the input, and the derivative $dS_R/dA=(dS_{BH}/dA)/(1+\\lambda S_{BH}/k_B)$ converts area dependence into a modified gravitational force; the same entropy feeds the Bekenstein-bound inequality and the Landauer mass-loss calculation. In the limit $\\lambda\\to 0$, the modified Rényi entropy reduces to $S_{BH}$, and the force law, temperature, Bekenstein bound, and Landauer mass loss all reduce to their standard black-hole forms.","core_discovery":"The central claim is that the modified Rényi entropy $S_R=(k_B/\\lambda)\\ln(1+\\lambda S_{BH}/k_B)$, obtained by conjecturing Tsallis entropy equals the Bekenstein–Hawking entropy, plugged into the entropic-force formula $F=(GMm/R^2)(4l_p^2/k_B)(dS/dA)$, yields the effective force $F_{\\rm eff}=GMm/(R^2(1+\\lambda\\pi R^2/l_p^2))$, which can be written as $ma_N/(1+a_0/a_N)$ with interpolating function $\\mu(a_N/a_0)=a_N/(a_0+a_N)$. The same entropy satisfies the Bekenstein bound in the form $S_R\\le(e^{\\lambda S_R}-1)/\\lambda$, and the Landauer principle gives a black-hole mass loss $\\Delta M=(\\ln 2/8\\pi)(1+4\\pi\\lambda M^2)/M$. The paper also emphasizes that the derived force decays as $1/R^4$ at large $R$, so the model is MOND-like only through the interpolating function and does not reproduce MOND's deep-infrared $1/R$ acceleration.","pith_inferences":["If the $1/R^4$ large-distance decay is taken literally, the model would predict galaxy rotation curves that fall off rather than flatten; existing low-acceleration rotation-curve data could test this directly, and the paper's own admission indicates the test would fail.","The microstate-count conjecture $W=(1+\\lambda S_{BH}/k_B)^{1/\\lambda}$ could be checked for consistency: for a given physical black hole, $W$ must be an integer for every allowed $\\lambda$, a constraint the paper does not examine.","The modified Rényi temperature has a minimum at finite mass, suggesting a phase transition and a heat-capacity sign change; this stability feature could be probed in analogue-gravity or condensed-matter realizations of nonextensive entropy, though the paper does not propose such tests."],"forward_implications":["For $\\lambda\\to0$, the modified Rényi entropy reduces to $S_{BH}$, and the force law, temperature, Bekenstein bound, and Landauer mass loss all reduce to their standard black-hole forms.","The interpolating function $\\mu(x)=x/(1+x)$ is exactly the simple MOND interpolating function, even though the underlying force does not share MOND's $1/R$ large-distance decay.","The Bekenstein bound holds under modified Rényi entropy for typical $\\lambda\\ge0$, with the ratio $R_R=(e^{\\lambda S_R}-1)/(\\lambda S_R)\\ge1$.","Landauer's principle applied to this entropy predicts a black-hole mass loss that, for $\\lambda>0$, has a minimum and then grows with mass, differing qualitatively from the standard $1/M$ Hawking behavior.","The framework connects nonextensive statistics to gravitational dynamics and information theory, giving a common origin for a MOND-like interpolation, an entropy bound, and an information-erasure cost."],"supporting_citations":[{"why":"Supplies Hawking radiation and the area theorem that motivate black-hole thermodynamics, the starting setting of the paper.","marker":"[1]"},{"why":"States the Landauer principle, the information-erasure energy cost used in Section 5.","marker":"[12]"},{"why":"Introduced the modified Rényi entropy for black-hole horizons, the object the paper adopts and extends.","marker":"[13]"},{"why":"Also introduced the modified Rényi entropy dual to Tsallis entropy, supporting the framework used here.","marker":"[15]"},{"why":"States the Bekenstein bound conjecture that Section 4 verifies for the modified entropy.","marker":"[17]"},{"why":"Defines the original Tsallis entropy, the starting point of the conjecture in Eq. (3).","marker":"[22]"},{"why":"Provides the entropic-force formula Eq. (13) that converts entropy variation into a gravitational force.","marker":"[42]"},{"why":"Also supplies the entropic-corrections formulation behind the force formula used in Eq. (13).","marker":"[43]"}],"fun_headline_variants":["Modified entropy yields MOND-like force but not 1/R falloff","Renyi entropy tweak gives MOND-like force, but decays as 1/R^4","Black hole entropy variant mimics MOND, yet force falls as 1/R^4","Tsallis-based entropy yields MOND-like law, but force decays 1/R^4","MOND-like force from modified entropy, but 1/R^4 at large R"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the original Tsallis entropy, $k_B(W^{1-q}-1)/(1-q)$, equals the Bekenstein–Hawking entropy $S_{BH}$ for a black hole; if that equality fails, the microstate count, the modified Rényi entropy, and all three derived results lose their foundation.","fun_headline_variants_meta":{"raw":{"variants":["Modified entropy yields MOND-like force but not 1/R falloff","Renyi entropy tweak gives MOND-like force, but decays as 1/R^4","Black hole entropy variant mimics MOND, yet force falls as 1/R^4","Tsallis-based entropy yields MOND-like law, but force decays 1/R^4","MOND-like force from modified entropy, but 1/R^4 at large R"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000774,"raw_usage":{"total_tokens":3443,"prompt_tokens":980,"completion_tokens":2463,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":596,"completion_tokens_details":{"reasoning_tokens":2351}},"tokens_in":596,"tokens_out":2463,"duration_ms":16680,"temperature":1.0,"reasoning_tokens":2351,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T00:00:23.833828+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a Schwarzschild black hole of mass $M$, compute $W=(1+\\lambda S_{BH}/k_B)^{1/\\lambda}$ with $S_{BH}=4\\pi k_B G M^2/(\\hbar c)$; the conjecture (3) requires $W$ to be an integer microstate count for any $\\lambda\\neq0$, and a non-integer result would falsify it.","supporting_citations":[{"cited_title":"Bekenstein [2] was among the first to identify thermodynamic properties in BHs, noting that their surface area behaves analogously to entropy","cited_arxiv_id":null,"evidence_quote":"Supplies Hawking radiation and the area theorem that motivate black-hole thermodynamics, the starting setting of the paper."},{"cited_title":"Milgrom, Astrophys","cited_arxiv_id":null,"evidence_quote":"States the Bekenstein bound conjecture that Section 4 verifies for the modified entropy."},{"cited_title":"Tsallis and H","cited_arxiv_id":null,"evidence_quote":"Provides the entropic-force formula Eq. (13) that converts entropy variation into a gravitational force."}],"review_version":1}