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Notes on Wasserstein distance and the JT gravitational ensemble

Primarily AI-generated textHuman understanding: some partsmath-ph — Mathematical Physicsmath.PR — Probabilityquant-ph — Quantum Physics

Contributed by Ville Samuli Keränen ↗

Version 1 / Nov 11, 2026 / CC BY 4.0

Abstract

This note completes the study of the Boltzmann–Wasserstein (BW) distance, an optimal-transport metric on quantum theories, begun in a companion paper (arXiv:2606.01296). There the distance was evaluated between macroscopically different theories; here we take the opposite limit, two independent draws of the JT-gravity ensemble, which are identical at the classical level. The distance is computable and is controlled by the connected spectral form factor: the ramp, the double-trumpet wormhole, regulated by the plateau, the spectral discreteness. The mean is exponentially small, ⟨W2⟩∼S0 e−S0\langle \mathcal{W}^2\rangle \sim S_0\, e^{-S_0}, whereas uncorrelated (Poisson) levels would give an unsuppressed O(1)O(1) distance: the self-distance is a sharp diagnostic of spectral chaos. As a metric, the BW distance renders the ensemble a point cloud: exponentially small, strongly anisotropic, its leading principal axis a coherent “breathing” deformation of the low-energy spectrum. Its effective dimension grows asymptotically as Deff≃0.42 S02D_{\mathrm{eff}} \simeq 0.42\, S_0^2, so the distance self-averages, its relative width shrinking as std/mean∼1/S0\mathrm{std}/\mathrm{mean} \sim 1/S_0. At low temperatures it localizes on the Airy edge, where a single Tracy–Widom mode dominates and self-averaging fails. We test the picture with exact-diagonalization GUE and SYK simulations: in GUE the effective dimension grows logarithmically at every temperature and is predicted at finite NN, with no free parameters, by the same covariance; unfolded SYK exhibits the predicted anisotropy and mode structure of the cloud. Together with the companion result, this fixes the resolution of the metric: the microstate rigidity computed here is the noise floor beneath a macroscopic difference between theories.

Provenance statement

This work was produced in a back-and-forth collaboration between VK and Claude (Anthropic; models Opus 4.8 and Fable 5). Claude did the calculations and the writing first; VK checked the main-text calculations by hand or by independent simulation and rewrote parts of the text. The appendix numerics (exact diagonalization of SYK and GUE) are Claude’s, cross-checked internally but not by hand. The manuscript was reviewed in several passes by referee-prompted Claude agents and by Claude Fable 5.2 in a separate session; that review contributed one quantitative correction, which the authors then reproduced. A detailed account, including the limits of the review, is in the paper’s closing “Note on the preparation of this work.”

Tools used

Anthropic
Claude FableVersion 5
Anthropic
Claude FableVersion 5.2
Anthropic
Claude OpusVersion 4.8

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Version history

  1. v1 · 37 pagesSubmitted by Ville Samuli KeränenInitial depositCurrentNov 10, 2026