{"id":"d9952445-4fcb-4ef1-88ec-d9b2d2d6affb","arxiv_id":"2506.03235","paper_version":2,"verdict":"REJECT","confidence":"HIGH","novelty_score":5.0,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Genetic-algorithm reconstruction of H(z) from Hubble data yields an f(T) gravity function close to ΛCDM, with a small quadratic correction, but the result depends on an imposed ΛCDM-like initial condition and on Ωm0=0.3.","lead":"The authors use genetic algorithms to fit the cosmic expansion history H(z) and then solve the f(T) gravity field equations to reconstruct the function f(T) from the data. The result looks like ΛCDM, but the reconstruction quietly assumes ΛCDM at z=0 and a fixed matter density, so the advertised model independence is overstated.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 'model-independent' reconstruction is conditioned on fT(0)=0 and Ωm0=0.3, pinning f(0) to ΛCDM; the 'mildly favored' quadratic claim has no model-selection statistic.","rationale":"The reader's weakest assumption identifies the same load-bearing concern, and it is the point on which the paper's central claim hinges. Eq. (32) is a first-order ODE, so every reconstruction requires one boundary condition. The paper's choice, Eq. (33), is not determined by the H(z) data; it imposes fT(0)=0 and a fiducial Ωm0. This forces f(T0) to exactly the ΛCDM value and forces a near-vanishing slope at T0, so agreement with ΛCDM near z=0 is tautological rather than a data-driven finding. The 1σ band shown in Fig. 2 is computed around this fixed-boundary solution and does not include the spread from marginalizing over the boundary condition or over Ωm0; including those degrees of freedom would widen the band and could remove the claimed consistency. The abstract's claim that the quadratic deviation is 'mildly favored' is also unsupported: the cited χ2 values are for HGA versus ΛCDM fits to H(z), not for the fitted quadratic f(T) model against ΛCDM, and no information criterion is reported. The GA prescription lacks the hyperparameters, grammar, and seeds needed for independent reproduction, but the more fundamental problem is that the advertised 'without any assumptions' reconstruction relies on a prior that pins the result to ΛCDM. A revision that marginalizes over f(0) and Ωm0 and reports a proper model-selection statistic could address the concern; as it stands, the central claim is not supported, so the reader's rejection is appropriate.","tokens_in":15542,"tokens_out":6294,"duration_ms":72502,"concrete_test":"Reconstruct f(T) using the same HGA(z) and the same H(z) covariance, but treat f(0) as a free parameter (equivalently, vary fT(0)) when solving Eq. (32), and repeat with Ωm0 = 0.25, 0.30, 0.35. If the resulting 1σ band no longer contains the ΛCDM curve across the reported redshift range, or if f(0) is only weakly constrained by the data, then the claimed consistency with ΛCDM is an artifact of Eq. (33).","verdict_should_be":"UNCHANGED","load_bearing_attack":"Eq. (33) fixes f(0) = -6H0^2(1-Ωm0) by imposing fT(z=0)≈0, and the Fig. 2 caption fixes Ωm0=0.3. Since Eq. (32) is a first-order ODE, this boundary condition selects one member of a one-parameter family of reconstructed f(T) curves. The chosen member is forced to pass through the ΛCDM value at T=T0 and to have a near-vanishing slope there (the quadratic fit gives β+2γ≈0.0028), so the reported 'consistency within 1σ' and the small quadratic deviation are partly baked into the solution rather than discovered from the data. The 1σ band in Fig. 2 is computed around this fixed-boundary solution and does not include the spread from marginalizing over f(0) or over Ωm0; including those degrees of freedom would widen the band and could remove the claimed consistency. Separately, the abstract's 'mildly favored' statement is not backed by any model-comparison statistic: the χ2 values 41.43 versus 42.47 compare HGA and ΛCDM fits to H(z), not the quadratic f(T) model against ΛCDM, and the GA expression has many functional degrees of freedom that are not penalized. The GA hyperparameters, grammar, and seeds are also not given, so exact reproduction is not possible. The derivation of Eq. (32) is sound, but the advertised 'without any assumptions' reconstruction is not.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper applies genetic algorithms (GA) to H(z) data from cosmic chronometers, BAO, and DESI to obtain an analytic reconstruction H_GA(z), then uses this in the f(T) Friedmann equation to solve a first-order ODE for f(T) as a function of redshift. The central result is a reconstructed f(T) curve that lies within 1σ of the ΛCDM prediction across a broad redshift range, with a quadratic parametrization f(T)=αT0+βT+γT0(T/T0)^2 reported to fit the reconstruction with R^2≈0.9999. The abstract further claims that the quadratic deviation from ΛCDM is mildly favored by the data, and the paper presents derived quantities (Ω_DE, q(z)) and a DESI-inclusion comparison.","tokens_in":15813,"tokens_out":5921,"duration_ms":68785,"significance":"If the reconstruction were genuinely assumption-free, the paper would be a novel and useful demonstration of GA-based reconstruction of modified-gravity functions, complementing existing Gaussian-process reconstructions. The derivation of the ODE (Eqs. 30–32) is algebraically correct, and the use of the path-integral error method is in line with the literature. However, the advertised 'model-independent' nature is not realized: the reconstruction requires the f(T) field equations, a flat FRW geometry, pressureless matter, and a boundary condition that assumes ΛCDM at z=0. Moreover, the 'mildly favored' quadratic deviation is not supported by any model-selection statistic, and the GA hyperparameters are not specified, so the results are not reproducible as presented. The paper's core claims therefore need substantial revision.","major_comments":[{"comment":"The boundary condition f_T(z=0)≈0, combined with Eq. (33), fixes f(0)=−6H0^2(1−Ωm0) and forces the reconstructed f(T) to pass through the ΛCDM value at T=T0 with near-vanishing slope. Since Eq. (32) is a first-order ODE, this initial condition selects one member of a one-parameter family of solutions; the reported consistency of f(T) with ΛCDM within 1σ is therefore partly built into the calculation rather than discovered from the data. The uncertainty band in the left panel of Fig. 2 is computed around this fixed-boundary solution and does not include the spread arising from marginalizing over f(0) or over Ωm0, which is fixed to 0.3 in the caption. A reconstruction that varies these degrees of freedom could have a substantially wider band and a different mean curve.","section":"Sec. 4, Eq. (33) and Fig. 2"},{"comment":"The claim that the quadratic deviation from ΛCDM is 'mildly favored by the data' is not supported by the statistics presented. The χ^2 values 41.43 and 42.47 in Sec. 3.3 compare H_GA(z) and the ΛCDM best fit to the H(z) data, not the quadratic f(T) model against ΛCDM. The coefficients α, β, γ in Eq. (34) are obtained by fitting the reconstructed f(T) curve, not by fitting the original data, and no uncertainties on these coefficients or a model-selection statistic (ΔAIC, ΔBIC, or a degrees-of-freedom-penalized Δχ^2) are reported. The GA expression has many functional degrees of freedom that are not penalized, so a raw Δχ^2 of 1.04 between H_GA and ΛCDM is not evidence that the quadratic f(T) deviation is favored.","section":"Abstract and Sec. 4, Eq. (34)"},{"comment":"The genetic-algorithm reconstruction is not reproducible because the paper does not specify the GA hyperparameters (population size, number of generations, mutation and crossover rates, tournament size, grammar, or random seeds) and does not provide code. Since GA is a stochastic method, the particular analytic form of H_GA(z) in Eq. (29) could depend on these choices, and the reader cannot verify convergence or robustness. Without this information, the subsequent f(T) reconstruction is not independently checkable, which is a significant omission for a machine-learning-based analysis.","section":"Sec. 3.1 and Eq. (29)"},{"comment":"The phrase 'without any assumptions' and 'completely model-independent' used in the abstract, Sec. 3.3, and the conclusion overstates the nature of the reconstruction. The derivation of Eq. (32) assumes the f(T) field equations (8), a spatially flat FRW metric, pressureless matter with ρ_m ∝ (1+z)^3, and the relation T=−6H^2. In addition, Eq. (33) imposes a ΛCDM-like condition at z=0, and Ωm0 is fixed by hand. These are physical and statistical assumptions; the reconstruction is model-independent only in the narrow sense that no dark-energy equation-of-state model is assumed for H(z). The text should state these assumptions explicitly and temper the model-independence claim.","section":"Sec. 4, Eqs. (31)–(33)"}],"minor_comments":[{"comment":"There is a typo, 'wehre', which should read 'where'.","section":"Sec. 4, text near Eq. (30)"},{"comment":"The path-integral error method is described correctly, but the statement that 'the first generation does not affect the final solution' is vague; it should clarify that the initial population influences only convergence speed, not the converged result, and even that is only true given sufficient generations and appropriate hyperparameters.","section":"Sec. 3.2, Eqs. (21)–(23)"},{"comment":"The text says the analysis uses 64 H(z) points with 32 from CC and 32 from BAO, and the BAO set includes five DESI points; the table lists 32 CC and 32 BAO entries including the DESI rows. This is consistent, but the sentence counting could be clearer to avoid the impression that DESI points are additional to the 32 BAO points.","section":"Sec. 3.3, Table 1"},{"comment":"The left panel's axes are labeled with T and f(T) divided by 10^5, but the units and the scaling factor are only mentioned in the caption; adding this directly to the axis labels would improve readability.","section":"Fig. 2"},{"comment":"The reported values α=0.69825, β=−0.00031, γ=0.00153 are given without uncertainties or a measure of the goodness of fit beyond R^2; reporting the covariance or confidence intervals would strengthen the parametrization claim.","section":"Sec. 4, Eq. (34)"},{"comment":"The conclusion repeats the 'completely model-independent' phrasing; this should be revised to reflect the assumptions identified in Sec. 4.","section":"Conclusion"}],"recommendation":"major_revision","confidential_remarks":"The paper addresses a timely topic, and the ODE derivation is sound, but the main advertised claims—'without any assumptions' and 'mildly favored'—are not supported by the analysis as presented. The boundary condition and fixed Ωm0 effectively bake in a ΛCDM-like solution, and the model-comparison claim lacks any information criterion or direct fit to data. The issues are fixable in principle: the authors could marginalize over f(0) and Ωm0, report the resulting f(T) band, and compute a proper model-selection statistic for the quadratic parametrization against ΛCDM. However, if such an analysis shows that the consistency with ΛCDM disappears or that the quadratic deviation is not statistically distinguishable, the paper's central conclusion would change substantially. I therefore view this as a borderline case; the current version overclaims, but the methodology is worth pursuing with the required corrections."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Quick take: the genuinely new piece is applying genetic algorithms to reconstruct f(T) from H(z) data, where earlier work used Gaussian processes or constrained parametric forms. The ODE route from H to f(T) (Eqs. 30–32) is correct, the H(z) fit is reasonable, and the DESI inclusion sharpens the reconstruction. That is worth a referee's time.\n\nWhere the paper oversells itself: Eq. (33) fixes the f(T) solution by imposing f_T(z=0)≈0 and assuming Ωm0=0.3, which pins f(0) to the ΛCDM value. The advertised 'without any assumptions' reconstruction is therefore conditioned on a ΛCDM-like initial condition. The 1σ band in Fig. 2 is computed around that chosen solution and does not include the spread from Ωm0 or the boundary condition, so the consistency with ΛCDM is partly baked in. That doesn't make the result useless, but it does mean the abstract and conclusions should say 'given these priors' rather than 'model-independent'.\n\nSecond soft spot: 'mildly favored' is not supported. The quoted χ² values compare H_GA and ΛCDM fits to H(z), not the quadratic f(T) model against ΛCDM. A real model-selection check (AIC/DIC or a proper evidence estimate) is needed before claiming the quadratic deviation is favored.\n\nMinor: GA hyperparameters, grammar, and random seeds are not given, so exact reproduction is not possible. The error estimate uses the path-integral method, which is known, but the uncertainty should propagate through the ODE and boundary condition.\n\nProportionately: for a subfield looking for new tools, this is honest incremental progress. The derivation is sound, the data handling is clear, and the DESI impact analysis is useful. The central qualitative message—that current H(z) data don't push f(T) far from ΛCDM—probably survives even with the caveats. What does not survive is the stronger claim of a fully assumption-free reconstruction and a mildly favored quadratic term.\n\nI would send it to a competent referee; with disclosure of the initial conditions and a proper model comparison, it could become a solid contribution. If I were working on f(T) reconstruction I'd cite it, mainly as the first GA application.","headline":"GA reconstruction of f(T) is a genuine but modest extension; the 'model-independent' claim overstates what the fixed ΛCDM boundary condition allows, and the 'mildly favored' quadratic term lacks a model-selection statistic.","tokens_in":16481,"tokens_out":1717,"would_cite":true,"duration_ms":19613,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["98.80.-k","95.36.+x","04.50.Kd"],"model":"deepseek-v4-flash","headline":"Genetic algorithms applied to 64 Hubble-rate measurements reconstruct the f(T) gravity function directly from data; the result is consistent with ΛCDM within 1σ and mildly prefers a small quadratic deviation.","keywords":["f(T) gravity","genetic algorithms","model-independent reconstruction","Hubble parameter","cosmic chronometers","baryon acoustic oscillations","DESI data","torsion scalar"],"falsifier":"Re-run the reconstruction with the same $H(z)$ compilation while allowing $f_T(z=0)$ to vary, or while varying $\\Omega_{m0}$ away from 0.3; if the 1$\\sigma$ band of the resulting $f(T)$ no longer contains the $\\Lambda$CDM line over the full redshift range, the reported consistency is an artifact of the boundary condition rather than a data-driven result.","tokens_in":15225,"feed_emoji":"🔭","tokens_out":13413,"duration_ms":125000,"temperature":0.7,"pith_summary":"This paper tries to establish that the functional form of $f(T)$ gravity can be pulled directly from Hubble-parameter measurements rather than chosen from a set of proposed ansätze. The tool is a genetic algorithm, which evolves analytic expressions for $H(z)$ against 64 cosmic-chronometer and BAO data points, including DESI. Because in a flat Friedmann universe the torsion scalar is $T=-6H^2$, a reconstructed $H(z)$ immediately provides $T(z)$, and the first Friedmann equation supplies a differential equation for $f'(z)$ whose integration yields $f(T)$. The resulting curve stays within the 1$\\sigma$ band of $\\Lambda$CDM over the sampled range, and its mean shape is almost exactly $f(T)=\\alpha T_0+\\beta T+\\gamma T_0(T/T_0)^2$ with $\\alpha=0.69825$, $\\beta=-0.00031$, $\\gamma=0.00153$. The paper's advertised 'without any assumptions' claim is qualified by an imposed boundary condition $f_T(z=0)\\approx 0$ and a fixed $\\Omega_{m0}=0.3$.","feed_headline":"Reconstructed f(T) gravity stays within 1σ of ΛCDM","feed_subtitle":"Genetic algorithms turn 64 Hubble-rate measurements into a data-driven f(T), with only a mild hint of quadratic deviation.","key_machinery":"The machinery is a genetic algorithm paired with a first-order reconstruction equation. The GA performs symbolic regression: starting from a grammar of simple functions, it evolves a population of $H(z)$ candidates by crossover and mutation, selecting on $\\chi^2$ against the data, until it converges to $H_{\\mathrm{GA}}(z)=H_0(1+z(0.682+0.222z-0.036z^2))^2$, with path-integral error bands. Because the torsion scalar in a flat FLRW universe is $T=-6H^2$, every $H(z)$ curve maps to a $T(z)$ curve; the Friedmann equation then gives $f'(z)=6(H'/H)(H^2-H_0^2\\Omega_{m0}(1+z)^3+f/6)$. Integrating this ODE with the boundary value $f(0)=-6H_0^2(1-\\Omega_{m0})$ yields $f(T)$, and the quadratic form $f(T)=\\alpha T_0+\\beta T+\\gamma T_0(T/T_0)^2$ is then fit to the mean curve.","core_discovery":"The central discovery, as the authors state it, is that $f(T)$ gravity can be reconstructed from data alone: a genetic algorithm converts 64 $H(z)$ points into the differentiable Hubble function $H_{\\mathrm{GA}}(z)=H_0(1+z(0.682+0.222z-0.036z^2))^2$, and Eq. (32) then converts this into an $f(T)$ curve via $T=-6H^2$. That curve agrees with the $\\Lambda$CDM constant $f(T)=-6H_0^2(1-\\Omega_{m0})$ within its 1$\\sigma$ region over a broad redshift range, and the mean curve is fit to a quadratic polynomial with dimensionless coefficients $\\alpha=0.69825$, $\\beta=-0.00031$, $\\gamma=0.00153$ ($R^2\\approx 0.9999$). The authors interpret this as the first demonstration that genetic algorithms can reconstruct modified-gravity functions, and as a mild preference for a small quadratic deviation from $\\Lambda$CDM in the background expansion.","pith_inferences":["A consequence the paper leaves implicit is that the low-redshift agreement with $\\Lambda$CDM is partly inherited from Eq. (33), which forces $f(0)$ to equal the $\\Lambda$CDM value; this is an input, not a piece of information extracted from the data.","Because the reconstruction uses only background expansion data, it cannot by itself adjudicate the $H_0$ or $\\sigma_8$ tensions; the mild quadratic preference could change once perturbation-level data are included.","A natural stress test is to leave $\\Omega_{m0}$ free and sample the boundary derivative $f_T(0)$, then add supernova and large-scale-structure data; if the quadratic term still appears, the preference would be a real feature of the data rather than a boundary artifact."],"forward_implications":["The GA-reconstructed $H(z)$ fits the 64-point compilation with $\\chi^2=41.43$, slightly better than the $\\Lambda$CDM best fit at $\\chi^2=42.47$, without assuming a dark-energy model.","The quadratic form $f(T)=\\alpha T_0+\\beta T+\\gamma T_0(T/T_0)^2$, with $R^2\\approx 0.9999$, can be substituted into the Friedmann equation and reproduces both the GA and $\\Lambda$CDM Hubble histories.","The effective dark-energy density and the deceleration parameter derived from the reconstructed $f(T)$ match $\\Lambda$CDM, including a late-time transition from deceleration to acceleration.","Adding DESI data shrinks the $H_0$ uncertainty by 13.9% (from $\\pm 9.98$ to $\\pm 8.59$ km/s/Mpc) and keeps the $f(T)$ reconstruction closer to $\\Lambda$CDM than the reconstruction without DESI.","The smallness of $\\beta$ and $\\gamma$ means that any genuine departure from $\\Lambda$CDM is sub-dominant in the background history."],"supporting_citations":[{"why":"Introduces genetic-algorithm symbolic regression for cosmological $H(z)$ reconstruction, the core method adapted here.","marker":"[48]"},{"why":"Supplies the path-integral error estimation that produces the 1$\\sigma$ band around the GA best-fit function.","marker":"[83]"},{"why":"Provides the compiled 64-point $H(z)$ dataset (cosmic chronometers plus BAO) on which the reconstruction is built.","marker":"[84]"},{"why":"Contributes the DESI BAO measurements that sharpen $H_0$ and stabilize the $f(T)$ reconstruction.","marker":"[59]"},{"why":"Derives the $f'(z)$ reconstruction equation and documents late-time quadratic $f(T)$ parametrizations that motivate Eq. (34).","marker":"[41]"},{"why":"Defines $f(T)$ gravity and the torsion-scalar cosmology that yields $T=-6H^2$.","marker":"[31]"},{"why":"Supplies the radial BAO method behind part of the Hubble data.","marker":"[57]"},{"why":"Supplies the cosmic-chronometer method for the other part of the Hubble data.","marker":"[58]"}],"fun_headline_variants":["Genetic algorithm reconstructs f(T) gravity from 64 data points","AI reconstructs f(T) gravity, matches ΛCDM with mild deviation","Data-driven f(T) gravity agrees with ΛCDM, hints quadratic twist","Genetic algorithm f(T) reconstruction: ΛCDM holds, mild deviation","Machine learning reconstructs f(T) gravity, stays near ΛCDM"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The reconstruction fixes the starting value of $f(T)$ by assuming $f_T(z=0)\\approx 0$ and fixing $\\Omega_{m0}=0.3$; if those choices are relaxed, the reconstructed curve and its agreement with $\\Lambda$CDM change.","fun_headline_variants_meta":{"raw":{"variants":["Genetic algorithm reconstructs f(T) gravity from 64 data points","AI reconstructs f(T) gravity, matches ΛCDM with mild deviation","Data-driven f(T) gravity agrees with ΛCDM, hints quadratic twist","Genetic algorithm f(T) reconstruction: ΛCDM holds, mild deviation","Machine learning reconstructs f(T) gravity, stays near ΛCDM"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000658,"raw_usage":{"total_tokens":3005,"prompt_tokens":936,"completion_tokens":2069,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":1972}},"tokens_in":552,"tokens_out":2069,"duration_ms":16920,"temperature":1.0,"reasoning_tokens":1972,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T11:09:56.600473+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the reconstruction with the same $H(z)$ compilation while allowing $f_T(z=0)$ to vary, or while varying $\\Omega_{m0}$ away from 0.3; if the 1$\\sigma$ band of the resulting $f(T)$ no longer contains the $\\Lambda$CDM line over the full redshift range, the reported consistency is an artifact of the boundary condition rather than a data-driven result.","supporting_citations":[{"cited_title":"Nesseris and A","cited_arxiv_id":null,"evidence_quote":"Introduces genetic-algorithm symbolic regression for cosmological $H(z)$ reconstruction, the core method adapted here."},{"cited_title":"Nesseris and J","cited_arxiv_id":null,"evidence_quote":"Supplies the path-integral error estimation that produces the 1$\\sigma$ band around the GA best-fit function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Provides the compiled 64-point $H(z)$ dataset (cosmic chronometers plus BAO) on which the reconstruction is built."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Contributes the DESI BAO measurements that sharpen $H_0$ and stabilize the $f(T)$ reconstruction."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Derives the $f'(z)$ reconstruction equation and documents late-time quadratic $f(T)$ parametrizations that motivate Eq. (34)."},{"cited_title":"Gaztanaga, A","cited_arxiv_id":null,"evidence_quote":"Supplies the radial BAO method behind part of the Hubble data."}],"review_version":1}