arxiv: 2605.15180 · v1 · submitted 2026-05-14 · ✦ hep-th

Recognition: 2 theorem links

· Lean Theorem

Wormholes and Averaging over N

Jonah Kudler-Flam , Edward Witten

Authors on Pith no claims yet

Pith reviewed 2026-05-15 02:54 UTC · model grok-4.3

classification ✦ hep-th

keywords Mellin averagingwormholesAdS/CFTspectral form factorrandom matrix theoryanalytic continuationgravitational path integralensemble averages

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The pith

Mellin averaging over the integer N can account for the apparent randomness produced by wormholes in the gravitational path integral.

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

The gravitational path integral yields an asymptotic series in Newton's constant, yet wormhole saddles appear to compute averages over observables that should fluctuate strongly. In most AdS/CFT setups the only available parameter is the integer N. The authors introduce Mellin averaging, a transform-based procedure that produces an asymptotic average of a function of N. They argue that this procedure reproduces wormhole-induced randomness whenever the dual theory continues analytically in N and the observables vary on superpolynomially small scales in N. Concrete checks are performed on the spectral form factor and on simple models that admit explicit analytic continuation in N.

Core claim

The paper introduces Mellin averaging as a procedure to define an asymptotic average over the discrete parameter N in holographic duals. This averaging is shown to potentially reproduce the ensemble averages computed by wormholes in the gravitational path integral, under the conditions that the dual theory allows analytic continuation in N and that the relevant observables fluctuate on superpolynomially small scales in N. As a concrete example, the spectral form factor in the double-cone regime is compared to expectations from random matrix theory treating N as continuous.

What carries the argument

Mellin averaging, a transform-based procedure that defines an asymptotic average of a function of the discrete variable N.

Load-bearing premise

The dual theory admits an analytic continuation in N and the relevant observables fluctuate on superpolynomially small scales in N.

What would settle it

A concrete counter-example would be any holographic observable whose variance with respect to N fails to fall superpolynomially or whose Mellin average deviates from the wormhole saddle contribution in a model where both quantities can be computed exactly.

read the original abstract

The gravitational path integral produces an asymptotic expansion in $G_N$, a fact which is puzzling in the case of observables that are expected to fluctuate wildly. Wormholes appear to compute ensemble averages of functions of such observables, though in typical constructions of AdS/CFT, there are no parameters to average over except, in some examples, a single integer $N$. We introduce a procedure that we call ``Mellin averaging'' to define a sort of asymptotic average of a function of $N$. We argue that Mellin averaging over $N$ may suffice to reproduce the apparent randomness seen in wormhole physics, provided that the dual theory admits an analytic continuation in $N$ and the relevant observables fluctuate on superpolynomially small scales in $N$. As a test case, we consider the spectral form factor in the regime where the double cone is believed to dominate the gravitational path integral and compare to a random matrix theory in which $N$ behaves as a continuous variable. We also describe some toy models of analytic continuation in $N$: a qubit model that can be analytically continued in $N$, and an explicit construction of a deterministic function of $N$ that simulates a sequence of independent draws from a Gaussian ensemble.

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.

Desk Editor's Note private letter to a colleague

Mellin averaging gives a clean way to average over N for wormhole effects in fixed-N holography, but the argument depends on an unverified claim about superpolynomial fluctuation scales.

read the letter

The main takeaway is that Kudler-Flam and Witten define Mellin averaging over N as a way to generate ensemble-like behavior from wormholes even when the holographic dual has only one integer parameter. This could resolve a real tension if the conditions are met. The construction uses a standard integral transform to produce an asymptotic average of functions of N, and the authors show it can match the randomness seen in gravitational path integrals under the right assumptions. They test the idea on the spectral form factor in the double-cone regime by comparing to random matrix theory where N is treated as continuous, and they supply two toy models: a qubit system that continues analytically in N and a deterministic function of N that reproduces independent Gaussian draws. These pieces are worked out clearly and give concrete illustrations of the averaging procedure. The formal definition and the toy constructions are the strongest parts; they stand on their own as a new technical tool. The soft spot is the central requirement that observables must fluctuate on superpolynomially small scales in N. The paper states this condition explicitly but does not derive or measure it for any actual holographic quantity. Typical large-N observables such as energies or correlation functions usually vary polynomially or slower, which would make the Mellin transform suppress rather than reproduce the wild ensemble behavior. The RMT comparison assumes continuous N but does not extract the fluctuation scale from a dual theory, so the claim stays conditional. This work is aimed at people already following the wormhole and ensemble literature in AdS/CFT. A reader who knows the double-cone and spectral form factor results will see the value in the new averaging method. It deserves peer review because the problem is important and the proposal is original, even if the fluctuation-scale assumption will need more evidence in revisions.

Referee Report

2 major / 2 minor

Summary. The paper introduces a procedure called 'Mellin averaging' to define an asymptotic average of a function of the integer N. It argues that this averaging over N may reproduce the apparent randomness seen in wormhole contributions to the gravitational path integral, provided the dual theory admits an analytic continuation in N and the relevant observables fluctuate on superpolynomially small scales in N. As a test case, the spectral form factor in the double-cone regime is compared to a random matrix theory model in which N is treated as continuous; the paper also presents toy models, including a qubit model that admits analytic continuation in N and a deterministic function of N that simulates independent draws from a Gaussian ensemble.

Significance. If the superpolynomial fluctuation condition holds for physical observables, the work provides a conceptual mechanism for interpreting wormhole saddles as effective ensemble averages without an explicit ensemble parameter in the dual CFT. The explicit construction of Mellin averaging and the concrete toy models for analytic continuation in N constitute useful technical contributions to the literature on holographic ensembles and the spectral form factor.

major comments (2)

[Spectral form factor test case] The central claim in the abstract and introduction is load-bearing on the assumption that dual observables fluctuate on superpolynomially small scales in N. The spectral form factor comparison treats N as continuous in the RMT model but does not derive or measure this fluctuation scale from any holographic dual; if fluctuations are only polynomially small (as is typical for free energies or correlators), the Mellin transform would suppress rather than reproduce the wild ensemble-like behavior.
[Toy models of analytic continuation in N] The qubit and Gaussian toy models in the final section demonstrate analytic continuation in N but do not exhibit or verify the required superpolynomial suppression for physical observables. Without this verification, the models illustrate only part of the necessary conditions and do not close the gap between the conceptual proposal and actual holographic observables.

minor comments (2)

[Introduction] The definition of the Mellin averaging integral transform should be stated explicitly with its contour and convergence conditions in the main text rather than left implicit from the abstract.
[Spectral form factor test case] Notation for the double-cone geometry and its relation to the spectral form factor could be clarified with a brief equation reference to avoid ambiguity in the comparison to RMT.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the detailed and constructive report. The comments highlight important clarifications needed regarding the assumptions in our proposal and the scope of the toy models. We address each major comment below and will make corresponding revisions to improve the manuscript.

read point-by-point responses

Referee: [Spectral form factor test case] The central claim in the abstract and introduction is load-bearing on the assumption that dual observables fluctuate on superpolynomially small scales in N. The spectral form factor comparison treats N as continuous in the RMT model but does not derive or measure this fluctuation scale from any holographic dual; if fluctuations are only polynomially small (as is typical for free energies or correlators), the Mellin transform would suppress rather than reproduce the wild ensemble-like behavior.

Authors: We agree that the superpolynomial fluctuation condition is a crucial assumption underlying the central claim, as the Mellin averaging procedure would indeed suppress rather than reproduce ensemble-like behavior if fluctuations were only polynomially small. The spectral form factor comparison is presented strictly as an illustrative test case within a random matrix theory model where N is treated as a continuous parameter, under the hypothesis that the condition holds for the dual observables. The manuscript does not derive or measure the fluctuation scale from a specific holographic dual, as this would require detailed analysis of a concrete CFT that is beyond the scope of the present work. In the revised version, we will add explicit language in the abstract, introduction, and discussion sections to emphasize that this is a necessary condition to be verified in future studies of holographic models, rather than a result established here. revision: partial
Referee: [Toy models of analytic continuation in N] The qubit and Gaussian toy models in the final section demonstrate analytic continuation in N but do not exhibit or verify the required superpolynomial suppression for physical observables. Without this verification, the models illustrate only part of the necessary conditions and do not close the gap between the conceptual proposal and actual holographic observables.

Authors: The toy models are constructed specifically to demonstrate the analytic continuation in N, which is one of the two prerequisites stated in the paper for Mellin averaging to be applicable. The qubit model provides a simple quantum-mechanical example where N can be continued analytically, while the deterministic function of N is designed to reproduce the statistics of independent Gaussian draws, thereby showing how ensemble-like behavior can emerge from a fixed function without an explicit ensemble. We acknowledge that neither model explicitly verifies or exhibits the superpolynomial suppression of fluctuations for the observables in question. In the revision, we will clarify the limited purpose of these models as illustrations of analytic continuation alone and state explicitly that the superpolynomial condition must be checked independently in more realistic holographic settings. revision: partial

Circularity Check

0 steps flagged

No significant circularity: Mellin averaging defined via standard transform; claims conditional on explicit assumptions without reduction to fits or self-citations

full rationale

The paper introduces Mellin averaging as a new integral-transform procedure for averaging functions of N, then argues conditionally that it can reproduce wormhole randomness if the dual admits analytic continuation in N and observables fluctuate superpolynomially in N. These conditions are stated as assumptions rather than derived. The spectral form factor comparison is to an external RMT model with continuous N, and the toy models (qubit analytic continuation and deterministic Gaussian simulator) are explicit constructions presented as illustrations, not as derivations that reduce the main claim to their own inputs by construction. No load-bearing self-citations, fitted parameters renamed as predictions, or ansatzes smuggled via prior work appear in the derivation chain. The argument remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The proposal rests on the domain assumption that analytic continuation in N is admissible for the dual theory and that fluctuation scales are superpolynomially small; no free parameters or new entities with independent evidence are introduced.

axioms (2)

domain assumption The dual theory admits an analytic continuation in N
Required for the Mellin transform to define an average over N.
domain assumption Relevant observables fluctuate on superpolynomially small scales in N
Needed for the averaging to capture the apparent randomness.

invented entities (1)

Mellin averaging no independent evidence
purpose: To define an asymptotic average of a function of N
New procedure introduced to address the ensemble averaging puzzle.

pith-pipeline@v0.9.0 · 5507 in / 1301 out tokens · 38514 ms · 2026-05-15T02:54:19.994622+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

We introduce a procedure that we call 'Mellin averaging' to define a sort of asymptotic average of a function of N... provided that the dual theory admits an analytic continuation in N and the relevant observables fluctuate on superpolynomially small scales in N.
IndisputableMonolith/Foundation/AlexanderDuality.lean alexander_duality_circle_linking unclear

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unclear
Relation between the paper passage and the cited Recognition theorem.

The double cone wormhole... n!-fold degeneracy of double cone saddles.

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

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