{"id":"73cc8fa7-6108-4f4f-a6ac-8d56af132bc0","arxiv_id":"2505.00112","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":5.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":10,"one_line_summary":"For a chosen parameter range, a massless tensor perturbation of a Euclidean scalar-tensor theory obeys a Lorentzian dispersion relation, while extra scalar/vector modes remain Euclidean or ghost-like and require boundary suppression.","lead":"A theory built on a purely Euclidean (space-like, no-time) manifold can, with the right parameters, produce a gravitational wave mode that behaves as if it lived in ordinary Lorentzian spacetime. The other modes in the same theory are not Lorentzian and must be silenced by boundary conditions, a step the authors argue for but do not prove.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Boundary suppression of non-Lorentzian modes is asserted, not proved; the calculation has no signature-change surface, and nonlinear sources can regenerate the modes.","rationale":"Good-faith reading: the explicit tensor-sector calculation in Eq. (12), together with conditions (14)–(15), does exhibit a massless mode obeying ω² = (µ2/µ3)k² with positive kinetic term in the long-distance limit. That part is plausible and deserves credit. The conditional aspect is the further claim required for the Euclidean theory to behave as an effective Lorentzian theory: all unwanted modes are suppressed. The paper itself labels this as an argument rather than a proof ('we argue'), and the reader's CONDITIONAL verdict is appropriate. My stress test converges on the same weak point, and adds a concrete mechanism: nonlinear couplings generically source the supposedly suppressed modes, and the boundary used in the suppression argument is not present in the actual constant-gradient background. The factor-of-two discrepancy between Eq. (34) and Eqs. (43)–(44) is a separate internal algebra issue; it does not change the sign of G′_1 for the sample parameters, so it does not by itself overturn the tensor existence claim, but it shows that the scalar-sector computation should be checked before the suppression claim is accepted. These considerations do not move the verdict: the result should remain CONDITIONAL pending an explicit boundary/matching treatment or a softened claim, plus the algebraic cleanup.","tokens_in":11599,"tokens_out":12370,"duration_ms":140028,"concrete_test":"Compute the cubic part of S′ in (6) on the background (8) and isolate the term that sources B at order h², e.g. h(B_y²+B_z²) or hψ². Then solve the linearized vector equation (16) on a slab −L < t < L with B = 0 on both boundaries and a nonzero tensor wavepacket h. If the sourced B is nonzero, the boundary-suppression claim fails at first nonlinear order. If the coupling vanishes by symmetry, recompute the determinant expansion (30) at O(k²) and resolve the factor-of-two discrepancy between Eq. (34) and Eq. (43).","verdict_should_be":"UNCHANGED","load_bearing_attack":"The most load-bearing step is not the tensor-sector calculation but the assertion (Sec. IV B: 'Euclidean tachyonic modes can be easily controlled by suitable boundary conditions'; Sec. IV C; Sec. V: 'we argue') that all non-Lorentzian modes—vector B, massive scalars, and the harmonic scalar ψ—can be set to zero by boundary conditions. This is not established. The perturbation analysis is performed on an infinite, constant-gradient Euclidean background (8), where no signature-change surface exists, so the boundary invoked in the argument is not part of the solved problem. In a finite region bounded by the degenerate effective metric, boundary data are fixed by matching conditions across the signature-change surface, not freely chosen. The quadratic action alone also cannot justify suppression at nonlinear order: the cubic terms of S′ generically couple h to B and ψ, so a nonzero tensor perturbation sources the elliptic equations for the unwanted modes, and zero Dirichlet data does not force a zero bulk solution when a source is present. An internal inconsistency reinforces the need for repair: for the stated sample parameters, Eq. (34) gives G′_1 = 2/5, while Eqs. (43)–(44) and the tabulated value give 1/5; the sign is unchanged, but the scalar-sector algebra is not yet reliable.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies a shift-symmetric scalar-tensor theory on a Euclidean-signature manifold, with a clock field phi, and asks whether the theory can, in the long-distance limit, produce degrees of freedom obeying Lorentzian dispersion relations. Around the flat background (8) with constant phi-gradient, the authors expand the action to quadratic order, separate the tensor, vector, and scalar sectors, and derive dispersion relations. The main positive result is in Sec. III B: under conditions (14)-(15), the tensor modes h_+ and h_x acquire a massless Lorentzian dispersion relation omega^2 = (mu2/mu3) k^2 with a positive kinetic term. The vector and scalar sectors are argued to be Euclidean or tachyonic and therefore suppressible by boundary conditions, leaving the tensor mode as the only long-distance Lorentzian degree of freedom. The paper presents two explicit parameter sets as existence demonstrations and concludes that a massless tensor degree of freedom can, with an appropriate choice of parameters, satisfy a Lorentzian dispersion relation.","tokens_in":11887,"tokens_out":9638,"duration_ms":100243,"significance":"If the central claim is sound, the paper provides an explicit proof-of-principle that a Euclidean, renormalizable gravity theory can yield a Lorentzian tensor dispersion relation at low momenta. The tensor-sector calculation is explicit and self-contained, and the conditions (14)-(15) are transparent; the paper is honest that this is an existence demonstration rather than a prediction. The significance is reduced, however, by two problems: the scalar-sector algebra contains an apparent numerical inconsistency and undefined symbols, and the claim that all non-Lorentzian modes can be suppressed by boundary conditions is asserted rather than demonstrated. Because the abstract and Sec. V present this suppression as part of the scenario, the paper's broader claim is not yet established even though the tensor-sector result appears sound.","major_comments":[{"comment":"For the sample parameters lambda=eta=alpha0=1, q=gamma0=3, l=2 used in Eq. (36), the formula for G'_1 in Eq. (34) gives G'_1 = -2 X0 mu2 nu1/(mu3 mu7) = 2/5, while the reduced Lagrangian in Eq. (43) gives G'_1 = - X0 nu1 mu2/(mu3 mu7) = 1/5 under the same dispersion convention as Eq. (30). The value tabulated in Eq. (36) is 1/5, so the two derivations are inconsistent by a factor of two. This is not a typographical nuance: the determinant method of Sec. IV B and the explicit integration-out of Sec. IV C must agree. Please locate the source of the discrepancy and recompute the scalar-sector sample values before the scalar-sector conclusions can be accepted.","section":"Sec. IV B, Eq. (34), Eq. (36), Eq. (43)"},{"comment":"The scalar-sector formulas are not reproducible as printed. Equation (34) uses the symbol mu9, which is not defined in Table I or anywhere in the text, and Table I itself contains two entries labeled mu6 with different expressions. Because G'_3 and G'_4 are presented numerically in Eqs. (36)-(37) and used to support the claim that all remaining modes are Euclidean or tachyonic, these missing or duplicate definitions block verification of a load-bearing part of the analysis. Please define every symbol and re-derive the scalar-sector matrix algebra, checking the duplicate entries.","section":"Sec. IV B, Eq. (34), Table I"},{"comment":"The suppression of the non-Lorentzian modes is asserted, not proved. The perturbation calculation is performed on the infinite, constant-gradient background (8), where there is no boundary and no signature-change hypersurface; the 'suitable boundary conditions' invoked for the vector modes after Eq. (16), for the scalar psi after Eq. (44), and in Sec. V are therefore not part of the solved problem. No explicit boundary-value construction is given, and the argument based on the quadratic action alone does not control nonlinear effects: cubic and higher-order terms in S' will couple h to B, psi, and the massive scalars, so a nonzero tensor perturbation can act as a source for the elliptic equations of the unwanted modes. This is load-bearing because the abstract and Sec. V present suppression as the mechanism by which the effective low-energy sector is Lorentzian. Please either construct the boundary-value problem, including the matching conditions across the degenerate effective-metric surface, or explicitly weaken the claim to the existence of a Lorentzian tensor sector within the unrestricted perturbation space.","section":"Sec. IV B, Sec. IV C, Sec. V"}],"minor_comments":[{"comment":"The equation lists G'_3 twice; the second entry should read G'_4.","section":"Eq. (36)"},{"comment":"The table contains two entries labeled mu6; please give the second one a distinct subscript so that formulas in the text referring to mu6 are unambiguous.","section":"Table I"},{"comment":"The text refers to vector degrees of freedom 'Bx and By', but the metric perturbation (9) contains By and Bz; the labels should be corrected.","section":"Sec. III C"},{"comment":"The expression for G'_2 is ambiguous as typeset; please use parentheses to show that the numerator is -3Z + X0(2 beta0 + gamma0) and the denominator is 3 mu3.","section":"Eq. (34)"},{"comment":"The phrase 'the massless dispersion relation in Eq. (34)' is ambiguous because Eq. (34) contains several dispersion-relation formulas; please refer explicitly to the formula for G'_1.","section":"Sec. IV C, text after Eq. (43)"}],"recommendation":"major_revision","confidential_remarks":"The tensor-sector result appears sound and could support a publishable proof-of-principle paper. The main risks are the scalar-sector algebra (factor-of-two discrepancy, undefined mu9, duplicate mu6) and the unproved boundary-suppression argument. I would not reject on the present conditional claim, but the authors need to fix the scalar-sector formulas and either provide a boundary-value construction or substantially weaken the suppression claim in the abstract and conclusions."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"I'll give it to you straight. The paper does a real calculation: the first SVT decomposition of the renormalizable ELST action from Mukohyama (2013) and shows that, on a flat background with constant clock gradient, the tensor mode gets a massless Lorentzian dispersion for a clean parameter region (eqs. 14-15). That is the new thing, and it is done carefully. The tensor-sector derivation is explicit, the parameter conditions are clear, and the paper is honest that the result is conditional and limited to linear order on flat space. It also correctly flags the scalar-sector dichotomy: the massless scalar psi is either a Lorentzian ghost or a harmonic function, depending on the sign of mu7.\n\nThe soft spots are where the reader guessed. The boundary-suppression argument in Secs. IV-V is asserted, not proved. The perturbation is on an infinite, constant-gradient background with no signature-change surface, so the boundary being invoked is not part of the solved problem. And even if you add Dirichlet data on some surface, the cubic terms in the action couple h to psi and the vector modes, so a nonzero tensor source regenerates the unwanted modes at nonlinear order. That is a genuine gap, and the paper should either fill it or soften the wording. The 'in principle' in the abstract is doing real work.\n\nThere is also a concrete algebra inconsistency: for the sample parameters below Eq. (35), Eq. (34) gives G'_1 = 2/5, while Eq. (43) and Eq. (36) give 1/5. The sign is unchanged, so the conclusion survives, but it suggests the heavy scalar algebra has not been fully cross-checked. Add to that the duplicate G'_3 in Eq. (36) and the repeated mu_6 in Table I, and you have a paper that needs a careful referee pass.\n\nThe circularity concern is overblown. The action is taken from prior work, but the calculation is not derived from the result; it is a perturbation analysis of a given theory. That is normal. The self-citations here point to where the action came from, which is legitimate.\n\nFor whom: anyone working on emergent time, signature change, or Euclidean quantum gravity. It is a concrete existence mechanism, not a full theory. Should an editor send it to referees? Yes. It is a real, nontrivial calculation with a clear claim, and the weaknesses are fixable. I would ask for a revised version that either proves the boundary suppression or explicitly restricts the conclusion to linear order with specified boundary data, and that fixes the G'_1 discrepancy.","headline":"A concrete tensor-sector existence result with an unproved mode-suppression argument; worth refereeing after cleanup.","tokens_in":12396,"tokens_out":3637,"would_cite":true,"duration_ms":35631,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"A Euclidean scalar-tensor theory can admit massless tensor modes with Lorentzian dispersion relations, so the low-energy gravity sector would look exactly Lorentzian.","keywords":["emergent Lorentz signature","clock field","Euclidean gravity","scalar-tensor theory","tensor perturbations","dispersion relations","tachyonic modes","boundary conditions"],"falsifier":"On a bounded Euclidean region around the flat background, with the sample parameters $\\lambda=\\eta=\\alpha_0=1$, $q=\\gamma_0=3$, $l=2$, impose zero boundary values for the vector modes and for $\\psi$, then solve the linearized equations; if any nonzero bulk mode survives, or if the nonlinear system develops such modes starting from generic data on one side of the signature-change surface, the suppression claim fails.","tokens_in":11383,"feed_emoji":"⏳","tokens_out":13252,"duration_ms":115682,"temperature":0.7,"pith_summary":"This paper asks whether a theory formulated on a Euclidean (all-plus-signature) manifold can nevertheless produce degrees of freedom that obey the Lorentzian rules of space and time. It studies small perturbations of a renormalizable shift-symmetric scalar-tensor theory around a flat background in which a clock scalar field has a constant nonzero gradient. It finds a parameter regime where the tensor perturbations, the analogues of gravitational waves, obey the massless Lorentzian dispersion relation $\\omega^2 = (\\mu_2/\\mu_3) k^2$ with a positive kinetic term. The remaining modes satisfy Euclidean dispersion relations or have large tachyonic masses, and the paper argues they can be suppressed by boundary conditions. The upshot is an explicit existence scenario in which time and light cones are emergent, not fundamental.","feed_headline":"Euclidean gravity can still produce Lorentzian gravitational waves","feed_subtitle":"A clock-field gradient gives long-wavelength tensor modes the same wave equation as gravity in a Lorentzian spacetime","key_machinery":"The central object is the quadratic perturbation action for the theory, split into tensor, vector, and scalar sectors. The load-bearing identity is the low-momentum dispersion relation $\\omega^2 = (\\mu_2/\\mu_3) k^2$ for tensor modes, with $\\mu_3 = X_0(\\beta_0+\\gamma_0)+Z$ controlling the kinetic sign and $\\mu_2 = X_0\\gamma_0 - Z$ controlling the spatial-gradient term; their product must be positive for Lorentzian propagation. For the scalar sector, the machinery is the matrix method in which the equation-of-motion matrix $E = \\omega^2 K + i\\omega(M-M^T) + V$ has determinant proportional to the product of dispersion relations, and expanding $\\det E$ in powers of $k^2$ yields the masses $m^2_a$ and effective gradient factors $G'_a$ that determine whether each degree of freedom is Lorentzian, Euclidean, or tachyonic.","core_discovery":"Expanding the Euclidean scalar-tensor action around the flat background with $\\bar\\varphi = \\sqrt{X_0}\\,t$, the quadratic tensor action is, in the long-distance limit $k \\ll M_{\\rm Pl}$, that of a massless field with dispersion relation $\\omega^2 = (\\mu_2/\\mu_3) k^2$. The conditions $\\mu_3 = X_0(\\beta_0+\\gamma_0)+Z > 0$ and $\\mu_2\\mu_3 = (\\gamma_0 X_0 - Z)\\mu_3 > 0$ ensure a positive kinetic term and a Lorentzian, rather than Euclidean, propagation. For the scalar sector, the paper computes the low-momentum gradient factors $G'_a$ from the determinant of the equation-of-motion matrix; with sample parameters such as $\\lambda=\\eta=\\alpha_0=1$, $q=\\gamma_0=3$, $l=2$, the massless scalar has $G'_1 = 1/5$ while the tensor mode has positive kinetic term. The paper concludes that a massless tensor degree of freedom can, with an appropriate choice of parameters, satisfy a Lorentzian dispersion relation, and it argues that the leftover Euclidean, tachyonic, and harmonic modes can be set to zero by boundary conditions.","pith_inferences":["We infer that the suppression of the harmonic scalar $\\psi$ is the fragile step: in a finite region with generic boundary data, the boundary value of $\\psi$ cannot necessarily be tuned to zero independently of the data sourcing the tensor modes, so a bounded-domain analysis may reveal residual Euclidean modes.","We infer a concrete observable consequence: if matter couples to the effective metric of the theory, the ratio of gravitational-wave speed to photon speed would generically equal $\\sqrt{\\mu_2/\\mu_3}$, so a precise measurement of that ratio would bound the parameter combination.","We infer that the full suppression argument could be tested by studying the coupled linearized system on a compact Euclidean region with the signature-change surface included; existence and uniqueness of solutions with zero boundary data would settle whether the Euclidean modes stay absent."],"forward_implications":["Long-wavelength tensor perturbations of the Euclidean theory would be indistinguishable from massless gravitational waves on a Lorentzian spacetime, so a low-energy observer could not tell that the underlying geometry has Euclidean signature.","The sector is not fully Lorentz invariant unless the scalar and vector modes are actually removed: the massless scalar is either a ghost with a Lorentzian dispersion relation or a harmonic function, and the paper argues the harmonic option is preferable because its boundary value can be set to zero.","Euclidean tachyonic modes with large masses are acceptable because they obey elliptic equations and are suppressed in the bulk once set to zero at the boundary, so they never propagate.","If the mechanism extends to curved backgrounds with curvature scales below the momenta considered here, the same Lorentzian tensor dispersion would hold for black-hole-like geometries, a direction the paper identifies for future work.","Adding matter coupled to the effective metric would force the tensor speed and matter speed to match if Lorentz invariance is to be recovered, which would impose additional constraints on the parameters."],"supporting_citations":[{"why":"introduces the clock-field mechanism by which a scalar gradient changes the effective metric signature; the present background relies on it.","marker":"[2]"},{"why":"defines the shift-symmetric Emergent Lorentz Signature Theory action and its parameter relations that the perturbation analysis starts from.","marker":"[6]"},{"why":"establishes renormalizability of the higher-derivative Euclidean theory, motivating its use as a UV-complete setting.","marker":"[7]"},{"why":"provides the first-order formalism used to put the scalar-sector quadratic action into kinetic-plus-mass form.","marker":"[13]"},{"why":"extends that formalism with the Lagrange-multiplier treatment that lets the paper integrate out auxiliary and massive scalar fields.","marker":"[14]"},{"why":"supplies the general quadratic decomposition of the scalar Lagrangian into kinetic, mass, and friction matrices used throughout the analysis.","marker":"[19]"}],"fun_headline_variants":["Euclidean gravity yields Lorentzian tensor modes","Scalar field trick gives Lorentzian waves from Euclidean theory","Lorentzian dispersion emerges from a Euclidean scalar-tensor theory","How a Euclidean theory can mimic Lorentzian gravity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The result rests on the assumption that every non-Lorentzian degree of freedom (the tachyonic vector and scalar modes and the harmonic scalar $\\psi$) can be set to zero by boundary conditions and that nonlinear effects or the signature-change surface do not regenerate them; the paper explicitly labels this an argument rather than a proof.","fun_headline_variants_meta":{"raw":{"variants":["Euclidean gravity yields Lorentzian tensor modes","Scalar field trick gives Lorentzian waves from Euclidean theory","Lorentzian dispersion emerges from a Euclidean scalar-tensor theory","How a Euclidean theory can mimic Lorentzian gravity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000638,"raw_usage":{"total_tokens":2936,"prompt_tokens":936,"completion_tokens":2000,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":552,"completion_tokens_details":{"reasoning_tokens":1935}},"tokens_in":552,"tokens_out":2000,"duration_ms":16882,"temperature":1.0,"reasoning_tokens":1935,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-16T04:52:22.476793+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On a bounded Euclidean region around the flat background, with the sample parameters $\\lambda=\\eta=\\alpha_0=1$, $q=\\gamma_0=3$, $l=2$, impose zero boundary values for the vector modes and for $\\psi$, then solve the linearized equations; if any nonzero bulk mode survives, or if the nonlinear system develops such modes starting from generic data on one side of the signature-change surface, the suppression claim fails.","supporting_citations":[{"cited_title":"0 0 β0k2 √ 2 1 2 −k2κ1 6λ −ηk2 8 √ 2 0 0 # , C =","cited_arxiv_id":null,"evidence_quote":"introduces the clock-field mechanism by which a scalar gradient changes the effective metric signature; the present background relies on it."},{"cited_title":"Signature change events: A challenge for quantum gravity?","cited_arxiv_id":"0812.3744","evidence_quote":"defines the shift-symmetric Emergent Lorentz Signature Theory action and its parameter relations that the perturbation analysis starts from."},{"cited_title":"Wald, General Relativity (University of Chicago Press, Chicago, 1984); S","cited_arxiv_id":null,"evidence_quote":"provides the first-order formalism used to put the scalar-sector quadratic action into kinetic-plus-mass form."},{"cited_title":"A new quasidilaton theory of massive gravity","cited_arxiv_id":"1410.1996","evidence_quote":"extends that formalism with the Lagrange-multiplier treatment that lets the paper integrate out auxiliary and massive scalar fields."},{"cited_title":"Aurilia and H","cited_arxiv_id":null,"evidence_quote":"supplies the general quadratic decomposition of the scalar Lagrangian into kinetic, mass, and friction matrices used throughout the analysis."}],"review_version":1}