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arxiv: 2507.23667 · v2 · submitted 2025-07-31 · ✦ hep-th · gr-qc

Universal Time Evolution of Holographic and Quantum Complexity

Masamichi Miyaji , Shan-Ming Ruan , Shono Shibuya , Kazuyoshi Yano This is my paper

Pith reviewed 2026-05-19 02:15 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords holographic complexityquantum complexityspectral representationpole structurelinear growthrandom matrix theoryquantum chaosspectral form factor
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The pith

Holographic complexity grows linearly at late times because its generating functions have a specific pole structure in the energy eigenbasis, proven necessary and sufficient by the residue theorem.

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

The paper establishes that both holographic and quantum complexity evolve with a universal slope-ramp-plateau pattern in time. This pattern arises because the generating functions for codimension-one and codimension-zero complexity measures possess a particular pole structure in their matrix elements when written in the energy eigenbasis. A sympathetic reader would care because the result supplies a first-principles account, rooted in random-matrix universality and the residue theorem, for why complexity measures exhibit long-time linear growth followed by saturation in chaotic systems and black-hole interiors. The late-time plateau itself is shown to follow directly from spectral level repulsion.

Core claim

We introduce the spectral representation for generating functions associated with codimension-one and codimension-zero holographic complexity measures. These generating functions exhibit a universal slope-ramp-plateau structure. We demonstrate that this universal behavior originates from random matrix universality in spectral statistics and from a particular pole structure of the matrix elements of the generating functions in the energy eigenbasis. Using the residue theorem, we prove that the existence of this pole structure is both a necessary and sufficient condition for the linear growth of complexity measures. Furthermore, we show that the late-time saturation plateau arises directlyfrom

What carries the argument

The pole structure of the matrix elements of the generating functions in the energy eigenbasis, which permits the residue theorem to establish necessary and sufficient conditions for linear growth of the associated complexity measures.

Load-bearing premise

The matrix elements of the generating functions possess a particular pole structure in the energy eigenbasis that is assumed to hold for the holographic complexity measures under consideration.

What would settle it

A direct calculation of the matrix elements for a concrete holographic complexity measure in a chaotic system that reveals the absence of the required pole structure would falsify the necessity claim.

Figures

Figures reproduced from arXiv: 2507.23667 by Kazuyoshi Yano, Masamichi Miyaji, Shan-Ming Ruan, Shono Shibuya.

Figure 1
Figure 1. Figure 1: FIG. 1. Left: Typical time evolution of generating functions [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Complexity=anything proposal. Left: Codimension [PITH_FULL_IMAGE:figures/full_fig_p002_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. The integration contour corresponding to both the [PITH_FULL_IMAGE:figures/full_fig_p003_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: FIG. 4. Euclidean wormholes in gravitational path integral [PITH_FULL_IMAGE:figures/full_fig_p005_4.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. The universal time evolution of holographic and quan [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. The integral contour defined on the complex plane of [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: FIG. 8. Analytical continuation of [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
read the original abstract

Holographic complexity, as the bulk dual of quantum complexity, encodes the geometric structure of black hole interiors. Motivated by the complexity=anything proposal, we introduce the spectral representation for generating functions associated with codimension-one and codimension-zero holographic complexity measures. These generating functions exhibit a universal slope-ramp-plateau structure, analogous to the spectral form factor in chaotic quantum systems. In such systems, quantum complexity evolves universally, displaying long-time linear growth followed by saturation at late times. By employing the generating function formalism, we demonstrate that this universal behavior originates from random matrix universality in spectral statistics and from a particular pole structure of the matrix elements of the generating functions in the energy eigenbasis. Using the residue theorem, we prove that the existence of this pole structure is both a necessary and sufficient condition for the linear growth of complexity measures. Furthermore, we show that the late-time saturation plateau arises directly from the spectral level repulsion, a hallmark of quantum chaos.

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

1 major / 1 minor

Summary. The paper introduces spectral representations for generating functions associated with codimension-one and codimension-zero holographic complexity measures. These are shown to exhibit a universal slope-ramp-plateau structure analogous to the spectral form factor. The authors employ the residue theorem to prove that a particular pole structure in the matrix elements of these generating functions (in the energy eigenbasis) is both necessary and sufficient for the observed linear growth of complexity, while late-time saturation is attributed to spectral level repulsion characteristic of random matrix universality in chaotic systems.

Significance. If the pole structure can be justified from the bulk geometric definitions, the result would provide a rigorous, universal explanation for the linear growth phase of holographic complexity measures, independent of specific details, and strengthen connections between the complexity=anything proposal, random matrix theory, and complex analysis techniques. The explicit use of the residue theorem to establish necessity and sufficiency constitutes a clear technical strength.

major comments (1)
  1. [Abstract and section introducing the spectral representation] The residue-theorem argument (as outlined in the abstract) establishes necessity and sufficiency of the pole structure for linear growth, but the manuscript assumes without explicit derivation that the matrix elements of the generating functions for the holographic measures possess this structure when expanded in the boundary energy eigenbasis. No computation from the bulk definitions (maximal volume, gravitational action, or similar) is provided to confirm the poles at E_i - E_j = 0 with the required time-dependent residues; this assumption is load-bearing for the central universality claim.
minor comments (1)
  1. Clarify the precise definition of the generating functions and their relation to the codimension-one versus codimension-zero measures at the outset to improve readability for readers unfamiliar with the complexity=anything framework.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for identifying this important point about the justification of the pole structure. We agree that explicitly connecting the assumed pole structure to the bulk definitions would strengthen the central universality claim, and we will revise the manuscript accordingly.

read point-by-point responses

  1. Referee: [Abstract and section introducing the spectral representation] The residue-theorem argument (as outlined in the abstract) establishes necessity and sufficiency of the pole structure for linear growth, but the manuscript assumes without explicit derivation that the matrix elements of the generating functions for the holographic measures possess this structure when expanded in the boundary energy eigenbasis. No computation from the bulk definitions (maximal volume, gravitational action, or similar) is provided to confirm the poles at E_i - E_j = 0 with the required time-dependent residues; this assumption is load-bearing for the central universality claim.

    Authors: We thank the referee for this observation. The current manuscript introduces the pole structure at vanishing energy differences (with time-dependent residues) as an input motivated by the complexity=anything proposal and by the general properties of holographic observables in the energy eigenbasis of chaotic systems. We do not provide an explicit derivation of this structure starting from the bulk geometric definitions such as the maximal volume or the gravitational action. This is a fair criticism, and the assumption is indeed central to the universality argument. In the revised version we will add a new subsection (or appendix) that sketches how the required poles arise from the bulk definitions, using the known late-time behavior of the volume and action proposals together with the spectral properties of the boundary Hamiltonian. If a fully rigorous bulk-to-boundary derivation proves technically involved, we will at minimum clarify the status of the assumption and its relation to existing literature on holographic complexity. revision: yes

Circularity Check

0 steps flagged

No significant circularity; linear growth follows from explicit pole assumption via standard residue theorem

full rationale

The paper introduces spectral representations for codimension-one and codimension-zero holographic complexity generating functions, assumes a particular pole structure in their energy-eigenbasis matrix elements, and applies the residue theorem to prove this structure is necessary and sufficient for linear growth (with late-time saturation from level repulsion). This chain is self-contained: the pole structure is stated as an assumption bridging to holographic measures under the complexity=anything proposal, the residue-theorem step is ordinary complex analysis, and random-matrix universality for spectral statistics is an external input. No step reduces by construction to a fitted parameter, self-definition, or load-bearing self-citation; the derivation does not rename a known result or smuggle an ansatz. The result is therefore an equivalence under stated assumptions rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claim rests on standard mathematical tools and the assumption of random matrix universality for chaotic systems; no free parameters or new invented entities are introduced in the abstract.

axioms (2)
  • standard math Residue theorem from complex analysis applies to the contour integrals of the generating functions
    Invoked to establish necessity and sufficiency of the pole structure for linear growth.
  • domain assumption Spectral statistics of the system follow random matrix universality due to quantum chaos
    Used to explain both the ramp and the late-time plateau from level repulsion.

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Forward citations

Cited by 4 Pith papers

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

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  2. Toward Krylov-based holography in double-scaled SYK

    hep-th 2025-10 unverdicted novelty 6.0

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  3. Stringy Effects on Holographic Complexity: The Complete Volume in Dynamical Spacetimes

    hep-th 2026-04 unverdicted novelty 5.0

    Gauss-Bonnet corrections to the complete volume introduce a competition effect in static cases and prolong the critical time in two-sided shocks while the complexity growth rate stays governed by conserved momentum.

  4. Stringy Effects on Holographic Complexity: The Complete Volume in Dynamical Spacetimes

    hep-th 2026-04 unverdicted novelty 5.0

    Gauss-Bonnet corrections to the complete volume proposal introduce a competition effect in static black holes while preserving momentum-governed growth rates and logarithmic scrambling times in dynamical Vaidya geometries.

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