{"id":"4edff940-002a-408c-80e6-2a8748999b53","arxiv_id":"2507.07425","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"Static spherical stellar models with vanishing complexity are constructed numerically in Rastall gravity, and all three satisfy standard viability criteria, with one model stable only for nonzero Rastall parameter.","lead":"This paper constructs three models of compact stars in Rastall gravity by imposing a vanishing-complexity condition on the interior fluid. It claims the models are physically viable and that one model is more stable in Rastall theory than in general relativity, offering a test of this modified gravity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Eq. (36) is inconsistent with Eq. (35): in the GR limit it gives Π = ρ − (2/r³)∫ρ, while Eq. (35) gives Π = ρ/2 − (3/(2r³))∫ρ; since all three models are built on Eq. (36), the vanishing-complexity claim is unsupported.","rationale":"The reader's weakest assumption was the Visser equivalence, an interpretive issue about whether Rastall is physically distinct from GR. The concern identified here is stronger and internal: the defining vanishing-complexity equation is algebraically inconsistent with the paper's own expression for Y_TF. This is not a peripheral detail: Eq. (36) is the methodological novelty advertised in the title and abstract, and it is used to derive the reduced equations for every model (e.g., Eq. (41) for model I, Eq. (44) for models II and III, and Eq. (49) in the dimensionless polytrope). If Eq. (36) is wrong, none of the reported configurations is verified to have zero complexity, and the comparison with GR, including the claimed stabilization of the polytropic model for α=0.1 and 0.2, is not established. The proposed check is a direct symbolic substitution plus a re-derivation of the Section 5 constraints; it settles the matter without relying on interpretation of the theory's physical status. Given that the central claim currently rests on an invalid equation, the verdict should move from CONDITIONAL to REJECT pending correction and re-analysis.","tokens_in":20205,"tokens_out":24008,"duration_ms":262663,"concrete_test":"Set α=0 in Eq. (35), impose Y_TF=0, and solve for Π analytically; compare the result with Eq. (36) at α=0 and with Herrera's formula Π = (1/(2r³))∫_0^r x³ ρ′(x) dx. If the two differ as the text indicates, replace Eq. (36) with the corrected condition, re-derive the reduced equations of Section 5, and re-run the numerical integrations to see whether the reported stability conclusions, especially Figure 8 for model II, survive. As an additional check, compare the α=0 solutions against the published GR vanishing-complexity solutions [73]; a mismatch would confirm that the displayed equation, not merely the typesetting, entered the numerics.","verdict_should_be":"REJECT","load_bearing_attack":"Equation (36), described as the vanishing-complexity condition that generates all three models in Section 5, does not follow from Equation (35). In the GR limit α=0, Equation (35) with Y_TF=0 yields Π = ρ/2 − (3/(2r³))∫_0^r ρ r̄² d r̄, which is Herrera's standard condition Π = (1/(2r³))∫_0^r r̄³ ρ′(r̄) d r̄ after integration by parts. Equation (36) at α=0 instead gives Π = ρ − (2/r³)∫_0^r ρ r̄² d r̄. The two expressions differ in both the local term (ρ/2 vs ρ) and the integral coefficient. Under the paper's own assumption of restoring GR, these should coincide. Because Eqs. (40), (41), (43), (44), and (51) are all stated consequences of Eq. (36), the numerical solutions in Figures 1–13 are not shown to satisfy Y_TF=0. The central claim that Rastall 'provides more suitable results in the case of model 2' therefore rests on an unvalidated constraint. The Visser-equivalence issue raised by the reader is real but secondary; the immediate problem is an algebraic inconsistency in the core construction.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper extends Herrera's complexity-factor formalism to Rastall gravity for static, spherically symmetric, anisotropic stellar interiors. It derives the field equations and mass function, performs an orthogonal splitting of the Riemann tensor to obtain structure scalars, and identifies Y_TF as the complexity factor. Three stellar models are constructed by imposing Y_TF = 0 together with, respectively, a vanishing radial pressure, a polytropic equation of state, and a non-local equation of state. The models are solved numerically for several values of the Rastall parameter α, and their physical viability is assessed through energy conditions, redshift bounds, Buchdahl limits, and cracking stability. The paper concludes that the three models are consistent with GR and that Rastall theory yields more suitable results than GR for the polytropic model, 'indicating its superiority over Einstein's gravity theory'.","tokens_in":20524,"tokens_out":5272,"duration_ms":53287,"significance":"If the construction were correct, the paper would provide a useful extension of Herrera's complexity method to a non-conservative gravity theory and would offer explicit anisotropic stellar models that satisfy a vanishing-complexity condition. The authors deserve credit for deriving the Rastall field equations, defining the mass function, and checking a battery of viability criteria for three distinct models. However, the central derivation is undermined by an algebraic inconsistency between the expression for Y_TF and the condition (36) used to build all three models. The paper does not include machine-checked proofs, reproducible code, or a derivation of the structure scalars, and the numerical methods are described only vaguely. As written, the results do not support the claim that the models satisfy Y_TF = 0 or the accompanying conclusion of Rastall superiority.","major_comments":[{"comment":"The condition called 'vanishing complexity' in Eq. (36) is not equivalent to setting Y_TF = 0 in Eq. (35). In the GR limit α = 0, Eq. (35) yields Π = ρ/2 − (3/(2r³))∫₀ʳ ρ r̄² d r̄, which is Herrera's standard condition after integration by parts, whereas Eq. (36) reduces to Π = ρ − (2/r³)∫₀ʳ ρ r̄² d r̄. These differ in both the local term and the integral coefficient. Since all three model-building equations in Section 5, including Eqs. (40), (41), (43), (44), and (51), are stated as consequences of Eq. (36), the numerical solutions are not shown to satisfy vanishing complexity. The paper's central claim about Rastall's superiority in model 2 therefore rests on an unvalidated constraint.","section":"Section 3, Eqs. (35)-(36)"},{"comment":"The four structure scalars are introduced with the comment 'simple but detailed calculations (which are not presented here)'. Because the identification of Y_TF as the complexity factor is foundational to the entire paper, the derivation should either be included in full or a direct reference to the Rastall-specific computation should be provided.","section":"Section 3, Eqs. (28)-(31)"},{"comment":"The numerical Buchdahl limits are presented without stating the method used to obtain them. The text only says that explicit analytical expressions cannot be derived because the equations are highly nonlinear; it does not specify the differential equations solved, the boundary conditions, or the numerical scheme. Without this information, the values 0.889, 0.772, 0.714, 0.652, and 0.636 cannot be reproduced or checked.","section":"Section 4, Table 1"},{"comment":"The 'superiority over GR' conclusion is based on the instability of the α = 0 case versus stability for α = 0.1 and α = 0.2 in model 2. Because the models are constructed from the incorrect Eq. (36), this conclusion is unsupported. Moreover, the dismissal of Visser's equivalence argument is an assertion rather than a quantitative rebuttal; if Rastall solutions are simply GR solutions with a redefined energy-momentum tensor, then the stability difference reflects a relabeled effective fluid rather than a new physical theory. The paper should either demonstrate the physical distinctness of the Rastall fluid explicitly or soften the superiority claim.","section":"Sections 2, 5.2, and 6"}],"minor_comments":[{"comment":"The notation 'η3 being a polytropic exponent' is unclear; the polytropic exponent should be defined consistently with the subsequent dimensionless variables in Eq. (45).","section":"Section 5.2, Eq. (42)"},{"comment":"The numerical integration is described as relying on 'carefully chosen initial conditions' without giving the actual values or solver details; please provide the initial conditions and numerical method used for each model.","section":"Section 5 (general)"},{"comment":"The figure captions contain placeholder symbols such as '/ScriptR' and '/ScriptE'; these should be replaced with standard mathematical notation (e.g., r, e^δ1, e^−δ2).","section":"Figures 1-13"},{"comment":"The abstract calls the vanishing complexity condition 'well-known', but Eq. (36) differs from the standard Herrera condition in the GR limit; this wording is misleading and should be revised.","section":"Abstract and Section 1"},{"comment":"There are numerous typographical and language errors, such as 'heavily systems' in Section 1, 'deriv[e]' in Section 5.2, and 'orthogonally to' in Section 6; a careful proofread is needed.","section":"Throughout"}],"recommendation":"major_revision","confidential_remarks":"The algebraic inconsistency between Eqs. (35) and (36) is a serious flaw that undermines all three models. The authors should be asked to correct the condition, redo the numerical solutions, and verify which stability claims survive. The paper would also benefit from including the derivation of the structure scalars, documenting the numerical methods, and engaging quantitatively with the Visser equivalence rather than dismissing it on the grounds of an analogy to f(R,T) gravity."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: the stress-test note is right, and it is the main story. Equation (36) does not follow from Equation (35). In the GR limit α=0, setting Y_TF=0 in (35) gives Π = ρ/2 – (3/(2r³))∫ρ r² dr, while (36) gives Π = ρ – (2/r³)∫ρ r² dr. These are different conditions. Since all three models are built on (36), the paper never actually imposes vanishing complexity.\n\nWhat is worth something: the Rastall field equations for a static anisotropic sphere, the mass function, the TOV equation, and the standard physical criteria are all laid out cleanly. The extension of Herrera's complexity factor to Rastall is natural, and three explicit stellar models are constructed and checked against the usual requirements (energy conditions, redshift, Buchdahl bound, cracking). Once the constraint equation is corrected, the models could be a useful worked example.\n\nThe soft spots are real. The structure scalars in Section 3 are stated without derivation—'simple but detailed calculations (which are not presented here)'. The numerical procedure is under-specified: initial conditions are 'carefully chosen', no code or data are given, and Table 1's Buchdahl values appear without a method. The dismissal of Visser's equivalence is a sentence, not an argument; Eq. (6) shows the reduction and the text gives no quantitative rebuttal. The 'superiority over GR' claim is based on one polytropic model with α=0.1,0.2 compared to α=0 in a single cracking plot—that is a parameter-scan anecdote, not a demonstration of theoretical superiority.\n\nWho is this for? Someone wanting the Rastall analogue of the standard complexity-free stellar construction. But as it stands the central derivation is wrong, so the paper cannot be used. It still deserves a serious referee: the error is isolated and fixable in principle, and the subfield routinely checks such constructions by hand. My recommendation is to send it to review, with a referee report that demands a corrected Eq. (36), a full derivation of the structure scalars, numerical reproducibility details, and a much more cautious conclusion.","headline":"The vanishing-complexity condition is algebraically wrong: Eq. (36) contradicts Eq. (35), so the models are not complexity-free and the Rastall-superiority claim is unsupported.","tokens_in":21047,"tokens_out":5961,"would_cite":false,"duration_ms":58089,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.20.-q","04.40.Dg","97.10.-q"],"model":"deepseek-v4-flash","headline":"The paper claims that anisotropic stellar models built from the vanishing complexity condition are physically viable in Rastall gravity, and that a polytropic model unstable in general relativity becomes stable for nonzero values of the…","keywords":["Rastall gravity","Vanishing complexity","Complexity factor","Anisotropic stellar models","Structure scalars","Polytropic equation of state","Energy conditions","Stability"],"falsifier":"Take the numerically generated solutions for the polytropic model and map the Rastall energy-momentum tensor to the effective Einstein tensor; if the resulting anisotropic fluid in general relativity satisfies the same energy and cracking conditions, then the stabilization is reproduced inside general relativity, undermining the claim that Rastall theory is distinct.","tokens_in":19987,"feed_emoji":"⭐","tokens_out":6300,"duration_ms":61754,"temperature":0.7,"pith_summary":"This paper tries to show that the complexity-factor approach, originally developed for general relativity, can be transplanted into Rastall's non-conservative gravity to build physically realistic anisotropic stellar models. It constructs three families of static spherical solutions by imposing the vanishing complexity condition $\\mathcal{Y}_{TF}=0$ together with one of three extra constraints: zero radial pressure, a polytropic equation of state, or a non-local equation of state. It then reports that all three models satisfy the standard viability checks for the chosen values of the Rastall parameter $\\alpha$, including energy conditions, redshift bound, Buchdahl limit, and cracking stability. The paper's central comparative claim is that the polytropic model, which suffers cracking at $\\alpha=0$ (general relativity), becomes stable for $\\alpha=0.1$ and $\\alpha=0.2$, which it reads as Rastall theory being superior to Einstein's theory.","feed_headline":"Vanishing complexity builds stable stars in Rastall gravity","feed_subtitle":"Models pass energy, redshift, and Buchdahl checks; the Rastall parameter stabilizes a polytropic star that cracks in GR.","key_machinery":"The central object is the structure scalar $\\mathcal{Y}_{TF}$, the complexity factor obtained from the orthogonal splitting of the Riemann tensor into Weyl and Ricci parts. It carries the whole construction: setting $\\mathcal{Y}_{TF}=0$ produces the vanishing-complexity condition that links the metric potentials with the fluid variables, and combining it with one of three constraints reduces the five-unknown system to solvable fourth-order differential equations in the metric potentials. The Rastall parameter $\\alpha$ enters through the effective energy-momentum tensor and controls how far the solutions depart from general relativity.","core_discovery":"In Rastall gravity, where the energy-momentum tensor has a non-zero divergence proportional to the Ricci scalar gradient, the complexity factor $\\mathcal{Y}_{TF}$ obtained from the orthogonal decomposition of the Riemann tensor still encodes the combined effect of density inhomogeneity and pressure anisotropy. Setting $\\mathcal{Y}_{TF}=0$ yields a non-local condition (Eq. 36) that, together with each of the three constraints, closes the under-determined field equations. The paper claims the resulting numerical solutions are non-singular, have energy density and pressures peaked at the center, meet all energy conditions, keep gravitational redshift below the observational bound, respect the Rastall-adjusted Buchdahl limit, and pass the cracking criterion except in the cases it identifies as unstable. Its headline result is that the polytropic model is unstable in general relativity but stable for nonzero Rastall parameter, which it presents as evidence that Rastall corrections improve stellar stability.","pith_inferences":["If Rastall gravity is equivalent to general relativity through an effective stress-energy redefinition, then the claimed 'superiority' of the polytropic model may describe a particular anisotropic fluid rather than a genuinely different theory; the paper offers no quantitative rebuttal to that equivalence.","The numerical scheme fixes the polytropic index and constant, so stability across a broader polytropic parameter space remains an open question that could be tested by repeating the calculation for other values.","Matching these interior solutions to observed neutron-star masses and radii would test whether the Rastall-stabilized models are observationally favored over their general-relativistic counterparts."],"forward_implications":["The complexity-factor program, developed in general relativity, remains workable in a non-conservative gravity theory: the same scalar $\\mathcal{Y}_{TF}$ organizes the stellar equations.","Polytropic anisotropic stars that develop cracking in general relativity can be stabilized by switching on the Rastall parameter, at least for $\\alpha=0.1$ and $alpha=0.2$ within the paper's numerical setup.","All three Rastall stellar models reproduce general-relativistic behavior at $\\alpha=0$ and remain consistent with the standard physical viability tests for the $\\alpha$ values considered.","The Buchdahl compactness limit is numerically smaller for larger $\\alpha$, meaning the Rastall corrections lower the maximum allowed mass-radius ratio for these solutions.","The non-local equation-of-state model is stable only for $\\alpha=0$ and $0.1$, while the zero-radial-pressure model is stable for all tested $\\alpha$, giving a concrete stability ordering across the three constructions."],"supporting_citations":[{"why":"Supplies the Rastall field equations and the non-conservation law on which the whole paper is built.","marker":"[10]"},{"why":"Defines the complexity factor whose vanishing is imposed in all three models.","marker":"[59]"},{"why":"Provides the orthogonal decomposition of the Riemann tensor used to derive the structure scalars.","marker":"[60, 61]"},{"why":"Presents the earlier general-relativistic anisotropic vanishing-complexity models that serve as the baseline for comparison.","marker":"[73]"},{"why":"Offers the equivalence argument between Rastall gravity and general relativity that the paper must dismiss to claim genuine novelty.","marker":"[74]"},{"why":"Introduces the non-local equation of state used as one of the three closing constraints.","marker":"[87]"}],"fun_headline_variants":["Rastall gravity stabilizes stars that crack in GR","Vanishing complexity yields stable stars in Rastall theory","Zero complexity factor stabilizes Rastall stellar models","Rastall parameter rescues polytropic stars from instability","Stellar stability improved by non-conserved Rastall gravity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that Rastall gravity is a genuinely distinct theory from general relativity, with the non-conserved energy-momentum tensor as the true matter content; if the theory is only a relabeling of general relativity through an effective stress tensor, the claimed superiority of the Rastall model reduces to a choice of which anisotropic fluid is being described.","fun_headline_variants_meta":{"raw":{"variants":["Rastall gravity stabilizes stars that crack in GR","Vanishing complexity yields stable stars in Rastall theory","Zero complexity factor stabilizes Rastall stellar models","Rastall parameter rescues polytropic stars from instability","Stellar stability improved by non-conserved Rastall gravity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000237,"raw_usage":{"total_tokens":1519,"prompt_tokens":966,"completion_tokens":553,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":582,"completion_tokens_details":{"reasoning_tokens":470}},"tokens_in":582,"tokens_out":553,"duration_ms":5930,"temperature":1.0,"reasoning_tokens":470,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T18:42:16.873988+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Take the numerically generated solutions for the polytropic model and map the Rastall energy-momentum tensor to the effective Einstein tensor; if the resulting anisotropic fluid in general relativity satisfies the same energy and cracking conditions, then the stabilization is reproduced inside general relativity, undermining the claim that Rastall theory is distinct.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Defines the complexity factor whose vanishing is imposed in all three models."},{"cited_title":"and Ramos, A.: Ann","cited_arxiv_id":null,"evidence_quote":"Presents the earlier general-relativistic anisotropic vanishing-complexity models that serve as the baseline for comparison."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Offers the equivalence argument between Rastall gravity and general relativity that the paper must dismiss to claim genuine novelty."},{"cited_title":"and Nunez, L.A.: Can","cited_arxiv_id":null,"evidence_quote":"Introduces the non-local equation of state used as one of the three closing constraints."}],"review_version":1}