{"id":"ab361a54-97ba-4fb2-a1a2-6869590ced9a","arxiv_id":"1908.09310","paper_version":3,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"high","formal_verification":"none","parameter_count":1,"one_line_summary":"For Lovelock black holes, the Complexity=Action growth rate at late times is a coupling-independent multiple of the mass, and the Schwarzschild limit is recovered only up to a constant under the authors' boundary-term prescription.","lead":"This paper computes how holographic complexity grows over time for Lovelock black holes, using the 'Complexity=Action' conjecture. It reports that the late-time growth rate per unit mass is a coupling-independent constant, so the Schwarzschild limit is approached only up to a constant offset.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"GHY boundary-term ambiguity documented in the paper itself undermines the coupling-independent late-time rate as a robust CA prediction.","rationale":"The reader and I identify the same weakest assumption, which is independently flagged by the manuscript in footnote 5. The derivations leading to Eqs. (45)-(47) are internally coherent, and the numerical implementation appears careful. However, the central physical claim — a coupling-independent late-time complexity growth rate that does not reduce to Schwarzschild — depends on choosing surface term (36) over the alternative (footnote 5). That choice is not protected by any derivation in the paper, only by citation to [86]. The paper's own admission of a different rate from [90] makes the ambiguity factual rather than speculative. Therefore conditional acceptance is appropriate: the calculation is valid under a stated convention, but the physical robustness of the headline result is not established.","tokens_in":27300,"tokens_out":1035,"duration_ms":9636,"concrete_test":"Recompute the late-time action growth rate for D=5 Gauss-Bonnet black holes using the Davis boundary term [90] in place of Eq. (36), keeping all other CA conventions fixed; if the result is 4M/3π (as stated in footnote 5) rather than 2M, then Eq. (46) is convention-dependent and the headline claim about coupling-independent rates is not robust.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The paper's central late-time claim, Eq. (46) with coupling-independent Δ, rests entirely on the generalized GHY term (36). Section 4, footnote 5, concedes that the alternative, also-established GB boundary term of [90] yields dC/dt = 4M/3π instead of 2(D-3)M/(D-4) for the same black holes. The late-time rate is thus not a prediction of the CA proposal plus Lovelock gravity; it is an artifact of the chosen surface-term convention. Since Eq. (45) and Δ are derivable only through the Noether-charge/GHY combination, the non-smooth λ→0 limit (Δ→2M) likewise has no invariant status once the boundary term is ambiguous. The authors explicitly frame Δ as the GHY contribution in Eqs. (36)-(37), and their own inequality (62) exposes the order-of-limits issue without resolving the underlying convention problem.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies the full time dependence of holographic complexity, under the CA proposal, for neutral and charged Lovelock black holes. The authors first derive, in Section 3, a general formula for the action growth rate in higher-order gravities by combining the bulk Noether-charge identity (33), a generalized GHY boundary term (36), and the joint/counter-term actions. For planar Lovelock black holes this leads to Eq. (45) and the late-time result Eq. (46), where the asymptotic rate is the constant Δ coming from the GHY term at the singularity. They evaluate Δ for Gauss-Bonnet and third-order Lovelock black holes and find that Δ/M is a pure number independent of the higher-order couplings (e.g., Δ = 2(D-3)M/(D-4) for Gauss-Bonnet). The paper emphasizes that, in the vanishing coupling limit, this rate does not reduce to the Schwarzschild value, that the two nevertheless differ by a constant during the whole evolution, and that the late-time limit is approached from above, violating the conjectured upper bound. For charged Lovelock black holes the authors recover the known late-time rate (98) from reference [34] and study numerically the approach to the uncharged limit, finding a mismatch with the neutral case. Numerical results for the full time dependence are presented for several families of Lovelock black holes.","tokens_in":27467,"tokens_out":3195,"duration_ms":34445,"significance":"If the late-time rate were a robust prediction of the CA proposal for Lovelock gravity, the paper would provide an interesting extension of previous results [39] and a concrete, falsifiable statement about how higher-curvature corrections affect complexity growth. The derivation is coherent, uses no fitted parameters, and is cross-checked against the independent late-time result [34] for charged black holes. However, the central claim is not stable under the choice of surface term: the paper itself notes in Section 4, footnote 5, that an alternative established Gauss-Bonnet GHY term [90] gives a different late-time rate (4M/3π instead of Δ = 2(D-3)M/(D-4)). The coupling-independent constant and the non-smooth Einstein limit are therefore properties of the convention (36), not robust CA predictions. This caveat does not destroy the value of the paper as a careful derivation under a specific prescription, but it requires the conclusions to be reframed and the ambiguity to be addressed in the main text.","major_comments":[{"comment":"The paper's central late-time claim, Eq. (46) with the coupling-independent Δ, rests entirely on the generalized GHY term (36). Footnote 5 concedes that the alternative, also-established Gauss-Bonnet boundary term of reference [90] gives dC/dt = 4M/3π for the same black holes, rather than Δ = 2(D-3)M/(D-4) from Eq. (61). This means that the constant Δ, the non-smooth λ→0 limit, and the statement that the rate 'does not reduce to Schwarzschild' are not invariants of the CA proposal but depend on the chosen surface-term convention. This issue should be moved from the footnote into the main text, and the corresponding claims in the abstract and Section 7 should be explicitly qualified as holding within the prescription (36).","section":"Section 4, footnote 5"},{"comment":"The argument that the non-commutativity of the limits ϵ→0 and λ→0 makes the result 'mathematically sound' explains the origin of the mismatch but does not resolve the physical ambiguity: the same order-of-limits reasoning applies to the alternative GHY term [90], which would give a different value of Δ. To make the central claim robust, the paper needs a physical criterion for selecting the surface term (36) over other admissible boundary terms, or it must explicitly present the result as one possible convention. Without such a criterion, Eq. (46) cannot be described as the complexity growth rate of Lovelock black holes.","section":"Section 4, Eq. (62)"},{"comment":"The statement that the difference from the Schwarzschild result is 'a constant during the whole time evolution' inherits the same surface-term ambiguity. Because Δ enters Eqs. (45), (61), and (77), and because all numerical plots in Sections 4 and 5 normalize by Δ, the curves in Figs. 4, 7, 8, and 9 would be different, including the constant offset from the Schwarzschild curve, if the boundary term of [90] were used. The convention-dependence of these results should be acknowledged explicitly wherever they are summarized.","section":"Abstract and Section 7"}],"minor_comments":[{"comment":"The numerical procedure is described only verbally; no code or detailed discretization/error estimates are provided. Given that the central results are numerical, it would improve reproducibility to include pseudocode or state the integration method and tolerances.","section":"Section 3.5 and Section 6.3"},{"comment":"The arbitrary normalization α of the null normals affects the logarithmic term in Eq. (45) and hence the early-time results such as Eqs. (70) and (81). The text states this only implicitly; it should be stated explicitly which observables (e.g., the late-time rate Δ) are α-independent and which early-time curves depend on the choice α.","section":"Section 3.4, Eqs. (43)-(45)"},{"comment":"The sentence 'In this section, we further examine the complexity growth rate for charged Lovelock black holes' appears in the introduction of Section 5, but the charged analysis is in Section 6; this organization error should be corrected.","section":"Section 5, paragraph 1"},{"comment":"The concluding paragraph states that the paper studied 'third order Lovelock gravities' and then refers to 'section 6' for the organizational structure; the previous organization lists section 6 as the charged case and section 7 as the conclusion. The cross-reference should be updated for consistency.","section":"Section 7, Conclusion"},{"comment":"The caption reports 'the complexity difference δC = C(t)−C(tc)' but the surrounding text and the left panel indicate that the plotted quantity is the rate dC/dt; please verify that the captions correctly label each panel.","section":"Section 4, Fig. 2 caption"}],"recommendation":"major_revision","confidential_remarks":"The boundary-term ambiguity is a well-known subtlety in the CA proposal for higher-curvature theories. The authors are aware of it (footnote 5) but bury it in a footnote and do not confront its consequences for the main claims. I believe the paper can be made publishable by reframing the conclusions as prescription-dependent, adding a discussion of the physical selection of the GHY term, and addressing the remaining presentation issues. I do not think rejection is warranted, because the Noether-charge derivation is coherent and the charged-section result (98) is robust. However, the abstract's unqualified headline claim about the 'characteristic constant' is not supported as stated."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Dear colleague,\n\nYou should know two things about Fan–Liang (arXiv:1908.09310). First, it does something genuinely useful: it derives a general Noether-charge formula for the time-dependent CA complexity of higher-curvature black holes and applies it to Lovelock, including Gauss–Bonnet and third-order Lovelock, neutral and charged. That extends the Einstein-gravity treatment of Carmi et al. and gives a clean framework. Second, the paper's headline claim — a coupling-independent late-time complexity rate, e.g. 2(D−3)M/(D−4) for GB — is not a robust prediction of the CA proposal as such. It is a consequence of the specific generalized GHY boundary term they adopt in Eq. (36). They document in footnote 5 that the alternative, also established GB boundary term of Davis gives 4M/3π instead. So the \"non-smooth λ→0 limit\" they emphasize is a property of their convention, not an invariant statement.\n\nWhat is actually good: the bulk action reduction via the Wald–Iyer identity is elegant; the resulting Eq. (45) is compact and should be useful. The charged late-time rate reproduces the earlier independent result of Cano–Hennigar–Marrochio, a genuine cross-check. The numerical work is described carefully, covers several families, and the early-time expansions match the numerics. No fitted parameters appear; the free normalization α affects early-time expressions but not late-time ones, a known CA feature rather than a new flaw.\n\nWhere it is soft: the GHY ambiguity is the main issue, and it is central rather than cosmetic. The late-time rate is the paper's headline result, so readers should not treat it as a unique CA prediction until the boundary-term question is settled. There is also no released code or independent numerical verification, though the algorithm is clearly specified. I call that minor.\n\nWho is it for? People working on holographic complexity in higher-curvature gravity. I would cite it for the general formula and for the charged-sector completeness, with a caveat on the convention dependence. It deserves a serious referee: the derivation is solid on its own terms and the ambiguity is honestly reported; a referee can require a fuller discussion of which GHY term is correct.\n\nMy recommendation: send it to peer review.","headline":"Useful general Noether-charge derivation of time-dependent CA complexity for Lovelock black holes, but the headline coupling-independent late-time rate is GHY-convention-dependent and the paper itself documents an alternative boundary term giving a different rate.","tokens_in":27964,"tokens_out":4098,"would_cite":true,"duration_ms":38281,"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":"The paper argues that for Lovelock black holes, Complexity=Action predicts a late-time complexity growth rate that is a coupling-independent constant multiple of mass and does not reduce to Schwarzschild when couplings vanish.","keywords":["holographic complexity","Complexity=Action","Lovelock black holes","Gauss-Bonnet gravity","Wheeler-DeWitt patch","action growth rate","Noether charge","higher-curvature corrections"],"falsifier":"Compute the late-time complexity growth for a Gauss-Bonnet black hole using the alternative boundary term cited in the paper's footnote 5 instead of Eq. (36); the paper reports this gives $4M/3\\pi$ rather than $(D-3)M/((D-4)\\pi)$, so choosing one boundary term decides whether the coupling-independent $\\Delta$ is physical.","tokens_in":27121,"feed_emoji":"🕳️","tokens_out":9217,"duration_ms":84873,"temperature":0.7,"pith_summary":"This paper asks how holographic complexity grows with time under the Complexity=Action proposal when the bulk gravity is a Lovelock theory rather than Einstein gravity. By combining the Wald-Iyer Noether charge with a generalized Gibbons-Hawking-York boundary term, it derives the full action growth rate and shows that at late times the rate is a fixed multiple of the black hole mass, independent of the higher-order couplings: $2(D-3)M/(D-4)$ for Gauss-Bonnet and $6M$ for third-order Lovelock in seven dimensions. Because that multiple is not Schwarzschild's $2M$, the growth rate does not reduce to the Einstein result when the couplings vanish, even though the metric and action do; the two normalized rates differ by a constant through the whole evolution. The paper also shows that the late-time limit is approached from above, that the critical time for the onset of growth decreases with the couplings, and that charged Lovelock black holes show a related mismatch between their uncharged limit and the neutral result.","feed_headline":"Complexity growth in Lovelock black holes is a fixed multiple of mass","feed_subtitle":"Higher-curvature corrections leave the late-time rate a constant that differs from Schwarzschild.","key_machinery":"The machinery has three pieces. The Noether-charge identity $\\partial_r(\\sqrt{-g}Q^{rt})=\\sqrt{-g}L$ rewrites every bulk-action integral as boundary terms, so time derivatives of the action become values of a temperature function times a Wald entropy function at the moving joints. The generalized Gibbons-Hawking-York boundary term, Eq. (36), evaluated at the singularity contributes the constant $\\Delta$. The joint term at the moving corner $r_m$, whose motion is fixed by $t=-2r_*(r_m)$, contributes the logarithmic term; the Wald entropy function $\\hat{S}(r)$ runs through both bulk and joint pieces and cancels except for the derivative term that vanishes at late times.","core_discovery":"The central claim is that for neutral Lovelock black holes the full time evolution of the gravitational action on the Wheeler-DeWitt patch is captured by $$\\frac{dI_{\\rm grav}}{dt} = \\$\\Delta$ - \\frac{\\hat{S}'(r_m)}{4\\pi}|f(r_m)|\\log\\left(\\frac{|f(r_m)|}{\\$alpha^{2}$}\\right),$$ where $r_m(t)$ is the past joint of the patch and $\\hat{S}$ is the Wald entropy function. At late times $r_m\\to r_h$, the logarithm vanishes, and the growth rate becomes $dI_{\\rm grav}/dt = \\Delta$. Evaluating the generalized Gibbons-Hawking-York term at the future singularity gives $\\Delta=2(D-3)M/(D-4)$ for Gauss-Bonnet black holes and $\\Delta=6M$ for the third-order Lovelock case, so $\\Delta/M$ is a pure number with no dependence on the higher-order couplings. Since the limits $\\lambda\\to 0$ and $r\\to 0$ do not commute, the $\\lambda\\to0$ limit of $\\Delta$ is not the Schwarzschild value $2M$; the paper shows numerically that the normalized growth rates differ by a fixed constant for the entire evolution after the critical time.","pith_inferences":["If a fundamental principle later fixes the boundary-term convention, the coupling-independent $\\Delta$ would become a sharp Complexity=Action prediction distinguishing it from other complexity proposals; until then, the absence of a smooth Einstein limit should be read as convention-dependent.","The derivation assumes a single central singularity, so Lovelock solutions with an alternative finite-radius singularity may behave differently; testing those backgrounds would show whether coupling independence survives.","The observed inequality $T_+\\le T_-$ for charged Lovelock black holes, if universal, predicts that all such charged solutions approach late-time complexity growth from above, a property that can be checked in other charged geometries.","The early-time growth laws, constant for five-dimensional Gauss-Bonnet and logarithmic for higher dimensions, could be compared with boundary circuit-complexity computations in CFTs with finite $N$ corrections, where Lovelock couplings correspond to subleading $1/N$ effects."],"forward_implications":["At late times, the action and therefore complexity growth rate for neutral Lovelock black holes is $\\Delta/\\pi$, with $\\Delta$ a coupling-independent constant multiple of the mass.","In the vanishing-coupling limit, the normalized growth rate differs from Schwarzschild by a fixed constant for every time after the critical time, so perturbative treatments around Einstein gravity miss a finite offset.","Because the next-to-leading-order late-time correction is positive, the growth rate reaches its limiting value from above, violating any conjectured upper bound set by that late-time rate.","The critical time $t_c$ measured in thermal time decreases with the Lovelock couplings, so higher-curvature corrections make complexity start growing earlier than in Schwarzschild.","For charged Lovelock black holes, the uncharged limit approaches the universal rate $2M/\\pi$, yet this remains offset from the neutral black hole's rate by a constant over the whole evolution."],"supporting_citations":[{"why":"Defines the Complexity=Action proposal: complexity equals the bulk action on the Wheeler-DeWitt patch, the duality being tested.","marker":"[9, 10]"},{"why":"Establishes the Einstein-gravity time-dependence framework and the Schwarzschild/Reissner-Nordstrom baselines against which the Lovelock results are compared.","marker":"[39]"},{"why":"Gives the generalized Gibbons-Hawking-York boundary term used to extract the late-time rate $\\Delta$.","marker":"[86]"},{"why":"Provide the Wald-Iyer Noether charge formalism and Wald entropy used in the bulk-boundary identity.","marker":"[83, 84]"},{"why":"Show that the bulk action can be rewritten as Noether-charge boundary terms, the identity at the core of the derivation.","marker":"[48, 49]"},{"why":"Supply the joint or corner term for higher-order gravities that produces the logarithmic time-dependent term.","marker":"[35, 87]"},{"why":"Earlier Lovelock complexity-growth result for charged black holes whose late-time universal rate the paper reproduces and extends.","marker":"[34]"},{"why":"Presents an alternative generalized GHY boundary term that gives a different late-time rate, marking the main convention dependence.","marker":"[90]"}],"fun_headline_variants":["Lovelock complexity rate is a fixed fraction of mass, independent of couplings","Late-time complexity in Lovelock black holes: constant per mass, not Schwarzschild","Higher-curvature effects absent in late-time complexity growth rate","Complexity growth in Lovelock: coupling-independent constant ratio to mass"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"Everything rests on Eq. (36) being the correct generalized Gibbons-Hawking-York boundary term for Lovelock gravity; the paper notes that another established boundary term yields a different late-time growth rate.","fun_headline_variants_meta":{"raw":{"variants":["Lovelock complexity rate is a fixed fraction of mass, independent of couplings","Late-time complexity in Lovelock black holes: constant per mass, not Schwarzschild","Higher-curvature effects absent in late-time complexity growth rate","Complexity growth in Lovelock: coupling-independent constant ratio to mass"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.00046,"raw_usage":{"total_tokens":2370,"prompt_tokens":1077,"completion_tokens":1293,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":693,"completion_tokens_details":{"reasoning_tokens":1212}},"tokens_in":693,"tokens_out":1293,"duration_ms":8959,"temperature":1.0,"reasoning_tokens":1212,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T11:15:23.304538+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the late-time complexity growth for a Gauss-Bonnet black hole using the alternative boundary term cited in the paper's footnote 5 instead of Eq. (36); the paper reports this gives $4M/3\\pi$ rather than $(D-3)M/((D-4)\\pi)$, so choosing one boundary term decides whether the coupling-independent $\\Delta$ is physical.","supporting_citations":[],"review_version":1}