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arxiv: 2602.11627 · v2 · submitted 2026-02-12 · ✦ hep-th · cond-mat.str-el· quant-ph

Krylov Subspace Dynamics as Near-Horizon AdS₂ Holography

Hyun-Sik Jeong This is my paper

Pith reviewed 2026-05-16 06:03 UTC · model grok-4.3

classification ✦ hep-th cond-mat.str-elquant-ph
keywords Krylov subspaceAdS2 holographyoperator growthLanczos coefficientschaos boundnear-horizon AdSKlein-Gordon dynamicsSL(2,R) representation
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The pith

The discrete Krylov chain becomes Klein-Gordon dynamics of a scalar field in the AdS2 throat in the continuum limit.

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

This paper links the abstract process of operator growth in quantum mechanics to the geometry near a black hole horizon. In the continuum limit, the steps along the Krylov subspace chain become the wave equation for a field living in two-dimensional anti-de Sitter space. The growth speed of the Lanczos coefficients then equals pi times the Hawking temperature, which automatically gives the fastest allowed rate of quantum chaos. The stability bound from AdS gravity appears as a necessary condition for this mapping to hold. A reader should care because it provides a geometric picture for how quantum information spreads, connecting two seemingly distant areas of physics.

Core claim

We show that in the continuum limit the discrete evolution on the Krylov chain transforms into the dynamics of a continuous field isomorphic to the Klein-Gordon equation for a scalar field in the AdS2 throat. This correspondence identifies the linear growth rate of Lanczos coefficients with the Hawking temperature, alpha equals pi T, thereby recovering the saturation of the maximal chaos bound. The Breitenlohner-Freedman bound emerges as a necessary consistency requirement for the dual description of Krylov subspace dynamics in a unified SL(2,R) representation.

What carries the argument

The continuum limit of the discrete Krylov chain, which maps isomorphically to the Klein-Gordon dynamics of a scalar field in the AdS2 near-horizon throat.

If this is right

  • The linear growth rate of Lanczos coefficients saturates the maximal chaos bound when set equal to pi T.
  • The Breitenlohner-Freedman bound becomes a necessary consistency condition for the dual description of Krylov dynamics.
  • The emergent geometry of the Krylov subspace reflects the near-horizon AdS spacetime in a unified SL(2,R) representation.
  • Operator growth in Krylov subspace admits a holographic gravitational dual in AdS2 gravity.

Where Pith is reading between the lines

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

  • If the mapping holds, Krylov complexity in thermal systems could serve as a probe of black hole interior dynamics.
  • Numerical checks in solvable models like the SYK chain could confirm the exact relation alpha equals pi T.
  • This unification suggests that other measures of operator growth might have similar geometric duals in higher-dimensional AdS spaces.

Load-bearing premise

The continuum limit of the discrete Krylov evolution must exist and map isomorphically to the AdS2 geometry without extra corrections or parameter inconsistencies.

What would settle it

Compute the Lanczos coefficients for a concrete system known to be dual to AdS2, such as the finite-temperature SYK model, and check if their asymptotic growth rate is exactly pi times the temperature or shows deviations.

Figures

Figures reproduced from arXiv: 2602.11627 by Hyun-Sik Jeong.

Figure 1
Figure 1. Figure 1: Holographic mapping between the AdS2 bulk and the Krylov chain. The AdS2 bulk spacetime, where a field Φ(t, ζ) propagates towards the horizon, is mapped onto a semi-infinite chain with sites ϕn(t) in the large-n limit: the continuum field ϕ(t, x) captures the near-horizon AdS2 dy￾namics. Large-n Krylov Subspace Near-horizon AdS2 Gravity Krylov wave equation Klein-Gordon equation Krylov chain index (n) Bulk… view at source ↗
read the original abstract

We establish a holographic gravitational dual for the fundamental dynamical equations governing operator growth in Krylov subspace. Specifically, we show that the deep interior of the Krylov subspace maps directly to the near-horizon regime of AdS$_2$ gravity. We demonstrate that, in the continuum limit, the discrete evolution on the Krylov chain transforms into the dynamics of a continuous field, which is isomorphic to the Klein-Gordon equation for a scalar field in the AdS$_2$ throat. This correspondence identifies the linear growth rate of Lanczos coefficients with the Hawking temperature, $\alpha=\pi T$, thereby recovering the saturation of the maximal chaos bound. Notably, the Breitenlohner-Freedman bound, a fundamental stability criterion in AdS gravity, emerges as a necessary consistency requirement for the dual description of Krylov subspace dynamics. Our results advance a Krylov-based holographic dictionary in a unified $SL(2, \mathbb{R})$ representation, revealing that the emergent geometry of Krylov subspace is a reflection of the near-horizon AdS spacetime.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

3 major / 2 minor

Summary. The paper claims to establish a holographic duality between Krylov subspace dynamics governing operator growth and near-horizon AdS₂ gravity. In the continuum limit, the discrete Krylov chain evolution maps isomorphically to the Klein-Gordon equation for a scalar field in the AdS₂ throat. This identifies the linear growth rate α of Lanczos coefficients with the Hawking temperature via α=πT, recovering saturation of the maximal chaos bound, with the Breitenlohner-Freedman bound arising as a consistency condition, all within a unified SL(2,ℝ) representation.

Significance. If the central mapping is rigorously established, the result would furnish a direct geometric interpretation of Krylov complexity in the simplest AdS₂ setting, linking quantum chaos and operator growth to black-hole thermodynamics. It could strengthen the emerging Krylov-holography dictionary and provide a controlled arena for testing how discrete recurrence relations generate emergent spacetime geometries.

major comments (3)
  1. [Continuum limit section] The continuum-limit procedure that replaces the three-term Krylov recurrence with the AdS₂ Klein-Gordon operator must explicitly demonstrate the absence of O(a) and O(a²) corrections. The scaling that sends the effective lattice spacing a→0 while holding the AdS radius and temperature fixed is not shown to preserve the exact AdS₂ metric; any residual deformation would simultaneously invalidate both the geometry and the α=πT identification.
  2. [Parameter identification and SL(2,ℝ) section] The identification α=πT is presented as a consequence of the duality, yet the manuscript does not derive the temperature scale independently from the Krylov parameters. Because the same SL(2,ℝ) generators are used both to construct the Krylov basis and to realize the AdS₂ isometries, it remains unclear whether the equality follows from the mapping or is imposed by normalization choices to recover the chaos bound.
  3. [Consistency conditions section] The emergence of the Breitenlohner-Freedman bound as a consistency requirement for the dual description is asserted, but the manuscript does not show that the effective mass squared in the emergent AdS₂ geometry lies above the BF bound for generic Lanczos coefficients; this step is load-bearing for the stability claim.
minor comments (2)
  1. [Introduction] The abstract refers to the 'deep interior of the Krylov subspace' without a precise definition in terms of the Lanczos coefficients b_n; the manuscript should specify the regime of n where the continuum approximation is controlled.
  2. [Notation and dictionary] Notation for the effective AdS₂ radius and the precise dictionary between Krylov chain parameters and gravitational quantities should be tabulated for clarity.

Simulated Author's Rebuttal

3 responses · 0 unresolved

We thank the referee for the careful reading and constructive comments. We address each major comment below and indicate the revisions we will make to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Continuum limit section] The continuum-limit procedure that replaces the three-term Krylov recurrence with the AdS₂ Klein-Gordon operator must explicitly demonstrate the absence of O(a) and O(a²) corrections. The scaling that sends the effective lattice spacing a→0 while holding the AdS radius and temperature fixed is not shown to preserve the exact AdS₂ metric; any residual deformation would simultaneously invalidate both the geometry and the α=πT identification.

    Authors: We agree that a more explicit demonstration of the continuum limit is required. In the revised manuscript we will add a dedicated subsection performing a systematic Taylor expansion of the discrete three-term recurrence in powers of the lattice spacing a. We will show that the O(a) and O(a²) terms vanish identically in the limit a→0 with fixed AdS radius and temperature, confirming that the emergent metric is exactly AdS₂ with no residual deformation. revision: yes

  2. Referee: [Parameter identification and SL(2,ℝ) section] The identification α=πT is presented as a consequence of the duality, yet the manuscript does not derive the temperature scale independently from the Krylov parameters. Because the same SL(2,ℝ) generators are used both to construct the Krylov basis and to realize the AdS₂ isometries, it remains unclear whether the equality follows from the mapping or is imposed by normalization choices to recover the chaos bound.

    Authors: The identification follows from the representation theory: the Lanczos coefficients fix the matrix elements of the SL(2,ℝ) generators on the Krylov chain, which map to the Killing vectors of AdS₂ in the continuum limit. The temperature is then fixed independently by the surface gravity of the emergent near-horizon geometry. We will expand the relevant section to derive this relation explicitly from the commutators and the metric identification, clarifying that it is not an imposed normalization. revision: partial

  3. Referee: [Consistency conditions section] The emergence of the Breitenlohner-Freedman bound as a consistency requirement for the dual description is asserted, but the manuscript does not show that the effective mass squared in the emergent AdS₂ geometry lies above the BF bound for generic Lanczos coefficients; this step is load-bearing for the stability claim.

    Authors: We acknowledge the need for an explicit verification. In the revision we will compute the effective mass squared directly from the continuum limit of generic Lanczos coefficients and demonstrate that it satisfies m² ≥ −1/4 (in units where the AdS radius is unity) as a necessary condition for stability of the dual description. revision: yes

Circularity Check

1 steps flagged

Continuum limit identification of Lanczos growth rate α with AdS₂ temperature T is imposed by coordinate normalization choice

specific steps
  1. fitted input called prediction [Abstract]
    "This correspondence identifies the linear growth rate of Lanczos coefficients with the Hawking temperature, α=πT, thereby recovering the saturation of the maximal chaos bound."

    The input α is the slope of the Lanczos coefficients b_n ≈ α n taken from the discrete Krylov chain. The continuum limit replaces the recurrence by a differential operator whose coefficient is then scaled to match the standard AdS₂ Klein-Gordon form (already containing T). The equality α=πT is therefore enforced by the normalization chosen in the limit, not derived from the discrete dynamics alone.

full rationale

The paper takes the linear Lanczos growth b_n ≈ α n as input from the Krylov recurrence and replaces the discrete three-term recurrence with a second-order differential operator in the continuum limit. The emergent equation is then matched to the standard Klein-Gordon operator in the AdS₂ throat by rescaling the continuous coordinate; this rescaling is chosen so that the coefficient of the resulting operator reproduces the known AdS₂ form whose scale is set by the Hawking temperature. Consequently the relation α = π T follows by construction of the dictionary rather than as an independent output. The saturation of the maximal chaos bound is recovered only after this identification is imposed. The BF bound appears as a consistency check on the same matched equation. No external benchmark or parameter-free derivation is supplied that would fix the temperature scale without reference to the target AdS₂ geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on the continuum-limit assumption and the SL(2,R) unification without introducing new free parameters or invented entities beyond standard holographic identifications.

axioms (2)
  • domain assumption Continuum limit of the discrete Krylov chain exists and yields a continuous field theory isomorphic to the Klein-Gordon equation in AdS2
    Invoked to transform discrete Lanczos evolution into the AdS2 throat dynamics.
  • domain assumption SL(2,R) representation provides a unified holographic dictionary
    Used to identify the emergent geometry of the Krylov subspace with near-horizon AdS spacetime.

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Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Holographic Krylov Complexity for Charged, Composite and Extended Probes

    hep-th 2026-04 unverdicted novelty 7.0

    Holographic Krylov complexity for charged composite and extended probes retains universal leading large-time growth but acquires structure-dependent subleading corrections.

  2. Cosmological brick walls & quantum chaotic dynamics of de Sitter horizons

    hep-th 2026-03 unverdicted novelty 6.0

    Brick-wall spectra in de Sitter space show long-range chaotic signatures via spectral form factor and Krylov complexity even when conventional level repulsion is absent.

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