{"id":"e187cc95-5eaf-4e68-9c3f-9f2b3dba7002","arxiv_id":"2507.09327","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":6,"one_line_summary":"In a modified gravity theory called F(Q,Lm,T), a wormhole shape built from a geometric embedding condition still requires exotic matter and shows only a narrow band of thermal stability.","lead":"This paper builds a mathematical model of a traversable wormhole, a tunnel through spacetime, inside a newly proposed modified theory of gravity. It checks whether such a wormhole can stay open and thermodynamically stable, and finds it still needs exotic matter with unusual properties near the throat.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Shape function (40) is not a solution of the Karmarkar condition; the added parameter P invalidates the claimed embedding-class-1 derivation, undermining the construction.","rationale":"The reader's weakest_assumption correctly identifies the ad hoc addition of the parameter P in Section 4 as the central flaw. My analysis sharpens this into a definite inconsistency: the modified shape function (40) cannot satisfy the Karmarkar condition for any nonzero P, so the paper's derivation of the wormhole solution via the embedding class-1 approach is mathematically invalid. This is the single most load-bearing concern because the entire paper—shape function, traversability checks, energy conditions, ANEC, and thermodynamics—depends on Eq. (40) being a legitimate solution of the theory. If this fails, the claimed result that 'exotic matter remains essential' is not supported by a valid construction, and the abstract's description of the method is false. The issue is correctable in principle: the authors could either find a nontrivial shape function that genuinely satisfies the Karmarkar condition, or explicitly abandon the Karmarkar derivation and present Eq. (40) as an arbitrary ansatz, in which case the energy-condition analysis would be a valid (but much weaker) example. Since the reader already recommended a conditional acceptance pending revision of this exact point, my stress-test does not move the verdict; it reinforces that the revision is essential. I do not find a stronger objection in the thermodynamic analysis (S=8\\pi r^2 rather than A/4, negative temperatures, and the narrow stability band) because those are secondary to the central construction and could be corrected without changing the paper's main conclusion.","tokens_in":28713,"tokens_out":6879,"duration_ms":75966,"concrete_test":"Substitute the shape function (40) into the Karmarkar condition (32), or equivalently check whether e^\\gamma = r/(r-b(r)) equals (r^4 + 4\\mu^2 D e^{-2\\mu/r})/r^4 for any constant D, with r_t=1, \\mu=-8, P=0.5. The equality fails at all r, analytically forcing P=0 as shown above. A second check: repeat the throat-condition elimination of D without the extra P in Eq. (39); this yields only the trivial solution b(r_t)=r_t, confirming that the nontrivial throat condition is an artifact of the ad hoc addition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The paper derives a shape function from the Karmarkar condition in Eq. (38), finds that the throat condition b(r_t)=r_t is trivial, and then 'resolves' this by adding a free parameter P in Eq. (39). This is not a harmless modification: the resulting shape function (40) no longer satisfies the Karmarkar condition (32)/(33). Indeed, with A = r_t^4(r_t-P) e^{2\\mu(1/r_t-1/r)}, substituting (40) into e^\\gamma = 1/(1-b/r) gives e^\\gamma = r(P r^4 + A)/[P(r^4(r-P)-A)]. The Karmarkar solution requires e^\\gamma = (r^4 + 4\\mu^2 D e^{-2\\mu/r})/r^4. Writing x = e^{-2\\mu/r} and A = B x, the constant term in x on the left is P r^9, while on the right it is P r^8(r-P). Equality forces P=0, contradicting 0<P<r_t. Therefore the wormhole spacetime built from Eq. (40) is not an embedding class-1 solution, and the central claim that the shape function was 'derived through the Karmarkar condition' is unsupported. Since every subsequent result—energy condition violations, ANEC, thermodynamics—rests on this unvalidated shape function, the core construction collapses unless Eq. (40) is re-derived or the Karmarkar claim is dropped and the solution is presented as a phenomenological ansatz. The paper provides no physical or geometric justification for P, so the traversability conditions are enforced rather than derived.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies static, spherically symmetric traversable wormhole solutions in the extended symmetric teleparallel gravity theory F(Q, L_m, T). It chooses a specific redshift function, derives a shape function from the Karmarkar condition, and then checks the standard traversability requirements (throat condition, flaring-out condition, asymptotic flatness). The authors plot embedding diagrams, compute the proper radial distance, analyze the energy conditions (NEC, WEC, SEC, DEC) and the averaged null energy condition via the volume integral quantifier, and carry out a thermodynamic analysis involving Hawking temperature, wormhole temperature, average pressure, work density, total energy, energy flux, and specific heat. The paper's central conclusion is that exotic matter remains essential for sustaining traversable wormholes even in F(Q, L_m, T) gravity.","tokens_in":28976,"tokens_out":7502,"duration_ms":84136,"significance":"If the construction were valid, the paper would provide a concrete wormhole model in a relatively new modified-gravity framework and would support the general expectation that traversable wormholes require energy-condition violations. The authors give explicit (though lengthy) expressions for the effective density and pressures, and the graphical treatment of the energy conditions is systematic. However, the validity of the central construction is undermined by a load-bearing error: the shape function used in all subsequent computations is not actually a solution of the Karmarkar condition, and the parameter P that fixes the throat condition is introduced ad hoc. The thermodynamic analysis also relies on an unjustified entropy definition. These issues must be resolved before the results can be accepted.","major_comments":[{"comment":"The shape function obtained by adding the free parameter P is not a solution of the Karmarkar condition. For the Karmarkar solution, e^γ = (r^4 + 4μ^2 D e^{-2μ/r})/r^4, while inserting Eq. (39) into e^γ = 1/(1 - b/r) gives e^γ = r(r^4 + 4μ^2 D e^{-2μ/r}) / [r^5 - P(r^4 + 4μ^2 D e^{-2μ/r})]. Equality of these two expressions forces P = 0, contradicting the stated range 0 < P < r_t. Thus Eq. (40) does not satisfy the Karmarkar condition, and the claim that the shape function was 'derived through the Karmarkar condition' is unsupported. Because all subsequent results—energy conditions, ANEC violation, and thermodynamics—are computed with Eq. (40), the central construction collapses unless Eq. (40) is re-derived or the Karmarkar claim is dropped and the solution is presented as a phenomenological ansatz. If the latter route is taken, the paper must state clearly that the traversability conditions are imposed by construction rather than derived from the embedding geometry.","section":"Section 4, Eqs. (38)-(40)"},{"comment":"The thermodynamic analysis depends on an unjustified entropy definition. The paper states S = 8πr^2 for the wormhole entropy without derivation or discussion. For a spherically symmetric throat of radius r, the standard geometric entropy would be S = A/4 = πr^2; the factor 8 is nonstandard and appears to be chosen ad hoc. The specific heat C_V = T_Hawk dS/dT_Hawk in Eq. (60) and the stability intervals in Table 2 are direct consequences of this choice. The paper's conclusion that the wormhole is thermodynamically stable in a narrow band near the throat is therefore not robust. In addition, the interpretation of negative Hawking temperature as 'thermal stability' is nontrivial and requires support beyond a reference to the exotic-matter literature; the negativity here is driven by the parameter choice μ = -8, so it is not a parameter-independent feature.","section":"Section 9, Eqs. (56)-(60) and Table 2"},{"comment":"The introduction of P is explicitly described as a remedy after the throat condition b(r_t) = r_t 'results in only a trivial solution' for the Karmarkar shape function. This exposes a circularity in the construction: the geometric traversability conditions are not derived from the embedding formalism but are enforced by hand through the free parameter P. The paper provides no physical or geometric justification for P, nor any independent constraint on it. Even if the Karmarkar claim is retracted and Eq. (40) is treated as a phenomenological ansatz, the authors need to demonstrate that the chosen range 0 < P < r_t is natural and that the results are not sensitive to the arbitrary choice of P within that range.","section":"Section 4, Eq. (37) and Section 3, conditions (1)-(5)"}],"minor_comments":[{"comment":"The integrands in Eqs. (45) and (47) contain garbled radical notation; they should be typeset clearly so that the integration is unambiguous.","section":"Section 5, Eq. (45)"},{"comment":"The caption lists P = 0.1 → ♠, P = 0.3 → ♠, P = 0.5 → ♠, P = 0.7 → ♠, P = 0.9 → ♠; all five cases are assigned the same marker, which makes the legend uninformative. Distinct markers should be used.","section":"Figure 2 caption"},{"comment":"The text says 'as illustrated in Fig. 1 for various selected values of the parameters β, α, m, and P', but the average pressure is plotted in Fig. 10, not Fig. 1; the cross-reference is incorrect.","section":"Section 9, paragraph before Fig. 10"},{"comment":"In the row for r = 0.8, the entry for P = 0.5 is marked 'Stable' while all other P columns at the same r are marked 'Unstable'. Since the left panel of Fig. 14 shows negative C_V for r < r_t across all P values, this entry appears to be a typo and should be corrected.","section":"Table 2"},{"comment":"The field equations are derived for a general F(Q, L_m, T), but the paper later specializes to F = f(Q) + αL_m + βT. It would help the reader if this specialization were announced before the long expressions in Eqs. (24)-(26), rather than only in the text, and if the lengthy reduced field equations were moved to an appendix.","section":"Section 2, Eq. (13)"},{"comment":"The concluding section states that the shape function b(r) 'derived using the Karmarkar condition Eq. (31)' proves viable, but this repeats the central issue that Eq. (40) does not satisfy the Karmarkar condition. The language should be revised to match the actual derivation, whether or not the solution is re-derived.","section":"Concluding remarks"}],"recommendation":"major_revision","confidential_remarks":"The paper is within the scope of the journal and addresses a topic of current interest. The main concern is not the physical conclusion (exotic matter remains necessary), which is plausible, but the invalid Karmarkar-based derivation of the shape function. This issue is fixable in principle—by re-deriving the shape function with a legitimate parameter or by presenting Eq. (40) as a phenomenological ansatz—so I recommend major revision rather than rejection. The self-citation pattern (refs. [23,35,85,89]) is noticeable but does not affect my recommendation. The thermodynamic analysis needs a more careful treatment of the entropy and the interpretation of negative temperature."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. The paper is the first wormhole construction in F(Q,Lm,T) gravity that I know of, and the shape function in Eq. (40) is new. It also does a careful job with the energy-condition and ANEC plots, and the main physical conclusion—that exotic matter is still required—comes out of the field equations rather than being put in by hand. But the central derivation doesn't hold up. The authors solve the Karmarkar condition, find that the throat condition gives only a trivial solution, and patch it by adding a free parameter P. I checked the stress-test's algebra: substituting Eq. (40) into e^γ = 1/(1 − b/r) gives a function that cannot be written in the Karmarkar form e^γ = 1 + D e^σ σ'^2 for any constant D; the coefficient matching forces P = 0. So the wormhole is not embedding class-1. The paper is honest about the P insertion, but then the conclusion that the shape function was 'derived through the Karmarkar condition' is simply false. At best, Eq. (40) is a phenomenological ansatz, and the authors would need to say so and give P some physical or geometric justification.\n\nThere are other, smaller problems. The entropy S = 8πr^2 doesn't match the standard A/4 = πr^2, and no justification is offered. The field equations (24)–(26) are long and are presented without derivation or numerical checks, so I'd want those verified. And the abstract says the thermodynamic quantities 'support the thermal and equilibrium stability,' but the specific heat is positive only in a narrow band just outside the throat (Fig. 14 and Table 2); that's a narrower claim. None of these are fatal by themselves, but they add up.\n\nWhat's genuinely useful: the field equations in this specific gravity theory, and the demonstration that the NEC/ANEC are violated for this whole family of parameters. That's a coherent, if unsurprising, result. The citation pattern is fine—self-citations are to relevant prior wormhole work.\n\nRecommendation: this deserves a serious referee, but not as-is. A revision that re-derives the shape function without the Karmarkar claim, or fixes the derivation, addresses the entropy and stability overstatements, and verifies the field equations could be acceptable. I would not cite it in its current form.","headline":"The shape function in Eq. (40) is not a Karmarkar-derived solution—the added parameter P breaks the embedding condition—so the paper's central construction is an ansatz, not a derivation.","tokens_in":29584,"tokens_out":5941,"would_cite":false,"duration_ms":58543,"reading_group":"no","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.20.Jb","04.50.Kd"],"model":"deepseek-v4-flash","headline":"Constructing traversable wormholes in $\\mathcal{F}(Q,\\mathcal{L}_m,\\mathcal{T})$ gravity via the Karmarkar embedding condition, this paper finds that the null and averaged null energy conditions remain violated near the throat, so exotic…","keywords":["wormhole","F(Q,Lm,T) gravity","traversable wormhole","energy conditions","exotic matter","Karmarkar condition","non-metricity gravity","thermodynamic stability"],"falsifier":"Check the original Karmarkar shape function in Eq. (38) for a non-trivial throat radius: the paper asserts that $b(r_t)=r_t$ gives only the trivial solution, which is why $P$ is inserted. If a non-trivial root exists for any allowed parameter range, then the ad hoc parameter is unnecessary and the constructed solution is not the unique embedding-class-1 wormhole; if no non-trivial root exists, the entire construction rests on the unconstrained parameter $P$.","tokens_in":28426,"feed_emoji":"🕳️","tokens_out":7925,"duration_ms":82503,"temperature":0.7,"pith_summary":"This paper tries to establish that static, spherically symmetric traversable wormholes can be constructed in $\\mathcal{F}(Q,\\mathcal{L}_m,\\mathcal{T})$ gravity — an extended symmetric-teleparallel theory where the Lagrangian depends on non-metricity $Q$, the matter Lagrangian $\\mathcal{L}_m$, and the trace $T$ of the energy–momentum tensor — and that these wormholes still demand exotic matter. Using the Karmarkar embedding condition and the redshift function $\\Phi(r)=-2\\mu/r$, the authors derive a shape function that satisfies the throat, flaring-out, and asymptotic-flatness requirements. They then show that the null energy condition and the averaged null energy condition are violated near the throat for every parameter choice explored, which they read as evidence that the $\\mathcal{F}(Q,\\mathcal{L}_m,\\mathcal{T})$ framework does not remove the need for exotic matter. A thermodynamic analysis reports negative wormhole temperature, positive average pressure and work density, and a narrow radial band of positive specific heat just outside the throat, which the authors interpret as local thermal and equilibrium stability.","feed_headline":"Wormholes still demand exotic matter in F(Q,Lm,T) gravity","feed_subtitle":"Karmarkar-derived shape function plus one free parameter yields traversable wormholes, but null energy conditions still fail at the throat.","key_machinery":"The load-bearing object is the shape function of Eq. (40) with $0<P<r_t$, derived from the Karmarkar (embedding class-1) condition $R_{2323}R_{1414}=R_{1224}R_{1334}+R_{1212}R_{3434}$ under the redshift choice $\\Phi(r)=-2\\mu/r$. The Karmarkar condition alone gives $b(r)=r-r^5/(r^4+4\\mu^2 D e^{-2\\mu/r})$, for which the throat condition $b(r_t)=r_t$ has only the trivial solution; the paper adds the free parameter $P$ to make the throat condition hold by construction. All subsequent results — flaring-out plots, embedding diagrams, proper radial distance, energy-condition inequalities, the volume-integral quantifier, and the thermodynamic quantities — are computed from this shape function in the model $\\mathcal{F}(Q,\\mathcal{L}_m,\\mathcal{T})=Q+mQ^2+\\alpha \\mathcal{L}_m+\\beta T$.","core_discovery":"In the authors' own terms, the central discovery is a traversable wormhole solution in $\\mathcal{F}(Q,\\mathcal{L}_m,\\mathcal{T})$ gravity whose shape function, $b(r)=P - P r^5/[P r^4 + r_t^4(r_t-P)e^{2\\mu(1/r_t-1/r)}] + r$ (with $0<P<r_t$), is obtained by applying the Karmarkar condition to the redshift function $\\Phi=-2\\mu/r$ and then forcing the throat condition $b(r_t)=r_t$ through an inserted free parameter $P$. This shape function meets the geometric traversability criteria, yet the associated matter fluid violates the null, weak, strong, and dominant energy conditions near the throat; the volume-integral quantifier is negative, so the averaged null energy condition is also violated. The authors take this to show that exotic matter remains indispensable for sustaining traversable wormholes even in this extended gravity theory. They further report that the solution's Hawking and wormhole temperatures are negative, the average pressure and work density are positive, the total energy and energy flux are negative, and the specific heat is positive only in a narrow band immediately outside the throat, which they interpret as locally stable thermal equilibrium supported by exotic matter.","pith_inferences":["The ad hoc parameter $P$ is doing the real work: because the throat condition on the Karmarkar-derived shape function is trivial, every conclusion below depends on an unconstrained insertion. A physically motivated derivation of $P$ (for example, from junction conditions or from demanding a specific asymptotic mass) would turn this construction into a genuine prediction rather than a curve-fit.","The conclusion that exotic matter is unavoidable may be tied to the particular choices $\\Phi=-2\\mu/r$ and $f(Q)=Q+mQ^2$; other non-minimal couplings in $\\mathcal{F}(Q,\\mathcal{L}_m,\\mathcal{T})$ could in principle mimic exotic matter at the throat, so a systematic scan over $\\mathcal{F}$ forms would be a direct test of how general the claim is.","The reported stability is purely thermodynamic and local; it does not address dynamical stability against perturbations, which is the standard requirement for an astrophysically viable wormhole. A linear-perturbation analysis around this background would settle whether the narrow stability band survives."],"forward_implications":["If the central claim is right, then modifying the gravitational sector to $\\mathcal{F}(Q,\\mathcal{L}_m,\\mathcal{T})$ does not bypass the Morris–Thorne conclusion: any traversable wormhole in this theory requires matter that violates the null energy condition near the throat.","The volume-integral-quantifier result gives an explicit, parameter-dependent measure of the amount of exotic matter needed, so different choices of $\\alpha$, $\\beta$, $m$, and $P$ can be ranked by how much NEC-violating fluid they require.","The thermodynamic results imply a locally stable configuration only in a narrow band just outside the throat (the specific heat is positive for ranges such as $(1, 1.21]$ for $P=0.1$), with unstable behavior away from the throat and at the throat itself.","Negative wormhole and Hawking temperatures, positive work density, and negative total energy are each consistent with a metastable exotic-matter-supported equilibrium, so the model offers a concrete starting point for dynamical perturbation studies."],"supporting_citations":[{"why":"Morris–Thorne traversable wormhole metric and the throat, flaring-out, and asymptotic-flatness conditions that define the geometry.","marker":"[50]"},{"why":"Karmarkar condition used to derive the embedding class-1 shape function.","marker":"[94]"},{"why":"Eisenhart's Gauss–Codazzi embedding conditions that justify the class-1 technique.","marker":"[95]"},{"why":"Volume integral quantifier approach used to evaluate the averaged null energy condition violation.","marker":"[118]"},{"why":"Hong–Kim treatment of negative-temperature wormhole thermodynamics that the paper invokes for stability interpretation.","marker":"[78]"},{"why":"Starobinsky form $f(Q)=Q+mQ^2$ adopted as the non-linear model.","marker":"[93]"}],"fun_headline_variants":["Exotic matter still key for wormholes in F(Q,Lm,T) gravity","Extended gravity still can't bypass exotic matter for wormholes","F(Q,Lm,T) gravity fails to remove exotic matter in wormholes","Even F(Q,Lm,T) gravity can't save wormholes from exotic matter","Wormhole thermodynamics in F(Q,Lm,T): still exotic matter required"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Every conclusion depends on the shape function that is obtained by adding a free parameter $P$ to the Karmarkar-derived formula, with no physical justification or independent constraint on $P$; if that insertion is invalid, the wormhole solution and all of the paper's results collapse.","fun_headline_variants_meta":{"raw":{"variants":["Exotic matter still key for wormholes in F(Q,Lm,T) gravity","Extended gravity still can't bypass exotic matter for wormholes","F(Q,Lm,T) gravity fails to remove exotic matter in wormholes","Even F(Q,Lm,T) gravity can't save wormholes from exotic matter","Wormhole thermodynamics in F(Q,Lm,T): still exotic matter required"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000844,"raw_usage":{"total_tokens":3741,"prompt_tokens":1075,"completion_tokens":2666,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":691,"completion_tokens_details":{"reasoning_tokens":2567}},"tokens_in":691,"tokens_out":2666,"duration_ms":20177,"temperature":1.0,"reasoning_tokens":2567,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T17:59:52.222054+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Check the original Karmarkar shape function in Eq. (38) for a non-trivial throat radius: the paper asserts that $b(r_t)=r_t$ gives only the trivial solution, which is why $P$ is inserted. If a non-trivial root exists for any allowed parameter range, then the ad hoc parameter is unnecessary and the constructed solution is not the unique embedding-class-1 wormhole; if no non-trivial root exists, the entire construction rests on the unconstrained parameter $P$.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Morris–Thorne traversable wormhole metric and the throat, flaring-out, and asymptotic-flatness conditions that define the geometry."},{"cited_title":"Karmarkar in Proceedings of the Indian academy of sciences-se ction A, Vol","cited_arxiv_id":null,"evidence_quote":"Karmarkar condition used to derive the embedding class-1 shape function."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Eisenhart's Gauss–Codazzi embedding conditions that justify the class-1 technique."},{"cited_title":"Visser, S","cited_arxiv_id":null,"evidence_quote":"Volume integral quantifier approach used to evaluate the averaged null energy condition violation."},{"cited_title":"Hong, S.-W","cited_arxiv_id":null,"evidence_quote":"Hong–Kim treatment of negative-temperature wormhole thermodynamics that the paper invokes for stability interpretation."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Starobinsky form $f(Q)=Q+mQ^2$ adopted as the non-linear model."}],"review_version":1}