{"id":"875f46e9-5941-4576-95f5-e0e6bdb988a4","arxiv_id":"2608.02715","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":4.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":1,"one_line_summary":"Along holographic RG flows, the acceleration of spread complexity and the covariant c-function are algebraically related: inversely in fixed-dimension domain walls and Dp-branes, co-monotonically in twisted compactifications.","lead":"The paper compares the acceleration of Krylov spread complexity with a covariant central charge along holographic RG flows, finding inverse relations in fixed-dimension flows and co-monotonic relations in compactification flows. It aims to show that complexity acceleration is a geometric diagnostic of how degrees of freedom are depleted or reorganised during renormalisation.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The paper's central claim rests on a dictionary (dCK/dt equals proper momentum) proven only in AdS3/CFT2 locally excited states, and the authors explicitly flag it as an extension; a failure of that dictionary would collapse the U–ccov relations into metric-function identities.","rationale":"The reader's weakest_assumption identifies exactly the same load-bearing concern: the proper-momentum/spread-complexity dictionary, established only for AdS3/CFT2 locally excited states, is being used as an unproven extension in higher-dimensional and non-conformal backgrounds. The paper itself acknowledges this in Section 6 ('established sharply for a particular class of AdS3/CFT2 states [5]; its use in more general top-down, non-conformal backgrounds is a physically motivated extension'), which confirms that the concern is a genuine gap, not a stylistic objection. The algebraic content of the paper is the derivation of relations between U and ccov; those relations are internally consistent and cleanly derived, and the paper does not claim a field-theoretic proof of the dictionary. The central claim—that U diagnoses RG reorganization of degrees of freedom—collapses to a statement about metric functions if the dictionary fails, because U would no longer be the acceleration of a boundary Krylov complexity. The concrete test I propose is the most direct settlement: derive the first Lanczos coefficients from the boundary spectral data (survival amplitude) of a specific flow and compare with the bulk prediction in Appendix A. This is a finite, computational check. If it succeeds in one nontrivial flow, the conditionality is lifted substantially; if it fails, the paper's main interpretation is unsupported. I therefore keep the verdict CONDITIONAL, in agreement with the reader, rather than moving to ACCEPT or REJECT, because the concern is about the interpretation layer rather than the internal algebraic consistency, and an honest non-finding is not possible given the explicitly flagged extension.","tokens_in":22415,"tokens_out":1949,"duration_ms":19261,"concrete_test":"Choose a flow with a known or computable boundary dual at strong coupling, e.g., the GPPZ flow or the compactified Klebanov-Witten flow, and independently derive the first few Lanczos coefficients (b1, b2, a1) from the boundary survival amplitude, using the moment problem and the recursion in Eq. (A.2). Then compare with the bulk prediction in Eq. (A.3): b1^2 = (m/4) sqrt(A(rUV)) U(rUV), and its higher-order generalization. If the boundary-derived b1, b2, a1 match the bulk-derived values (after fixing the single mass parameter m), the dictionary is supported and the central claim stands. If they disagree, the extension of [5] to these flows is invalid and the U–ccov relations must be reinterpreted as geometric identities.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The central claim is that the normalized second derivative U of Krylov spread complexity is physically related to the covariant c-function. Every algebraic relation in Sections 3–5 follows from Eq. (2.13), which itself follows from Eqs. (2.10)–(2.11): dCK/dt equals the proper radial momentum. The authors explicitly state in Section 6 that the proper-momentum/spread-complexity equality is established only for a class of AdS3/CFT2 locally excited states [5], and that its use in non-conformal, top-down, higher-dimensional backgrounds is a physically motivated extension. Thus the paper's strongest claim is conditional on an unproven dictionary. If the dictionary fails, U is merely a combination of metric functions A'(r)/sqrt(A^3 B), and the 'co-monotonicity reversal' and 'exponent 8' results reduce to kinematic properties of geodesics in the bulk, telling us nothing about boundary Krylov complexity or RG irreversibility. The manuscript itself flags this limitation (Section 6), so the concern is not manufactured. The weakest point is not any internal inconsistency—the metric manipulations are algebraically sound—but the leap from a proven AdS3 result to arbitrary flows, including flows across dimensions. Appendix A computes the short-time Lanczos coefficient b1^2 from U, but this is a one-sided check: it expresses bulk data in terms of boundary Lanczos data without deriving the Lanczos sequence from the dual QFT. The proposed test below would close that gap, at least in a controlled example.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"The manuscript studies Krylov spread complexity along holographic RG flows, following the proposal of Caputa et al. that the growth rate of spread complexity is the proper radial momentum of an infalling massive probe. It defines a normalized acceleration U(t) = (2/m) sqrt(A(r_UV)) \\ddot{C}_K(t) and compares it with the covariant holographic c-function c_cov of Ref. [34]. The paper derives explicit relations for Lorentz-invariant domain-wall flows (c_cov = 1/(G_N U^{d-1})), for the Dp-brane family (c_cov U^8 = constant), and for two flows across dimensions (compactified Klebanov-Witten and wrapped M5), where c_cov and U are found to be co-monotonic. The authors interpret the fixed-dimension anti-correlation as depletion of degrees of freedom and the across-dimension co-monotonicity as reorganization of degrees of freedom. The paper explicitly flags that the proper-momentum/spread-complexity dictionary is established only for a class of AdS3/CFT2 states and that its use in more general backgrounds is an extension.","tokens_in":22722,"tokens_out":10523,"duration_ms":95558,"significance":"If the dictionary holds, the paper provides a simple and elegant geometric link between the acceleration of Krylov spread and a holographic central function, with explicit monotonicity statements and universal exponents. The metric manipulations are internally consistent, the paper is transparent about validity windows (Dp-branes, singular endpoints, internal-angle choices), and Appendix A gives short-time Lanczos formulas that could serve as falsifiable checks. The central physical claim, however, is conditional on an unproven extension of the proper-momentum dictionary, and the across-dimensional interpretation would require an independent boundary-side calculation of an IR central charge. As a set of algebraic relations among geometric quantities, the paper is sound; as a statement about boundary Krylov complexity, it is not yet established.","major_comments":[{"comment":"The equality dC_K/dt = proper radial momentum is imported from Ref. [5], where it was derived for a class of AdS3/CFT2 locally excited states. The present paper applies it to ten- and eleven-dimensional, non-conformal, and across-dimensional flows and explicitly labels this a physically motivated extension. Because every relation in Sections 3-5 is an algebraic consequence of U = A'/sqrt(A^3 B), if this dictionary fails then U is merely a combination of metric functions and none of the statements about boundary Krylov complexity follow. The conditional nature is acknowledged, but the central claim is not yet supported. I would ask for at least one independent boundary-side consistency check, for example using Eqs. (A.3)-(A.4) to compute b_1^2 and the combination 2 b_2^2 - 4 b_1^2 - (a_1-a_0)^2 from a known spectral function or an explicit CFT calculation, and comparing with the bulk U and its derivative. Without such a check, the paper's headline result remains conditional.","section":"Sec. 2, Eqs. (2.10)-(2.11); Sec. 6"},{"comment":"The claim that co-monotonicity of c_cov and U in across-dimensional flows reflects reorganization rather than depletion is interpretive and is not backed by a field-theoretic computation. The quantity c_cov is computed from the higher-dimensional covariant formula (2.17) with the UV spacetime dimension held fixed (d=4 for the compactified Klebanov-Witten flow, d=6 for the wrapped-M5 flow), so its monotonic increase is a property of a geometric functional in the higher-dimensional description, not a direct measure of the number of degrees of freedom in the IR CFT. To make the 'reversal' substantive, the authors should compare c_cov(IR) with an independent IR central charge (e.g., the Brown-Henneaux central charge for the AdS3 IR or the a-central charge for the four-dimensional SCFT) and show that the co-monotonic relation tracks that physical quantity. Without such a comparison, the interpretation is not forced by the calculations.","section":"Sec. 5, Eqs. (5.7)-(5.8), (5.21)-(5.26)"}],"minor_comments":[{"comment":"The Newton constant should appear in the denominator: from Eq. (3.5), d c_cov/dr = (1-d) 2^{d-1} a''/(G_N^{(d+1)} (a')^d). As written with G_N in the numerator, the expression is dimensionally inconsistent, although the sign conclusion is unaffected.","section":"Sec. 3, Eq. (3.7)"},{"comment":"The UV expansion of U(z) should tend to the AdS7 value 2, but the displayed factor 3 U_0 2^{1/3} with U_0 = 2^{4/3}/3 evaluates to 2^{5/3}, not 2. The exponent appears to have the wrong sign; the correct UV limit is 3 U_0 2^{-1/3}.","section":"Sec. 5.1, Eq. (5.23)"},{"comment":"The final relations depend on the chosen stationary internal angles (theta_0, psi_0) = (0,0). The paper states that other choices change the value of \\tilde{\\Delta}(r, theta_0, psi_0), but it does not check whether the co-monotonicity and the parametric relation (5.22) are robust to the other allowed stationary points.","section":"Sec. 5.1, text after Eq. (5.14)"},{"comment":"The figure captions refer to U(t) as the 'second derivative of complexity'. By Eq. (1.3), U is the normalized second derivative, so the captions should say 'normalized acceleration of spread complexity' to avoid confusion.","section":"Figs. 1, 3-5"},{"comment":"The 'conservation law' Q + U = 4 is a simple algebraic rewriting of Eq. (5.8) and should be presented as such, rather than as an independent dynamical constraint.","section":"Sec. 5.2, Eqs. (5.29)-(5.30)"}],"recommendation":"major_revision","confidential_remarks":"The paper is a systematic application of an existing proposal; its main novelty is a set of algebraic relations between U and c_cov. The authors are honest about the conditional dictionary, which is a credit. However, the central physical claim is not yet established, and the across-dimensional interpretation would need an independent boundary-side check. I would not reject the paper, but I would require either such a check or a clear reframing of the results as 'geometry of U under the proper-momentum proposal' before publication. The work seems within the scope of JHEP."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Worth reading if you work on holographic complexity. The paper computes, for several holographic RG flows, the normalized second derivative U of Krylov spread complexity under the proper-momentum prescription, and compares it with the covariant c-function ccov of Jokela et al. The explicit results are new: for fixed-dimensional domain walls ccov U^{d-1} = const; for Dp-branes ccov U^8 = Υ(p); and for two across-dimension compactification flows ccov and U switch from anti-correlation to co-monotonicity, with a clean algebraic relation in the Klebanov-Witten case and a parametric relation for wrapped M5. The metric manipulations are straightforward and internally consistent; there is no numerical fitting and no invented machinery.\n\nThe main soft spot is exactly what the authors flag in Section 6: the identification dCK/dt = proper radial momentum is proven for a class of AdS3/CFT2 locally excited states, and its use in non-conformal, higher-dimensional or across-dimension backgrounds is an extension, not a derivation. Every relation in Sections 3–5 rests on that step. If the dictionary fails, U is just a kinematic combination of metric functions and the paper's conclusions about boundary Krylov complexity do not follow. That is a load-bearing assumption, but the paper states it clearly; I do not think it is being hidden. Secondary soft spots: the covariant c-function is adopted from a paper co-authored by one of the authors, so the 'relations' are partly mutual consequences of the definitions; and the wrapped-M5 example chooses representative internal angles (θ0, ψ0), which the paper acknowledges. The short-time Lanczos check in Appendix A is one-sided, as the stress-test note says: it writes bulk data in terms of boundary Lanczos coefficients, but does not derive the Lanczos sequence from the dual QFT.\n\nTaken as a holographic probe comparison, this is solid and useful work. The co-monotonicity reversal across dimensions is genuinely worth discussing; even if its physical interpretation is not yet settled, the explicit formulas will be cited. It deserves a serious referee, with the request that the dictionary extension be attacked in at least one controlled example beyond AdS3 rather than just asserted. I would send it for review.","headline":"Clean holographic comparison of spread-complexity acceleration with covariant c-functions; conditional on an unproven dictionary, but honest and with new explicit relations.","tokens_in":23225,"tokens_out":2712,"would_cite":true,"duration_ms":25594,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"The acceleration of Krylov spread complexity is directly tied to the covariant c-function along holographic RG flows, with the correlation reversing sign for flows that change dimension.","keywords":["Krylov complexity","spread complexity","holographic RG flow","covariant c-function","proper radial momentum","Lanczos coefficients","Dp-branes","twisted compactification"],"falsifier":"For a chosen flow, for example the GPPZ domain wall, compute the survival amplitude's moments from the spectral density of the dual field theory and reconstruct the Lanczos coefficients; then verify whether the resulting spread complexity and its second derivative reproduce U = A'(r)/$A^{{3/2}}$(r) exactly along the radial geodesic. Any short-time discrepancy between the reconstructed and the proper-momentum-computed complexity would falsify the dictionary on which all the relations depend.","tokens_in":22210,"feed_emoji":"⚛️","tokens_out":6850,"duration_ms":58681,"temperature":0.7,"pith_summary":"This paper tries to establish that the acceleration of Krylov spread complexity—the second time derivative of how far a state has spread along its Krylov chain—is a geometric diagnostic that is directly tied to the holographic covariant c-function, the standard geometric count of effective degrees of freedom along a renormalization-group flow. Using the proposal that the growth rate of spread complexity equals the proper radial momentum of an infalling massive probe in the dual geometry, the authors normalize that acceleration to a quantity U that is local in the redshift and radial metric functions. They find simple algebraic relations in every class of flows considered: an inverse power law for fixed-dimensional domain walls, a universal product relation for Dp-branes, and co-monotonicity for flows across dimensions induced by twisted compactification. The paper argues that the reversal of the correlation across dimensions is the signature of a reorganization, rather than a mere reduction, of the degrees of freedom.","feed_headline":"Krylov complexity acceleration tracks the holographic c-function","feed_subtitle":"A single metric quantity links information spreading to the count of active degrees of freedom along RG flows.","key_machinery":"The central object is U(t) = (2/m)√A(r_UV) C̈_K(t), the normalized second time derivative of Krylov spread complexity; for a radial metric this equals the local combination A'(r)/(A(r)^{3/2} B(r)^{1/2}). The comparison quantity is the covariant c-function of Ref. [34], built from the extrinsic curvature of a spacelike slice and given by eq. (2.17). The argument is carried by the dictionary of Ref. [5], which equates the growth rate of spread complexity with the proper radial momentum of an infalling probe, together with the radial geodesic equations that turn U into a purely geometric expression. The energy condition a''(r) < 0 of Ref. [36] then simultaneously yields monotonicity of c_cov and of U in the fixed-dimensional cases.","core_discovery":"The paper establishes that the covariant c-function c_cov, defined through the extrinsic curvature of a spacelike slice of the holographic background, obeys a direct algebraic or parametric relation with the normalized Krylov acceleration U. In fixed-dimensional Lorentz-invariant domain walls, c_cov = 1/(G_N $U^{{d-1}}$), so the same energy condition that makes c_cov decrease toward the infrared makes U increase along the falling trajectory. For the Dp-brane family, c_cov $U^{8}$ = Υ(p), with the exponent eight universal while the normalization carries the brane dimension, the Yang–Mills coupling and the number of colors. For two flows across dimensions—a twisted compactification from four to two dimensions and wrapped M5-branes from six to four dimensions—c_cov and U are co-monotonic rather than anti-correlated. The authors interpret this reversal as evidence that U distinguishes the depletion of degrees of freedom from their reorganization into lower-dimensional sectors.","pith_inferences":["If the proper-momentum dictionary is trusted beyond its proven AdS3/CFT2 setting, the same comparison could serve as a quick diagnostic for other families of flows: a sign change in the c_cov–U correlation would flag a flow that reorganizes rather than merely depletes its degrees of freedom.","The co-monotonic reversal suggests a block-Lanczos picture in which compactification splits the spectrum into zero modes and Kaluza–Klein towers, so co-monotonicity could arise from probability transfer between coupled Krylov blocks even when each block individually behaves like a fixed-dimensional flow.","A testable extension would be to compute the first few Lanczos coefficients from the short-time expansion of the complexity in these backgrounds and compare them with direct spectral calculations in the dual field theory, checking whether the combinations selected by c_cov are universal at large N.","The results raise the possibility of a general theorem: an energy condition on the bulk, monotonicity of the covariant central function, and a consistent one-dimensional truncation of the probe dynamics may together fix the sign of the correlation between the radial derivative of the central function and the divergence of the Krylov probability current."],"forward_implications":["For Lorentz-invariant domain-wall flows, decreasing effective degrees of freedom are accompanied by an increasing acceleration of the Krylov spread along the falling trajectory.","Across the Dp-brane family, c_cov U^8 = Υ(p) is a radial constant, so the exponent is universal while the normalization encodes the brane dimension, the Yang–Mills coupling and the number of colors.","In the twisted compactification from four to two dimensions, c_cov and U obey the conservation law Q + U = 4 with Q ∝ c_cov^{1/3}, so both grow together toward the infrared.","In the wrapped-M5 flow from six to four dimensions, c_cov and U are co-monotonic, with explicit ultraviolet and infrared expansions showing the same increase, indicating the reorganization of degrees of freedom.","The short-time expansion of the spread complexity relates the first Lanczos coefficient to the value of U at the release point, connecting the geometric relations to microscopic spectral data."],"supporting_citations":[{"why":"Establishes the dictionary equating the spread complexity growth rate with the proper radial momentum of an infalling probe, which is the starting point of the calculation.","marker":"[5]"},{"why":"Provides the holographic formulas for the spread complexity and its time derivatives that Section 2 is built on.","marker":"[13]"},{"why":"Defines the covariant c-function used for all the comparisons in the paper.","marker":"[34]"},{"why":"Proves the energy condition (a'' < 0) that makes the fixed-dimensional domain-wall c-function monotonic and also drives the monotonic increase of U.","marker":"[36]"},{"why":"Supplies the compactified Klebanov–Witten background and its c-function, used for the flow from four to two dimensions.","marker":"[40]"},{"why":"Gives the Dp-brane near-horizon supergravity solutions and their range of validity, used for the Dp-brane family in Section 4.","marker":"[56]"},{"why":"Supplies the wrapped M5-brane solution used for the flow from six to four dimensions.","marker":"[64]"}],"fun_headline_variants":["Krylov acceleration flips c-function relation across dimensions","Universal Dp-brane link: Krylov acceleration and c-function","In twisted compactifications, Krylov and c-function move together","Complexity acceleration diagnoses RG degree-of-freedom reorganization"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The whole comparison rests on taking the equality between the growth rate of spread complexity and the proper radial momentum of the falling probe, which was proven only for a class of AdS3/CFT2 locally excited states, to be valid for all holographic RG flows.","fun_headline_variants_meta":{"raw":{"variants":["Krylov acceleration flips c-function relation across dimensions","Universal Dp-brane link: Krylov acceleration and c-function","In twisted compactifications, Krylov and c-function move together","Complexity acceleration diagnoses RG degree-of-freedom reorganization"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001865,"raw_usage":{"total_tokens":7326,"prompt_tokens":957,"completion_tokens":6369,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":573,"completion_tokens_details":{"reasoning_tokens":6300}},"tokens_in":573,"tokens_out":6369,"duration_ms":45089,"temperature":1.0,"reasoning_tokens":6300,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T00:09:17.726052+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"For a chosen flow, for example the GPPZ domain wall, compute the survival amplitude's moments from the spectral density of the dual field theory and reconstruct the Lanczos coefficients; then verify whether the resulting spread complexity and its second derivative reproduce U = A'(r)/$A^{{3/2}}$(r) exactly along the radial geodesic. Any short-time discrepancy between the reconstructed and the proper-momentum-computed complexity would falsify the dictionary on which all the relations depend.","supporting_citations":[{"cited_title":"Covariant unification of holographic c-functions","cited_arxiv_id":"2605.18942","evidence_quote":"Defines the covariant c-function used for all the comparisons in the paper."}],"review_version":1}