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A full-spectrum Krylov complexity stops growing a little past the Heisenberg time in every low-energy state, not at the Hilbert-space dimension.

Reviewed by Pith at T0; open to challenge. T0 means a machine referee read the full paper against a public rubric. the ladder, T0–T4 →

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2026-08-02 02:47 UTC pith:SMA3PA7H

load-bearing objection A real theorem and a promising application, but the printed derivation has a lambda-scaling error and the quantitative claim rests on an unverified spectral-rigidity assumption. the 3 major comments →

arxiv 2607.14220 v2 pith:SMA3PA7H submitted 2026-07-15 hep-th math-phmath.MPquant-ph

Non-perturbative saturation of Krylov complexity, and its implications in quantum gravity

Andrew Lucas , Amit Vikram This is my paper
classification hep-th math-phmath.MPquant-ph
keywords Krylov complexitydouble-scaled SYKJT gravitycomplexity equals volumeorthogonal polynomialsspectral rigidityHeisenberg timewormhole length
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

This paper proves that Krylov state complexity defined over the entire spectrum—the minimalist, β=0 version used in earlier double-scaled SYK to quantum-gravity comparisons—does not keep growing until it has explored the full Hilbert space. Instead, the orthogonal polynomials that build the Krylov basis crowd away from spectral regions with low density of states, so near any low-energy eigenstate the basis is essentially complete after about e^{S(E)} Krylov states, where S(E) is the thermodynamic entropy at that energy. In the DSSYK-model description of JT gravity, where the Krylov index maps to a wormhole length, this means the length saturates at x_sat(E) ∼ √E e^{S(E)} times a mild logarithmic factor. This matches the expected black-hole interior volume saturation and the ergodicity-based length estimate, without changing the seed state or resorting to microcanonical or finite-temperature variants.

Core claim

The central claim is that even a Krylov complexity built from a seed state spread uniformly over all eigenstates still saturates at the Heisenberg scale in every low-energy state. The mathematical mechanism is a theorem about orthogonal polynomials on discrete spectra: near the spectral edge, the first q polynomials attain near-completeness once q is roughly the number of levels in an edge window times a spectral-rigidity factor, rather than the full dimension d. Applying this to double-scaled SYK with the mapping x=λn and z=z0+λE, the saturation length obeys x_{K,sat}(E) ≲ √(C+1)E e^{S0+2π√(2(C+1)E)}[√(J/(CE)){ln(4J/λ²)−L_edge(E)} + λ]; choosing C=J/E gives x_{K,sat}(E) ≈ √E e^{S(E)}[ln(4J/

What carries the argument

The engine is the Christoffel variational principle, which controls the Christoffel–Darboux kernel K_q(z,z)=Σ_{n≤q}|p_n(z)|²: any polynomial Q of degree q with Q(z_k)=1 must satisfy ∫|Q|² dμ ≥ 1/K_q(z_k,z_k). Using a trial polynomial that vanishes at all edge states except z_k and then decays via a Chebyshev residual polynomial outside the edge window, the paper extracts an explicit upper bound on the index q at which K_q(z_k,z_k) ≥ 1/(1+ε²). The input that converts this into a clean magnitude is the logarithmic spectral rigidity potential ν_{k,edge}, whose thermodynamic limit is evaluated under random-matrix rigidity assumptions on the edge spectrum.

Load-bearing premise

The quantitative bound depends on the low-energy edge spectrum of DSSYK obeying random-matrix rigidity—namely that minimum level spacings are no smaller than a power of 1/d and that level counts in an interval grow with interval size—which the authors assert is overwhelmingly reasonable but do not prove.

What would settle it

Numerically diagonalize the DSSYK Hamiltonian at moderate λ and, for each low-energy eigenstate with JT energy E, compute the cumulative Krylov overlap Σ_{n≤λ^{-1}x_{K,sat}(E)} |⟨z_k|K_n⟩|²; if any eigenstate retains significant overlap at Krylov indices beyond the bound from Eq. (4.12), the claimed saturation scale is falsified.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Every low-energy state, not only the thermofield double, has its Krylov components effectively confined to indices n ≲ e^{S(E)} log(1/λ).
  • No modification of the Krylov seed to microcanonical or finite-temperature Gibbs states is needed; the β=0 seed that reproduces the semiclassical JT Hamiltonian already saturates at the physically correct scale.
  • The saturation length matches, up to a logarithmic factor, the ergodicity-based length operator and the expected black-hole interior volume √E e^{S(E)}.
  • The bound is an upper bound, so actual saturation may occur even closer to the ideal N(E) than the proven estimate.
  • The result is state-independent in the sense that it constrains each energy component of any initial state, rather than depending on how that state was prepared.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Theorem 1 is a generic statement about orthogonal polynomials on discrete spectra, so the same edge-crowding should appear in any bounded-spectrum chaotic Hamiltonian, not only DSSYK, suggesting a universal relationship between spectral rigidity and Krylov saturation.
  • If the logarithmic factor ln(1/λ) is tight, it predicts a concrete finite-λ correction to wormhole-length saturation that could be compared to direct numerical simulation of double-scaled SYK at moderate λ.
  • The identification x=λn is verified semiclassically only at small x; if this mapping fails by the time x reaches e^{S(E)}, the gravitational interpretation of the saturation length could shift even though the underlying spectral theorem remains intact.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. The paper studies Krylov state complexity in finite-dimensional quantum systems, with the Krylov basis generated from the uniform seed (2.4). The central mathematical result, Theorem 1 (Section 3.1, proved in Appendix A), asserts that for a spectral edge window of d_edge levels, the first q_k Krylov polynomials nearly saturate the completeness relation at an edge level z_k, with q_k bounded by d_edge times a logarithmic spectral rigidity potential ν_{k,edge} and an inverse-square-root factor in a width ratio. The authors argue that this “crowding away” of orthogonal polynomials from low-density spectral edges implies that the Krylov complexity of any low-energy state saturates only slightly beyond the Heisenberg scale, despite the Krylov basis being defined over the full spectrum. They then apply this to DSSYK/JT gravity via the identification x = λn, obtaining the quantitative bound (4.12): x_{K,sat}(E) ≲ √((C+1)E) e^{S_0+2π√(2(C+1)E)} [√(J/(CE)){ln(4J/λ²) − L_edge(E)} + λ], which for C=J/E and E≫J reduces to x_{K,sat}(E) ≲ √E e^{S(E)}[ln(4J/λ²) − L_edge(E) + λ]. This is presented as evidence that the β=0 Krylov proposal is consistent with the complexity=volume expectation, without modifying the seed as in Refs. [18,19].

Significance. If the result holds, it is a significant step toward reconciling the minimalist full-spectrum Krylov complexity proposal with non-perturbative expectations for black-hole interior volume. The proof of Theorem 1 is a genuine derivation from the spectrum; it contains no fitted parameters and the bound is a theorem consequence rather than a fit labeled as a prediction. The application to DSSYK/JT gives a concrete, falsifiable quantitative prediction for the saturation scale, and it sharpens the connection between Krylov complexity and ergodic notions of length. However, the advertised quantitative form of the DSSYK bound, in particular the clean logarithm ln(4J/λ²)−L_edge(E), rests on a set of spectral-rigidity assumptions in Appendix B that are asserted but not verified. There is also a displayed-factor inconsistency between the intermediate Eq. (4.8) and the final Eq. (4.12). These issues are local and correctable, but they currently make the strongest gravitational claim conditional.

major comments (3)
  1. [§3.2, Eq. (3.10); §4, Eq. (4.8)] The width-ratio simplification in Eq. (3.10) appears to invert η_k. With z_{d_edge}−z_0 ≈ (C+1)(z_k−z_0), one has η_k = (z_{d_edge}−z_k)/(z_{d−1}−z_{d_edge}) ≈ C(z_k−z_0)/(z_{d−1}−z_0), so the correct factor from Eq. (3.8) is 1/(2√η_k), i.e. √{(z_{d−1}−z_0)/(4C(z_k−z_0))}, not 1/(2√{(z_{d−1}−z_0)/(C(z_k−z_0))}). This propagates to Eq. (4.8), where the printed denominator yields a term scaling as λ², while the corrected expression gives d_edge ν_{k,edge}√(J/(CE)) — which is what Eq. (4.12) ultimately uses. The final result may be correct, but the displayed intermediate equations are inconsistent and should be corrected and re-audited through Eqs. (4.8)–(4.13).
  2. [Appendix B, Eqs. (B.11)–(B.16); Eq. (4.12)] The quantitative saturation scale in Eq. (4.12) depends crucially on the continuum estimate ν_{k,edge} ≈ ln(4J/λ²) − L_edge(E). This estimate is not a consequence of Theorem 1, which bounds q_k in terms of the realization-dependent ν_{k,edge}; the clean logarithm is obtained in Appendix B from the assumptions δz_min=Ω(d^{−a}), Δ̃₃=O(d^{−ε₃}), and N(z_k,ε_J)=O(d_edge ε_J/(z_{d_edge}−z_0)). The authors state these are “overwhelmingly reasonable” for DSSYK, but no numerical or analytic verification is provided. If the low-energy edge spectrum has non-universal clustering, ν_{k,edge} can be parametrically larger and the claimed saturation at e^{S(E)} log(1/λ) can fail. Because this is the bridge from the rigorous theorem to the gravitational prediction, the paper should either prove these rigidity estimates for DSSYK (for example from the matrix-integral/dual description), provide numerical
  3. [§2.2, Eq. (2.17); §3.1, Theorem 1] Theorem 1 gives, for each edge energy z_k, a small tail weight ∑_{m>q_k}|⟨z_k|K_m⟩|² from the completeness relation. To conclude that the Krylov complexity “stops growing” at q_k, one needs to control the m-weighted tail in the time-averaged identity (2.17). For fixed ε, the tail contribution to ⟨X⟩ can be as large as d ε², which is not small. The paper mentions in passing that ε≪1/√d can force completeness, but the main statement is not phrased with this precision. A rigorous statement would choose ε=O(1/d) and phrase the conclusion as a bound on the long-time average of ⟨X⟩, or as a bound that holds for most times. Please clarify this step so that the abstract’s “stops growing” claim is formally supported.
minor comments (4)
  1. [General notation] The notation oscillates between d_edge, dedge, and d_edge in the main text and Appendix B; please unify. Also define clearly whether d_edge denotes the number of levels in the edge window or the index of the last level in the window.
  2. [§2.1] The statement that “for any reasonable (not strongly oscillating) choice of log ρ(z)” the Lanczos asymptotics depend only on the tail (2.15) is informal. Please either cite the precise theorem in the orthogonal-polynomials literature or qualify the statement as heuristic; the later rigorous argument does not actually rely on this asymptotic claim.
  3. [§4, Eq. (4.8)] In addition to the factor issue in the major comment, the sentence introducing Eq. (4.8) should make explicit which definition of E_edge is used, since Eq. (4.8) and Eq. (4.12) use different choices of C and E_k. The text would be easier to follow if the two choices were displayed separately.
  4. [Acknowledgements] The acknowledgements describe substantial LLM assistance in constructing the variational ansatz and correcting typos. If the journal has a policy on disclosure of AI assistance, please ensure the statement complies with it; otherwise this is fine.

Circularity Check

0 steps flagged

No significant circularity: Theorem 1 is derived from the input spectrum via a variational principle; no fitted quantity is relabeled as a prediction.

full rationale

The paper's central derivation is self-contained and non-circular. Theorem 1 (Appendix A) is proved from the Christoffel variational principle and a Chebyshev residual-polynomial ansatz; the bound (3.8) is written directly in terms of the input spectrum through d_edge and ν_k,edge, not in terms of the target saturation scale. The application to DSSYK/JT in Eq. (4.12) substitutes the DOS and estimates ν_k,edge with the continuum expression in Appendix B. That estimate is conditional on explicit spectral-rigidity assumptions (Eqs. (B.11)–(B.16)) and on the stated expectation that DSSYK obeys CUE-type rigidity; this is an unverified input assumption, which is a correctness risk, but it is not circular because the rigidity assumptions do not contain the conclusion that Krylov complexity saturates at the Heisenberg scale. The x=λn identification is imported from Refs. [11–13] (not self-citations) and used only after the index bound is derived. Self-citations [21,22] are used for comparison and a conjecture, not as load-bearing proof of the saturation bound. No parameter is fitted to data and then 'predicted'; the saturation length is a theorem-consequence of the assumed spectrum, with the Heisenberg scale arising from the integrated DOS N(E), which is the physical input rather than an assumed output.

Axiom & Free-Parameter Ledger

3 free parameters · 6 axioms · 0 invented entities

The central claim rests on standard orthogonal-polynomial math (Christoffel variational principle, Chebyshev/residual polynomials), two domain assumptions imported from prior work (the DSSYK–JT dictionary x=λn and the JT entropy formula), and one paper-specific assumption: CUE-type spectral rigidity for the DSSYK edge window (Appendix B). No free parameters are fitted to data; the hand-chosen parameters C and ε affect the bounds' tightness but not the existence of a bound. No new physical entities are postulated.

free parameters (3)
  • C (edge window cutoff) = C = J/E chosen in Sec. 4 (E-dependent); C = O(1) otherwise
    Defines the edge window z_{d_edge} − z_0 ≈ (C+1)(z_k − z_0) in the application of Theorem 1. The clean saturation form (4.13) specifically requires the E-dependent choice C = J/E; for C = O(1) the bound degrades to e^{S((C+1)E)}, exponentially larger than e^{S(E)}. Hand-chosen, not fitted.
  • ε (completeness resolution) = any ε > 0; small value needed for tight tails
    Sets the near-completeness threshold 1/(1+ε²) in Eq. (3.7). The physical claim that complexity 'stops growing' requires the residual weight ε²/(1+ε²) to be small. Hand-chosen.
  • Spectral-rigidity scaling exponents a, ε₃, c_ε = a > 0, ε₃ ∈ (0,1), c_ε > 0
    Assumed in Appendix B for the DSSYK spectrum (CUE-like behavior). They set the rate at which the continuum approximation to ν_{k,edge} holds (Eqs. (B.13)–(B.16)). Not verified for DSSYK in the paper.
axioms (6)
  • standard math Christoffel Variational Principle (Simon, Theorem 9.2 of Ref. [32])
    Core engine of the proof of Theorem 1, used at Eq. (A.2): for all trial polynomials with Q(z_0)=1, ∫dµ|Q|² ≥ 1/K_q(z_0,z_0).
  • standard math Chebyshev/residual polynomial sup-norm bound |T_m(z)| ≤ 1 on [−1,1] and the cosh form outside
    Used in Eqs. (A.7)–(A.10) to construct the rapidly decaying factor R_r(z) in the variational ansatz.
  • domain assumption DSSYK spectrum is a non-degenerate random draw from DOS ρ(z)=e^{S0}ρ_0(z) with CUE-type level rigidity
    Appendix B (Eqs. (B.11)–(B.17)) requires δz_min = Ω(d^{−a}), Δ̃₃ = O(d^{−ε₃}), and N(z_k,ε_J) = O(d_edge ε_J/width). The authors call this 'overwhelmingly reasonable' for DSSYK but provide no check. Without it, ν_{k,edge} is realization-dependent and the magnitude in Eq. (4.12) is unproven.
  • domain assumption Identification x = λn (Krylov index ↔ JT wormhole length) and the β=0 seed reproduce semiclassical JT Hamiltonian (4.1) at small x
    Taken from Refs. [11–13]; the saturation estimate extrapolates this dictionary to x ~ e^{S(E)}, beyond the regime where it was checked.
  • domain assumption JT entropy S(E) = S_0 + 2π√(2E) and DSSYK bandwidth z_{d−1} − z_0 ~ 4J/λ
    Standard JT/DSSYK inputs used in Eqs. (4.2), (4.8), (4.12).
  • domain assumption Level count in the JT window: d_edge = N(E) ~ √E e^{S(E)}
    Sets the size of the edge window d_edge in Eqs. (4.12)–(4.15).

pith-pipeline@v1.3.0-alltime-deepseek · 14791 in / 39632 out tokens · 341816 ms · 2026-08-02T02:47:25.740455+00:00 · methodology

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read the original abstract

We study the dynamics of Krylov state complexity in finite-dimensional quantum systems. Orthogonal polynomials built on a discrete energy spectrum crowd away from regions where the density of states is low at large Krylov index. The physical implication of this result is that, even using a minimalist Krylov state complexity which is defined over the entire spectrum, the Krylov complexity provably stops growing a little past the Heisenberg length in every low-energy state. This has interesting implications for the proposal that Krylov complexity is related to the wormhole length in 2D quantum gravity.

Figures

Figures reproduced from arXiv: 2607.14220 by Amit Vikram, Andrew Lucas.

Figure 1
Figure 1. Figure 1: Demonstration of the “withdrawal” of the Krylov basis polynomials [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗

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