{"id":"a14efc0c-5715-4670-84ec-ee1034653b50","arxiv_id":"2512.11583","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"A fractional-kernel Sáez–Ballester action in n dimensions yields exact FLRW solutions whose effective equation of state can reproduce all cosmic epochs for hand-chosen values of α and C.","lead":"This paper builds a cosmological model in n dimensions by multiplying a scalar-tensor action by a time-dependent fractional kernel, then derives exact solutions for the scale factor and matter variables. It claims one family of solutions can reproduce inflation, radiation, matter and late-time acceleration without a cosmological constant or an ad hoc potential.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Internal contradiction on diffeomorphism invariance: §VI first says time-reparametrization symmetry is broken, then says the action remains diffeomorphism invariant; the Noether/Bianchi consistency claims in §V D therefore do not follow.","rationale":"The reader's weakest assumption identified the same fragile foundation: the kernel ξ(t) is treated as an external background while diffeomorphism invariance is simultaneously invoked. I elevate this to a direct textual contradiction. It is load-bearing because the strongest claim includes consistency with Bianchi identities, Noether's theorems, and first-order perturbation dynamics; if the symmetry is broken, those derivations are not consequences of the action's symmetries. The concern is about internal consistency, not disagreement with consensus: Section VI explicitly asserts both symmetry breaking and invariance. The suggested variation test will settle whether the action is invariant under the model's own assumptions. If it fails, the perturbation equations and Noether identities must be either derived from the broken-symmetry action or removed from the strongest claim. This does not necessarily invalidate the background exact solutions, but it means the paper's full consistency claim is not currently supported. I therefore keep the reader's CONDITIONAL verdict rather than moving to accept or reject.","tokens_in":23879,"tokens_out":30401,"duration_ms":272422,"concrete_test":"Vary action (1) under an infinitesimal time reparametrization t→t′=t+ε(t), treating ξ as a fixed background (δξ=0), as assumed in Section V. Compute δS to first order in ε for ξ=(tbar−t)^{α−1}. If δS≠0 — it will contain terms proportional to ˙ξ — then the action is not diffeomorphism invariant and Noether's second theorem cannot be applied; if δS=0, the contradiction disappears. A secondary check: recompute the exchange law (83) directly from the Euler-Lagrange equations of action (1) with ξ fixed, and verify whether the identity holds off-shell or only after imposing the fractional Klein-Gordon equation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section VI contains a direct contradiction. It states that 'due to the presence of the time-dependent kernel in the action, the resulting field equations contain time-dependent terms that clearly break this symmetry' and that 'the freedom to reparametrize time is lost.' Later, it asserts 'the action remains diffeomorphism invariant, and therefore the second Noether theorem can still be applied.' Both cannot hold for the same fixed ξ(t). The perturbation calculation in Section V assumes δξ=0, i.e., ξ is an external background. Under an infinitesimal time reparametrization t→t+ε(t), a fixed ξ(t) would change if it were a scalar; imposing δξ=0 instead makes the action non-invariant whenever ˙ξ≠0. Hence the Noether identities (82)-(87), the exchange law (83), and the claim that the effective EMT is covariantly conserved do not follow from diffeomorphism invariance. The exact background solutions may still satisfy the equations of motion, but the central claim of consistency with Bianchi identities and Noether's theorems — which is part of the paper's strongest claim — is unsupported. Moreover, if time reparametrization symmetry is genuinely broken, the first-order perturbation equations in Newtonian gauge and the Mukhanov-Sasaki variable z in Eqs. (78)-(81) require a gauge-invariance justification that the paper does not provide.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper proposes a fractional generalization of Sáez–Ballester scalar-tensor cosmology in n dimensions by inserting a Riemann–Liouville-type time kernel into the action. It derives the background field equations (Eqs. (14)–(16)), a generalized continuity equation (Eq. (21)), and a first-order ODE for H (Eq. (22)), whose exact solution is given in Eqs. (23)–(30), together with the associated a(t), ρφ(t), pφ(t), V(t), and kinetic-term expressions. The paper then constructs two dynamical-systems formulations, presents qualitative plots of the effective equation-of-state and deceleration parameters, and derives first-order scalar perturbation equations, including a fractional Mukhanov–Sasaki equation. It claims that the time-dependent kernel preserves diffeomorphism invariance and that the Bianchi identities and Noether’s second theorem remain applicable, so that the effective energy–momentum tensor is conserved.","tokens_in":24287,"tokens_out":11327,"duration_ms":114383,"significance":"If the central claims hold, the paper provides a rare example of a fractional cosmological model with explicit exact background solutions and a unified, parameter-controlled description of early inflation, radiation, matter domination, and late acceleration without an ad hoc scalar potential. The explicit formulas (23)–(30), the transparent α=1 reduction, and the first-order perturbation framework are useful and nontrivial contributions. However, the significance is currently limited by unresolved issues: the claimed independence of the field equations is not proven; the treatment of time reparametrization symmetry is internally contradictory, which undermines the Bianchi/Noether consistency claims; and the advertised comparison with observational data is absent. These are not cosmetic deficiencies but affect the paper’s central claims of consistency and observational viability.","major_comments":[{"comment":"The treatment of time reparametrization invariance is self-contradictory. §VI states that due to the time-dependent kernel, the field equations contain time-dependent terms that clearly break time-reparametrization symmetry, and that 'the freedom to reparametrize time is lost'; item (ii) of the same section then states that 'the action remains diffeomorphism invariant, and therefore the second Noether theorem can still be applied.' For a fixed, non-dynamical ξ(t) both cannot hold. Under t→t+ε(t) with δξ=0, the action changes by a term proportional to ˙ξ ε; if ξ is instead allowed to transform as a scalar, then the background ξ(t) is not fixed and an equation of motion for ξ is missing. Consequently, the Noether identities (82)–(87), the exchange law (83), and the conservation law (85) do not follow from diffeomorphism invariance. Because time reparametrizations are broken, the Newtonian","section":"§VI and §V D"},{"comment":"The central route to exact solutions rests on the assertion in §III A that the fractional system has 'three independent equations for three unknowns.' This is stated without proof. No constraint or Frobenius analysis is provided, and the assertion is in tension with §V D, where the effective EMT is claimed to be conserved as a consequence of a Bianchi identity; if such an identity holds, one of the Friedmann equations should be redundant once the scalar equation is satisfied. A concrete test would be to compute the rank of the system (14)–(16) and to verify that the exact solution (23)–(30) does not simply use one equation as a definition of V or φ. Without this, the 'no-ad-hoc-potential' exact solution may be a specially selected trajectory rather than the general solution, and the claimed elimination of supplementary assumptions is not established.","section":"§III A; Eqs. (14)–(22)"},{"comment":"The abstract and §VI state that the model's predictions are compared with observational data and that the solutions can be consistent with observations. The body contains only qualitative plots for hand-picked parameter values: Fig. 1 uses α=0.35, 0.45, 0.55 with C=-475; Fig. 2 uses α=1.35, 1.45, 1.65 and C=5, 15, 25; Fig. 3 varies n. No observational dataset, likelihood, parameter constraint, or quantitative comparison appears anywhere in §IV C or elsewhere. The phase sequence (early inflation, radiation, matter, late acceleration) is therefore a demonstration that such epochs can be mimicked for chosen parameters, not a comparison with data. Either add a real data comparison (e.g., H(z) or distance moduli, with a scan over α and C) or remove the observational claims from the abstract and conclusions.","section":"Abstract; §IV C and Figs. 1–3"},{"comment":"The exact solution is not a complete scalar-field solution. Equations (29) and (30) give V(t) and the combination ωφ^r ˙φ² as functions of t, but φ(t) itself is never integrated. For general r and ω, the paper does not show that φ(t) can be obtained in closed form, and Section IV A admits that the functional dependence V(φ) is 'possibly unattainable.' This makes the comparison with standard potentials (Sec. IV A) and the reduction to ordinary single-field inflation (Sec. V C) formal rather than explicit. The special case r=0, ω=1/2 should be worked out; for general r, the status of the solution as an exact scalar-field configuration should be stated precisely.","section":"§III B, Eqs. (29)–(30); §IV A"}],"minor_comments":[{"comment":"The substitution t'− ¯t ≡ t makes t≤0 on the integration domain as written, which is inconsistent with the later use of t>0 and with fractional powers t^A. Please define t unambiguously (likely t = ¯t−t').","section":"§II, Eq. (12)"},{"comment":"The caption reads 'black curves' as 'back curves'; typographical correction needed.","section":"Fig. 3 caption"},{"comment":"The phrase 'second Noether theorem []' contains an empty bracket; the reference should be supplied.","section":"§I"},{"comment":"Equation numbering skips (26), leaving a blank line. Renumber or remove.","section":"Eqs. (25)–(26)"},{"comment":"The constant C in h(t)=(t^A−C)/(t^A+C) carries dimensions of t^A. The text never explains how C is made dimensionless in the figures; a clear scaling convention is needed.","section":"Eqs. (23)–(24)"},{"comment":"The statement that the model is 'expected to be ghost-free' is given without a proof from the quadratic action; either provide the positivity conditions or label it as heuristic.","section":"§VI"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits the scope of the journal and contains useful explicit background solutions and a perturbation framework. The main obstacles are the unresolved contradiction in the treatment of diffeomorphism invariance, the unproven independence count, and the overstatement about observational data. These are fixable in a revision, so I do not recommend rejection at this stage."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here is the short version: this paper does contain a genuine exact-solution exercise in fractional Sáez–Ballester cosmology, and the algebra mostly holds up, but its headline claims—three independent equations, natural emergence of all epochs, Noether-consistent perturbations—do not survive contact with the paper's own equations. The §VI contradiction is real: first they say time reparametrization is broken, then they say the action remains diffeomorphism invariant so the second Noether theorem applies. Both cannot be true for a fixed ξ(t). The Noether/Bianchi conservation claims and the gauge-invariance of the Mukhanov–Sasaki variable need a clean symmetry-breaking framework or a dynamical treatment of ξ; the paper gives neither.\n\nWhat is actually new is the specific combination: a kernel-weighted SB action in n dimensions with a potential, the exact FLRW solutions (23)–(30) for H, a, ρ, p, V, and a first-order perturbation system with a modified Mukhanov–Sasaki variable. The derivation of (22) from the Friedmann equations is transparent, the explicit H(t) looks like a real solution (I did not check every algebra step, but the α=1 limit correctly reduces to the standard model), and the reconstructed potential is non-trivial. That is a solid piece of exact-solution work.\n\nThe problems are in the interpretation. The “three independent equations” claim in §III A is not supported: equation (21) is a direct consequence of (14) and (15) once ρφ and pφ are expressed in terms of H and its derivatives; it is not an independent closure. This means the exact solution is effectively a potential reconstruction for a chosen H(t), a standard technique in scalar-field cosmology, not a no-ad-hoc-potential miracle. The “all epochs” claim in §IV C is supported only by hand-picked α and C (e.g., C=-475 in Figure 1) and qualitative plots; the abstract promises observational comparison, but the body gives none. That is a significant gap between the abstract and the actual content.\n\nThe load-bearing flaw is the Noether/Bianchi analysis in §V D. The exchange law (83) and effective-EMT conservation (82) are asserted, but if time reparametrization is genuinely broken, the effective EMT is not covariantly conserved in general; the background conservation (85) is an identity of the Friedmann equations, not a Noether theorem. The linearized conservation equations and the Mukhanov–Sasaki equation need a proper treatment of the broken gauge symmetry. This is not a minor typo.\n\nWho is this for? Readers working on fractional cosmology or exact solutions in modified gravity will find the exact solution useful and the perturbation formalism a starting point. It deserves a serious referee, but the authors need to fix the independence count, do a real data comparison, and resolve the diffeo contradiction before the stronger claims can be trusted. I would send it to referees with the clear expectation of major revision.","headline":"Genuine exact solutions in fractional Sáez–Ballester cosmology, but the three-equation independence claim and the Noether/Bianchi consistency claims overreach; the §VI diffeomorphism contradiction is real.","tokens_in":24711,"tokens_out":13100,"would_cite":false,"duration_ms":110680,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A single time-dependent fractional kernel inserted into a generalized scalar-tensor action yields exact cosmological solutions that pass through inflation, radiation, matter, and late acceleration without a cosmological constant or an ad ho","keywords":["Fractional Cosmology","Sáez–Ballester theory","FLRW cosmology","Extended theories of gravity","Dynamical system","Nonlocality and memory effect","Noether theorem","Exact solutions"],"falsifier":"Compute the Jacobian determinant (or numerical rank) of the three fractional equations — (14), (15), and either (16) or (21) — treated as equations for H, ρφ, pφ at generic α, n, t; if the rank drops below 3, the central independence claim fails. A second check: search the admissible parameter space (α, n, C) for a point where V(t) from Eq. (29) vanishes or where ρ_eff is negative during the purported radiation/matter phases; either would contradict the paper's claim that all epochs emerge without ad hoc input.","tokens_in":23797,"feed_emoji":"🌌","tokens_out":7910,"duration_ms":79033,"temperature":0.7,"pith_summary":"The paper tries to show that replacing the time measure in a generalized Sáez–Ballester scalar-tensor action with a Riemann–Liouville-type kernel changes the structure of the field equations: three unknowns (H, ρφ, pφ) can be solved from three independent equations, so the scalar potential emerges from the dynamics instead of being imposed by hand. It derives exact solutions for a flat FLRW universe in n dimensions and claims that, for appropriate values of the fractional parameter and one integration constant, the effective equation of state runs through early inflation, radiation domination, matter domination, and late-time acceleration. The same fractional structure is argued to satisfy Bianchi identities and Noether's second theorem, with the total effective energy-momentum tensor conserved while energy is exchanged between the scalar and kernel sectors. If right, this is a closed, analytic, parameter-controlled cosmology that reproduces the standard sequence of cosmic eras without dark energy or exotic fluids.","feed_headline":"Fractional kernel runs the whole cosmic history without dark energy","feed_subtitle":"Exact n-dimensional solutions emerge from a time kernel alone, generating the scalar potential instead of assuming it.","key_machinery":"The load-bearing object is the kernel-weighted action S(α)_SB = (1/Γ(α)) ∫ d^n x √−g ξ(t) L_SB with ξ(t) = (t̄ − t')^{α−1}, equivalent to a Riemann–Liouville fractional time integral applied to the whole Sáez–Ballester Lagrangian. Varying it produces a fractional-sector energy-momentum tensor T^{(f)}_{μν} = (1/(κ_n ξ))(∇_μ∇_ν ξ − ∇²ξ), whose FLRW specialization gives the 1/t and 1/t² terms in the Friedmann equations. The exact solution itself is organized around h(t) = (t^A − C)/(t^A + C), with A = √((α−3)² n² + 2(3α−5)n + 1), which solves a Riccati-type equation for H(t); this h(t) is what lets the potential be reconstructed rather than assumed.","core_discovery":"On the paper's own terms, the central discovery is that fractionalization via a time-dependent kernel changes the counting of independent equations. In the standard Sáez–Ballester model only two of the four field equations are independent, so a potential must be guessed; in the fractional version the kernel adds explicit 1/t and 1/t² terms that make three equations independent, allowing H(t), ρφ(t) and pφ(t) to be determined exactly without specifying V(φ). The resulting closed-form solution contains only α, n and one integration constant C, and its effective equation of state runs through initial acceleration, radiation-like, matter-like, and late accelerated phases, while the scalar potent","pith_inferences":["If the independence count is confirmed, an immediate testable extension is to convert the reconstructed V(t) into V(φ) and compare its shape with standard inflation/dark-energy potentials (e.g., m²φ², λφ⁴, plateau forms); the paper leaves this reconstruction open.","The fractional sector's effective density scales like H/t, which behaves as a time-dependent dark-radiation-like component; fitting α, C, n to supernova or CMB data could constrain the model, but the paper does not perform that quantitative fit.","The modified Mukhanov–Sasaki variable implies a calculable tilt and amplitude for the primordial spectrum; deriving n_s and r for the inflationary branch would let the model be tested against forthcoming CMB polarization data.","Because the kernel breaks time reparametrization invariance, the model effectively selects a preferred foliation; comparing its predictions in different time gauges would clarify whether the claimed gauge-independence of the perturbation results survives beyond Newtonian gauge."],"forward_implications":["If the system is genuinely three-dimensional for generic α, the model supplies exact, parameter-controlled solutions for the whole cosmic history with no potential input — a new way to generate scalar-field cosmologies.","The standard α = 1 model is recovered as a limit, so the fractional framework doubles as a solution-generating device for standard scalar field cosmology: it produces a particular potential that would be nearly impossible to guess by hand.","First-order perturbation theory is consistent with the background: the fractional Mukhanov–Sasaki equation with c_s² = 1 governs curvature perturbations, meaning scalar modes propagate at the speed of light and the spectral index will differ from the standard single-field prediction through the modified z² = a² Q_s.","Energy conservation is preserved only for the total effective sector; the scalar field continuously exchanges energy with the kernel whenever ξ̇ ≠ 0, which is a distinctive observational signature distinguishing this class from minimally coupled scalar field models.","Because V(t) never vanishes for admissible parameters, the framework cannot describe a potential-free (kinetic-dominated) phase, and bounce solutions are excluded in the present single-fluid setup — both are concrete, checkable predictions."],"fun_headline_variants":["Exact n-D fractional cosmology without a guessed scalar potential","Time kernel flips equation count, yields exact cosmic solutions","Fractional kernel gives closed-form cosmology, no dark energy needed","n-D fractional model solves fields exactly, potential emerges","Kernel-driven fractional cosmos: exact H, ρ, p without V(φ)"],"cache_read_input_tokens":2304,"weakest_assumption_plain":"The assertion that the fractional field equations contain three independent equations for the three unknowns H, ρφ, pφ is stated without proof; if one of the Friedmann equations is actually implied by the others for generic α, the exact solution is a specially selected trajectory rather than the general no-ad-hoc-potential solution.","fun_headline_variants_meta":{"raw":{"variants":["Exact n-D fractional cosmology without a guessed scalar potential","Time kernel flips equation count, yields exact cosmic solutions","Fractional kernel gives closed-form cosmology, no dark energy needed","n-D fractional model solves fields exactly, potential emerges","Kernel-driven fractional cosmos: exact H, ρ, p without V(φ)"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00025,"raw_usage":{"total_tokens":1391,"prompt_tokens":745,"completion_tokens":646,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":489,"completion_tokens_details":{"reasoning_tokens":569}},"tokens_in":489,"tokens_out":646,"duration_ms":7837,"temperature":1.0,"reasoning_tokens":569,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-03T16:51:03.148998+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Jacobian determinant (or numerical rank) of the three fractional equations — (14), (15), and either (16) or (21) — treated as equations for H, ρφ, pφ at generic α, n, t; if the rank drops below 3, the central independence claim fails. A second check: search the admissible parameter space (α, n, C) for a point where V(t) from Eq. (29) vanishes or where ρ_eff is negative during the purported radiation/matter phases; either would contradict the paper's claim that all epochs emerge without ad hoc input.","supporting_citations":[],"review_version":1}