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Codimension-Zero Holographic Complexity and Spectral Three-Point Correlators

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Masamichi Miyaji, Shan-Ming Ruan, Shono Shibuya, Kazuyoshi Yano

High Energy Physics - Theory · Submitted 9 Oct 2026 · 83 references

Pith's verdict: Solid extension of the spectral mechanism to codimension-zero, but a load-bearing factorization assumption about matrix elements and spectral fluctuations is acknowledged but untested.

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arXiv ID2610.13289 (version 1)
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CitationKazuyoshi Yano, Masamichi Miyaji, Shan-Ming Ruan, Shono Shibuya. "Codimension-Zero Holographic Complexity and Spectral Three-Point Correlators." arXiv:2610.13289 (2026). https://pith.science/paper/JFLYQLMY
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The paper

The paper claims

Codimension-zero holographic complexity saturates at the Heisenberg time due to level repulsion, with the generating function governed by three-point spectral correlators.

The reading

The paper studies holographic complexity, a measure of how difficult it is to prepare a black hole state. Earlier work showed that for simple hypersurfaces (codimension-one), complexity grows linearly in time and then plateaus because of quantum chaos effects. This paper extends that analysis to more general bulk regions (codimension-zero), such as the entire Wheeler-DeWitt patch. The main finding is that this more general complexity also shows linear growth followed by a plateau, and it equals the sum of two simpler hypersurface complexities. The key new ingredient is that the calculation now involves three-point correlations of energy levels, which organize the quantum corrections. The late-time plateau is a direct consequence of level repulsion, a universal feature of chaotic quantum systems. The paper also constructs a well-defined complexity operator by orthogonalizing the set of fixed-complexity states, which are overcomplete due to nonperturbative effects. This gives a clear quantum-mechanical interpretation of the saturation: the finite number of black hole microstates prevents indefinite growth.
Extended reading notes

Core claim

The codimension-zero holographic complexity reduces to the sum of two codimension-one contributions and exhibits a universal time evolution characterized by an exponentially long regime of linear growth followed by saturation at the Heisenberg time.

Load-bearing premise

The matrix element of the complexity generating operator must have a specific first-order pole structure in the energy difference (eq 2.12), which is taken as a necessary condition for linear growth and not derived from a more fundamental microscopic principle. The derivation relies on this pole structure as a load-bearing input.

Original abstract

Holographic complexity is expected to capture the linear growth of black hole interiors, with quantum corrections ultimately leading to a late-time saturation. We develop a spectral formulation of holographic complexity measures associated with codimension-zero bulk regions in the framework of the complexity=anything proposal. The corresponding generating function of complexity is governed by three-point correlation functions of the density of states, establishing a direct connection between bulk geometric observables and nonperturbative Euclidean spacetime wormholes. We find that the codimension-zero holographic complexity reduces to the sum of two codimension-one contributions and exhibits a universal time evolution, characterized by an exponentially long regime of linear growth followed by saturation at the Heisenberg time. The late-time plateau is a direct result of the spectral level repulsion, a generic feature of quantum chaotic systems, thereby providing a universal mechanism for the saturation of holographic complexity. We also develop a complementary operator interpretation, in which Euclidean wormhole corrections render the semiclassical basis of fixed-complexity states overcomplete. Constructing a complete orthonormal basis then yields well-defined holographic complexity operators that reproduce the same universal time evolution. Our results establish a unified connection among black hole interiors, chaotic spectral statistics, and Euclidean wormholes in quantum gravity.

The compiled record

The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.

Compiled by classifier-1.13.0 via TypeSafe API, question bank abstract-p1-20260923, 2026-10-12: 1 request, 6,358 input tokens.

What the abstract itself states, as probabilities: claims first 0.05 · states limitation 0.09 · human participants 0.01 · controlled comparison 0.02 · randomized assignment 0.01 · states quantitative result 0.19 · compares with prior results 0.23 · mentions code or data availability 0.02.

The abstract, sentence by sentence

arXiv listing abstract v1. Results and the main claim are underlined; .

Holographic complexity is expected to capture the linear growth of black hole interiors, with quantum corrections ultimately leading to a late-time saturation. We develop a spectral formulation of holographic complexity measures associated with codimension-zero bulk regions in the framework of the complexity=anything proposal. The corresponding generating function of complexity is governed by three-point correlation functions of the density of states, establishing a direct connection between bulk geometric observables and nonperturbative Euclidean spacetime wormholes. We find that the codimension-zero holographic complexity reduces to the sum of two codimension-one contributions and exhibits a universal time evolution, characterized by an exponentially long regime of linear growth followed by saturation at the Heisenberg time. The late-time plateau is a direct result of the spectral level repulsion, a generic feature of quantum chaotic systems, thereby providing a universal mechanism for the saturation of holographic complexity. We also develop a complementary operator interpretation, in which Euclidean wormhole corrections render the semiclassical basis of fixed-complexity states overcomplete. Constructing a complete orthonormal basis then yields well-defined holographic complexity operators that reproduce the same universal time evolution. Our results establish a unified connection among black hole interiors, chaotic spectral statistics, and Euclidean wormholes in quantum gravity.

background contribution method result interpretation main claim

What the classifier reports for each sentence. The number is the probability it gave the label.
#sentencerolethe authors' own workstates the main conclusionstrength
s0 Holographic complexity is expected to capture the linear growth of black hole interiors, with quantum corrections ultimately leading to a late-time saturation. background 1.00 0.06 0.06 hedged possibility 0.89
s1 We develop a spectral formulation of holographic complexity measures associated with codimension-zero bulk regions in the framework of the complexity=anything proposal. contribution 0.98 0.95 0.06 asserted 1.00
s2 The corresponding generating function of complexity is governed by three-point correlation functions of the density of states, establishing a direct connection between bulk geometric observables and nonperturbative Euclidean spacetime wormholes. result 0.65 0.75 0.19 asserted 1.00
s3 We find that the codimension-zero holographic complexity reduces to the sum of two codimension-one contributions and exhibits a universal time evolution, characterized by an exponentially long regime of linear growth followed by saturation at the Heisenberg time. result 1.00 0.95 0.48 asserted 1.00
s4 The late-time plateau is a direct result of the spectral level repulsion, a generic feature of quantum chaotic systems, thereby providing a universal mechanism for the saturation of holographic complexity. interpretation 0.94 0.77 0.26 asserted 1.00
s5 We also develop a complementary operator interpretation, in which Euclidean wormhole corrections render the semiclassical basis of fixed-complexity states overcomplete. interpretation 0.51 0.92 0.07 asserted 1.00
s6 Constructing a complete orthonormal basis then yields well-defined holographic complexity operators that reproduce the same universal time evolution. result 0.38 0.91 0.10 asserted 1.00
s7 Our results establish a unified connection among black hole interiors, chaotic spectral statistics, and Euclidean wormholes in quantum gravity. interpretation 0.92 0.86 0.83 asserted 1.00

What kind of paper the classifier reports this is: theoretical result 0.96 · evidence from proof or derivation 0.98.

What was examined, and what was not

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  • figures and tables
  • references and citing sentences
  • relations to other papers
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Every quoted sentence on this record is the authors'. Every label is from Pith's closed vocabulary, with the probability shown beside it. Every number in the measurements table is the authors', checked by code. Record as JSON

Figures
Figure 1. Schematic illustration of the Complexity=Anything proposal. The left panel rep￾resents the codimension-one observable evaluated on an extremal hypersurface Σ anchored at the boundary times tL and tR. The right panel defines codimension-zero observables associated with a spacetime region, whose future and past boundaries are the extremal hypersurfaces Σ+ and Σ−. WDW-patch observables arise as the special limit of this more general construction. 1 Introduction and Summary Holographic complexity has become an important probe of the black hole interior. Quantum complexity provides a…
Figure 2. Universal time evolution of holographic complexity. The early-time linear regime is controlled by a characteristic pole structure in the matrix elements of the holographic complexity operator. The late-time plateau is controlled by level repulsion in the discrete spectrum of chaotic systems. The crossover occurs at t ∼ e S0 , equivalently at the Heisenberg time TH. the Wheeler-DeWitt patch. These proposals all exhibit the expected late-time linear growth for two-sided AdS black hole spacetimes. At the same time, they also contain conventional ambiguities. For example, the CV and CV2
Figure 3. Discrete spectrum of holographic CFT and black hole microstates. The left part depicts the large-N hierarchy: low-spin operators lie near ∆ = O(1), higher-spin operators are separated by a large gap, and black hole microstates appear at energies of order N #. The right part zooms into a microcanonical window E0 − ∆E/2 < E < E0 + ∆E/2, containing ρ(E0)∆E ∼ e SBH states. The discreteness of the spectrum is expected to be visible at the Heisenberg time TH. Its spectral representation takes the form 1
Figure 4. Schematic illustration of Euclidean wormhole saddles suggested by the decom￾position of the two-point correlation function of the spectral density in eq. (2.15). Each diagram represents a distinct topology of connectivity between bra and ket. This limit is performed by subtracting the unphysical divergence while imposing that the generating operator reduces to the identity in the limit α → 0, i.e., lim α→0 e[−αC = 1ˆ = X complete basis |C⟩⟨C| . (2.19) Interestingly and crucially, the particular pole located Eij = −iα˜ therefore reduces to Pole: Ei → Ej , with α → 0 . (2.20) See se
Figure 5. Left: Schematic time evolution of the generating function ⟨e[−αC⟩, displaying the characteristic slope-ramp-plateau structure. Right: The corresponding expectation value of the complexity measure is extracted from the generating function in the limit α → 0. the Heisenberg time, spectral correlations gradually suppress the growth and eventually cancel the classical linear growth. The complexity then saturates at a plateau of order TH. Since TH ∼ e S0 , the linear-growth regime persists for a time exponentially long in the entropy, in agreement with the expected behavior of complexity…
Figure 6. Schematic illustration of the Euclidean wormhole sectors suggested by the de￾composition of the three-point spectral correlator in eq. (2.73). The diagrams represent the different connectivity patterns among the bra, operator insertion, and ket sectors. Schemat￾ically, the operator-ket sector can be obtained by inverting the bra-operator sector. To characterize the semiclassical limit, it is useful to interpret the sine kernel as a distribution. When integrated against test functions that vary smoothly on energy scales much larger than the mean level spacing, one has sin(Λx) πx ⇀Λ→∞…
Figure 7. Contributions to the codimension-zero generating function G(0)(α±;t) (2.84) for ∆E = 1, TH = 100 and αe± = 0.1. Green: the total generating function; red: Gclassical; blue: Gbra−ket; orange: Gbra−op and Gop−ket contributions; purple: Gbra−op−ket. For these finite control parameters α±, Gclassical and Gbra−ket provide the dominant contributions to the slope-ramp-plateau profile of the codimension-zero generating function G(0)(α±;t). The dashed lines indicate the leading ramp and plateau approximations in the regime ∆E ≫ αe±. where the exact integration domain is determined by the…
Figure 8. Total probability distribution (3.22) for tC+ − tC− = 0. We choose S0 = 3, TH ≃ 856, ∆E = 0.05, and M± = 1. The sharp peak at tC+ = tC− = t originates from Pclassical. The broad constant plateau is dominated by Pbra−ket. The bra–operator contribution follows by exchanging the two codimension-one variables, Pbra−op(tC+ , tC− , t) = Pop−ket(tC− , tC+ , t), (3.27) while the bra–ket contribution is obtained by the corresponding permutation of the three time arguments, Pbra−ket(tC+ , tC− , t) = Pop−ket(tC+ , t, tC− ). (3.28) These relations make transparent the permutation structure…
Figure 9. Total probability distribution away from the near-diagonal regime. Left: 1/∆E ≲ |tC+ −tC− | < TH. Right: |tC+ −tC− | > TH. The remaining parameters are the same as in figure 8. The operator–ket and bra–operator sectors generate peaks at tC+ = t and tC− = t, respectively. Their amplitudes grow with |tC+ − tC− | until this separation reaches the Heisenberg scale, after which they saturate. The fully connected bra–operator–ket con￾tribution vanishes at both peak centers and approaches a constant sufficiently far from both peaks. As illustrated in figure 8, the distribution is dominated…

Figures reproduced from arXiv:2610.13289 by the authors.

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