{"id":"593fb752-ff45-4efa-b5b4-1211d6da8207","arxiv_id":"2508.11592","paper_version":2,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Reconstructed AdS-Rindler bulk operators have three-point functions scaling as N^{-1} exp((pi/2)(|lambda|-omega)), so they break down above a (2/pi) ln N momentum scale.","lead":"This paper shows that in a large but finite N holographic CFT, certain reconstructed black-hole-interior operators grow exponentially with momentum and stop being well defined beyond a log N scale. The result suggests a fundamental limit on reconstructing the black hole interior from boundary data.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The 1/N-correction argument is unsupported: a double-trace correction's disconnected four-point contribution can share the 3pt coordinate dependence, so cancellation is not ruled out.","rationale":"The paper's strongest technical content is the saddle-point evaluation in Appendix A; I do not see an obvious error there, and the claimed exponential factor is internally consistent with the normalization and the two-point cancellation. The genuine soft spot is the last section's argument against finite-N corrections. The assertion that a four-point function cannot have the same coordinate dependence as the three-point function is questionable: in the OPE limit, the disconnected part of a double-trace correction factorizes into products of two-point functions that carry the same powers of (u1-u2) and (v2-v1). Therefore the possibility of a 1/N correction with a Rindler-enhanced kernel canceling (3.17) is not excluded. Uniqueness of the N=∞ Rindler reconstruction fixes the leading term but not the subleading coefficients; those are determined by bulk interactions and may contain the same 1/Nω factors that produce the exponential. An explicit first-order computation is needed. This does not refute the exponential growth of the N=∞ operator, but it blocks the strong conclusion that no well-defined finite-N operator exists. The reader's CONDITIONAL verdict remains appropriate.","tokens_in":19219,"tokens_out":41413,"duration_ms":518687,"concrete_test":"Construct the first 1/N correction to the AdS-Rindler reconstructed operator for a bulk φ^3 interaction using the Kabat-Lifschytz algorithm [9,10], and compute its leading contribution to ⟨0|O1 δφR_l O2|0⟩ in the OPE limit u→u2, v→v1 used in (3.16). If the disconnected double-trace term is of order (1/N)e^{π/2(|λ|-ω)} with the same (u1-u2)^{-Δ/2}(v2-v1)^{-Δ/2} structure and opposite sign, the exponential growth is cancelled and the central claim fails. A quicker analytic version: determine whether a kernel K(u,v;u',v') can make ∫K ⟨O1O⟩⟨OO2⟩ match the coordinate dependence of (3.16) for all insertions; if yes, the paper's 'cannot be the same form' assertion is refuted.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that the exponential growth in (3.17) cannot be removed by finite-N corrections rests on the last section's assertion that a double-trace correction δφR_l would give a four-point function that 'cannot be the same form' as the three-point function. That assertion is too strong. At leading order in 1/N, the disconnected part of ⟨0|O1 (∫ K OO) O2|0⟩ factorizes into products of two-point functions of the form (u-u1)^{-Δ/2}(v-v1)^{-Δ/2}(u'-u2)^{-Δ/2}(v'-v2)^{-Δ/2}; after integrating against a kernel localized at u≈u2, v≈v1, this reproduces precisely the (u1-u2)^{-Δ/2}(v2-v1)^{-Δ/2} coordinate dependence of the leading term in (3.16). Thus a 1/N double-trace correction with a Rindler-enhanced kernel is not excluded by the paper's coordinate-dependence argument. The uniqueness of the N=∞ reconstruction fixes only the leading term; it does not constrain the 1/N coefficient, which may carry the same 1/Nω factors as (3.5). Without an explicit first-order (or all-orders) construction showing that such corrections are absent, the claim that the smeared operator cannot be defined is not established.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19533,"tokens_out":10519,"duration_ms":125265,"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":[{"comment":"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","section":"Sec. 3, 'Corrections to the reconstructed operator'"},{"comment":"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.'","section":"Sec. 3, Eqs. (3.5)-(3.7) and (3.17)"}],"minor_comments":[{"comment":"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.","section":"Eq. (2.23)"},{"comment":"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.","section":"Abstract and Sec. 1"},{"comment":"There is a minor grammatical issue: the paper appears to have a single author, but the acknowledgement reads 'The authors would like to thank...'.","section":"Acknowledgements"},{"comment":"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.","section":"Sec. 2.2"}],"recommendation":"major_revision","confidential_remarks":"The manuscript relies heavily on the author's previous work (refs. [15]-[19]) for the broader interpretation, and the incremental contribution should be made clearer. The main issue for publication is the unsupported no-cancellation argument for 1/N corrections; an explicit first-order calculation, even in a simplified model, would considerably strengthen the paper. The editor may also wish to consider whether the strong claims about EWR and black hole physics are justified by the d=2 AdS-Rindler calculation alone."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"The paper does something real: it evaluates the AdS-Rindler HKLL operator at finite N and shows its three-point function grows as N^{-1} exp(pi/2(|lambda|-omega)), giving a critical smearing scale ~ (2/pi) ln N. Appendix A's saddle-point analysis of the Rindler mode functions and the contour argument in Section 3 that two-point functions cancel the exponential while three-point functions don't are concrete and mostly plausible. The paper is also honest about restrictions (d=2, operator ordering, generic insertion points) and doesn't oversell the exact prefactors. This is a genuine new result in the author's program: the explicit exponential formula and the ln N cutoff are not in the earlier papers. Credit where due: the calculation is worth engaging with.\n\nBut the central physical conclusion — that the operator cannot be defined beyond this cutoff — rests on the final section's claim that 1/N corrections can't remove the growth. That argument is too quick. The paper says a double-trace correction's four-point function 'cannot be the same form' as the three-point function. That's just false. The disconnected part of <O1 (integral K OO) O2> factorizes into two two-point functions, and after integrating against a kernel localized at the appropriate points it can reproduce the same (u1-u2)^{-Delta/2}(v2-v1)^{-Delta/2} dependence. The N=infinity uniqueness fixes only the leading term; it doesn't constrain the 1/N coefficient, which could carry the same enhancement. So the paper has not ruled out an improved reconstructed operator. This is a load-bearing gap, not a cosmetic one.\n\nThe saddle-point and contour steps themselves also drop power-law prefactors and Gaussian factors. That's probably survivable, but it means the exponential estimate is not a proof of a sharp cutoff. The real fix is to either construct the 1/N corrections explicitly and show they fail, or soften the claim to: 'the leading-order HKLL operator has exponential growth, and a consistent finite-N reconstruction is not known.'\n\nWho is this for? Holographers working on bulk reconstruction, EWR, and quantum error correction. If the obstruction holds, it matters. But in current form it's a conditional result. I'd send it to a good referee, because the calculation is non-trivial and the flaw is addressable. A serious referee would ask for a rigorous treatment of 1/N corrections, or at least an honest statement of what is and isn't proven.","headline":"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.","tokens_in":19995,"tokens_out":5475,"would_cite":false,"duration_ms":59453,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":[],"pacs":[],"model":"deepseek-v4-flash","headline":"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","keywords":["AdS/CFT","bulk reconstruction","AdS-Rindler wedge","finite N","1/N expansion","entanglement wedge reconstruction","horizon-to-horizon modes","black hole information paradox"],"falsifier":"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.","tokens_in":19134,"feed_emoji":"🕳️","tokens_out":12554,"duration_ms":119782,"temperature":0.7,"pith_summary":"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.","feed_headline":"Bulk reconstruction breaks down at a ln N cutoff, far below Planck","feed_subtitle":"Three-point functions grow exponentially; 1/N expansion and subregion reconstruction fail above a cutoff far below Planck.","key_machinery":"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","core_discovery":"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-","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the HKLL bulk reconstruction formula for AdS-Rindler that defines the CFT operator $\\phi_R^l$ whose correlators are analyzed.","marker":"[8]"},{"why":"Fixes the Rindler mode normalization $N_{\\omega,\\lambda}$, identifies the horizon-to-horizon ('tachyonic') modes, and gives the Bogoliubov-transformation cancellation behind the two-point function.","marker":"[17]"},{"why":"Provides the subregion-complementarity setup in which $\\delta_l = \\phi_R^l - \\phi_G^l$ annihilates the vacuum and has vanishing two-point functions, so the reconstructed operator is small on the vacuum but can be large on other states.","marker":"[18]"},{"why":"Supplies the contour-shift and $i\\epsilon$-pole evaluation of wave-packet correlators used to compute both the two- and three-point functions.","marker":"[29]"},{"why":"Extends the wave-packet holographic dictionary to AdS4/CFT3, supporting the same contour techniques used here.","marker":"[30]"},{"why":"States the entanglement-wedge reconstruction result at $N=\\infty$ whose extension to finite $N$ the paper argues fails.","marker":"[13]"},{"why":"Identifies null-geodesic/horizon-horizon modes in subregion AdS/CFT, the modes that carry the exponential enhancement.","marker":"[26]"},{"why":"Justifies treating bulk local fields only after smearing and supplies boundary-to-bulk maps for AdS causal wedges, setting up the UV regulator $\\Lambda$.","marker":"[23]"}],"fun_headline_variants":["Bulk reconstruction dies at a ln N cutoff, far below Planck","ln N wall stops bulk reconstruction far below Planck","Finite N truncates bulk reconstruction at log scale","Bulk reconstruction's logarithmic cutoff is far below Planck","Black hole interior reconstruction fails at a ln N scale"],"cache_read_input_tokens":2816,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Bulk reconstruction dies at a ln N cutoff, far below Planck","ln N wall stops bulk reconstruction far below Planck","Finite N truncates bulk reconstruction at log scale","Bulk reconstruction's logarithmic cutoff is far below Planck","Black hole interior reconstruction fails at a ln N scale"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000609,"raw_usage":{"total_tokens":2670,"prompt_tokens":739,"completion_tokens":1931,"prompt_tokens_details":{"cached_tokens":256},"prompt_cache_hit_tokens":256,"prompt_cache_miss_tokens":483,"completion_tokens_details":{"reasoning_tokens":1867}},"tokens_in":483,"tokens_out":1931,"duration_ms":14116,"temperature":1.0,"reasoning_tokens":1867,"cache_read_input_tokens":256,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-05T19:50:27.785311+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[],"review_version":1}