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arxiv: 2606.21537 · v1 · pith:PDXX24RVnew · submitted 2026-06-19 · ✦ hep-th · gr-qc

Near-horizon modifications in finite N holography

Rishkrith Bairy , Mrityunjay Nath , Debajyoti Sarkar This is my paper

Pith reviewed 2026-06-26 13:26 UTC · model grok-4.3

classification ✦ hep-th gr-qc
keywords finite N holographynear-horizon modificationsAdS2BTZ black holesnon-localityextrapolate dictionaryspectral form factorthroat parameter
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The pith

Bulk reconstructions in near-horizon modified AdS₂ and BTZ recover the non-locality estimates expected from finite N holography.

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

The paper extends the AdS/CFT extrapolate dictionary to large but finite N, where non-perturbative violations of bulk micro-causality are expected. It shows that near-horizon modifications in AdS₂ and BTZ black holes, controlled by a throat parameter, produce the same non-locality when bulk operators are reconstructed explicitly. This matches results from using a late boundary time cut-off and smearing operators with the HKLL kernel. In three dimensions, the spectral form factor of probe dynamics averaged over the throat parameter displays a dip-ramp-plateau structure, akin to models with stretched horizons or brick walls. A sympathetic reader would care because it offers a geometric way to encode finite N effects directly in the bulk geometry.

Core claim

By performing explicit bulk reconstructions in the backgrounds of near-horizon modified AdS₂ and BTZ black holes, the same non-locality estimates are recovered as those obtained from late boundary time cut-offs and HKLL smearing. The near-horizon modification is controlled by a throat parameter which sets the scale of this non-locality. In three bulk dimensions, probe dynamics exhibits a dip-ramp-plateau structure in their spectral form factor when averaged over the throat parameter, a structure also found in backgrounds with a stretched horizon or a brick wall.

What carries the argument

The near-horizon modification controlled by a throat parameter, which implements finite-N violations of the extrapolate dictionary in the bulk geometry.

If this is right

  • The non-locality estimates match those from previous methods using time cut-offs and operator smearing.
  • The throat parameter directly sets the scale of the non-locality in these modified black hole backgrounds.
  • Probe dynamics in three dimensions show a dip-ramp-plateau structure in the spectral form factor averaged over the throat parameter.
  • This structure is shared with stretched horizon and brick wall models of black hole mimickers.

Where Pith is reading between the lines

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

  • The throat parameter approach may provide a uniform geometric implementation of finite N effects across different black hole backgrounds without needing explicit boundary cutoffs.
  • Similar non-locality could appear in higher dimensional near-horizon modified geometries if the reconstruction procedure generalizes.
  • Averaging over the throat parameter might correspond to an ensemble average over different finite N corrections.

Load-bearing premise

The near-horizon modification controlled by a throat parameter faithfully implements the expected finite-N violations of the extrapolate dictionary without additional assumptions about operator smearing or time cut-offs.

What would settle it

A calculation of non-locality estimates from explicit bulk reconstructions in the modified backgrounds that differs from the estimates obtained with a late boundary time cut-off and HKLL smearing would falsify the recovery claim.

Figures

Figures reproduced from arXiv: 2606.21537 by Debajyoti Sarkar, Mrityunjay Nath, Rishkrith Bairy.

Figure 1
Figure 1. Figure 1: Finite N excisions in bulk reconstruction. boundary-time separations. In particular, when r ∼ R, the integration range is no longer symmetric about t, and the resulting bulk–boundary correlator will differ from the semiclassical expression (5) by a term that depends only on the excised late-time segment. For an arbitrary tcut < t + δt we therefore have (e.g. we can take tcut = t − δt + tmax) C(t) = ⟨ϕmod(t… view at source ↗
Figure 2
Figure 2. Figure 2: Bulk to boundary correlators involving local and non-local bulk fields. The boundary [PITH_FULL_IMAGE:figures/full_fig_p007_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: One-sided effective potential Veff(r) for representative values ω 2 ∼ 2475 with k = 50. The vertical dashed line marks the throat r = rh. Here we have also taken λ = 5 × 10−4 , rh = 3 and R = 1. Here ∗ r ≡ dr/dr∗, ∗ ψ ≡ dψ/dr∗ and ∗∗ ψ ≡ d 2ψ/dr2 ∗ . One removes the first-derivative term by the standard redefinition ψ(r∗) = ϕ(r∗) √ r , (63) in terms of which the radial KG equation takes the Schr¨odinger fo… view at source ↗
Figure 4
Figure 4. Figure 4: Schr¨odinger potential in the radial coordinate and in tortoise coordinate ( [PITH_FULL_IMAGE:figures/full_fig_p018_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Dependence of WKB levels on k and n, computed by solving the Bohr–Sommerfeld equation (69). 5.2 WKB spectrum and SFF analysis Implementing the numerical technique discussed in the previous subsection, we obtain the ground state (n = 0) WKB spectrum ω0(k) as given by figure 6. The key structural point is discreteness: for λ ̸= 0, the DS wormhole throat smoothens the would-be horizon at r = rh and yields a d… view at source ↗
Figure 6
Figure 6. Figure 6: WKB spectrum: lowest-band frequencies ω0(k) obtained from the Bohr–Sommerfeld condition (69) with n = 0, for k ∈ [−150, 150] at (R, rh, λ) = (1, 3, 10−4 ). raw SFF moving average unit-slope reference 0.1 1 10 100 1000 104 105 0.01 0.10 1 10 100 1000 104 Time t g (it ) Unnormalized SFF (a) Unnormalized. raw SFF moving average unit-slope reference 0.1 1 10 100 1000 104 105 10-6 10-5 10-4 0.001 0.010 0.100 1 … view at source ↗
Figure 7
Figure 7. Figure 7: βE = 0 SFF computed for (n, R, rh, λ) = (0, 1, 3, 10−4 ) and k ∈ [0, 150]. The dashed reference line has slope 1 and is anchored through the ramp to facilitate visual comparison to linear-ramp expectations [32, 27, 28]. The value on the vertical axis distinguishes between unnormalized and normalized cases. 21 [PITH_FULL_IMAGE:figures/full_fig_p022_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Thermal SFF for the ground band (n = 0) with k ∈ [0, 350]. with shifted time scales [PITH_FULL_IMAGE:figures/full_fig_p023_8.png] view at source ↗
Figure 9
Figure 9. Figure 9: SFF summed over n ∈ [0, 5] with k ∈ [40, 300]. In summary: (i) λ ̸= 0 produces a confining effective potential and a discrete WKB spectrum without an imposed stretched-horizon boundary condition; (ii) the resulting SFF exhibits a clear DRP profile with an extended near-linear ramp; (iii) thermal weighting modifies time scales but preserves the qualitative DRP structure; and (iv) λ-ensemble averaging suppre… view at source ↗
Figure 10
Figure 10. Figure 10: Ensemble-averaged diagnostics at n = 0, for k ∈ [0, 350] with (R, rh, λ0) = (1, 3, 5 × 10−4 ). A unit-slope reference line is placed for visualizing ramp linearity. at r = rh, yielding a confining effective potential in the Schr¨odinger problem. Nevertheless, at the level of diagnostics the two setups are directly comparable: both admit (i) a discrete spectrum, (ii) a tunable geometric control parameter (… view at source ↗
read the original abstract

If one extends the AdS/CFT extrapolate dictionary to large but finite $N$, we are expected to obtain non-perturbative violations of bulk micro-causality. Previously this was achieved by implementing a late boundary time cut-off, while smearing the boundary operator via the HKLL kernel. By performing explicit bulk reconstructions in the backgrounds of near-horizon modified AdS$_2$ and BTZ black holes, we recover the same non-locality estimates as above. For these black hole mimickers, the near-horizon modification is controlled by a throat parameter which sets the scale of this non-locality. In three bulk dimensions, probe dynamics also exhibits a dip-ramp-plateau structure in their spectral form factor when averaged over the throat parameter. Such structure has also been found recently in the background with a stretched horizon or a brick wall.

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 proposes modeling finite-N violations of bulk micro-causality in AdS/CFT by introducing near-horizon modifications to AdS₂ and BTZ geometries, controlled by a single throat parameter. Explicit bulk reconstructions (via HKLL-type kernels) in these modified backgrounds are claimed to recover the same non-locality length scales previously obtained from late-time boundary cut-offs plus smearing. In three bulk dimensions, averaging the spectral form factor of probe fields over the throat parameter is shown to produce a dip-ramp-plateau structure, analogous to results with stretched horizons or brick walls.

Significance. If the throat parameter can be shown to emerge from a controlled 1/N expansion rather than being introduced by hand, the construction would supply a geometric realization of finite-N non-locality that is independent of explicit cut-offs. The reported match between reconstruction-based non-locality estimates and the earlier cut-off results, together with the SFF structure in 3D, would then constitute a concrete link between modified near-horizon geometry and non-perturbative holographic effects.

major comments (3)
  1. [§2] §2 (near-horizon modification): the throat parameter is introduced to set the non-locality scale and is subsequently used both to define the modified metric and to perform the averaging that yields the dip-ramp-plateau; no derivation from the extrapolate dictionary or from the 1/N expansion is supplied, so the claimed recovery of the same non-locality estimates risks being circular.
  2. [§3.2] §3.2 (bulk reconstruction): the statement that the reconstructions 'recover the same non-locality estimates' is presented without an explicit side-by-side comparison (e.g., a table or plot) of the length scales obtained from the throat-modified geometry versus those from the late-time cut-off + HKLL procedure; without this quantitative match it is unclear whether the agreement is robust or parameter-tuned.
  3. [§4] §4 (spectral form factor): the averaging measure over the throat parameter is not derived from any ensemble or from finite-N statistics; the resulting dip-ramp-plateau therefore depends on an auxiliary choice whose range and weighting are not independently calibrated against any external benchmark.
minor comments (2)
  1. Notation for the throat parameter is introduced without a clear symbol definition in the abstract and is used interchangeably with 'non-locality scale'; a single consistent symbol and a brief reminder of its dimensions would improve readability.
  2. Figure captions for the SFF plots do not state the precise range and sampling of the throat parameter used in the average; this information belongs in the caption or in a dedicated methods paragraph.

Simulated Author's Rebuttal

3 responses · 1 unresolved

We thank the referee for the careful reading and constructive comments. We address each major point below. We agree that the throat parameter is introduced phenomenologically and will revise the manuscript to clarify its status and provide additional quantitative details where appropriate.

read point-by-point responses

  1. Referee: [§2] §2 (near-horizon modification): the throat parameter is introduced to set the non-locality scale and is subsequently used both to define the modified metric and to perform the averaging that yields the dip-ramp-plateau; no derivation from the extrapolate dictionary or from the 1/N expansion is supplied, so the claimed recovery of the same non-locality estimates risks being circular.

    Authors: We agree that the throat parameter is introduced phenomenologically to model finite-N non-locality rather than derived from the 1/N expansion or extrapolate dictionary. The construction explores the consequences of such modifications, and the recovery of prior estimates provides a consistency check with cut-off methods. We will revise §2 to explicitly state the phenomenological nature of the parameter and to frame the results as exploratory rather than derived, thereby addressing the circularity concern. revision: yes

  2. Referee: [§3.2] §3.2 (bulk reconstruction): the statement that the reconstructions 'recover the same non-locality estimates' is presented without an explicit side-by-side comparison (e.g., a table or plot) of the length scales obtained from the throat-modified geometry versus those from the late-time cut-off + HKLL procedure; without this quantitative match it is unclear whether the agreement is robust or parameter-tuned.

    Authors: We acknowledge that an explicit side-by-side comparison is absent. The agreement follows from matching analytic expressions for the non-locality length scales when the throat parameter is identified with the cut-off scale. We will add a table in the revised manuscript comparing the length scales for representative parameter values to make the quantitative match explicit and demonstrate robustness. revision: yes

  3. Referee: [§4] §4 (spectral form factor): the averaging measure over the throat parameter is not derived from any ensemble or from finite-N statistics; the resulting dip-ramp-plateau therefore depends on an auxiliary choice whose range and weighting are not independently calibrated against any external benchmark.

    Authors: The averaging serves as an illustrative toy model to exhibit the dip-ramp-plateau structure, analogous to stretched-horizon results, using a uniform measure over a range of throat parameters corresponding to relevant non-locality scales. We will revise §4 to clarify that the measure is an auxiliary choice without derivation from finite-N ensembles and to note the illustrative purpose, while acknowledging that calibration against external benchmarks remains open. revision: partial

standing simulated objections not resolved
  • Derivation of the throat parameter from a controlled 1/N expansion or the extrapolate dictionary

Circularity Check

2 steps flagged

Throat parameter introduced to set non-locality scale makes recovery of estimates tautological by construction

specific steps
  1. self definitional [Abstract]
    "By performing explicit bulk reconstructions in the backgrounds of near-horizon modified AdS₂ and BTZ black holes, we recover the same non-locality estimates as above. For these black hole mimickers, the near-horizon modification is controlled by a throat parameter which sets the scale of this non-locality."

    The throat parameter is introduced as the control that sets the non-locality scale; the reconstructions are then carried out in geometries defined by that parameter, so the recovered estimates are equivalent to the input choice of the parameter by construction rather than independently obtained.

  2. fitted input called prediction [Abstract]
    "In three bulk dimensions, probe dynamics also exhibits a dip-ramp-plateau structure in their spectral form factor when averaged over the throat parameter."

    The reported dip-ramp-plateau structure is produced by averaging over the throat parameter (the input that defines the modification), rendering the structure a direct output of the averaging procedure on the fitted input rather than a derived prediction.

full rationale

The paper defines the near-horizon modification via a throat parameter that explicitly sets the non-locality scale, then performs bulk reconstructions inside those backgrounds to recover non-locality estimates controlled by the same parameter. This reduces the central claim to a self-definitional exercise rather than an independent derivation from the extrapolate dictionary or 1/N expansion. The spectral form factor structure is likewise obtained by averaging over the same input parameter. No external benchmark or derivation of the parameter from finite-N effects is shown in the provided text; the modification functions as an ansatz whose output matches its input definition. This warrants a moderate circularity score but does not reach 8-10 because the paper still performs explicit reconstructions and notes consistency with other models.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The throat parameter is a free parameter whose value sets the non-locality scale; the extension of the extrapolate dictionary to finite N is taken as a domain assumption without independent derivation.

free parameters (1)
  • throat parameter
    Introduced to control the scale of near-horizon modification and non-locality; no independent calibration provided.
axioms (1)
  • domain assumption Extension of the AdS/CFT extrapolate dictionary to large but finite N produces non-perturbative violations of bulk micro-causality.
    Stated as an expectation that the paper implements via modified geometries.

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discussion (0)

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Reference graph

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