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The quadratic growth of Krylov spread complexity in the BTZ black hole

T0 review · 3 major / 5 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that Krylov spread complexity reconstructed from black-hole thermodynamics is quadratic in the BTZ case, not linear.

desk verdict A serious paper with a strong DSSYK benchmark and an honest but unproven late-time claim; worth a real referee. read the letter →

arxiv 2608.09922 v1 pith:A732PZLF submitted 2026-08-10 hep-th cond-mat.str-elgr-qcquant-ph

Aranya Bhattacharya , Mario Flory , Michal P. Heller , Emiliano Rizza , Tim Schuhmann This is my paper · ORCID
classification hep-thcond-mat.str-elgr-qcquant-ph
keywords KrylovspreadcomplexityholographicBTZblackholethermofielddoublestateLanczoscoefficientspartitionfunctionequalsanythingPadéapproximation
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper develops a partition-function construction of Krylov spread complexity for thermofield-double states and uses it to make a concrete prediction for the BTZ black hole. The central claim is that, in a large-central-charge CFT2 above the Hawking-Page temperature, the classical Krylov spread complexity is quadratic at short times, dips away from quadratic growth at intermediate times, and is compatible with returning to quadratic growth in the semiclassical late-time window, rather than showing the linear growth of volume and of standard finite-function complexity=anything proposals. A key methodological claim is that the semiclassical saddle is sufficient only when the classical limit is taken after the full complexity is assembled, because subleading Lanczos coefficients feed the leading answer through cancellations. The paper also constructs a bulk functional on maximal-volume slices, an infinite series of extrinsic-curvature invariants whose resummation diverges on the final slice, as a candidate geometric dual.

What carries the argument

The load-bearing object is the reconstruction chain from partition function to Krylov dynamics: $Z(\beta) \to m_k = (-\partial_\beta)^k Z/Z \to$ Lanczos coefficients $a_n,b_n \to$ early-time series $C_K(t)=\sum_n c_{2n}t^{2n}$. The moment method guarantees that $c_{2n}$ depends only on the first $n$ Lanczos coefficients, so a large finite number of derivatives of the saddle partition function fixes the series. Beyond the convergence radius the paper uses diagonal Pad\'e approximants of the instantaneous power $\gamma(t)=t\,C_K'(t)/C_K(t)$, whose late-time plateau fixes the growth exponent. On the bulk side the new object is a maximal-volume slice weighted by $F_1 = \bar\lambda\big[1 + \sum_{n\ge1} \lambda_n (K_{ab}K^{ab}/2)^{n+1}\big]$, with $\lambda_n$ fixed order by order from boundary data; a $1/n$ tail resums to a logarithmic divergence at the final slice and converts linear late-time growth into quadratic growth.

What would settle it

A reader could go look at the exact thermal partition function of a large-central-charge CFT2, or a modular-invariant continuation of the Cardy saddle, compute the early-time series of Krylov complexity to significantly higher order, and form higher-order diagonal Pad\'e approximants: if the instantaneous power $\gamma(x)$ with $x=t/\beta$ heads toward 1 instead of back up toward 2 in the window $1 \ll x \ll c$, the predicted return to quadratic growth is wrong.

Watch

Extended reading notes

Core claim

The discovery is that Krylov spread complexity reconstructed from the Cardy saddle $Z = \exp(\mathfrak{c}/\beta)$ has an early-time series, and the Pad\'e continuation of this series indicates an intermediate departure from quadratic growth followed by a turn upward toward quadratic asymptotics, in contrast to the linear late-time behavior of the volume and of the standard finite-functional complexity=anything class. In the double-scaled Sachdev-Ye-Kitaev model the paper establishes the order-of-limits rule: the classical limit must be taken after the complexity is assembled, since the leading classical complexity receives contributions from subleading orders in the Lanczos coefficients. The bulk counterpart is a generalized complexity=anything observable with a local functional built from an infinite series of powers of $K_{ab}K^{ab}$, whose coefficients are fixed state-independently by matching the boundary series order by order; the infinite resummation can diverge on the accumulation surface, and this divergence is the mechanism that supports quadratic pre-saturation growth.

Load-bearing premise

The late-time conclusion rests on assuming that rational-function (Pad\'e) approximations of the truncated early-time series, with a power-law late-time form imposed by hand, tell us what the true complexity does beyond the series' convergence radius.

Editorial extensions

If this is right

  • The same partition-function construction applies to any holographic theory where the thermal partition function is dominated by a saddle, so higher-dimensional AdS black holes inherit a concrete boundary prediction for Krylov complexity.
  • In the semiclassical window $1 \ll t/\beta \ll c$, Krylov complexity of the BTZ thermofield-double state is not described by any finite functional from the original complexity=anything class; a dual must involve an infinite, resummable series that becomes singular on the final slice.
  • Finite-order bulk functionals reproduce the boundary early-time series to high order but eventually exhibit linear growth; only the infinite-order completion avoids this crossover, so truncating the bulk series changes the qualitative late-time prediction.
  • The crossover timescale is set by the inverse temperature $\beta$, not by the scrambling time $\beta\log c$, meaning the effect is encoded in the equilibrium saddle and in the unperturbed thermofield-double state.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • If the quadratic asymptotics survives a full finite-$c$ computation, the late-time growth exponent $\gamma$ becomes a holographic order parameter that distinguishes Krylov complexity from circuit complexity and from all finite complexity=anything functionals.
  • Applying the same construction below the Hawking-Page temperature, or to rotating or charged black holes, would test whether the state-independent coefficients $\lambda_n$ remain universal; the paper's own internal-consistency criterion suggests this is where the proposal lives or dies.
  • The order-of-limits lesson likely generalizes to other asymptotic expansions used in Krylov computations: truncating Lanczos coefficients before assembling the complexity can discard contributions that are subleading pointwise but leading after cancellations.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

3 major / 5 minor

Summary. The manuscript develops a 'complexity from partition function' (CfZ) construction: for a thermofield-double reference state, Hamiltonian moments are derivatives of the thermal partition function, and the Lanczos recursion plus early-time expansion yields Krylov spread complexity. The authors benchmark CfZ in double-scaled SYK, comparing the exact partition function with its semiclassical saddle, and find that the classical (q→1) limit must be taken only after assembling the complexity, because subleading orders of individual Lanczos coefficients contribute to the leading classical complexity through cancellations. They then apply the saddle-point prescription to the high-temperature CFT2 Cardy partition function Z=exp(c/β), dual to BTZ. The resulting early-time series deviates from eternal quadratic growth within its convergence radius; diagonal Padé continuation shows a dip in the instantaneous power followed by an upward turn compatible with a return to quadratic growth, but with visible order dependence. Finally, they construct a bulk 'generalized CAny' functional, an infinite series of extrinsic-curvature invariants on maximal slices, with coefficients fixed by boundary matching; a fitted 1/n tail produces a logarithmic singularity on the final slice that supports quadratic late-time growth, whereas every finite truncation reverts to linear growth.

Significance. If established, the result would be significant: it would give a dimension-independent route from semiclassical partition functions to Krylov complexity, show that the classical limit does not commute with Lanczos reconstruction, and indicate that Krylov spread complexity for the BTZ TFD state lies outside the standard finite CAny class, with quadratic rather than linear growth in the semiclassical window. The DSSYK validation is a genuine strength: the explicit cancellation in Eq. (45), the β=0 limit reproducing 2 log cosh(t/2), and the low-temperature Schwarzian limit provide nontrivial checks. The manuscript also reports open data and code, and includes a toy model isolating the non-commutativity of truncation and late-time limits. These strengths justify careful consideration, but the central late-time claim rests on unconverged Padé continuations and fitted bulk tails, so the published version must either strengthen those arguments or frame the claims as conditional.

major comments (3)
  1. [Section V.C, Fig. 7] The paper's central physical claim—a return toward quadratic growth—is not established by the evidence presented. The diagonal Padé approximants of γ(t) are constructed from the truncated early-time series with M=N, which by construction enforces γ→constant (power-law growth) but does not determine the constant; the approximants still show order dependence among orders 10–100, as the authors acknowledge. Within the convergence radius |x|≲0.8 the series establishes only the departure from γ=2; the upward turn is a property of the chosen continuation, not a demonstrated property of the true complexity. Direct Krylov-chain evolution from the computed Lanczos coefficients, or a converged sequence of Padé approximants with explicit error estimates, is needed to support the title's claim of quadratic growth.
  2. [Section VI.C, Eqs. (77)-(78), Fig. 8] The bulk argument is not independent evidence for quadratic late-time growth. The coefficients λ_n are fixed order-by-order by matching the same boundary early-time series (64) to the bulk expansion (74); the claim that the resummed functional diverges on the final slice and yields γ→2 depends entirely on the fitted large-n ansätze λ_n∼A_o/n and λ_n∼A_e/n+B_e/n^{3/2}. The even branch is explicitly unstable under fit-window changes and changes sign at n≥54, so the fits do not determine the true tail. The toy model of App. H shows that a harmonic tail can produce the observed pattern, but it does not show that the actual λ_n sequence is such a tail. Without an independent derivation of the large-n asymptotics, or a proof that all coefficient tails consistent with the matching data give the same late-time exponent, the bulk construction cannot support the quadratic-growth conclusion.
  3. [Section VI.C, parameter matching] The claim that the coefficients λ_n are state-independent is not demonstrated. The text states that 'based on structural agreement of (74) and (75), the lambdas we find are independent of β', but the matching is presented for a single representative β (with c=3.1·10^15 in the numerical implementation). To establish that the bulk functional is the same for all BTZ temperatures, the authors should repeat the order-by-order matching at several β values, such as those in Fig. 5, and show that the resulting λ_n coincide within numerical precision. As written, the state-independence assertion is an assumption, not a verified property of the construction, and it is load-bearing for the internal-consistency test emphasized in the conclusions.
minor comments (5)
  1. [Title and abstract] The title states 'The quadratic growth of Krylov spread complexity in the BTZ black hole', while the abstract and Section V.C only claim behavior 'compatible with a return toward asymptotically quadratic growth'. Please align the title with what is actually established, or provide the missing convergence evidence.
  2. [Section V.A, Eq. (48)] The notation uses both c (central charge) and \mathcal{c} (Cardy coefficient) with c = π c V/6. This is easy to confuse in later sections, especially where 'large c' is used for both. Please clarify the notation consistently.
  3. [Eq. (79)] In the expression for F_fit(X), the term Li_{3/2}(X^2) is finite at X=1, unlike the arc-tanh and log terms. The text should clarify whether the claimed divergence on the final slice comes only from the log(1-X) pieces; as written, a reader may infer that Li_{3/2} itself diverges at X=1, which is not the case for this order.
  4. [Appendix H] The toy model is helpful, but the parameter choices µ1, µ2, κ are presented without discussion. A sentence explaining that these parameters only set time scales and do not affect the structural conclusion would make the analogy to the bulk construction clearer.
  5. [Section VI.C, fits (77)-(78)] The text says 'the exact numerical values are not crucial for the late-time argument', yet the hybrid resummation (82) uses precisely those numerical values for n≤99 and the fitted values for the tail. Please reconcile these statements and state explicitly which aspects of the late-time conclusion survive if the fitted tail is replaced by a different subleading behavior.

Circularity Check

2 steps flagged · score 6.0 of 10

The bulk functional is fixed by matching the boundary series, and its resummed 1/n tail is chosen to produce γ→2; the boundary Padé return toward quadratic growth is additionally assumption-dependent, so the quadratic-growth claim is only partially established.

  1. fitted input called prediction [Section VI.C, Eqs. (74)-(76) and Conclusions]
    "The triangular matching fixes one new bulk coefficient at every accessible order and thereby defines the candidate functional recursively. ... Our gravitational construction reproduces the boundary behavior by construction, and it is certainly not unique."

    The parameters λ̄, λ1, λ2, ... are fixed by equating the bulk early-time derivative (74) to the boundary Krylov derivative (75) order by order. Hence the bulk observable equals the boundary series by construction; any statement that the bulk computation reproduces or matches the boundary result is a restatement of the fitting condition. The paper's own admission that the construction is not unique confirms that the bulk object carries no independent predictive content beyond the data used to define it. This is the fitted-input-called-prediction pattern: the functional is constructed from the target and then presented as a candidate dual of that target.

  2. fitted input called prediction [Section VI.C, Eqs. (77)-(82) and Fig. 9]
    "Over the available large- n range, the odd- n coefficients are consistent with the asymptotic form λn ≈ Ao/n with Ao ≈ 1.066. ... A plausible asymptotic ansatz is λn ≈ Ae/n + Be/n^{3/2} ... The critical asymptotic scaling λn ∼ 1/n is singled out by the requirement of late-time quadratic growth."

    The λn entering the asymptotic fits (77)-(78) are themselves outputs of the order-by-order matching to the boundary series (75); the fits are not independent bulk data. The selected harmonic 1/n tail is then resummed to produce the −log(1−X) divergence on the final slice, and this divergence is the mechanism that drives γ→2 in the hybrid curve of Fig. 9. Thus the quadratic late-time behavior of the bulk observable is an extrapolation of coefficients fitted to the boundary early-time expansion, with the singular tail form chosen precisely because it yields the desired γ→2. The subsequent agreement with the boundary Padé curve is therefore a consistency check on the fit, not an independent derivation.

full rationale

The boundary part of the paper is largely self-contained: moments are obtained from the Cardy partition function Z = exp(c/β), the Lanczos recursion is exact, and the early-time series (64) is a genuine calculation with no circularity. The DSSYK benchmark in Sec. IV provides an external check and does not rely on a self-citation chain. The circular content resides in the bulk construction of Sec. VI: the coefficients of the generalized CAny functional are fixed order by order by matching the bulk expansion (74) to the boundary series (75), so the bulk observable reproduces the boundary result by construction; the paper explicitly concedes this non-uniqueness. The additional late-time quadratic behavior in the bulk is generated by a fitted 1/n tail whose resummation produces a logarithmic singularity on the final slice, i.e., the conclusion is built into the chosen fit form rather than independently derived. The boundary Padé continuation of the instantaneous power γ(x) also imposes a constant late-time exponent through the diagonal choice M=N, and the paper itself notes that order dependence remains visible and that the strict late-time limit is not established; hence the 'return toward quadratic growth' is a compatibility statement under an ansatz, not a controlled derivation. No load-bearing self-citation or imported uniqueness theorem was found; the self-citations used are benchmarks or future-work references and do not force the central claim. Overall, the central boundary series is independent, but the bulk 'prediction' and the late-time quadratic return are partially circular or assumption-dependent, giving a partial circularity score of 6.

Assumptions & free parameters 4 free parameters · 6 assumptions · 1 invented entities

The boundary construction uses only standard moment methods plus a Cardy saddle assumption. The bulk construction introduces a large set of free coefficients, all fixed by matching the boundary expansion, plus a fitted asymptotic tail. This makes the bulk dual an engineered observable rather than an independent derivation. The paper is transparent about this, but the ledger shows that the central bulk claim rests on roughly 100 fitted parameters and an ad hoc tail ansatz.

free parameters (4)
  • lambda_bar (overall bulk normalization) = 1/6 from Eq. (76)
    Fixed by matching the leading bulk coefficient to the boundary complexity early-time series; a free normalization of the candidate functional.
  • lambda_n for n = 1..99 = lambda_1 = 9/10, lambda_2 = 1/14, lambda_3 = 43/130, remaining values numerical from Fig. 8
    Each lambda_n is fixed order by order by matching the bulk early-time expansion (74) to the boundary series (75). These are not predicted quantities.
  • A_o (odd tail amplitude) = approximately 1.066
    Fitted to the computed odd-n lambda_n values; used for the asymptotic tail resummation in Eq. (77).
  • A_e and B_e (even tail amplitudes) = A_e approximately -0.036, B_e approximately 0.26
    Fitted to the computed even-n lambda_n values; used for the asymptotic tail resummation in Eq. (78).
assumptions (6)
  • standard math Thermofield-double moments satisfy m_k = (-d_beta)^k Z(beta) / Z(beta).
    Derived in Appendix B; a standard identity for TFD states, not a paper-specific postulate.
  • standard math The Lanczos moment method reconstructs Lanczos coefficients from the first moments.
    Standard recursion stated in Appendix A; no new content is imported.
  • domain assumption The Cardy partition function Z = exp(pi c V / (6 beta)) above the Hawking-Page temperature is the dominant saddle and can be differentiated to arbitrarily high order to define Lanczos data in the classical limit.
    Invoked in Sec. V.A, Eq. (48). This treats the asymptotic high-temperature density of states as an exact generating function for all moments.
  • domain assumption Diagonal Pade approximants of the instantaneous power, with M=N, fix the late-time behavior to be a power law with unspecified exponent, and faithfully continue the complexity beyond the radius of convergence.
    Stated explicitly in Sec. IV.B.2 and used in Sec. V.C. The late-time form is therefore an assumption of the continuation method, not derived from the series.
  • ad hoc to paper The large-n tail of the bulk coefficients follows the parity-resolved fits lambda_n ~ A_o/n for odd n and lambda_n ~ A_e/n + B_e/n^(3/2) for even n.
    Used to resum the infinite series in the hybrid bulk completion, Eq. (82). These fits are based on 99 computed coefficients and are presented as evidence compatible with a harmonic tail, not as a derivation.
  • domain assumption BTZ is locally AdS3, so all bulk curvature invariants are constants and all higher contractions of the extrinsic curvature reduce to powers of X = P_v^2 / r^4.
    Used in Sec. VI.B and Appendix G to justify the ansatz for the local functional F1 in Eq. (73).
invented entities (1)
  • Singular infinite-order CAny functional F1(X) = lambda_bar (1 + sum_n lambda_n X^(n+1))
    purpose: Candidate bulk observable dual to boundary Krylov spread complexity in AdS3/CFT2.
    The coefficients are fixed by matching to the boundary early-time series, and the divergence on the final slice is engineered through the 1/n tail. There is no falsifiable handle outside the boundary series used to fit it.

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Pith. "Pith review of The quadratic growth of Krylov spread complexity in the BTZ black hole." pith.science (2026). https://pith.science/paper/A732PZLF

@misc{pith2026260809922,
  author       = {Pith},
  title        = {Pith review of: The quadratic growth of Krylov spread complexity in the BTZ black hole},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/A732PZLF}},
  note         = {Machine review of arXiv:2608.09922}
}
read the original abstract

The boundary quantity that captures the growth of black-hole interiors quantified by holographic complexity remains unknown beyond 2d dilaton gravity. We provide a critical analysis of a partition-function construction of Krylov spread complexity for thermofield-double states that provides a dimension-independent boundary reconstruction from semiclassical holographic partition functions, while developing the present dynamical and bulk construction for the BTZ saddle. In the double-scaled Sachdev-Ye-Kitaev model, where exact and semiclassical results can be compared, we show that the classical limit is reliable only when taken after the complexity has been reconstructed; taking this limit at the level of individual Lanczos coefficients discards essential information. Applying the construction to a large-central-charge two-dimensional conformal field theory above the Hawking-Page temperature dual to a Ba\~nados-Teitelboim-Zanelli black hole, we find an intermediate departure from early-time quadratic growth followed by behavior compatible with a return toward asymptotically quadratic growth, rather than the linear late-time behavior of the volume and the standard finite-functional complexity = anything class. We then match this boundary behavior to a generalized complexity = anything bulk object built from an infinite series of extrinsic-curvature invariants. The construction provides a systematic route from black-hole thermodynamics to Krylov dynamics and can naturally be extended to higher-dimensional holographic black holes.

Figures

Figures reproduced from arXiv: 2608.09922 by the authors.

Figure 1
Figure 1. FIG. 1. Lanczos coefficients [PITH_FULL_IMAGE:figures/full_fig_p006_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Instantaneous power [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Comparison of Lanczos coefficients [PITH_FULL_IMAGE:figures/full_fig_p008_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Visualization of the temperature dependence ( [PITH_FULL_IMAGE:figures/full_fig_p009_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. First 100 Lanczos coefficients [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Instantaneous power [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Numerical values of the parameters [PITH_FULL_IMAGE:figures/full_fig_p017_8.png]
Figure 9
Figure 9. Figure 9: FIG. 9 [PITH_FULL_IMAGE:figures/full_fig_p018_9.png]

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Reviewed August 11, 2026 · model on record in the stance chip above.