{"id":"bf70e6a3-a481-4ede-8a05-e859c9dcf16b","arxiv_id":"2607.23403","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":3.5,"correctness_risk":"high","formal_verification":"none","parameter_count":4,"one_line_summary":"Fractional holographic dark energy is used as the source for Morris–Thorne wormholes with φ=-0.1/r, yielding shape functions whose traversability and reduced exoticity are claimed to improve with the fractional parameter γ.","lead":"The authors build Morris–Thorne wormhole geometries whose energy density is fractional holographic dark energy, with an ad hoc redshift, and catalog geometry, energy conditions, and thermodynamic plots. A smart generalist might care only if fractional cosmology genuinely reduces the exotic matter needed for traversable tunnels; the paper is a parameter-scan model exercise, not an observational or foundational advance.","discovery_kind":"extension","skeptic_critique":{"model":"moonshotai/kimi-k3","headline":"The shape function (eq. 22) is derived from a density μ∝r^{2/γ−3} (eq. 17) that does not follow from the paper's own fractional-HDE density ρ=3d²L^{2−3γ}/γ (eq. 2) under L→r; the two agree only at γ=1 or γ=2/3, and eq. 2 also fails its own γ=2 limit against eq. 1.","rationale":"The reader identified the two right problems — the unmotivated L→r holographic-cutoff identification (weakest_assumption) and the failure of asymptotic flatness for γ≤2 (in the rationale) — and correctly set correctness_risk high with a CONDITIONAL verdict. I agree with that verdict level, so UNCHANGED. However, the reader's weakest_assumption is not the most load-bearing spot: the L→r substitution is a known, if poorly justified, community practice, whereas the mismatch between eq. 2 and eq. 17 is an internal inconsistency in the derivation itself — the shape function does not follow from the paper's own stated density for any of the plotted parameter values, and eq. 2 fails its own γ=2 recovery claim against eq. 1. This is checkable in two lines of algebra and is independent of any community convention. I therefore agree only partially with the reader's diagnosis. I also strengthen the flatness objection from \"mathematically false\" to concretely fatal for the γ<2 branch: 1−Ψ/r changes sign at a finite second root (≈34 for the plotted parameters), so the metric loses its Lorentzian wormhole character there, and the VIQ diverges for all plotted γ — both contradicting specific summary claims. None of this requires bad faith to explain; it reads as a propagated typo in eq. 2 plus overclaimed asymptotics. The fix is well-defined: reconcile eqs. 1, 2, and 17, restrict the solution family to γ>2 (or repair the density), and restate the flatness/VIQ conclusions accordingly. That is exactly the reader's CONDITIONAL with one added mandatory condition, so the verdict should not move.","tokens_in":18744,"tokens_out":5229,"duration_ms":151822,"concrete_test":"Two-line check: (i) substitute L=r into eq. 2 and compare with the density actually used in eq. 17; they match only at γ=1, 2/3, and eq. 2 at γ=2 yields 3d²L^{−4}/2 ≠ eq. 1. If eq. 2 is a typo, re-derive Ψ from the corrected ρ and recompute lim Ψ/r. (ii) Numerically solve Ψ(r)=r for γ=1.25, d=0.05, r0=1; a second root near r≈34 where 1−Ψ/r changes sign confirms the γ<2 branch has no asymptotically flat exterior, so flatness and VIQ claims must be restricted to γ>2.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Everything in the paper — the shape function, all figures, NEC/SEC analysis, thermodynamics, complexity — rests on Ψ(r) = r0 + 12πd²γ(r^{2/γ} − r0^{2/γ}), obtained by inserting μ = 3d²r^{−3+2/γ} into the (t,t) field equation (eqs. 16–18). But the stated source is eq. 2, ρ_FHDE = 3d²L^{2−3γ}/γ, which under the paper's own identification L→r gives ρ ∝ r^{2−3γ}, not r^{−3+2/γ}. The exponents coincide only where 2−3γ = −3+2/γ, i.e. γ=1 or γ=2/3 — none of the plotted values (1.25–3). Moreover eq. 2 fails its own consistency check: at γ=2 it gives 3d²L^{−4}/2, not the standard HDE density 3d²M_p²L^{−2} of eq. 1 that the text claims is recovered. So either eq. 2 is a typo (and the real model is ρ∝L^{2/γ−3}, which still doesn't reduce to eq. 1 at γ=2 without a factor), or the derived Ψ is not sourced by the advertised fractional HDE. This is distinct from, and sharper than, the L→r motivation problem the reader flagged: it is an internal algebraic break in the derivation itself. Compounding this, the flatness claim fails in closed form: lim_{r→∞} Ψ/r = 0 only for γ>2; at γ=2 it tends to 24πd² ≈ 0.19 (d=0.05), and for γ<2 it diverges, with 1−Ψ/r acquiring a second zero (for γ=1.25, d=0.05, r0=1, near r≈34), so the second asymptotic region and the claimed signature do not exist for most of the advertised parameter range, and the volume-integral quantifier (eq. 51, integrand ~ r^{2/γ−1}) diverges for every γ≥1 plotted, contradicting the \"finite exotic matter\" claim.","agreement_with_reader":"partial"},"referee_report":{"model":"moonshotai/kimi-k3","summary":"The authors construct Morris–Thorne wormhole solutions in GR by inserting a fractional holographic dark energy (FHDE) density into the (t,t) Einstein equation with redshift function ϕ(r) = −0.1/r, obtaining the closed-form shape function Ψ(r) = r0 + 12πd²γ(r^{2/γ} − r0^{2/γ}) (Eq. 22). They then survey throat/flare-out conditions, embeddings, energy conditions, EoS parameters, active mass, compactness, exoticity, volume-integral quantifier, TOV equilibrium, Kretschmann scalar, complexity factor, and a thermodynamic analysis (Hawking/wormhole temperature, first law with S = 4πr², specific heat), concluding that increasing the fractional parameter γ yields less exotic, more regular, thermodynamically stable wormholes over the plotted range γ = 1.25–3 with d = 0.05, r0 = 1.","tokens_in":19388,"tokens_out":4893,"duration_ms":46831,"significance":"If the solution were internally sound, the work would be a competent example of the \"DE-sourced wormhole\" genre: the integration yielding Eq. (22) is elementary and given in closed form, the throat and flare-out checks are explicit, and the survey of diagnostics is unusually complete, so the parameter dependence of NEC violation is cleanly falsifiable within the model. However, as detailed below, the derivation contains an internal algebraic break at its starting point (Eq. 2 vs. Eq. 17), and the asymptotic-flatness and finite-exotic-matter claims fail in closed form for most of the advertised parameter range. As written, the central conclusions of the abstract, Summary, and §VII do not hold for three of the five γ values plotted. The results as stated are therefore not currently reliable; a corrected version restricted to the genuinely asymptotically flat regime (γ > 2) might still be of modest interest to the wormhole-phenomenology community.","major_comments":[{"comment":"The derivation is internally inconsistent at its starting point. Eq. (2) defines ρ_FHDE = 3d²L^{2−3γ}/γ, so under the paper's own identification L → r (Eq. 16) the source density should scale as r^{2−3γ}. But Eq. (17) instead inserts μ = 3d²r^{−3+2/γ}. The exponents agree only where 2−3γ = −3+2/γ, i.e. γ = 1 or γ = 2/3 — none of the plotted values (γ = 1.25…3). Everything downstream (Ψ in Eq. 22, all figures, NEC/SEC, thermodynamics, complexity) is sourced by Eq. 17, not by the advertised Eq. 2. Furthermore Eq. (2) fails its own stated consistency check: at γ = 2 it gives (3d²/2)L^{−4}, not the standard HDE density 3d²M_p²L^{−2} of Eq. (1) which the text claims is recovered; note that Eq. 17's density r^{2/γ−3} does reduce to r^{−2} at γ = 2, suggesting Eq. (2) may be a transcription error for the actual model of Ref. [33] (Trivedi et al.). The authors must (i) state the FHDE density exa","section":"§II, Eqs. (2), (16)–(18)"},{"comment":"The asymptotic-flatness claim fails in closed form for three of the five plotted γ values. From Eq. (22), Ψ/r = r0/r + 12πd²γ(r^{2/γ−1} − r0^{2/γ}/r). The limit as r→∞ is 0 only for γ > 2; at γ = 2 it tends to 24πd² ≈ 0.19 (d = 0.05), and for γ < 2 it diverges. Consequently, for γ = 1.25 and 1.5 the function 1 − Ψ/r acquires a second zero at finite r (for γ = 1.25, d = 0.05, r0 = 1, near r ≈ 34), beyond which g_rr changes sign and the spacetime is not Lorentzian in the advertised way — there is no second asymptotic region at all. The figures only extend to r ≈ 3 (Fig. 1) and r ≈ 7 (Fig. 3), so the failure is invisible in the plots, but the claims in §III.A ('monotonically grows up to unity… spacetime becomes asymptotically flat'), §III.C ('two asymptotically flat universes'), and the Summary are false for γ ≤ 2. Related quantities inherit the failure: the active mass M(r) ~ r^{2/γ} and c","section":"§III.A–C, Fig. 1, Eq. (6) vs. Eq. (22)"},{"comment":"The volume-integral quantifier claim ('finite amount of exotic matter', §V.D, Fig. 9b) is contradicted by the asymptotics of the solution itself. Using Eqs. (9)–(10) with Eq. (22), μ + p_rd ~ r^{2/γ−3} at large r (the ϕ′ term is subleading), so the integrand of Eq. (51) behaves as r^{2/γ−1} and the integral to ∞ diverges for every γ ≥ 1 — i.e., for the entire plotted range. The finite-looking curves in Fig. 9(b) are artifacts of truncating the integral at r = 3. Either the VIQ must be computed and reported as divergent (removing the 'finite exotic matter' selling point), or the model must be changed. Note this point is logically tied to Major Comment 2: for γ > 2 the VIQ still diverges (2/γ − 1 > −1 requires γ < 1 for convergence), so no value of γ in the advertised family yields a convergent VIQ.","section":"§V.D, Eq. (51), Fig. 9(b)"},{"comment":"The replacement of the cosmological IR cutoff L by the static radial coordinate r (Eq. 16) is load-bearing and unjustified. In HDE, L is a horizon scale (e.g., the future event horizon) of a cosmological spacetime; in a static MT geometry there is no such horizon, and no argument is given for why a local radial coordinate should play that role. At minimum the authors should (i) acknowledge this is an ad hoc identification following (and citing) prior literature that does the same, and (ii) discuss sensitivity: e.g., whether L = r vs. L = proper radial distance ∫dr/√(1−Ψ/r) changes the conclusions. As it stands the identification is presented as definition rather than assumption.","section":"§II, Eq. (16)"},{"comment":"The thermodynamic section imports black-hole identities into a horizonless spacetime without justification, and one result is internally inconsistent. Eq. (31) defines a 'Hawking temperature' T_HK = √(1−Ψ/r)ϕ′/(2π); for a wormhole with no horizon there is no Hawking radiation, and the cited source (Ref. [37], an undergraduate research journal) does not establish the formula. Moreover, since 1−Ψ/r = 0 at the throat, Eq. (31) gives T_HK(r0) = 0, contradicting the text's claim that the temperature 'attains its maximal value close to the WH throat' (Fig. 4a shows the maximum at r ≈ 1.2, off the throat). Similarly, S = 4πr² (Eq. 37) is an area law imported without argument, and C_v via Eq. (42) inherits both issues. The authors should either provide a derivation of these identifications for horizonless geometries (there is literature on wormhole thermodynamics that could be engaged, e.g., Ref","section":"§IV.A–D, Eqs. (31), (36)–(37), (42)"}],"minor_comments":[{"comment":"Eq. (11) mixes notation: it contains both Ψ (used elsewhere) and b ('b′r−b' terms) for the shape function; also the last term −b′r−b/2r²(r−b) appears dimensionally/structurally inconsistent with the standard MT tangential-pressure equation. Please rewrite in uniform Ψ notation.","section":"§II, Eq. (11)"},{"comment":"§III.C, first sentence after Eq. (30): 'The integration of Eq. (1)' should read Eq. (30).","section":"§III.C"},{"comment":"Eq. (34) and the Fig. 4 caption write the holographic parameter as 'c' (12πc²γ…, 'c = 0.05') whereas everywhere else it is d = 0.05.","section":"§IV.A–B, Fig. 4"},{"comment":"Typos: 'fractonal' (§I roadmap and §IV.C), 'density of HFR' (opening of §III) presumably meaning FHDE, 'paramtric' (Fig. 7 caption), 'SEC is quite happy with the given domain' (§V.A) is informal for a journal article.","section":"§I, §III, §IV.C, Fig. 7"},{"comment":"Eq. (2) omits the factors of M_p present in Eq. (1); even after the exponent issue in Major Comment 1 is fixed, units and the γ = 2 limit should be made explicit.","section":"§II, Eq. (2)"},{"comment":"All figures are confined to r ≤ 3 (or r ≤ 7 for embeddings); given the closed-form solution, extending the radial range (e.g., to r ~ 50) would immediately reveal the γ ≤ 2 pathology of Major Comment 2 and should be standard practice.","section":"Figs. 1–11"},{"comment":"Fig. 10(d) plots the force balance for a single (unspecified) γ while panels (a)–(c) show all γ; please state which curve is shown, or show the sum for each γ.","section":"§V.E, Fig. 10"},{"comment":"The data-availability statement says no data exist, yet the paper's results are entirely numerical plots from a 'reliable Python library' (§III.C); depositing the plotting scripts would materially improve reproducibility at negligible cost.","section":"Data Availability"}],"recommendation":"major_revision","confidential_remarks":"The manuscript fits a recognizable template from this group (identical diagnostic pipeline, parameter values d = 0.05, r0 = 1, ϕ = −0.1/r appear in several of the cited self-references, e.g. Refs. [19–21, 23–27]), which raises a salami-slicing concern the editor may wish to weigh. More substantively, two of the major comments are checkable in closed form by any reader (the Eq. 2 vs. Eq. 17 exponent mismatch, and the Ψ/r limit), which suggests the manuscript was not internally stress-tested before submission. If the authors can show Eq. (2) is a typo and that Eq. (17) matches Ref. [33], and they honestly restrict to the γ > 2 regime with the VIQ and thermodynamic claims revised accordingly, a publishable (if narrower) paper could result; as submitted it should not appear."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The one thing to know: this is a standard Morris–Thorne build that imports a fractional-HDE density, integrates for Ψ(r), and runs the full community checklist (embeddings, NEC/SEC, VIQ, TOV, Kretschmann, complexity, thermo). The increment is real but narrow—an explicit shape function with φ=−0.1/r and a γ scan.\n\nWhat they do well: given the working density μ∝r^{−3+2/γ}, the integration to Ψ(r)=r0+12πd²γ(r^{2/γ}−r0^{2/γ}) is elementary and consistent. Throat and flare-out hold near r0. The figures are systematic, γ is tracked cleanly, and there is no data-fitting circularity. Self-citations sit inside their prior wormhole line; that is normal here.\n\nSoft spots, in proportion. First, and load-bearing as written: eq. (2), ρ=3d²L^{2−3γ}/γ, does not recover standard HDE at γ=2 and does not equal the μ they plug into G_{tt}. The exponents match only at γ=1 or 2/3, none of the plotted values. This is almost certainly a typesetting error (they meant something like L^{2/γ−3}), but on the page the advertised source and the derived Ψ are algebraically disconnected. Second: lim Ψ/r=0 only for γ>2. At their d=0.05, γ=2 leaves a nonzero floor ~0.19; for γ<2, Ψ/r diverges and 1−Ψ/r can acquire another zero, so the second asymptotic region fails for most of the advertised range. They still claim flatness and plot γ=1.25–3. Third: L→r is unmotivated (common in this subfield, still soft), φ=−0.1/r is arbitrary, and the VIQ integrand does not obviously converge for the plotted γ, which undercuts the “finite exotic matter” claim.\n\nWho it is for: people already writing DE-sourced wormhole families who want another γ knob. Not for anyone after DE microphysics or observables.\n\nI would send it to referees. The γ>2 slice plus a corrected density formula is a legitimate formal family after major revision. Engage if you work this niche; otherwise you can skip.","headline":"Solid checklist construction once you accept their working density, but eq. (2) does not match what they integrate, and asymptotic flatness only holds for γ>2—not the full range they advertise and plot.","tokens_in":20202,"tokens_out":615,"would_cite":false,"duration_ms":39743,"reading_group":"no","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.20.Gz","04.20.Jb","04.50.Kd","95.36.+x","98.80.-k"],"model":"grok-4.5","headline":"Fractional holographic dark energy can source traversable Morris–Thorne wormholes that need less exotic matter as the fractional order rises.","keywords":["traversable wormholes","fractional holographic dark energy","Morris–Thorne geometry","energy conditions","wormhole thermodynamics","shape function","Kretschmann scalar","complexity factor"],"falsifier":"Compute or measure whether the radial null-energy-condition violation and the volume-integral quantifier of exotic matter actually decrease with rising γ for the derived shape function; if they do not, or if the Kretschmann scalar diverges at the throat, the central claim fails.","tokens_in":19602,"feed_emoji":"🕳️","tokens_out":882,"duration_ms":15304,"temperature":0.7,"pith_summary":"This paper builds a new family of traversable wormholes in ordinary Einstein gravity by taking fractional holographic dark energy as the matter source and a simple non-constant redshift function. From the Einstein equations it obtains an explicit shape function controlled by the fractional parameter γ, then checks that the throat, flare-out and asymptotic-flatness conditions all hold. The same solutions violate the radial null energy condition only near the throat (so a finite amount of exotic matter is still required), stay curvature-regular, and remain hydrostatically balanced. Raising γ weakens the energy-condition violation, reduces the volume of exotic matter, softens the phantom equation of state, and leaves the thermodynamic quantities (Hawking and local temperatures, internal energy, specific heat) smooth and largely stable. A sympathetic reader cares because the construction links a quantum-inspired cosmological density to concrete wormhole geometry and shows that one extra fractional degree of freedom can systematically dial down the exotic-matter burden while preserving traversability.","feed_headline":"Fractional dark energy builds wormholes that need less exotic matter","feed_subtitle":"Raising the fractional order weakens energy-condition violations while keeping the tunnels traversable and nonsingular.","key_machinery":"The fractional holographic density ρ_FHDE=3d² L^{2−3γ}/γ (recovered as ordinary holographic dark energy when γ=2), identified with the wormhole energy density μ(r) and integrated once to give the explicit shape function Ψ(r)=r₀+12π d² γ (r^{2/γ}−r₀^{2/γ}). That shape function carries every subsequent geometric, physical and thermodynamic claim.","core_discovery":"A new class of Morris–Thorne wormholes sourced by fractional holographic dark energy, with redshift φ(r)=−0.1/r, satisfies the throat, flare-out and asymptotic-flatness conditions, remains free of curvature singularities, and becomes progressively less exotic and more thermodynamically regular as the fractional parameter γ is increased.","pith_inferences":["If the radial-cutoff identification can be derived from a proper holographic screen in static spherical symmetry, the same fractional density could be ported to other modified-gravity wormhole models with less ad-hoc input.","Observational signatures proposed in the outlook (shadows, photon rings, gravitational-wave echoes) would scale with γ, offering a concrete target for future imaging or ringdown searches.","The reduction of exoticity with γ suggests a broader pattern: non-integer calculus corrections may systematically lower the energy-condition price of other exotic spacetimes."],"forward_implications":["Larger fractional order γ yields traversable wormholes that require a smaller finite amount of exotic matter while still flaring out.","The same solutions remain curvature-nonsingular and hydrostatically balanced for the explored range of γ.","Thermodynamic quantities (temperatures, internal energy, specific heat) stay positive and smooth, indicating local thermal stability near the throat.","The fractional parameter acts as a continuous regulator that softens phantom behaviour without destroying traversability.","The construction supplies an explicit analytic bridge between fractional cosmology and static wormhole geometry inside Einstein gravity."],"fun_headline_variants":["Fractional holographic dark energy sources less-exotic wormholes","Higher γ weakens energy violations in FHDE wormholes","FHDE Morris-Thorne wormholes stay nonsingular as γ rises","Fractional order cuts exotic matter need in dark energy wormholes","FHDE wormholes gain thermodynamic regularity with larger γ"],"cache_read_input_tokens":128,"weakest_assumption_plain":"The cosmological infrared cutoff that defines holographic dark energy is simply replaced by the static radial coordinate of the wormhole, with no derivation that justifies using a horizon-scale bound as a local radial density.","fun_headline_variants_meta":{"raw":{"variants":["Fractional holographic dark energy sources less-exotic wormholes","Higher γ weakens energy violations in FHDE wormholes","FHDE Morris-Thorne wormholes stay nonsingular as γ rises","Fractional order cuts exotic matter need in dark energy wormholes","FHDE wormholes gain thermodynamic regularity with larger γ"]},"model":"grok-4.5","effort":"low","cost_usd":0.004466,"raw_usage":{"total_tokens":1334,"prompt_tokens":830,"num_sources_used":0,"completion_tokens":65,"cost_in_usd_ticks":44664000,"prompt_tokens_details":{"text_tokens":830,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":439,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":830,"tokens_out":65,"duration_ms":7392,"temperature":1.0,"reasoning_tokens":439,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-30T23:06:44.268945+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"Compute or measure whether the radial null-energy-condition violation and the volume-integral quantifier of exotic matter actually decrease with rising γ for the derived shape function; if they do not, or if the Kretschmann scalar diverges at the throat, the central claim fails.","supporting_citations":[],"review_version":1}