{"id":"3c5ddb03-a340-4fdb-9446-9303af3656b0","arxiv_id":"2608.09379","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"The fractional holographic dark energy model with the future event horizon as its length scale survives DESI DR2 plus CMB distance-prior constraints, while the Hubble-horizon and particle-horizon versions are ruled out.","lead":"A dark energy model built from 'fractional' entropy and the future event horizon as its length scale fits current cosmological data about as well as the standard model, but only after paying an extra-parameter penalty that leaves the standard model preferred. The two other horizon choices are strongly rejected once cosmic microwave background data are added.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The strong rejection of FHDEH/FHDEP rests on CMB distance priors evaluated at fixed Planck z_* and z_drag; those redshifts should be recomputed for each model's own expansion history before concluding Δχ² > 200.","rationale":"The reader's weakest-assumption identification, Eq. (1), is legitimate: if the fractional-entropy density formula is invalid, all three models collapse. However, that formula is an input inherited from prior work, and the paper's own contribution is the observational comparison. The most load-bearing assumption introduced by this paper itself is the CMB distance-prior implementation: the shift parameters are evaluated at fixed Planck z_* and z_drag while the sound horizon is recomputed within each model. Because the headline rejection of FHDEH and FHDEP is based on Δχ² values above 200, a small systematic error in the compressed CMB likelihood could change the central verdict, even if the direction of the effect is not certain. The concrete test I propose would settle whether the fixed-redshift approximation is responsible. I credit the paper for using standard public datasets, reporting best-fit statistics, and applying AIC/BIC consistently; the FHDEF result is also more robust because its best fit is close to LambdaCDM. The reader's decision to condition acceptance on clarification and reproducibility remains appropriate, and my concern does not require changing that verdict.","tokens_in":15755,"tokens_out":13366,"duration_ms":138893,"concrete_test":"Re-run the MCMC for FHDEH and FHDEP with the SN+OHD+DESI DR2+CMB likelihood, but at each sample recompute z_* and z_drag from the model's own background (e.g., using the Hu-Sugiyama fitting formulae or a Boltzmann code with the same Ω_b h²) and evaluate R and l_A at the recomputed redshifts. If Δχ²_min relative to LambdaCDM drops below roughly 10, the 'strongly ruled out' conclusion is an artifact of the fixed-redshift CMB prior; if it remains above 200, the conclusion is robust.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper's central statistical conclusion is that FHDEH and FHDEP are 'strongly ruled out' by the SN+OHD+DESI DR2+CMB dataset, with Δχ²_min of 200.2 and 250.5. This rejection is driven by the CMB distance priors in Section III A, Eqs. (39)-(40), where R and l_A are evaluated at a fixed Planck 2018 value z_* = 1089.92 and r_d is derived at fixed z_drag = 1059.94. For non-LambdaCDM backgrounds the decoupling and drag redshifts depend on the model's own expansion history, H0, Ω_m, and Ω_de. The FHDEH and FHDEP best fits have H0 ≈ 62 km/s/Mpc and Ω_m ≈ 0.36-0.37, substantially different from the LambdaCDM values used to calibrate those redshifts. Evaluating the shift parameters at fixed redshifts while computing r_s in the model's own background can distort both the theoretical R and l_A by an amount that is not quantified. If those distortions are large, the enormous Δχ² values could be an artifact of the compressed-prior approximation rather than a genuine failure of the models. The claim that FHDEF 'survives' is less exposed because its best fit is close to LambdaCDM, but the same fixed-redshift approximation affects all three FHDE models.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper extends the fractional holographic dark energy (FHDE) framework, originally proposed with the Hubble horizon as the infrared cutoff (FHDEH), to the future event horizon (FHDEF) and particle horizon (FHDEP). Using Pantheon+ SNe Ia, cosmic chronometer OHD data, DESI DR2 BAO, and Planck 2018 CMB distance priors, the authors perform MCMC fits of the three models and compare them with ΛCDM via χ²_min, AIC, and BIC. They find that with low-redshift data alone all three models fit comparably to ΛCDM, with no significant preference after penalizing extra parameters. Adding the CMB distance priors strongly disfavors FHDEH and FHDEP (Δχ²_min ≈ 200 and 250), while FHDEF remains close to ΛCDM in χ² but is penalized by BIC. The paper then studies the background evolution of these models and performs a dynamical-systems analysis of FHDEF, identifying a stable attractor and a future phantom behavior leading to a big rip.","tokens_in":16192,"tokens_out":16830,"duration_ms":133121,"significance":"Should the constraints hold, the paper would provide a clear observational ranking of IR-cutoff choices in FHDE and a concrete dynamical distinction from ΛCDM. The model equations appear internally consistent: the EoS parameters (15), (18), (23) follow from differentiating the energy density, and the dynamical systems (11)–(12), (20), (25) are algebraically correct. The MCMC pipeline is standard, the autocorrelation-based convergence checks are described, and the AIC/BIC statistics are reported consistently. However, the headline exclusion of FHDEH/FHDEP is only as reliable as the compressed CMB distance priors at fixed Planck redshifts; the paper does not yet demonstrate that this approximation is valid for backgrounds whose best-fit H0 and Ωm differ substantially from ΛCDM. The FHDEF 'survival' conclusion is also weaker than the abstract suggests, since ΔBIC = 14 still strongly favors ΛCDM.","major_comments":[{"comment":"The strong claim that FHDEH and FHDEP are ruled out by the SN+OHD+DESI DR2+CMB dataset (Δχ²_min = 200.2 and 250.5) rests on the CMB distance priors being evaluated at the fixed Planck 2018 values z_* = 1089.92 and z_drag = 1059.94, as stated in Section III.B. For non-ΛCDM backgrounds, the decoupling and drag redshifts should be recomputed from each model's own expansion history, especially since the FHDEH/FHDEP best fits have H0 ≈ 62 km s⁻¹ Mpc⁻¹ and Ωm ≈ 0.36–0.37, far from the ΛCDM calibration of those priors. The resulting distortion of R and l_A is not quantified; without a model-consistent computation of z_* and z_drag (or a full CMB likelihood), the enormous Δχ² values may be an artifact of the compressed-prior approximation. Please repeat the analysis with self-consistent redshifts and report how Δχ², ΔAIC, and ΔBIC change.","section":"III.B, Eqs. (39)–(40)"},{"comment":"The definition of the auxiliary variable relating F (and P) to the model parameters appears to have the ratio inverted. For FHDEF, substituting α = 2 into the displayed relation F = 1/C^{α/(2−3α)} with C = 1/(κ² C² H^{(2−α)/α} Ωde) yields F = 1/(κ C Ωde^{1/2}), whereas direct calculation from ρde = 3C² R_F^{-2} and Ωde = κ² ρde/(3H²) gives F = Ωde^{1/2}/(κ C). The same issue appears in Eq. (24) for the FHDEP model. Please correct the definition (e.g., C = Ωde H^{(2−α)/α}/(κ² C²)) and ensure the dynamical equations and numerical code are consistent with the corrected formula.","section":"II.B, Eq. (19); II.C, Eq. (24)"}],"minor_comments":[{"comment":"Upper limits such as α < 1.164 and α < 1.002 are reported without specifying the confidence level; please state whether these are 68% or 95% limits and describe how the limits are derived from the posterior.","section":"Table II"},{"comment":"Equation (26) includes Ωr,0, but the text never gives its value or whether it is fixed; please state the value used (e.g., from Planck) and describe how Ωb h² is handled in the CMB prior and in the computation of r_s and r_d.","section":"Section III.A, Eq. (26)"},{"comment":"The word 'survives' for FHDEF is misleading given ΔBIC = 14; please rephrase to avoid implying that the model is preferred, or explicitly qualify it as 'not catastrophically excluded'.","section":"Abstract and Section V"},{"comment":"The auxiliary variable in Eq. (19) is typeset with the same symbol C as the model parameter; use a distinct symbol (e.g., mathcal{C}) to avoid the confusion identified in the major comment.","section":"Section II.B"},{"comment":"Minor grammar: 'its deceleration parameter q deviate' should be 'its deceleration parameter q deviates'.","section":"Abstract"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of a cosmology/gravity journal. The main result is conditional on the validity of the compressed CMB distance priors at fixed Planck redshifts; the authors should recompute z_* and z_drag self-consistently or quantitatively justify the approximation. The typo in Eq. (19) is a load-bearing reproducibility issue and must be corrected. I see no grounds for rejection, but the revision is substantive."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Read the paper. It's a straightforward constraints paper on a niche model: fractional holographic dark energy with three different IR cutoffs, constrained with Pantheon+ SN, OHD, DESI DR2 BAO, and CMB distance priors. The genuinely new bit is applying the future-event-horizon and particle-horizon cutoffs to the FHDE energy density from Trivedi et al.; that's a routine substitution but it hasn't been done in print.\n\nWhat the paper does well: the Friedmann equations and EoS derivations are internally consistent; the MCMC setup is standard; the tables report ΔAIC and ΔBIC honestly; and the authors are upfront that every FHDE variant is BIC-disfavored relative to ΛCDM. The phase-space analysis of FHDEF is competent and confirms the future big-rip behavior.\n\nThe soft spots are real. The headline result that FHDEH and FHDEP are 'strongly ruled out' rests on CMB shift parameters R and l_A evaluated at fixed Planck 2018 values of z_* and z_drag. For non-ΛCDM backgrounds, those redshifts should be recomputed from the model's own expansion history. The best-fit FHDEH and FHDEP parameters (H0 ~62, Ωm ~0.36–0.37) are far from ΛCDM, so the fixed-redshift approximation could distort R and l_A substantially. The paper gives no estimate of that error, so the Δχ² > 200 numbers are not yet credible as stated. The FHDEF constraint is less exposed because that model's best fit is close to ΛCDM, but the same approximation affects it.\n\nTwo smaller issues. The posteriors push α to the theoretical boundary (α < 1.002) for FHDEH and FHDEP, which deserves discussion rather than a one-line verdict. And no code or data artifacts are provided, so the MCMC results are hard to reproduce.\n\nThe deeper question about whether Eq. (1) is a valid consequence of fractional entropy is inherited from the prior literature; the paper doesn't re-derive it. That makes the whole model family conditional on a framework the paper doesn't defend.\n\nVerdict: worth a serious referee, but the referee should push for a self-consistent CMB-prior calculation and better reproducibility. As is, the strong disfavor claim is unverified; the more modest claim that FHDEF is the only FHDE variant not catastrophically inconsistent with CMB is probably right, though it's still statistically disfavored.","headline":"Honest DESI DR2 constraints on three FHDE cutoffs, but the 'strongly ruled out' claim for FHDEH/FHDEP is weakened by fixed Planck redshifts in the CMB distance priors.","tokens_in":16614,"tokens_out":4205,"would_cite":false,"duration_ms":40368,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["83F05"],"pacs":["95.36.+x","98.80.Es"],"model":"deepseek-v4-flash","headline":"Among the three fractional holographic dark energy variants, only the future-event-horizon one survives CMB distance priors, and even it is statistically disfavored compared with ΛCDM.","keywords":["fractional holographic dark energy","DESI DR2","CMB distance priors","future event horizon","particle horizon","Hubble horizon","phantom crossing","attractor analysis"],"falsifier":"Re-run the MCMC analysis with the full Planck 2018 CMB likelihood instead of the three distance priors (R, l_A, Ω_b h²); if the FHDEH and FHDEP best-fit χ² come within about 10 of ΛCDM, the claimed strong exclusion is an artifact of the compressed prior. Alternatively, measure ω_de across z ≳ 2 with future BAO data; if ω_de never drops below −1, the FHDEF phantom crossing and big-rip future are ruled out.","tokens_in":15578,"feed_emoji":"🌌","tokens_out":7806,"duration_ms":67725,"temperature":0.7,"pith_summary":"This paper tests whether fractional holographic dark energy, originally formulated with the Hubble horizon as its infrared cutoff, can survive current cosmological data when the cutoff is instead the future event horizon or the particle horizon. Combining supernova, cosmic-chronometer, DESI DR2 BAO, and optionally CMB distance-prior data, the authors find all three variants fit low-redshift data about as well as ΛCDM, but adding CMB priors rules out the Hubble-horizon and particle-horizon versions very strongly. The future-event-horizon variant remains the only one that survives, matching ΛCDM's past and present expansion history while predicting a phantom future that ends in a big rip; model-selection criteria still favor ΛCDM. If the paper is right, the choice of horizon is what decides whether fractional holographic dark energy is viable, and its observational signature is a future divergence rather than a present-day tension.","feed_headline":"Only one fractional dark energy model survives CMB data","feed_subtitle":"The future-event-horizon variant mimics ΛCDM in the past but predicts a phantom big-rip future; the other two variants fail.","key_machinery":"The central object is the fractional holographic dark-energy density ρ_de = $3C²L^{{(2-3α)/α}}$, Eq. (1), inherited from fractional entropy, with L the infrared cutoff and α ∈ (1,2] the fractional parameter; choosing L = 1/H, the future event horizon R_F, or the particle horizon R_P gives the three models FHDEH, FHDEF, and FHDEP. The argument runs by substituting this density into the Friedmann and conservation equations to solve for each model's equation of state ω_de, which fixes the expansion history E(z) used in the SN, OHD, DESI DR2 BAO, and CMB distance-prior likelihoods, and also feeds the autonomous dynamical system whose critical points deliver the attractor structure.","core_discovery":"The central claim is that viability of fractional holographic dark energy depends sharply on which infrared cutoff defines the horizon length in the density formula. With only low-redshift data (SN+OHD+DESI DR2), all three variants—Hubble-horizon FHDEH, future-event-horizon FHDEF, and particle-horizon FHDEP—fit about as well as ΛCDM, each giving a marginally lower χ²_min. Adding Planck CMB distance priors changes the picture: FHDEH and FHDEP are strongly excluded (Δχ²_min = 200.2 and 250.5) because their comoving sound horizon at recombination deviates badly from the Planck value, while FHDEF remains statistically compatible (Δχ²_min = −0.8) yet is still penalized by BIC (ΔBIC = 14) for its extra parameters. Dynamically, FHDEF reproduces the ΛCDM matter and dark-energy density evolutions across cosmic history, but its equation of state has recently crossed the phantom divide, and the attractor analysis shows the universe passes through a Λ-like saddle point before heading to a different stable late-time attractor and, ultimately, a big rip.","pith_inferences":["The sharp divide between horizon choices suggests that the infrared cutoff, not the fractional entropy itself, is what makes or breaks agreement with CMB data; other entropy-based dark energy models may show a similar cutoff-dependent pattern.","With low-redshift data alone, the fractional parameter α is essentially unconstrained for FHDEF, so higher-redshift BAO or CMB lensing data could either confirm or destroy the model's apparent viability.","If the fractional entropy density formula is correct, the FHDEF phantom future gives a distinctive, in-principle distinguishable fate from ΛCDM, even though the two models are currently indistinguishable in the observed past.","An interacting FHDEP model, not tested here, could plausibly evade the strong CMB exclusion; its parameter space would be a direct observational target for the same datasets."],"forward_implications":["Including CMB distance priors excludes the Hubble-horizon and particle-horizon variants at very high significance, because their predicted comoving sound horizon at recombination is incompatible with Planck.","The future-event-horizon variant is the only one of the three that remains compatible with the full combined dataset, although the Bayesian Information Criterion still strongly prefers ΛCDM.","The FHDEF model is dynamically nearly indistinguishable from ΛCDM in the past and present, but its equation of state has crossed the phantom divide, so its future evolution diverges and ends in a big rip.","An attractor analysis identifies a stable late-time dark-energy-dominated point distinct from the Λ-like point, meaning the universe passes through a ΛCDM-like stage without settling there.","Introducing an interaction between dark energy and pressureless matter is suggested by the authors as a possible way to rescue the excluded Hubble-horizon and particle-horizon variants."],"supporting_citations":[{"why":"supplies the fractional entropy from fractional quantum mechanics on which the FHDE energy density formula is based.","marker":"[25]"},{"why":"proposes the fractional holographic dark energy model and the density formula ρ_de = 3C²L^{(2-3α)/α} that all three variants inherit.","marker":"[24]"},{"why":"establishes that the choice of infrared cutoff determines the HDE energy density and evolution, motivating the three-cutoff comparison.","marker":"[15]"},{"why":"provides the DESI DR2 BAO measurements used in the χ² likelihood.","marker":"[52]"},{"why":"supplies the observational Hubble parameter data used in the χ² likelihood.","marker":"[29]"},{"why":"provides the Pantheon+ SN Ia sample used as the low-redshift distance indicator.","marker":"[27]"},{"why":"provides the CMB distance priors R, l_A, and Ω_b h² used in the high-redshift likelihood.","marker":"[76]"},{"why":"supplies the covariance matrix for the CMB distance priors incorporated into the χ² calculation.","marker":"[77]"},{"why":"the authors' earlier interacting FHDEH result that motivates proposing interactions as a rescue for the excluded variants.","marker":"[81]"}],"fun_headline_variants":["Fractional dark energy: only future-event-horizon variant passes CMB","CMB kills two fractional dark energy models, spares one","Phantom big-rip future for the one fractional dark energy model","DESI and CMB constrain fractional dark energy to one viable model"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"All three models inherit the energy density formula ρ_de = $3C²L^{{(2-3α)/α}}$, which the paper takes as given from the fractional-entropy framework of Refs. [24] and [25] rather than re-deriving from fractional quantum mechanics; if that formula does not apply to a cosmological horizon, every constraint and conclusion in this paper collapses.","fun_headline_variants_meta":{"raw":{"variants":["Fractional dark energy: only future-event-horizon variant passes CMB","CMB kills two fractional dark energy models, spares one","Phantom big-rip future for the one fractional dark energy model","DESI and CMB constrain fractional dark energy to one viable model"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000823,"raw_usage":{"total_tokens":3627,"prompt_tokens":999,"completion_tokens":2628,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":615,"completion_tokens_details":{"reasoning_tokens":2552}},"tokens_in":615,"tokens_out":2628,"duration_ms":17594,"temperature":1.0,"reasoning_tokens":2552,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-11T18:25:23.763988+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Re-run the MCMC analysis with the full Planck 2018 CMB likelihood instead of the three distance priors (R, l_A, Ω_b h²); if the FHDEH and FHDEP best-fit χ² come within about 10 of ΛCDM, the claimed strong exclusion is an artifact of the compressed prior. Alternatively, measure ω_de across z ≳ 2 with future BAO data; if ω_de never drops below −1, the FHDEF phantom crossing and big-rip future are ruled out.","supporting_citations":[{"cited_title":"Jalalzadeh, F","cited_arxiv_id":null,"evidence_quote":"supplies the fractional entropy from fractional quantum mechanics on which the FHDE energy density formula is based."},{"cited_title":"Trivedi, A","cited_arxiv_id":null,"evidence_quote":"proposes the fractional holographic dark energy model and the density formula ρ_de = 3C²L^{(2-3α)/α} that all three variants inherit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"establishes that the choice of infrared cutoff determines the HDE energy density and evolution, motivating the three-cutoff comparison."},{"cited_title":"Abdul Karim, J","cited_arxiv_id":null,"evidence_quote":"provides the DESI DR2 BAO measurements used in the χ² likelihood."},{"cited_title":"Cao and B","cited_arxiv_id":null,"evidence_quote":"supplies the observational Hubble parameter data used in the χ² likelihood."},{"cited_title":"Brout, D","cited_arxiv_id":null,"evidence_quote":"provides the Pantheon+ SN Ia sample used as the low-redshift distance indicator."},{"cited_title":"Zhai and Y","cited_arxiv_id":null,"evidence_quote":"provides the CMB distance priors R, l_A, and Ω_b h² used in the high-redshift likelihood."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"supplies the covariance matrix for the CMB distance priors incorporated into the χ² calculation."},{"cited_title":"Huang, H","cited_arxiv_id":null,"evidence_quote":"the authors' earlier interacting FHDEH result that motivates proposing interactions as a rescue for the excluded variants."}],"review_version":1}