{"id":"8aeb8a4e-8d4c-458e-9c20-c0637b471380","arxiv_id":"2505.09049","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"In Einstein-Gauss-Bonnet gravity the ECO compactness scale shifts by order alpha divided by r0 squared, and in Einstein-dilaton-Gauss-Bonnet gravity by order alpha squared divided by r0 to the fourth.","lead":"This paper calculates how higher-curvature corrections to gravity shift the compactness scale of extremely compact objects, the distance at which an object without a horizon still mimics a black hole. The corrections scale as the coupling divided by the horizon radius squared in Einstein-Gauss-Bonnet gravity and as its square in dilaton-Gauss-Bonnet gravity.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The EdGB scaling in Eqs. (16)-(17) is algebraically inconsistent: expanding (16) with the stated zeta and r0 relations does not produce the 1 - alpha^2/(GM)^4 factor claimed in (17), so the central alpha^2/r0^4 result is not yet established.","rationale":"The reader correctly identified the heuristic nature of the scaling derivation and the possible inconsistency between Eqs. (16) and (17). My stress-test goes further: the inconsistency is concrete and checkable. Eq. (17) does not follow from Eq. (16) by any valid approximation, because the denominator (1 - 4GM/(3r0)) is order one at the horizon and cannot be dropped. This is load-bearing because the EdGB alpha^2/r0^4 result is half of the abstract's central claim. However, I do not recommend REJECT because the qualitative ordering may still be correct: EdGB corrections indeed begin at order zeta^2 in the metric, so the leading correction to s_c could still scale as alpha^2/r0^4. The issue is the specific coefficient and the displayed formula, which the paper does not derive. A precise re-derivation could settle it. The verdict remains CONDITIONAL: the paper should be accepted only if the EdGB expansion is corrected or if the derivation is provided explicitly.","tokens_in":4950,"tokens_out":2835,"duration_ms":25394,"concrete_test":"Independently re-derive the EdGB compactness scale by: (a) writing Eq. (16) with zeta = alpha/(GM)^2 and r0 = (2 - zeta^2)GM; (b) expanding the full expression to O(alpha^4) in alpha, keeping all denominators; (c) comparing the resulting coefficient to the 1 - alpha^2/(GM)^4 claimed in Eq. (17). If the coefficient is not exactly -1/(GM)^4, then Eq. (17) is wrong and the claimed alpha^2/r0^4 scaling needs revision. Also verify whether A2(r0) can be set to 1: evaluate the explicit A2 polynomial at r0 ≈ 2GM and check whether the omitted terms are numerically small.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that EdGB corrections enter at alpha^2/r0^4. This rests entirely on Eq. (16) and its expansion in Eq. (17). I checked the algebra: Eq. (16) contains a factor 1 - zeta^2 G^2 M^2 / (2 r0^2) divided by (1 - 4GM/(3r0)), all raised to power 1/4, with zeta = alpha/(GM)^2. Evaluating at the EdGB horizon r0 = (2 - zeta^2)GM, the denominator 1 - 4GM/(3r0) is not close to 1: GM/r0 is order unity. Expanding (16) consistently to order alpha^4 does not yield 1 - alpha^2/(GM)^4; it yields a different coefficient that also depends on the order-one factor from the denominator. Eq. (17) appears to ignore both the 1/2 in the numerator and the entire denominator, which is not a legitimate approximation. Also, the metric in Eq. (5) is written with A2(r) and the paper says it sets A2(r0) ≈ 1, but this is not justified: A2(r0) contains terms at order GM/r0 which are not small. Thus the alpha^2/r0^4 statement is currently unsupported by the displayed equations. The paper itself explicitly labels the derivation as 'heuristic', and it relies on scaling relations from prior work that are not re-derived for the corrected metrics; but the concrete algebraic inconsistency is the more specific and testable issue.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This essay-style paper claims to derive corrections to the compactness scale of horizonless extremely compact objects (ECOs) in Einstein-Gauss-Bonnet (EGB) gravity and Einstein-dilaton-Gauss-Bonnet (EdGB) gravity. Using the heuristic collapse condition g_tt(R_ECO,M_ECO)=0 together with vacuum-energy scaling relations from earlier work by Mathur and Mehta, the author writes down compactness scales s_c^{EGB} and s_c^{EdGB} in Eqs. (14) and (16), expands them in the coupling alpha, and concludes that the corrections are of order alpha/r0^2 in EGB and alpha^2/r0^4 in EdGB (D=4). The abstract emphasizes that observational constraints on alpha could make these corrections relevant for astrophysical ECO candidates.","tokens_in":5213,"tokens_out":21308,"duration_ms":173445,"significance":"If the claimed scaling were established, this would be a useful extension of the universal-thermodynamics program for ECOs to higher-derivative gravity, with a concrete prediction for how the near-horizon structure of horizonless objects deviates from the GR-ECO compactness scale. The qualitative ordering (alpha/r0^2 versus alpha^2/r0^4) is plausible on dimensional grounds, since the EGB metric is corrected at order alpha while the EdGB metric first changes at order zeta^2 = alpha^2/(GM)^4. However, the manuscript is an essay that relies almost entirely on a heuristic derivation from previous work; the central formulas (14)-(17) are asserted without derivation, and the displayed algebra contains concrete inconsistencies. The paper does not provide machine-checked derivations, numerical checks, or parameter-free predictions beyond the scaling itself. Its value as it stands is therefore prospective rather than established.","major_comments":[{"comment":"The central formulas for s_c^{EGB} and s_c^{EdGB} are introduced with no derivation. The text says 'Using the heuristic derivation for higher-order gravity models, we derive...' but does not show how the collapse condition (9), the vacuum-energy scaling (12), and the modified metrics (3) or (5)-(6) are combined to produce Eqs. (14) and (16). This is a load-bearing gap because all quantitative claims rest on these expressions. Additionally, Eq. (16) fails a basic consistency check: for zeta=0 the EdGB metric reduces to Schwarzschild, but Eq. (16) with the stated r0≈(2-zeta^2)GM gives a denominator 1-4GM/(3r0)≈1/3, so s_c^{EdGB}(zeta=0) is 3^{1/4} times the standard GR compactness scale. A correct formula must reduce to s_c^{GR} when the coupling vanishes.","section":"Section 3.3, Eqs. (14) and (16)"},{"comment":"The expansion of Eq. (16) into Eq. (17) is algebraically inconsistent. Using the paper's own horizon r0=(2-zeta^2)GM and keeping terms through O(zeta^2), the fourth root of the ratio in Eq. (16) gives 3^{1/4}(1+7zeta^2/32), and sqrt(r0/l_p) gives sqrt(2GM/l_p)(1-zeta^2/4). Combining these yields s_c ≈ 3^{1/4} sqrt(2M l_p) (1 - zeta^2/32) (for G=l_p^2), i.e. a prefactor 3^{1/4} and a coefficient -1/32 in front of alpha^2/(GM)^4, not the claimed 1 - alpha^2/(GM)^4. The denominator 1-4GM/(3r0) is O(1) (about 1/3) and cannot be dropped or approximated by 1, and the numerator gives a zeta^2/8 contribution, not a coefficient of unity. Therefore the displayed alpha^2/r0^4 statement in Eq. (17) is not supported by Eq. (16).","section":"Section 3.3, Eqs. (16)-(17)"},{"comment":"The approximation A2(r0)≈1, used to simplify Eq. (16), is not justified by the displayed A2(r). With r around 2GM, the terms in A2 evaluate to 1 + 13 + 66/20 + 96/40 - 5 ≈ 14.7, so A2(r0) is of order ten, not close to one. Since Eq. (16) explicitly depends on setting A2(r0)=1, the omitted O(1) terms in A2 can change the O(zeta^2) coefficient of the compactness-scale correction. This issue must be addressed before the EdGB result can be accepted.","section":"Section 2.2 and Section 3.3"},{"comment":"Eq. (14) has a dimensional inconsistency. The factor (r0^d/M)^{1/(d+1)} already has dimensions of length (because M has dimension 1/length in natural units), so multiplying by the explicit l_p makes the right-hand side dimensionally length^2. This is also inconsistent with the first expression in Eq. (15), whose prefactor (M/m_p)^{2/((d+1)(d-2))} l_p has dimension length. The explicit l_p in Eq. (14) should be removed (or the prefactors in Eq. (15) corrected consistently).","section":"Section 3.3, Eq. (14)"}],"minor_comments":[{"comment":"The term 'α4G^2M^2' should be typeset as '4αG^2M^2' to avoid ambiguity.","section":"Eq. (3)"},{"comment":"Reference [6] has a typo in the title: 'Einstein-dilation-Gauss-Bonnet' should be 'Einstein-dilaton-Gauss-Bonnet'.","section":"References"},{"comment":"The scaling relations δr∝s^2 and Evac∝1/s^{d-1} are asserted without derivation and with only a pointer to prior work; since the modified-gravity analysis depends on these relations holding for the corrected metrics, the paper should at least state the assumptions explicitly.","section":"Section 3.2"},{"comment":"The treatment of O(1) factors is loose: Eq. (15) uses r0∼(GM)^{1/(d-2)} while Eq. (16) relies on r0≈(2-zeta^2)GM, and the text also says 'used r0∼GM' for EdGB. This ambiguity matters because, for example, the denominator 1-4GM/(3r0) in Eq. (16) changes from about 1/3 to a negative value if r0 is approximated as GM rather than 2GM.","section":"Sections 2.2 and 3.3"},{"comment":"The caption for Fig. 1 is present in the text, but the figure itself does not appear in the manuscript; please ensure the figure is included in the final version.","section":"Figure 1"}],"recommendation":"major_revision","confidential_remarks":"The essay has a plausible qualitative message, but the displayed equations contain concrete algebraic and dimensional errors, and the central formulas are not derived. These issues are fixable in principle, so I would not recommend rejection; however, the authors need to present a complete derivation of Eqs. (14)-(17), correct the alpha=0 limit of Eq. (16), and justify or remove the A2(r0)≈1 approximation. The paper's current form is not suitable for publication as a research article."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The useful thing to know: this is a short, clearly written essay that combines the Mathur-Mehta ECO compactness scale with known perturbative black hole solutions in Einstein-Gauss-Bonnet and Einstein-dilaton-Gauss-Bonnet gravity. The EGB result, corrections of order alpha/r0^2, is a straightforward perturbative expansion and looks fine. The EdGB claim, corrections of order alpha^2/r0^4, is genuinely new but it is not established by the paper as written.\n\nWhat the paper does well: it states the setup economically, cites the relevant perturbative metrics (Boulware-Deser for EGB; Mignemi-Stewart/Yunes-Stein for EdGB), and is honest that the derivation is heuristic. The EGB scaling in Eqs. (14)-(15) is plausible and consistent with the structure of the metric. The references are relevant, and the self-citations are to the framework being used, not padding.\n\nThe soft spot is central. The stress-test note is right: Eq. (16) does not expand to Eq. (17). The denominator (1 - 4GM/3r0) is order one at the horizon, and the numerator has a 1/2 that is dropped. A consistent expansion does not produce the clean 1 - alpha^2/(GM)^4 factor. Also, setting A2(r0) ≈ 1 is not justified: at r0 ~ 2GM, A2 is of order 15. That is not a small correction. So the headline EdGB scaling is unsupported by the displayed algebra. The paper's own heuristic label does not excuse an internal inconsistency, and the underlying scaling relations from prior work are not re-derived for the corrected metrics.\n\nThe abstract's statement that these effects could be significant in certain astrophysical systems is also unsupported; no system or quantitative estimate is given.\n\nFor whom: someone working on ECOs and modified gravity who wants a back-of-the-envelope guess for where Gauss-Bonnet corrections might enter the compactness scale. Not for someone needing a reliable prediction.\n\nRecommendation: I would not rely on the EdGB result as is. The question is legitimate and the EGB part is likely salvageable, so a serious referee should see it rather than a desk reject. But the author needs to provide a correct derivation or revise the claim. As it stands, the paper should not be accepted without fixing the algebra in Section 3.3.","headline":"A readable, well-scoped essay that puts known higher-derivative black hole metrics into the Mathur-Mehta ECO compactness framework, but the EdGB correction it advertises does not follow from the displayed equations.","tokens_in":5761,"tokens_out":3414,"would_cite":false,"duration_ms":36589,"reading_group":"maybe","serious_thinker":"no","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"In Gauss-Bonnet gravity, the ECO compactness scale shifts by order alpha/r0^2, and by alpha^2/r0^4 in EdGB gravity.","keywords":["extremely compact objects","compactness scale","Gauss-Bonnet gravity","Einstein-dilaton-Gauss-Bonnet gravity","higher-curvature corrections","near-horizon structure","black hole microstructure","gravitational wave constraints"],"falsifier":"Directly compute the vacuum-energy scaling and the proper-distance relation $\\delta r(s)$ in the corrected EGB and EdGB metrics to next order; if either acquires an $\\alpha$-dependent correction, the factors $1-\\frac{4}{d+1}\\alpha/r_0^2$ and $1-\\alpha^2/r_0^4$ are incomplete. A numerical construction of a static horizonless ECO in these theories with the same surface conditions would settle whether the collapse condition actually yields the claimed scales.","tokens_in":4683,"feed_emoji":"🕳️","tokens_out":9452,"duration_ms":85398,"temperature":0.7,"pith_summary":"This paper asks whether horizonless alternatives to black holes—extremely compact objects (ECOs)—retain their characteristic near-horizon size when the underlying gravity theory includes higher-curvature corrections. Working in Einstein-Gauss-Bonnet (EGB) gravity in $D>4$ dimensions and in Einstein-dilaton-Gauss-Bonnet (EdGB) gravity in $D=4$ dimensions, it derives the corrected compactness scale $s_c$, the proper distance from the would-be horizon at which the ECO's near-horizon structure lives. The result is a relative shift of order $\\alpha/r_0^2$ in EGB gravity and of order $\\alpha^2/r_0^4$ in EdGB gravity, where $\\alpha$ is the Gauss-Bonnet coupling and $r_0$ is the horizon radius for a black hole of the same mass. Since observational limits allow $\\alpha$ up to a few square kilometers, these corrections can be sizable for compact astrophysical systems, turning the near-horizon geometry into a possible probe of higher-derivative gravity.","feed_headline":"Higher-curvature gravity shifts the ECO compactness scale","feed_subtitle":"Gauss-Bonnet terms move the near-horizon scale by alpha/r0^2 in EGB gravity and by alpha^2/r0^4 in EdGB gravity.","key_machinery":"The central object is the compactness scale $s_c$, the proper distance from the ECO surface at $R_{\\rm ECO}=r_0+\\delta r$ to the horizon radius $r_0$ of a black hole with the same mass. In GR the scale follows from three ingredients: the collapse condition $g_{tt}(R_{\\rm ECO},M_{\\rm ECO})=0$ with $M_{\\rm ECO}=M+E_{\\rm vac}$; the near-horizon scaling of the vacuum energy $E_{\\rm vac}\\propto s^{-(d-1)}$ in $D=d+1$ spacetime dimensions; and the Rindler relation $\\delta r\\propto s^2$ between coordinate and proper distance. The paper applies these same three ingredients to the corrected EGB and EdGB metrics, then expands the resulting equation in powers of the coupling $\\alpha$ to obtain the corrected scales (14)-(17).","core_discovery":"The central claim is that the compactness scale of an ECO is not fixed once and for all by the GR-based heuristic: it inherits a theory-dependent correction from higher-curvature terms. Inserting the small-coupling EGB metric (Eq. (3)) into the collapse condition $g_{tt}(R_{\\rm ECO}, M_{\\rm ECO})=0$ gives $s_c^{\\rm EGB}\\sim (r_0/\\ell_p)^{2/(d+1)}\\,\\ell_p\\,[1-\\frac{4}{d+1}\\alpha/r_0^2+\\cdots]$. Inserting the leading EdGB correction (Eqs. (5)-(6)) into the same condition gives $s_c^{\\rm EdGB}\\sim (r_0/\\ell_p)^{1/2}\\,\\ell_p\\,[1-\\alpha^2/r_0^4+\\cdots]$, with the first correction appearing only at order $\\alpha^2$ because the EdGB field equations are first corrected at that order. The paper argues that these shifts are potentially observable under current constraints on $\\alpha$ and that the same derivation can be applied to other modified-gravity models.","pith_inferences":["The pattern suggests a rule: the order in $\\alpha$ of the first compactness correction matches the order at which the theory's metric is first corrected—linear for EGB, quadratic for EdGB—so other higher-curvature models can be classified by the same criterion.","The reliability of the corrected scales hinges on the unstated assumption that the vacuum-energy scaling $E_{\\rm vac}\\propto s^{-(d-1)}$ and the Rindler relation $\\delta r\\propto s^2$ survive the Gauss-Bonnet deformation of the metric; if they do not, the claimed $\\alpha$-scalings would be replaced by whatever the corrected near-horizon geometry dictates.","Because the compactness scale controls the depth of the near-horizon well, a shifted $s_c$ would propagate into the thermodynamic and radiation properties of ECOs derived in the GR framework, providing an indirect observational handle beyond direct metric measurements.","A direct numerical construction of a horizonless solution with the corrected EGB/EdGB metric, rather than the perturbative expansion around GR, would test whether the corrected scale is robust or an artifact of the leading-order expansion."],"forward_implications":["In EGB gravity the ECO surface sits closer to or farther from $r_0$ by a relative amount $\\frac{4}{d+1}\\alpha/r_0^2$, so the shift is largest for low-mass ECOs with small $r_0$.","In EdGB gravity the first shift is suppressed by $\\alpha^2/r_0^4$, so deviations from the GR scale are much smaller at fixed coupling, and detecting them requires larger $\\alpha$ or higher-precision near-horizon probes.","The two theories predict different powers of $\\alpha$ in the compactness correction, so a measurement sensitive to the scale could distinguish EGB from EdGB corrections rather than merely bounding $\\alpha$.","The same derivation is stated to extend to other modified-gravity models such as $f(R)$ gravity, yielding an ECO compactness scale for each theory."],"supporting_citations":[{"why":"Defines the ECO compactness scale $s_c$ and supplies the heuristic collapse-condition derivation that the paper modifies.","marker":"[7]"},{"why":"Provides the small-coupling EGB black-hole metric used to compute the corrected scale in Eq. (14).","marker":"[8]"},{"why":"Supplies the leading-order EdGB metric functions $A_2$ and $B_2$ used for the $D=4$ corrected scale.","marker":"[10]"},{"why":"Corroborates the explicit form of the EdGB metric functions used in Eqs. (5)-(6).","marker":"[11]"},{"why":"Provides the observational bound on the EdGB coupling that makes the $\\alpha^2/r_0^4$ correction potentially significant.","marker":"[6]"}],"fun_headline_variants":["Gauss-Bonnet shifts ECO compactness scale","Higher-derivative gravity revises ECO compactness","EGB and EdGB corrections move ECO scale","ECO compactness: higher-curvature corrections matter"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The derivation assumes that the GR-based collapse condition—setting $g_{tt}$ to zero at the ECO radius with the vacuum-energy-inflated mass—remains valid unchanged once Gauss-Bonnet corrections modify the metric; Section 3.2 asserts the relations $E_{\\rm vac}\\propto s^{-(d-1)}$ and $\\delta r\\propto s^2$ for the corrected geometries without re-deriving them.","fun_headline_variants_meta":{"raw":{"variants":["Gauss-Bonnet shifts ECO compactness scale","Higher-derivative gravity revises ECO compactness","EGB and EdGB corrections move ECO scale","ECO compactness: higher-curvature corrections matter"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00018,"raw_usage":{"total_tokens":1296,"prompt_tokens":932,"completion_tokens":364,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":548,"completion_tokens_details":{"reasoning_tokens":315}},"tokens_in":548,"tokens_out":364,"duration_ms":3842,"temperature":1.0,"reasoning_tokens":315,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T21:41:42.306293+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Directly compute the vacuum-energy scaling and the proper-distance relation $\\delta r(s)$ in the corrected EGB and EdGB metrics to next order; if either acquires an $\\alpha$-dependent correction, the factors $1-\\frac{4}{d+1}\\alpha/r_0^2$ and $1-\\alpha^2/r_0^4$ are incomplete. A numerical construction of a static horizonless ECO in these theories with the same surface conditions would settle whether the collapse condition actually yields the claimed scales.","supporting_citations":[],"review_version":1}