{"id":"18a6c2d0-99dd-41c5-afba-c2afe0427fbe","arxiv_id":"1908.10832","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":4,"one_line_summary":"In the AdS3 Vaidya geometry, the extremal volume defining holographic subregion complexity is genuinely x-dependent during the quench, so the standard x-independent ansatz fails at intermediate times; early and late time approximations are derived.","lead":"This paper computes how subregion complexity, a quantum information measure of a strongly coupled 2D field theory, changes during a sudden thermalization quench, using its 3D gravity dual. It shows that a previously used x-independent approximation to this quantity is inconsistent at intermediate times and provides the first full numerical computation of the true extremal volume surface.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The l^2 peak scaling is not yet established: it rests on an unproven lower-bound property of the pseudosolution and on extrapolation beyond the numerically solved range rhl≤6.","rationale":"The paper's clean analytical result—that the x-independent ansatz is inconsistent with the HRT boundary condition for the time-dependent Vaidya geometry—is checkable and convincing, and the early-time growth rate matching Ref. [33] plus the Gauss-Bonnet final-time value provide independent support. The numerical solutions for rhl≤6 are cross-checked with two solvers and a cell-size check, so the qualitative volume evolution is credible. The residual weak point is the logical bridge from those finite-l computations to the asymptotic 'at least l^2' maximum: the bridge relies on an unproven lower-bound property of a pseudosolution whose admissibility as a comparison surface is nontrivial when both HRT branches are present. The reader's weakest assumption captures part of this, but the sharper issue is the unproven inequality and the extrapolation beyond rhl=6 rather than solver error alone. A large-l computation or a rigorous proof of the lower bound would settle the matter, so the CONDITIONAL verdict is appropriate.","tokens_in":19752,"tokens_out":20157,"duration_ms":199771,"concrete_test":"Solve the extremal-volume PDE for rhl=8 and rhl=10 with an independent high-order (e.g., spectral or AMR) solver using the same HRT boundary data, and at rhl=6 run a convergence study with maximum cell sizes 1e-3, 1e-4, and 1e-5 while checking the sign of the second variation around the numerical solution. If the peak regularized volume at l=8,10 continues to grow like l^2 and the second variation is negative definite, the extrapolated claim is supported; if the growth flattens, stalls, or a boundary-respecting perturbation increases the volume, the lower-bound argument and the l^2 peak claim fail.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Section 3.6 asserts without proof that 'the pseudosolution in any case provides a lower bound of the volume of the solution,' and Section 4 uses this to conclude that the maximum of the solution's volume 'scales at least as l^2' via eq. (3.25). That inequality follows only if the numerical PDE solution is the global maximum of the volume functional and if the pseudosolution is an admissible comparison surface with the same HRT boundary. Neither condition is established. In particular, for rs<rh/√2 the two branches of the HRT geodesic carry different v(r) at the same r, so the object whose volume is computed in eqs. (3.13)–(3.15) is not obviously a single smooth admissible surface; a lower-bound argument for it needs a proof, not an expectation. Additionally, the numerical solutions stop at rhl=6, a regime the paper itself says is numerically challenging, and the large-l peak behavior is an extrapolation. No second-variation analysis is given to confirm that the computed branch is the relevant maximum.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper studies holographic subregion volume complexity for a line segment in AdS3 Vaidya spacetime, working within the complexity-equals-volume proposal. The authors argue that the previously used x-independent ansatz for the extremal volume surface is inconsistent with the Hubeny-Rangamani-Takayanagi boundary condition except near the initial and final times, and they solve the extremal-volume equation numerically as a boundary-value problem. They compute the subregion volume as a function of time for subregion sizes up to rhl=6, compare with the pseudosolution defined by the x-independent configuration, and derive approximate analytic expressions at early and late times. The main quantitative claim is that the volume grows to a peak whose maximum scales at least as l^2 before returning to the vacuum value.","tokens_in":19959,"tokens_out":13856,"duration_ms":140649,"significance":"If the main claims hold, the paper provides a useful correction to earlier calculations based on the x-independent ansatz, and it gives the first numerical determination of the time-dependent extremal subregion volume in Vaidya spacetime. The derivation in Sec. 3.2 that the x-independent ansatz is inconsistent whenever the HRT geodesic has nonzero E is clean and checkable. The numerical work is strengthened by the use of two independent solvers and by consistency checks against known limits: the early-time growth rate matches Ref. [33], and the late-time return to the vacuum value follows from the Gauss-Bonnet theorem. However, the headline l^2 peak scaling is not yet established, because it relies on an unproven lower-bound property of the pseudosolution and on extrapolation beyond the numerically solved range. The paper would be significantly stronger if this point were addressed.","major_comments":[{"comment":"The claim that the maximum of the solution volume scales at least as l^2 is load-bearing, and it rests on the assertion that 'the pseudosolution in any case provides a lower bound of the volume of the solution.' This assertion is not proven. For rs < rh/sqrt(2), the pseudosolution is assembled from geodesic branches carrying different v(r) at the same r, as seen in eqs. (2.21) and (3.13)-(3.15), and it is not demonstrated that the resulting object is a single admissible codimension-one surface with the HRT surface as its boundary; in fact, the volume formula appears to add contributions from overlapping r-intervals. Since the numerical solution is obtained only up to rhl=6, eq. (3.25), which is derived for the pseudosolution, does not by itself establish the large-l peak scaling of the solution. A proof of admissibility of the comparison surface, or a clearly stated weaker claim, is needed.","section":"Sec. 3.6, eq. (3.25); Sec. 4"},{"comment":"The complexity functional in eq. (1.1) is defined through a maximum, but the paper solves only the Euler-Lagrange equation (3.19) and states in Sec. 3.2 that 'the real solution is expected to be a local maximum of the volume functional' without providing a second-variation analysis or any other check that the numerically computed branch is the global maximum. If the computed branch is a saddle or a local minimum, its volume cannot be identified with the subregion complexity, and the comparison with the pseudosolution does not by itself repair this. A second-variation test, or at least numerical evidence that the branch maximizes the functional, is required to support the interpretation of the plotted volumes as complexities.","section":"Sec. 3.2; Sec. 3.4"},{"comment":"The numerical results are reported with a finite-element cell size of order 10^{-4} and a check against an independent linearized solver, but no convergence data, error bars, or discretization-error estimates are provided, and no public code is made available. The computational domain has non-smooth boundaries at the null shell, and the calculation stops at rhl=6 because larger values are described as numerically challenging. The large-l behavior used in the peak-scaling claim comes from the analytic pseudosolution formula, eq. (3.25), not from the full numerical solution. A convergence study and an estimate of numerical uncertainty for the volume curves in Fig. 7 are needed before the quantitative large-l conclusions can be regarded as established.","section":"Sec. 3.4; Fig. 7"}],"minor_comments":[{"comment":"There are several typos in the introduction, for example 'the x-independent ansatzisnotconsistent' and 'thex-independent', which should be corrected.","section":"Sec. 1"},{"comment":"The expression for rs/rh in eq. (2.23) appears to have unbalanced parentheses; please check the formula and its presentation.","section":"Sec. 2.3, eq. (2.23)"},{"comment":"The claim that the geodesic in eq. (2.13) satisfies the reduced equation (3.9) only for E=0 is stated verbally; writing out the residual of eq. (3.9) evaluated on the geodesic for a nonzero E would make this central step more transparent and easier to verify.","section":"Sec. 3.2"},{"comment":"The value of the finite shell-thickness parameter \\tilde v used in the numerical geodesics of Sec. 2.4 is not specified for the volume calculations; a statement about how the \\tilde v to 0 limit was taken would aid reproducibility.","section":"Sec. 3.4"},{"comment":"Given the numerical nature of the results, the plots in Fig. 7 would benefit from error bars or a separate convergence panel showing the dependence of the volume on the grid resolution.","section":"Fig. 7"}],"recommendation":"major_revision","confidential_remarks":"The central technical point about the inconsistency of the x-independent ansatz appears sound and is a useful contribution. The l^2 peak-scaling claim, however, is currently supported by an unproven lower-bound assertion and by extrapolation from moderate values of rhl; in my view this is fixable in revision and does not warrant rejection, but it must be addressed before the paper can be accepted."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The two things you should know: first, the paper gives a clean, checkable proof that the x-independent ansatz used in all previous subregion-complexity-in-Vaidya papers is inconsistent with the HRT boundary condition except at t=0 and t=l/2. That is a real correction to the literature, and it is the paper's core contribution. Second, the quantitative headline — that subregion complexity peaks at a value scaling at least as l^2 — is not actually established. It rests on an unproven lower-bound property of the pseudosolution and on extrapolation beyond the numerically solved range rhl ≤ 6.\n\nWhat is good: Section 3.2 is the heart. Plugging the HRT geodesic into the x-independent Euler-Lagrange equation shows violation unless E=0, which is clean. The numerical work is also solid as far as it goes: the PDE is solved with two independent methods, the early-time growth rate matches Chapman et al., and the late-time return to the vacuum value is a topological result from Gauss-Bonnet, not a numerical artifact. The authors are honest about the limits of their numerics and explicitly flag that the large-l conjecture from the pseudosolution should be revisited.\n\nThe soft spots are real but concentrated. The claim that the pseudosolution provides a lower bound for the volume of the true solution is asserted in Section 3.6 without proof. That matters because the l^2 scaling of the maximum is derived by maximizing the pseudosolution volume and then invoking that bound. The numerics only reach rhl=6, and the paper itself says the large-l regime is where deviations from the pseudosolution become largest, so the extrapolation to l^2 is speculative. There is also no second-variation analysis, so we are taking it on faith that the computed branch is the global maximum of the volume functional, not a saddle. The non-smooth boundary at the shell could cause solver drift, though the two-solver check mitigates that worry.\n\nI don't think any of this sinks the paper. The inconsistency result stands independently of the numerics, and the numerical solution is a genuine first. The gap is between what is shown and what the abstract claims. If the authors can either prove the lower bound (or at least state it as an assumption) and clearly label the l^2 scaling as an extrapolation, the paper would be in good shape.\n\nThis paper is for people working on holographic complexity and thermalization. It deserves a serious referee, but the referee should push on the large-l claim. I would send it to peer review, likely with a request for revision rather than acceptance as is.","headline":"A clean demonstration that the x-independent ansatz fails in time-dependent Vaidya, with a first numerical solution of the full extremal-volume PDE; the l^2 peak scaling is a reasonable conjecture but not proven.","tokens_in":20524,"tokens_out":1632,"would_cite":true,"duration_ms":19443,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":["04.70.-s","11.25.Tq"],"model":"deepseek-v4-flash","headline":"In AdS3 Vaidya, the standard x-independent ansatz for the extremal volume surface is inconsistent with the HRT condition, and the true subregion complexity peaks at least as l^2 before returning to the vacuum value.","keywords":["holographic complexity","subregion complexity","Complexity=Volume conjecture","Vaidya spacetime","AdS3/CFT2 correspondence","HRT surface","thermalization","extremal surfaces"],"falsifier":"Compute the second variation of the volume functional on the numerical solution, or run an independent high-resolution solver at $r_h l = 6$ and with the zero-thickness shell: if any deformation increases the volume, or if the finer grid changes $(V_{\\mathrm{sol}} - V_{\\mathrm{AdS}})/l$ beyond the stated deviations, the claimed extremum and the $l^2$ peak are not established.","tokens_in":19540,"feed_emoji":"📈","tokens_out":14369,"duration_ms":121612,"temperature":0.7,"pith_summary":"The paper tests the holographic Complexity=Volume proposal for a line segment in the AdS3 Vaidya geometry, the gravitational dual of a sudden quench that drives a strongly coupled field theory to thermalization. It shows that the translationally invariant ansatz used in earlier work for the extremal volume surface fails to satisfy both the Euler–Lagrange equation and the Hubeny-Rangamani-Takayanagi (HRT) boundary condition, except exactly at the initial and final times. Solving the extremal-volume equation as a partial differential equation with the HRT geodesic as boundary condition, the authors compute the subregion complexity as a function of time. They find that the complexity grows to a peak scaling at least as $l^2$ and then returns to the vacuum value once the system thermalizes, with approximate analytic expressions at early and late times.","feed_headline":"After a quench, subregion complexity peaks at least as l^2","feed_subtitle":"The flat slice fails in dynamical Vaidya; the true extremal surface returns to the AdS value after thermalization.","key_machinery":"The load-bearing object is the extremal codimension-one volume surface $z(x,v)$ stretching between the boundary segment and its HRT geodesic, whose volume functional is $V = \\int dv\\,dx\\, \\frac{\\sqrt{-(2\\partial_v z + f(v,z)) - (\\partial_x z)^2}}{z^2}$, with the HRT geodesic as boundary condition. The argument turns on the Euler–Lagrange equation derived from this functional, solved numerically by finite elements with two independent solvers, and on the comparison with the 'pseudosolution'—the $x$-independent configuration obtained by attaching the volume ansatz to the HRT surface—which is not an extremum of the action except at $t=0$ and $t=l/2$. The late-time return to the vacuum value rests on the Gauss-Bonnet identity $V_{\\mathrm{BTZ}} = V_{\\mathrm{AdS}}$ in AdS3, and the early-time regime is fixed by the geodesic junction conditions from the Vaidya geometry.","core_discovery":"The central claim is that in the AdS3 Vaidya spacetime the $x$-independent ansatz for the extremal subregion volume surface is not consistent with the HRT boundary condition and the Euler–Lagrange equation, so the true extremal surface is $x$-dependent. The regularized subregion complexity $\\Delta C_V$ starts at zero, rises to a maximum that scales at least as $l^2$, and decreases to exactly zero for $t \\ge l/2$, when the HRT surface lies entirely inside the BTZ black hole; in three bulk dimensions the final and initial volumes coincide by the Gauss-Bonnet theorem. The $x$-independent 'pseudosolution' is a lower bound on the true volume and a good approximation only near $t=0$ and $t=l/2$ (and at all times for small segments), while deviations grow with $l$ at intermediate times. The early-time growth rate saturates the conjectured Lloyd bound, matching the rate previously found for the one-sided Vaidya black hole.","pith_inferences":["Editorial inference: a peak-then-return profile with final value equal to the vacuum complexity is a sharp qualitative signature of CV-type subregion complexity in 2d CFTs during a global quench; any proposed mixed-state complexity measure should reproduce this non-monotonic behaviour.","Editorial inference: the same PDE formulation with HRT boundary condition can be applied in higher-dimensional Vaidya backgrounds, where the paper notes the final complexity does not return to zero; comparing peak scaling across dimensions would separate dynamical features from the topological AdS3 equality.","Editorial inference: the expected disagreement at intermediate times with the concurrent $x$-independent computation could be settled by an independent high-resolution solver on the zero-thickness shell; a solver that reproduced the pseudosolution would undermine the lower-bound argument and the at-least-$l^2$ peak."],"forward_implications":["Earlier quench calculations of subregion complexity built on the $x$-independent ansatz are reliable only near $t=0$ and $t=l/2$, or for small segments with $l \\ll 1/T$; at intermediate times and large $l$ they miss the true extremal surface.","The regularized subregion complexity is not monotonic: it grows to a peak scaling at least as $l^2$ and then returns to exactly zero after thermalization, in contrast with the monotonic approach of entanglement entropy.","At early times the growth rate saturates the conjectured Lloyd bound, $dC/dt = 8\\pi M$, reproducing the known result for a one-sided Vaidya black hole.","The conjectured linear-increase regime of complexity at intermediate times for large $l$ must be revisited, because deviations from the $x$-independent pseudosolution grow with $l$."],"supporting_citations":[{"why":"It supplies the analytic space-like geodesic solutions in AdS3 Vaidya and the junction conditions used to build the HRT boundary condition for the volume surface.","marker":"[30]"},{"why":"It proposes the subregion Complexity=Volume conjecture, the quantity the paper computes.","marker":"[40]"},{"why":"It defines the Hubeny-Rangamani-Takayanagi surface that anchors the extremal volume region.","marker":"[42]"},{"why":"It performs the earlier subregion complexity calculation using the x-independent ansatz that the paper shows is inconsistent at intermediate times.","marker":"[54]"},{"why":"It is the concurrent work whose early-time results agree and whose intermediate-time x-independent results the paper expects to be inaccurate.","marker":"[57]"},{"why":"It proves the Gauss-Bonnet identity $V_{\\mathrm{BTZ}} = V_{\\mathrm{AdS}}$ in AdS3 that fixes the late-time return to the vacuum value.","marker":"[46]"},{"why":"It provides the early-time complexity growth rate for the one-sided Vaidya black hole that the paper matches.","marker":"[33]"}],"fun_headline_variants":["Quench resets subregion complexity after l^2 peak","Flat slice fails: Vaidya subregion complexity peaks and resets","Subregion complexity after quench: peaks at least l^2, then vanishes","Vaidya complexity: x-dependent surface corrects naive estimate","Lloyd bound saturated in early-time growth of subregion complexity"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The numerical finite-element solution of the extremal-volume PDE is the true global maximum of the volume functional, converged to controlled accuracy, so that the reported time-dependence and the peak growing at least as $l^2$ follow from the lower-bound pseudosolution.","fun_headline_variants_meta":{"raw":{"variants":["Quench resets subregion complexity after l^2 peak","Flat slice fails: Vaidya subregion complexity peaks and resets","Subregion complexity after quench: peaks at least l^2, then vanishes","Vaidya complexity: x-dependent surface corrects naive estimate","Lloyd bound saturated in early-time growth of subregion complexity"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.001687,"raw_usage":{"total_tokens":6641,"prompt_tokens":854,"completion_tokens":5787,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":470,"completion_tokens_details":{"reasoning_tokens":5695}},"tokens_in":470,"tokens_out":5787,"duration_ms":40977,"temperature":1.0,"reasoning_tokens":5695,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T10:33:17.086397+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Compute the second variation of the volume functional on the numerical solution, or run an independent high-resolution solver at $r_h l = 6$ and with the zero-thickness shell: if any deformation increases the volume, or if the finer grid changes $(V_{\\mathrm{sol}} - V_{\\mathrm{AdS}})/l$ beyond the stated deviations, the claimed extremum and the $l^2$ peak are not established.","supporting_citations":[{"cited_title":"Holographic Subregion Complexity in General Vaidya Geometry","cited_arxiv_id":"1908.06432","evidence_quote":"It is the concurrent work whose early-time results agree and whose intermediate-time x-independent results the paper expects to be inaccurate."}],"review_version":1}