{"id":"1a2dc83b-8aaf-452e-9fbb-3cfa22891fe2","arxiv_id":"2608.10060","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":2,"one_line_summary":"In holographic B8 gauge theories, complexity growth flattens near a walking conformal regime, while Krylov oscillation periods track the infrared scale and persist in screened, non-confining phases.","lead":"This paper computes two quantum complexity measures in a family of three-dimensional gauge theories using their gravitational descriptions. It shows that complexity can detect near-conformal 'walking' dynamics and can help tell an ordinary infrared scale apart from genuine confinement.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The Krylov results rest on the proper-momentum dictionary, Eq. (1.12), which the paper itself flags as extrapolated beyond its AdS derivation; if that dictionary fails, the spread-complexity diagnostics of walking and confinement do not follow.","rationale":"After reading the full text, the CV computation is internally consistent: the extremal-surface calculation, the late-time growth G, and the walking plateau are all derived cleanly and are not controversial beyond the standard CV conjecture. The Krylov section is where the central, distinctive claims live: oscillations as a signature of a smooth cap rather than confinement, and the sensitivity of the oscillation frequency to the confining limit. Every one of those statements is downstream of Eq. (1.12). The paper is admirably explicit about this, but an explicit caveat does not make the assumption secure. The reader identified the same weakest assumption, and I agree; no additional flaw was found in the bulk computations or in the interpretation of the CV data. Because the paper is framed as part of a research program testing this dictionary, and because the verdict is already MODERATE confidence, I do not think the concern changes the reader's ACCEPT recommendation. However, if the dictionary were to fail, the impact would be substantial, so this is a genuine load-bearing concern rather than a cosmetic caveat. The proposed check---a direct Lanczos reconstruction from the bulk two-point function---is the cleanest way to settle it.","tokens_in":39484,"tokens_out":9989,"duration_ms":98185,"concrete_test":"Reconstruct the boundary survival amplitude S(t) for the operator dual to the massive particle in the B8 zero-temperature background by solving the scalar wave equation (or using the geodesic approximation) and computing the spectral density from the discrete normal modes. Apply the standard Lanczos algorithm to S(t) to obtain the Krylov spread complexity CK(t) directly, without invoking the momentum dictionary. Then compare the resulting \\dot C_K(t)---including its oscillation period and the number of zeroes per period---with the momentum-prescription results in Section 4 (e.g., Eqs. (4.11), (4.18), and Figs. 7-10). If the two agree, the extrapolated dictionary is supported; if they differ, the central spread-complexity claims are not established and the Krylov sections would need to be rederived from the survival-amplitude approach.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The load-bearing assumption is the identification \\dot C_K(t) = -P_{\\bar y}(t), Eq. (1.12), proposed for asymptotically AdS backgrounds in Refs. [64-67]. The B8 geometries are D2-brane-like in the UV and have a warped ten-dimensional description that is singular for 0<b0<1; the paper applies the prescription to both massive particles and D0-branes in these backgrounds. Section 4 explicitly states: 'our ultraviolet geometries are D2-brane-like rather than asymptotically AdS, so this prescription is being extrapolated beyond the setting in which it was derived.' No derivation, order-one estimate, or independent check is provided for this extrapolation. If the proper-momentum prescription receives corrections or fails away from AdS, then the oscillatory \\dot C_K(t), the identification of its period with an infrared scale, and the claimed contrast between the screened and confining windows are not established. Since the CV analysis does not rely on this dictionary, the concern is specific to the Krylov half of the paper, but that half carries the paper's most distinctive conclusions.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper computes two holographic complexity observables in the B8 family of three-dimensional quiver gauge theories: the complexity=volume (CV) growth of thermofield-double states and the Krylov spread complexity of locally excited states. For CV, the authors derive the maximal-volume growth rate, normalize it by the energy, and show that it flattens in the walking region near the Ooguri-Park fixed point while remaining qualitatively insensitive to the onset of Wilson-loop confinement. For local excitations, they adopt the holographic momentum-Krylov dictionary of Refs. [64-67], apply it to a massive particle and a D0-brane in the ten-dimensional geometries, and find oscillatory complexity growth whose period is controlled by the infrared scale. They argue that these oscillations diagnose a smooth infrared cap rather than confinement, since they persist throughout the screened window 0<b0<1, and they compare their periods with the lightest spin-2 mass, the screening length, and the entanglement-entropy saturation scale. The analysis is corroborated by an eleven-dimensional uplift and by a qualitative comparison with lattice results on the transverse-field Ising model.","tokens_in":39722,"tokens_out":4345,"duration_ms":46405,"significance":"If the results hold, the paper provides a genuinely useful disentangling of three logically distinct infrared properties—approximate conformality, a discrete massive spectrum, and Wilson-loop confinement—within a single holographic family. The CV computation is a clean application of a standard conjecture, with an analytic check at the Ooguri-Park fixed point (Eq. (3.30)) and consistent numerical behavior across the parameter space. The local-probe part is more conjectural but is strengthened by internal cross-checks: the D0-brane computation avoids the type IIA singularity, and the eleven-dimensional uplift reproduces the main qualitative features. The paper also offers concrete falsifiable statements, such as the period scales of the oscillations and the presence of a walking-induced maximum in the D0-brane period near b0=0. The authors are transparent about the main assumption, explicitly flagging that the momentum-Krylov prescription is being extrapolated beyond its original AdS setting. The central limitation is that this extrapolation is not independently verified, which makes the spread-complexity conclusions conditional rather than established.","major_comments":[{"comment":"The identification \\dot C_K(t)=-P_{\\bar y}(t) is the load-bearing assumption of the entire Krylov half of the paper. It was proposed for asymptotically AdS backgrounds in Refs. [64-67], whereas the B8 geometries are D2-brane-like in the UV and, for 0<b0<1, have a singular ten-dimensional description (Eq. (2.10)). The paper explicitly acknowledges that this prescription is being extrapolated beyond its derivation, but it does not provide an order-of-magnitude estimate, a derivation in the D2-brane regime, or an independent check. Because the oscillatory behavior of \\dot C_K, the identification of its period with an infrared scale, and the claimed contrast between screened and confining windows all follow from this dictionary, I ask the authors either to supply such a check (for example, by computing Lanczos coefficients or the survival amplitude from the boundary data in a limit where the dictionary is expected to hold) or to explicitly reframe the spread-complexity claims as conditional on this correspondence. Without this, the most distinctive conclusions of Section 4 are not established.","section":"Section 4.2 / Eq. (4.18)"},{"comment":"The period T in Eq. (4.18) for the ten-dimensional massive particle is computed using a reflective boundary condition at the singular end of the geometry. The manuscript states this prescription and later shows that the D0-brane and eleven-dimensional computations are qualitatively similar, which is reassuring. However, because the bounce occurs at a point where the geodesic equation is not defined, the period itself is prescription-dependent, and the main-text statement that the oscillation frequency is set by the infrared scale inherits this dependence. I request a more quantitative statement of how the reflective prescription affects the period: the comparison between Fig. 9 and Fig. 21 is only qualitative, and the difference between the ten-dimensional and eleven-dimensional periods for the same b0 is not quantified. Adding such a comparison would strengthen the claim that the oscillations and their periods are robust features rather than artifacts of the chosen boundary condition.","section":"Section 4.2 / Eq. (4.18) and Appendix B.2"}],"minor_comments":[{"comment":"The claim that G/E vanishes with energy whenever a mass gap is present is supported by a fit to the last five numerical points with two free parameters, but no error estimate is given and the text says this is a belief. Adding an error bar or an analytic argument would strengthen the walking-regime characterization.","section":"Section 3.3 / Eq. (3.36)"},{"comment":"The comparison with the transverse-field Ising model is qualitative. It would be helpful to state explicitly which of the holographic results could be falsified by the lattice calculation and which are expected to differ because b0 is not a string-tension parameter.","section":"Section 4.4"},{"comment":"There is a typo in the caption: 'the end o of space' should read 'the end of space'.","section":"Figure 12 caption"},{"comment":"The footnote marker for the monotonicity of A(r) appears without a visible footnote; please check that footnotes are rendered correctly.","section":"Section 4.2, after Eq. (4.14)"},{"comment":"The list of key differences in Appendix C is useful but would be clearer if the corresponding periods were tied back to Eqs. (4.18) and (4.30) for the screened case.","section":"Section 5, first paragraph after Eq. (C.13)"},{"comment":"Reference [63] is listed as 'to appear(2026)' without an arXiv number; please update it if a preprint exists.","section":"References"}],"recommendation":"major_revision","confidential_remarks":"The CV analysis is solid and would be publishable on its own. The Krylov half is interesting and likely correct, but the momentum-Krylov extrapolation is explicitly unverified, and the most distinctive conclusions depend on it. The authors should be asked to either provide a concrete check or to recast the spread-complexity conclusions as conditional on that dictionary. This is a fixed-point within the manuscript's scope, so major revision rather than rejection seems appropriate."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a solid, honest application of the CV and proper-momentum prescriptions to the B8 family, and it gives the first systematic separation of walking, screening, and confinement in complexity observables. The CV half is the stronger of the two: the general formula, the OP fixed-point check, and the numerical late-time growth-to-energy curves are clean, and the walking flattening of G/E is a genuine new diagnostic. The Krylov half is more conditional. The identification \\dot C_K = -P_{\\bar y} was derived for asymptotically AdS settings, and the B8 UV is D2-brane-like; Section 4 concedes this explicitly. So the oscillation periods and their claimed sensitivity to confinement inherit an unproven extrapolation. That is a real caveat, but not a hidden one, and the paper's own discussion is careful: it claims oscillations diagnose a smooth cap and an IR scale, not confinement itself, and it contrasts with the lattice Ising result rather than overstating a universal signature.\n\nWhat I think holds regardless: the geometric fact that probes bounce in a capped geometry is independent of the Krylov dictionary. Even if the momentum-rate identification needs correction away from AdS, the qualitative contrast between the conformal endpoint (no oscillations), the screened window (oscillations with IR-scale periods), and the confining limit (periods that survive but with different IR behavior) is a robust feature of the backgrounds. The eleven-dimensional uplift and the D0-brane computation both corroborate the 10D particle picture, which is a genuine cross-check.\n\nMinor soft spots: the numerical pipeline is not released, so the curves in Figs. 5 and 6 are not independently reproducible, and the comparison of periods to glueball masses in Fig. 18 is admittedly only qualitative. The reflective boundary condition at the 10D singularity is an effective prescription, but the authors flag it and the 11D result supports it, so I would not call it a flaw. The citation pattern is appropriate: the many self-citations are part of an active program, and the cited previous results are real and relevant.\n\nWho should read it: anyone working on holographic complexity or on the B8 family. It deserves a serious referee: the CV section is a full derivation with explicit checks, and the conceptual claim about what complexity can and cannot see is worth getting into the literature. My recommendation is to engage with it; the Krylov caveat should stay in the referee report, but it is not a ground for rejection.","headline":"A clean, honest application of two complexity prescriptions to the B8 family that cleanly separates walking, screening, and confinement; the Krylov half rests on an extrapolated dictionary, but the geometric core holds up.","tokens_in":40237,"tokens_out":1998,"would_cite":true,"duration_ms":20669,"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":"Two complexity measures in a holographic family of gauge theories separate walking dynamics from genuine confinement: the volume growth of the wormhole detects the walk, while the frequency of Krylov oscillations of locally excited states…","keywords":["holographic complexity","complexity=volume","Krylov spread complexity","spread complexity rate","walking dynamics","confinement","Chern-Simons-matter theories","holographic RG flows"],"falsifier":"Compute the Krylov spread complexity of a local operator directly in the boundary theory from the survival amplitude $S(t)$, without using the momentum dictionary, and compare the oscillation frequency with the radial-probe period predicted from the geometry, for example $T_{\\rm UV} \\simeq 7.88\\, q_c/\\rho_0^2$ at $b_0=1$; a mismatch would falsify the dictionary. Alternatively, a fully regular eleven-dimensional computation with a massive particle, which needs no reflective boundary condition, should reproduce the smooth D0-brane behaviour; if the oscillations or their period differ, the ten-dimensional prescription is the wrong effective description.","tokens_in":39273,"feed_emoji":"⏱️","tokens_out":12250,"duration_ms":107310,"temperature":0.7,"pith_summary":"Using the gravitational duals of the B8 family of three-dimensional Chern-Simons-matter quiver theories, the paper asks whether two different holographic complexity measures can separate three infrared phenomena that often occur together: approximate conformality (walking), a discrete massive spectrum supported by a smoothly capped geometry, and genuine Wilson-loop confinement. The authors compute the late-time complexity=volume growth for thermofield-double states and the Krylov spread complexity of locally excited states, the latter through the radial motion of a falling point particle or D0-brane. Near $b_0=0$, where the flow passes close to the Ooguri-Park conformal fixed point, both notions exhibit walking behaviour: the volume-growth-to-energy ratio flattens over a wide energy range, while the Krylov oscillation period grows, becoming a maximum at an intermediate smearing scale. Near $b_0=1$, the volume growth approaches the confining result without any new feature, whereas the Krylov oscillations persist throughout the screened window $0<b_0<1$ with a period controlled by the infrared scale. The main conclusion is that oscillations in the complexity growth of locally excited states are a signature of the smooth end of the dual geometry, not of confinement per se, while the frequency they set is sensitive to the confining nature of the ground state.","feed_headline":"Krylov oscillations, not wormhole volume, track confinement","feed_subtitle":"Spread-complexity oscillation periods are set by the infrared scale; complexity=volume growth shows no confinement signature.","key_machinery":"The machinery is threefold. The B8 family of type IIA backgrounds, with parameter $b_0$ interpolating between the Ooguri-Park conformal fixed point and a confining theory at $b_0=1$, provides the arena: for $0<b_0<1$ the ten-dimensional metric is singular in the infrared but its eleven-dimensional uplift caps off smoothly, producing a discrete massive tower while Chern-Simons terms screen fundamental charges. For thermal states, the complexity=volume prescription identifies complexity with the volume of a maximal Einstein-Rosen-bridge slice, whose late-time growth rate is computed from the geometry inside the horizon. For locally excited states, the dictionary $\\dot C_K(t) = -P_{\\bar y}(t)$ converts the proper radial momentum of a falling probe into the growth rate of Krylov spread complexity; a massive particle and a D0-brane are used as probes, with the D0-brane's dilaton coupling generating an effective potential well that prevents it from reaching the singular end of space. The smooth cap is the mechanism that makes the radial motion periodic, and the infrared scale sets the oscillation frequency.","core_discovery":"The central claim is that the two notions of complexity function as complementary filters on the same renormalization-group flow. In the walking regime the complexity=volume growth rate, normalized by energy, stays close to its constant Ooguri-Park value over a parametrically large energy window; this is the complexity-side manifestation of walking dynamics. In the same regime the period of Krylov oscillations grows and, for a D0-brane, develops a maximum at an intermediate release position, which the authors interpret as a qualitative modification of the smearing-scale dependence induced by the nearby conformal fixed point. Near the confining endpoint, by contrast, the complexity=volume ratio smoothly approaches the confining curve with no new feature at the transition, so wormhole growth is largely blind to whether the ground state confines. The Krylov oscillations, however, persist for every non-confining value $0<b_0<1$ and their period is controlled by the infrared scale; they therefore diagnose the smooth cap and discrete massive spectrum, not the area law. At $b_0=1$ the geometry becomes regular and the probe dynamics changes character, so spread complexity is markedly more sensitive than wormhole growth to genuine confinement.","pith_inferences":["If the momentum-Krylov dictionary extends beyond anti-de Sitter, the B8 results suggest a practical diagnostic: measure the dependence of the Krylov oscillation period on the smearing scale of the boundary operator; a maximum at intermediate smearing indicates a nearby walking region, while a monotonic decrease identifies a generic massive flow.","The discontinuity in $T_{\\rm UV}$ when the Chern-Simons level is rescaled to zero hints that the $b_0\\to 1$ limit and the truly confining theory are distinct physical theories that happen to share a smooth geometry; boundary observables that are continuous in $b_0$ may be blind to the difference, while spread complexity is not.","One could test the smooth-cap interpretation by engineering a holographic model with the same smooth cap but with confining Wilson loops restored, and checking whether the oscillation period still scales with the same infrared scale; the paper's logic predicts the period tracks the cap scale, not the string tension.","The walking-induced maximum of the D0-brane period at intermediate release position is a sharp, quantitative prediction that could be checked in lattice or tensor-network models of spreading, where the proximity of a critical point is tuned by a parameter."],"forward_implications":["Oscillations in the Krylov complexity growth of locally excited states can no longer be cited by themselves as evidence of confinement; in any holographic model with a smoothly capped infrared geometry and a discrete spectrum they will appear even when fundamental charges are screened.","In walking theories, the period of these oscillations is a direct probe of approximate conformality: it grows as the flow approaches a conformal fixed point, and the D0-brane period develops a maximum at intermediate operator smearing, giving a boundary-observable signature of walking.","The late-time complexity=volume growth-to-energy ratio is a reliable detector of walking but not of confinement; near the confining endpoint it approaches the confining result smoothly and is insensitive to the phase transitions of the system.","The comparison with the lattice Ising chain implies that an infrared mass scale acts as an oscillation clock for spread complexity regardless of its origin, so the period, not the mere presence of oscillations, carries the information about the scale.","The divergence of $\\dot C_K$ for a ten-dimensional massive particle, absent for D0-branes and in eleven dimensions, indicates that singular ten-dimensional descriptions must be handled with care: only regular or uplifted computations give finite spread-complexity rates."],"supporting_citations":[{"why":"Constructs the B8 ground-state geometries and establishes the mass-gap-without-confinement property that the paper's central distinction relies on.","marker":"[4]"},{"why":"Computes the discrete mass spectrum of the gapped non-confining theories and supplies the lightest spin-2 mass used as an infrared scale.","marker":"[6]"},{"why":"Shows the exact Goldstone mode that coexists with the discrete spectrum, so the theories are not strictly gapped.","marker":"[7]"},{"why":"Provides the finite-temperature solutions, energy, free energy, and phase diagram used in the complexity=volume calculation.","marker":"[11]"},{"why":"Established the late-time growth of the Einstein-Rosen bridge as holographic complexity, the basis of the CV computation.","marker":"[21]"},{"why":"Defines Krylov/spread complexity and the optimal-basis theorem on which the state-complexity observable rests.","marker":"[42]"},{"why":"Proposes the identification of the spread-complexity growth rate with the proper radial momentum of a bulk particle, the dictionary applied here.","marker":"[64]"},{"why":"Earlier holographic Krylov computation in confining backgrounds that found oscillating complexity and is the comparison point for the smooth-cap interpretation.","marker":"[68]"},{"why":"Reported that the oscillation frequency is set by the confinement scale and proposed it as a possible universal confinement signature, which this paper qualifies.","marker":"[70]"},{"why":"Lattice transverse-field Ising computation of Krylov complexity under confinement, used as a boundary benchmark for oscillations as an infrared clock.","marker":"[107]"}],"fun_headline_variants":["Spread complexity sees confinement; wormhole volume doesn't","Krylov oscillations expose IR scale; wormhole growth blind","Confinement shows up in spread complexity, not complexity=volume","Wormhole volume misses confinement; Krylov period captures it"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The Krylov results stand on the assumption that the growth rate of spread complexity equals the radial momentum of a falling object even though the ultraviolet geometry is D2-brane-like rather than asymptotically anti-de Sitter, and that a probe reaching the singular end of space bounces back rather than stopping or behaving otherwise.","fun_headline_variants_meta":{"raw":{"variants":["Spread complexity sees confinement; wormhole volume doesn't","Krylov oscillations expose IR scale; wormhole growth blind","Confinement shows up in spread complexity, not complexity=volume","Wormhole volume misses confinement; Krylov period captures it"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000503,"raw_usage":{"total_tokens":2460,"prompt_tokens":954,"completion_tokens":1506,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":570,"completion_tokens_details":{"reasoning_tokens":1436}},"tokens_in":570,"tokens_out":1506,"duration_ms":11801,"temperature":1.0,"reasoning_tokens":1436,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:14:41.825440+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the Krylov spread complexity of a local operator directly in the boundary theory from the survival amplitude $S(t)$, without using the momentum dictionary, and compare the oscillation frequency with the radial-probe period predicted from the geometry, for example $T_{\\rm UV} \\simeq 7.88\\, q_c/\\rho_0^2$ at $b_0=1$; a mismatch would falsify the dictionary. Alternatively, a fully regular eleven-dimensional computation with a massive particle, which needs no reflective boundary condition, should reproduce the smooth D0-brane behaviour; if the oscillations or their period differ, the ten-dimensional prescription is the wrong effective description.","supporting_citations":[],"review_version":1}