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Holography at Finite N: Breakdown of Bulk Reconstruction for Subregions

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Pith's one-line read Reconstructed AdS-Rindler bulk operators in a large but finite $N$ CFT acquire three-point functions growing like $(1/N)e^{\pi(|\lambda|-\omega)/2}$, so they cannot be defined once the smearing scale exceeds $\Lambda=(2/\pi)\ln N$, far belo

desk verdict A clean computation of exponential growth in the leading HKLL Rindler three-point function, but the argument that 1/N corrections can't cancel it is not rigorous — the breakdown conclusion is premature. read the letter →

arxiv 2508.11592 v2 pith:UWWGCIG7 submitted 2025-08-15 hep-th

Seiji Terashima This is my paper · ORCID
classification hep-th
keywords AdS/CFTbulkreconstructionS-RindlerwedgefiniteN1/Nexpansionentanglementhorizon-to-horizonmodesblackholeinformationparadox
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

This paper tries to establish that the standard bulk-reconstruction story in AdS/CFT—which works at $N=\infty$—does not survive when $N$ is large but finite, at least inside the AdS-Rindler wedge. It shows that the CFT operator built by the HKLL construction from boundary data on a subregion, which should represent a smeared bulk field, has three-point functions that grow like $(1/N)e^{\pi(|\lambda|-\omega)/2}$ with bulk momentum. Once the smearing scale $\Lambda$ exceeds $(2/\pi)\ln N$, this growth overwhelms the $1/N$ suppression, the large-$N$ expansion becomes unreliable, and the would-be bulk operator ceases to be a well-defined observable in the full quantum theory. Because the Rindler wedge describes the near-horizon region of black holes, a correct result here would impose a specific logarithmic cutoff on reconstructing operators across horizons, with direct bearing on the black hole information paradox.

What carries the argument

The central object is the ratio of the Rindler radial mode function $\tilde{\psi}_{\omega,\lambda}(\xi_0)$ to its normalization $N_{\omega,\lambda}$. In the horizon-to-horizon regime $|\lambda|>\omega$ with a bulk point satisfying (3.4), the mode function is only power-law suppressed while the normalization contains $e^{-\pi(|\lambda|-\omega)/2}$, so the reconstructed operator $\phi_R^l\sim (1/N_{\omega,\lambda})O_{\omega,\lambda}$ carries an uncompensated factor $e^{\pi(|\lambda|-\omega)/2}$. The argument then uses contour shifts in the light-cone coordinates $(u,v)$: in two-point functions the $i\epsilon$ poles cancel this factor, but in the ordered three-point function the poles do not. T

What would settle it

Numerically evaluate the ratio $\tilde{\psi}_{\omega,\lambda}(\xi_0)/N_{\omega,\lambda}$ from (2.12) for large $\omega,|\lambda|$ with $|\lambda|-\omega=(2/\pi)\ln N$ and $\xi_0$ satisfying (3.4); the claim requires this ratio to grow like $e^{\pi(|\lambda|-\omega)/2}$ times a power law, and fails if the two saddle points in (A.11)-(A.12) cancel the exponential. Equivalently, compute the $O(1/N)$ double-trace correction to $\phi_R^l$ and check whether it cancels the $(1/N)e^{\pi(|\lambda|-\omega)/2}$ term in the three-point function.

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Extended reading notes

Core claim

The central discovery is a quantitative obstruction: in the AdS-Rindler patch of a large but finite $N$ holographic CFT, the HKLL-reconstructed operator for a smeared bulk wave packet carries a normalization factor $1/N_{\omega,\lambda}\sim e^{\pi(|\lambda|-\omega)/2}$ for horizon-to-horizon modes ($|\lambda|>\omega$). Two-point functions hide this factor because contour shifts bring down a compensating $e^{-\pi(|\lambda|-\omega)/2}$; the ordered three-point function $\langle 0|O_1\phi_R^l O_2|0\rangle$ leaves it uncompensated, giving $\sim (1/N)e^{\pi(|\lambda|-\omega)/2}$. A bulk local operator smeared at scale $\Lambda$ contains modes with $|\lambda|-\omega$ up to $\Lambda$, so the large-

Load-bearing premise

The load-bearing premise is that, for horizon-to-horizon modes ($|\lambda|>\omega$) at a bulk point satisfying (3.4), the Rindler wavefunction is only power-law suppressed while the normalization contains $e^{-\pi(|\lambda|-\omega)/2}$, and that neither the choice of saddle point nor any $1/N$ correction to the reconstructed operator removes the resulting $e^{\pi(|\lambda|-\omega)/2}$ growth in the ordered three-point function.

Editorial extensions

If this is right

  • The standard HKLL reconstruction for the AdS-Rindler wedge, and with it entanglement-wedge reconstruction as stated at $N=\infty$, does not extend to finite $N$ once operators are smeared above $(2/\pi)\ln N$.
  • Any low-energy effective bulk description of the Rindler (near-horizon) region must have a UV cutoff of order $\ln N$, far below the Planck scale $M_{pl}\sim N^{2/(d-1)}$, rather than a Planck-scale cutoff.
  • Bulk operators that exist in the perturbative gravity theory—including gravitational Wilson-line dressings—fail to be well-defined observables in the full quantum theory above the threshold.
  • Holographic quantum error correction and subregion duality, which are built on the $N=\infty$ structure, cannot hold at finite $N$ without significant modification; the horizon-to-horizon modes are precisely the operator excitations that do not exist.
  • The black-hole information paradox is directly affected: near-horizon (Rindler) physics has a sharp $\ln N$ cutoff for reconstructing operators across the horizon, constraining how interior information can be encoded.

Reading between the lines

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

  • The same logarithmic obstruction plausibly applies to any subregion whose causal wedge has a bifurcate horizon, not just AdS-Rindler; the paper notes that gravitational-collapse black holes may differ, so this extension is mine.
  • A clean testable extension is to compute the three-point function with the stress tensor in place of $O_1$ (as the paper sketches in a footnote): if the same $(1/N)e^{\pi(|\lambda|-\omega)/2}$ growth appears, the obstruction is universal rather than an artifact of the scalar $O\,O\,O$ coupling.
  • If the claim holds, the effective UV cutoff of semiclassical gravity near horizons is set by $1/N$ corrections, which suggests that bulk locality in subregions is an emergent, $N$-dependent property rather than a fixed feature of the gravitational description.
  • One could try to construct an improved finite-$N$ reconstruction kernel (for example including double-trace corrections) and check directly whether the exponential three-point growth cancels; the paper argues it cannot, but the explicit construction would settle the point.
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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

2 major / 4 minor

Summary. This paper studies AdS/CFT bulk reconstruction in the AdS-Rindler wedge at large but finite N. It considers a smeared bulk wave-packet operator with large momentum (omega_0, lambda_0) with |lambda_0| > omega_0, and its HKLL reconstructed CFT expression. The paper shows that in a three-point function with the reconstructed operator in the middle, the exponential normalization factor exp[(pi/2)(|lambda_0|-omega_0)] is not canceled, unlike in the two-point function, giving Eq. (3.17): the correlator is of order (1/N) exp[(pi/2)(|lambda_0|-omega_0)]. Hence, for a UV cutoff Lambda >~ (2/pi) ln N, the 1/N expansion fails and the bulk operator is claimed not to be definable. The technical derivation involves saddle-point asymptotics of the Rindler mode function in Appendix A and contour/pole analysis of CFT correlators in Section 3.

Significance. If correct, the paper would imply a logarithmic, rather than Planckian, UV cutoff for subregion bulk reconstruction and would challenge entanglement wedge reconstruction and holographic quantum error correction at finite N. The Appendix's saddle-point calculation and the explicit two-point cancellation are useful technical steps. However, the paper's central conclusion is not yet supported because the argument that 1/N corrections cannot cancel the exponential enhancement is incomplete and, as stated, too strong. The strength of the claimed obstruction therefore remains conditional.

major comments (2)
  1. [Sec. 3, 'Corrections to the reconstructed operator'] The proof that 1/N corrections cannot remove the growth in (3.17) rests on the assertion that a double-trace correction delta phi^R_l would give a four-point function that 'cannot be the same form' as the three-point function. This is not correct as stated. If delta phi^R_l contains an integral of K(u,v) O(u,v) O(u,v), the disconnected part of <0|O1 delta phi^R_l O2|0> factorizes as <O1 O><O O2>, i.e. products like (u-u1)^{-D/2}(v-v1)^{-D/2}(u-u2)^{-D/2}(v-v2)^{-D/2}. After integrating against a kernel localized near u ~ u2, v ~ v1, which are the poles selected in (3.16), this gives exactly the (u1-u2)^{-D/2}(v2-v1)^{-D/2} coordinate dependence of the leading term, with a coefficient that may carry the same 1/N and exp[(pi/2)(|lambda|-omega)] factors. The N=infinity uniqueness of the Fourier reconstruction fixes only the leading term; it does not constrain the 1/N coefficient. Without an
  2. [Sec. 3, Eqs. (3.5)-(3.7) and (3.17)] The paper treats the N=infinity HKLL operator as the candidate finite-N operator and argues that corrections cannot help. But the conclusion that 'such an operator does not exist' is stronger than what is shown. The calculation demonstrates a property of the particular N=infinity reconstructed kernel; it shows that the leading-order 1/N expansion of that correlator grows. It does not rule out a finite-N definition of the bulk operator with the same two-point function but modified three-point functions, unless the missing 1/N-correction analysis is supplied. The wording should at least distinguish 'the HKLL N=infinity operator fails' from 'no bulk-local operator exists in the full theory.'
minor comments (4)
  1. [Eq. (2.23)] In the second line of (2.23), for omega^2 < lambda^2 the prefactor should be ((lambda^2-omega^2)/4)^{nu/2}, not ((omega^2-lambda^2)/4)^{nu/2}; the formula as printed is non-real for non-integer nu unless a modulus is intended.
  2. [Abstract and Sec. 1] The statement that the three-point function is 'fixed by conformal symmetry except for an overall constant' should be qualified: the computed object is an integral transform of a CFT three-point function, not itself a three-point function of local primary operators. The conformal symmetry statement is not the load-bearing step, but the current wording is misleading.
  3. [Acknowledgements] There is a minor grammatical issue: the paper appears to have a single author, but the acknowledgement reads 'The authors would like to thank...'.
  4. [Sec. 2.2] The gravitational Wilson line discussion is heuristic and not necessary for the main calculation. It would be clearer to present it as a separate conjecture or to make its assumptions explicit.

Circularity Check

0 steps flagged · score 2.0 of 10

No significant circularity: the exponential growth is computed from an explicit saddle-point evaluation, not fitted; self-citations are background.

full rationale

The central derivation is self-contained. The reconstructed operator (3.5) is obtained by substituting the mode relation (2.19) into the explicit wave packet (3.1); the large factor 1/N_{omega,lambda} is the standard Rindler normalization written out in (2.14), whose asymptotic (2.23) follows from Gamma-function asymptotics and is not an ansatz that assumes the target result. Appendix A evaluates the hypergeometric radial function by saddle point and identifies the contributing saddles via explicit descent-path/streamline analysis; no parameter is fitted to reproduce the three-point growth. The three-point result (3.15)-(3.17) is a direct contour integration of the fixed conformal three-point function (3.13); the O(1/N) coefficient C_OOO is an input, but the exponential e^{pi/2(|lambda0|-omega0)} is not inserted by hand: it emerges from Im(u0), Im(v0) and survives the pole contribution. Citations [15]-[19] are used for the interpretive framing (subregion complementarity, horizon-to-horizon modes, indications of 1/N-expansion breakdown), not as the derivation of (3.17). The citation to [17] for N_{omega,lambda} and to [8] for uniqueness of the N=infinity reconstruction are to independently checkable/standard results, so they do not make the argument circular. The only genuinely weak point is the final paragraph's assertion that a double-trace 1/N correction cannot cancel the growth because a four-point function 'cannot be the same form'; this is an unsupported (and possibly false) claim about the disconnected part of the four-point function, but it is a correctness risk, not a circularity. Hence score 2 rather than 0, reflecting the paper's reliance on the authors' earlier framework, but with no reduction of the central claim to its inputs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No fits to data or hand-tuned parameters enter the central estimate; omega0, lambda0 and a are kinematic wave-packet parameters. The result uses standard AdS/CFT assumptions and a technical saddle-point lemma. The strongest secondary argument, about gravitational Wilson lines, is not central and is excluded from the ledger.

assumptions (4)
  • domain assumption AdS/CFT with an interacting bulk scalar dual to a CFT primary O_Delta holds, and at N=infinity the HKLL reconstruction in the AdS-Rindler wedge is valid and unique.
    Invoked throughout Section 2.1; based on [4,8,13] and the author's earlier work [17,18].
  • domain assumption The three-point function of the primary has the conformal form (3.13) with nonzero coefficient C_OOO = O(1/N).
    Used in Section 3 after Eq. (3.13); the claimed N^{-1} exp(pi/2(|lambda|-omega)) scaling is directly proportional to C_OOO.
  • standard math The large-omega,|lambda| saddle point evaluation of the hypergeometric function in Appendix A, with both saddles contributing, gives (A.11) and (A.12).
    Supported by an Euler integral and streamline plots, but no formal verification; a wrong saddle selection removes the enhancement.
  • domain assumption The i epsilon prescription and contour deformations in Section 3 select the poles at u=u2+i epsilon and v=v1-i epsilon, leaving the exponential factor uncancelled.
    Standard CFT ordering conventions are applied; this is where the exponential enhancement survives in the three-point function.

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Pith. "Pith review of Holography at Finite N: Breakdown of Bulk Reconstruction for Subregions." pith.science (2026). https://pith.science/paper/UWWGCIG7

@misc{pith2026250811592,
  author       = {Pith},
  title        = {Pith review of: Holography at Finite N: Breakdown of Bulk Reconstruction for Subregions},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/UWWGCIG7}},
  note         = {Machine review of arXiv:2508.11592}
}
abstract

Within AdS/CFT, focusing on the AdS-Rindler wedge, we show that when $N$ is large but finite, correlation functions of reconstructed bulk operators grow exponentially with bulk momentum, overwhelming the usual $1/N$ suppression. The growth starts when the smeared operator's ultraviolet scale goes beyond a critical value $\Lambda_{crit} = \frac{2}{\pi}\ln N$, which is far below the Planck scale. Above this logarithmic threshold, the large $N$ expansion ceases to be reliable, and the would-be bulk operators cannot be consistently defined as observables in the full quantum gravity theory. Since the AdS-Rindler wedge describes the near-horizon region of black holes, this result implies a sharp $\ln N$ cutoff for reconstructing bulk operators across horizons. This has a direct impact on whether and how information from the black hole interior is encoded-a central question in the black hole information paradox.

Figures

Figures reproduced from arXiv: 2508.11592 by the authors.

Figure 1
Figure 1. Bulk operator in AdS-Rindler wedge MA. Claim of this paper Let us take the bulk operator ϕ as a bulk local operator smeared by a UV regulator Λ, for example ϕ ∼ R dx e− Λ 2 2 (x−x0) 2 ϕ(t = t0, x). Then, we can show that the corresponding reconstructed CFT operator ϕ R in A satisfies ⟨0|O1ϕ RO2|0⟩ ∼ 1 N e βΛ , (2.4) where O1, O2 are generic CFT operators and β is a non-negative constant, as we later find β = π/2. Le… view at source ↗
Figure 2
Figure 2. Bulk smeared local field ϕ in the AdS-Rindler wedge, dressed by a gravitational Wilson line that ends on the boundary subregion A on the tR = 0 slice. The line provides the diffeomorphism-invariant dressing, so that ϕ becomes gauge-invariant. The Rindler horizons bound the wedge (schematic). Let us consider a bulk (smeared) local field ϕ with the gravitational Wilson line attached to it, such that it becomes a gauge… view at source ↗
Figure 3
Figure 3. Unitary operator U = e iϕ with the gravitational Wilson line ending on the very small boundary subregion A. This illustrates that, while such dressed operators can exist in the bulk description, for a large UV smearing scale Λ, they cease to be well-defined within the low energy effective theory. that ⟨s|T00|s⟩ = ⟨s|e −iϕT00(x)e iϕ|s⟩ if T00(x) is in the subregion A¯ and |s⟩ is an arbitrary low energy state. Here, T… view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: Streamline plot of the complex vector field defined by the derivative [PITH_FULL_IMAGE:figures/full_fig_p022_4.png]
Figure 5
Figure 5. Figure 5: Streamline plot shown over the domain x ∈ [−0.2, 1.6], y ∈ [−0.5, 0.5] with parameters ϵ = 8, x = 1 3 . 21 [PITH_FULL_IMAGE:figures/full_fig_p022_5.png]
Figure 6
Figure 6. Figure 6: Streamline the plot shown over the domain x [PITH_FULL_IMAGE:figures/full_fig_p023_6.png]

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Forward citations

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