{"id":"b34b8f84-7343-4c37-adcb-0e4e38519372","arxiv_id":"2505.15169","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"On one non-degenerate family of teleparallel connections, the ghost-free condition requires 2c2+c5=0 in addition to the previously known b1=0.","lead":"This paper tests whether a previously found fix for a dangerous instability in a parity-violating modified gravity model still works on more general cosmic backgrounds. It finds one family of backgrounds where the old fix is insufficient and an extra constraint on the model's parameters is needed.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The extra ghost-free condition 2c2+c5=0 depends on the unshown reduction from the vector action (66) to the kinetic coefficient (68); an error in that Fourier-space algebra would invalidate the main new result.","rationale":"We checked the perturbation expansion of the connection, Eq. (33), against the appendix derivation and found it consistent. We also attempted an independent reduction from Eq. (66) to the kinetic coefficient: the leading large-k contribution plausibly originates from the (2c2+c5)F B_{j,i}B_k term after solving the B_A constraint, so the qualitative existence of a ghost is credible. However, the exact normalization and sign of z_A^2 in Eq. (68) are convention-sensitive and the paper does not display the decisive algebra. Since this coefficient is the sole basis for the paper's main new prediction (the extra ghost-free condition), the omission is load-bearing. The paper itself acknowledges in Sec. V that the additional condition is unnecessary if family 2 is not physically permissible, which further highlights that the quantitative content rests on this reduction. No definite error was found, but the missing derivation justifies a conditional verdict pending an independent check. This matches the reader's assessment and does not require changing the verdict.","tokens_in":14294,"tokens_out":58313,"duration_ms":441881,"concrete_test":"Use a symbolic algebra system to compute the quadratic vector action for background (29) from Eq. (66). Fourier-decompose B_i and E_i into circular-polarization modes, vary with respect to B_A to obtain the constraint, solve for B_A, substitute back, and extract the coefficient of |E'_A|^2. Verify that it equals Eq. (68), including the denominator 2a^2k + 4λ_A(2c2+c5)Fφ'^2. As a starting point, test the limit k→∞, where Eq. (68) gives z_A^2 ≈ λ_A(2c2+c5)Fφ'^2 k/2; a mismatch of sign or coefficient here would invalidate the ghost condition.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central new claim for background family 2 is that b1=0 is insufficient and the additional condition 2c2+c5=0 is required. This rests entirely on the step from Eq. (66) to Eqs. (67)-(68). The paper states the vector action (66) after 'tedious calculations' and then says 'conduct the same discussion as in the previous subsections', but does not display the Fourier decomposition, the constraint equation for B_A, the solution for B_A, or the substitution that yields z_A^2 in Eq. (68). A misprint in the B_{j,i}B_k coefficient in (66), a sign error in the helicity projection, or a missing factor in the constraint/solution would change the sign or form of z_A^2 and hence the ghost condition. The paper itself flags that the added condition is unnecessary if family 2 is not physical, so the quantitative content of the paper is exactly this reduction. A preliminary direct reduction indicates the leading large-k kinetic coefficient plausibly comes from the |B_A|^2 term in (66) after substituting B_A ≈ -E'_A, so the qualitative ghost is credible; however, the exact coefficient and denominator in (68) remain unchecked and must be independently verified.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a symmetric teleparallel gravity model with parity-violating couplings between gravity and a scalar field, extending earlier work by the same group. It classifies flat FRW background solutions into three families of affine connections, labelled degenerate, non-degenerate family 1, and non-degenerate family 2, according to the solutions of Eqs. (24)-(26). The paper then analyzes linear scalar, vector, and tensor perturbations around each family. It claims that scalar perturbations are unchanged from GR, tensor perturbations show velocity birefringence with no ghost, and vector perturbations are non-dynamical on the degenerate and non-degenerate family-1 backgrounds once the previous ghost-free condition b1=0 holds. The central new claim is that on non-degenerate family 2 the vector modes become dynamical and ghost-like unless an additional condition 2c2+c5=0 is imposed, and that the tensor dispersion relation acquires an additional F-dependent term.","tokens_in":14593,"tokens_out":29432,"duration_ms":260747,"significance":"If the central claim were correct, the paper would be a substantive extension of the authors' earlier ghost-free analysis: it would show that the previously derived condition b1=0 is background-dependent and is insufficient on a whole family of classically allowed cosmological backgrounds, thereby reducing the viable parameter space of the model. The background classification in Section III is a useful and internally consistent contribution, and the paper is honest about the assumption that the affine connection respects the cosmological principle. However, the main new result is undermined by a concrete algebraic issue: the F-dependent terms in the displayed quadratic actions on family 2 vanish identically by a vector identity. As written, the paper therefore does not establish that vector modes become dynamical on family 2 or that 2c2+c5=0 is needed. The manuscript also does not display the crucial Fourier-space reduction from the vector action to the claimed kinetic coefficient, so the central conclusion is not independently checkable.","major_comments":[{"comment":"The central claim is unsupported as written because the F-dependent terms that are supposed to generate the new ghost condition vanish identically. For any smooth vector field V, epsilon^{ijk} V_{j,i} V_k = V dot (curl V) = 0. Therefore the term (2c2+c5) F epsilon^{ijk} B_{j,i} B_k in Eq. (66) is identically zero, and the term (2c1+c4) F epsilon^{ijk} E_{j,i l} E_{k,l} is zero as well because, for each l, it equals (partial_l E) dot [curl(partial_l E)]. Once b1=0, the action in Eq. (66) is actually independent of F, so the claimed reduction to Eqs. (67)-(68) with z_A^2 proportional to (2c2+c5) F cannot follow. Unless these F terms are misprints with a genuinely different index structure, the paper's main new result is invalid.","section":"Sec. IV.D.3, Eqs. (65)-(66)"},{"comment":"The same vector identity invalidates the claimed F correction to the tensor dispersion on family 2. The term epsilon^{ijk} h_{jl,i} h_{kl} vanishes because, for each fixed l, it is V^l dot (curl V^l) with V^l_j = h_{jl}. Consequently the extra term -8 lambda_A F (2c1+c4) phi'^2/(a^2 k) in Eq. (64) is spurious as written, and the tensor perturbation result on family 2 should coincide with the degenerate and family-1 results. This is a secondary issue, but it indicates a systematic error in the reduction of the parity-violating terms on background (29).","section":"Sec. IV.D.2, Eqs. (61)-(64)"},{"comment":"Even if the F-dependent terms were not identically zero, the decisive algebra is omitted. The text says 'conduct the same discussion as in the previous subsections' and then jumps from the position-space action (66) to the Fourier-space result (67) with the specific coefficients z_A^2 and w_A^2 in (68) and (70). The revision must display the Fourier decomposition of Eq. (66), the constraint equation obtained by varying B_A, the explicit solution for B_A, the substitution back into the action, and the cancellation of the E'_A terms. Without these steps, the sign and the exact denominator in Eq. (68) cannot be verified by the reader.","section":"Sec. IV.D.3, Eqs. (66)-(68)"}],"minor_comments":[{"comment":"The scalar-field equation of motion is written as phi'' + 2 H phi' + a^2 phi = 0, but the potential term should be a^2 V_phi (the derivative of V with respect to phi) rather than a^2 phi.","section":"Sec. III.A, Eq. (30)"},{"comment":"The text refers to 'the constraint equation (46)', but the constraint equation is actually Eq. (45); Eq. (46) is the Fourier-mode expansion. This cross-reference should be corrected.","section":"Sec. IV.B.3, after Eq. (46)"},{"comment":"The statement that the hyper-potential sum vanishes automatically when U^rho is proportional to nabla^rho phi would benefit from a one-sentence explanation: the contraction with epsilon and the symmetry of the product of gradients makes the expression vanish. The current wording 'It is obvious' understates a step that is central to satisfying the connection equations.","section":"Sec. III.A, Eq. (19)"},{"comment":"Several quadratic actions are presented after the phrase 'after tedious calculations' (e.g., Eqs. (43), (56), and (65)). Given that the main new result depends on the detailed form of these actions, at least the key intermediate expressions for the perturbed non-metricity tensor should be provided or placed in an appendix so that the displayed actions can be checked.","section":"Secs. IV.B-IV.D"}],"recommendation":"reject","confidential_remarks":"The manuscript's central new condition is based on F-dependent terms in Eqs. (65)-(66) that are identically zero by epsilon^{ijk} V_{j,i} V_k = V dot (curl V) = 0. This is not a presentation gap; as written the claimed ghost-free condition 2c2+c5=0 is not derived. The authors may be able to correct the calculation if the displayed F terms are typos, but that would require a substantial recomputation of the perturbation actions on background family 2."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Colleague, here's my read on Tong's paper.\n\nThe genuinely new thing is the classification of three families of connection backgrounds and the discovery that on one of them, family 2, the previously derived ghost-free condition b1=0 is not enough; you also need 2c2+c5=0. The tensor sector on that family also gets a modified dispersion relation. That's a real extension of the author's earlier work, not a routine rerun.\n\nWhat the paper does well: the background classification from the curvature-free conditions is clean and internally consistent. The scalar and tensor sectors on the two 'safe' families match the earlier results, which is a good consistency check. The paper is also honest about its limitations: it states the connection is assumed to respect the cosmological principle (so the classification may not be exhaustive), and it explicitly says the new condition is unnecessary if family 2 is not physically admissible. That caveat is not buried.\n\nThe soft spot is where it matters: the step from the vector action (66) to the kinetic coefficient (68). The paper says 'conduct the same discussion as in the previous subsections' and does not display the Fourier decomposition, the constraint equation for B_A, its solution, or the substitution that gives z_A^2. The new ghost condition hangs entirely on that reduction. A sign error or a missed factor would change the condition. The stress-test note's preliminary direct reduction suggests the qualitative ghost is plausible, but the exact coefficient in (68) is unchecked. This is a genuine gap, though not necessarily fatal.\n\nThe other qualification is that the paper does not settle whether family 2 is physically allowed. If it isn't, the new condition is moot. So the result is conditional, but the condition is stated clearly.\n\nWho should read this: researchers working on symmetric teleparallel gravity, parity-violating gravity models, or cosmological perturbations in non-Riemannian geometries. It is a technical paper with a narrow audience.\n\nMy recommendation: send it to peer review. A referee with enough patience to redo the vector-sector algebra (or ask the author to expand it) can verify the central claim. The paper is coherent, the limitations are explicit, and the new condition is worth checking. I wouldn't cite it in my own work without an independent check, but it deserves a serious referee.","headline":"A substantive extension of a parity-violating STG model with a new ghost-free condition on one background family, but the decisive vector-sector algebra is compressed and needs verification.","tokens_in":15062,"tokens_out":3994,"would_cite":false,"duration_ms":32425,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.50.Kd","98.80.-k"],"model":"deepseek-v4-flash","headline":"This paper shows that the ghost-free condition previously derived for a parity-violating symmetric teleparallel gravity model must be supplemented by an additional coefficient relation, 2c2+c5=0, on one of the three allowed cosmological…","keywords":["symmetric teleparallel gravity","parity violation","ghost instability","cosmological perturbations","non-metricity","vector modes","modified gravity"],"falsifier":"Recompute the quadratic vector action on the non-degenerate family 2 backgrounds without imposing $b_1=0$ and explicitly integrate out $B_i$; if the resulting coefficient of $|E'|^2$ does not match Eq. (68), the additional condition $2c_2+c_5=0$ is an artifact. Alternatively, check whether family-2 backgrounds with $F\\neq0$ satisfy the full connection equations of motion when $U^\\rho$ is not assumed parallel to $\\nabla^\\rho\\varphi$.","tokens_in":14067,"feed_emoji":"🌌","tokens_out":6699,"duration_ms":57027,"temperature":0.7,"pith_summary":"The paper extends an earlier study of a symmetric teleparallel gravity model that adds parity-violating couplings between a scalar field and the non-metricity tensor to the Einstein-equivalent action. It identifies three families of flat FRW backgrounds consistent with the cosmological principle, each tied to a different affine connection, and analyzes linear scalar, vector, and tensor perturbations on each. The central result is that on two of the families the previously found ghost-free condition b1 = 2c1+2c2−c4−c5 = 0 remains sufficient, while on the third family the vector perturbations become dynamical and one polarization turns into a high-energy ghost unless an additional condition, 2c2+c5 = 0, is imposed. If correct, the viable parameter space of the model is smaller than previously thought whenever the third background family is physically admissible; the tensor sector still exhibits velocity birefringence with no ghosts on all three backgrounds.","feed_headline":"New background forces extra ghost-free condition on parity gravity","feed_subtitle":"On two of three allowed connections the old tuning holds; on the third it fails, shrinking the model's parameter space.","key_machinery":"The analysis rests on the three families of background non-metricity tensors derived from the cosmological-principle constraint $\\mathcal{L}_\\varsigma Q=0$ (Eq. 15), which force the non-metricity into the form (17) with three time-dependent functions $A,B,C$; solving the curvature-free and torsionless conditions (24–26) yields the degenerate family (27) and two non-degenerate families (28, 29). The perturbed connection is then expanded to second order as $\\Gamma^\\lambda_{\\mu\\nu}=\\bar\\Gamma^\\lambda_{\\mu\\nu}+\\bar\\nabla_\\mu\\bar\\nabla_\\nu u^\\lambda-\\bar\\nabla_\\mu\\bar\\nabla_\\nu u^\\rho\\bar\\nabla_\\rho u^\\lambda$ (Eq. 33), and the quadratic actions for scalar, vector, and tensor perturbations are computed in the 'coincident gauge on the perturbation level' (36). On family 2, the vector action (66) acquires $E_i$-dependent terms proportional to $F$ that, after solving the constraint for $B_i$, produce the kinetic coefficient $z^2_A = \\lambda_A (2c_2+c_5)\\bar\\varphi'^2 a^2 k^2 / (2 a^2 k + 4\\lambda_A (2c_2+c_5)\\bar\\varphi'^2)$ (Eq. 68), whose sign determines the ghost.","core_discovery":"The paper's central claim is that the consistency of this parity-violating symmetric teleparallel gravity model depends on which cosmological background the connection chooses. The author constructs three families of flat FRW background solutions from the requirement that the non-metricity tensor respect the cosmological principle, and shows that the ghost-free combination b1=2c1+2c2−c4−c5=0 — derived previously for the simplest, degenerate background — remains sufficient on the degenerate family and on the first non-degenerate family. On the second non-degenerate family, however, the vector modes of the metric become propagating and acquire a kinetic term whose sign flips for one circular polarization at high wavenumber, so a new condition, 2c2+c5=0 (with φ′ not zero), is needed to eliminate the ghost. The paper also finds that tensor perturbations stay ghost-free on all three backgrounds with helicity-dependent dispersion relations, and that scalar perturbations coincide with those of a minimally coupled scalar in GR.","pith_inferences":["A natural step not taken in the paper is to check whether the condition $2c_2+c_5=0$ also removes ghosts for non-FRW backgrounds, which would indicate a deeper structure in the parity-odd coefficient space.","If the early universe ever passed through a family-2 phase, the dynamical vector modes could source primordial vector perturbations or leave imprints in CMB polarization spectra — a testable consequence of taking the third background seriously.","The paper leaves open the fate of the seventh parity term that vanishes at quadratic order on FRW backgrounds; testing it on anisotropic or spherical backgrounds might reveal whether the three-family classification is part of a larger pattern.","A complementary derivation of the ghost condition directly from the equations of motion, rather than from the quadratic action, would independently confirm or refute the necessity of the extra condition."],"forward_implications":["If the third background family is physically allowed, the model's parameter space shrinks: parity-odd couplings must satisfy both $b_1=0$ and $2c_2+c_5=0$.","On the degenerate and first non-degenerate backgrounds the previously derived ghost-free condition is unchanged, so earlier phenomenological constraints on $c_1$ and $c_5$ remain valid there.","On all three backgrounds tensor perturbations are ghost-free and exhibit velocity birefringence, but the family-2 dispersion relation gains an extra $F$-dependent term that could distinguish the background through gravitational-wave observations.","Scalar perturbations are unaffected by the parity-violating terms at quadratic order, so the model keeps the standard GR-plus-scalar evolution for curvature perturbations.","If one judges the non-degenerate family 2 to be unphysical, the additional condition is unnecessary; otherwise it is essential to avoid ghost modes."],"supporting_citations":[{"why":"Defines the model and derives the ghost-free condition $b_1=0$ on the simplest background; this paper tests its universality.","marker":"[22]"},{"why":"Shows that the simplest parity-violating STEGR term gives velocity birefringence in tensor perturbations; the model under study extends this construction.","marker":"[20]"},{"why":"Finds that the simple parity-violating extension introduces ghost instability for vector perturbations, motivating the generalized couplings used here.","marker":"[21]"},{"why":"Establishes the two identities (7) that reduce the seven parity-odd terms to five independent ones.","marker":"[24]"},{"why":"Derives the cosmological-principle form of non-metricity (Eq. 16) used to classify the three background families.","marker":"[26]"},{"why":"Supplies the STEGR Bianchi identity used to drop Q-terms from the connection equations of motion.","marker":"[25]"}],"fun_headline_variants":["Ghost-free parity gravity needs extra condition on new background","New background adds condition to kill ghost in parity gravity","Parity gravity: third background requires extra ghost-free tuning","One background flips the ghost rule in parity-violating gravity","Extra ghost-free condition emerges on new parity gravity background"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result's load-bearing premise is that the second-order expansion of the perturbed connection (Eq. 33) and the unshown reduction from the vector action to the kinetic coefficient (68) are both correct, together with the assumption that the affine connection respects the cosmological principle.","fun_headline_variants_meta":{"raw":{"variants":["Ghost-free parity gravity needs extra condition on new background","New background adds condition to kill ghost in parity gravity","Parity gravity: third background requires extra ghost-free tuning","One background flips the ghost rule in parity-violating gravity","Extra ghost-free condition emerges on new parity gravity background"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000202,"raw_usage":{"total_tokens":1342,"prompt_tokens":864,"completion_tokens":478,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":480,"completion_tokens_details":{"reasoning_tokens":399}},"tokens_in":480,"tokens_out":478,"duration_ms":4572,"temperature":1.0,"reasoning_tokens":399,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T15:23:18.521928+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Recompute the quadratic vector action on the non-degenerate family 2 backgrounds without imposing $b_1=0$ and explicitly integrate out $B_i$; if the resulting coefficient of $|E'|^2$ does not match Eq. (68), the additional condition $2c_2+c_5=0$ is an artifact. Alternatively, check whether family-2 backgrounds with $F\\neq0$ satisfy the full connection equations of motion when $U^\\rho$ is not assumed parallel to $\\nabla^\\rho\\varphi$.","supporting_citations":[{"cited_title":"Gao and X","cited_arxiv_id":null,"evidence_quote":"Finds that the simple parity-violating extension introduces ghost instability for vector perturbations, motivating the generalized couplings used here."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Establishes the two identities (7) that reduce the seven parity-odd terms to five independent ones."},{"cited_title":"Aldrovandi and J","cited_arxiv_id":null,"evidence_quote":"Derives the cosmological-principle form of non-metricity (Eq. 16) used to classify the three background families."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the STEGR Bianchi identity used to drop Q-terms from the connection equations of motion."}],"review_version":1}